Arithmetic Refresher: Improve your working knowledge of arithmetic

DoverBooksonMathematics

HANDBOOKOFMATHEMATICALFUNCTIONSWITHFORMULASGRAPHSANDMATHEMATICALTABLESEditedbyMiltonAbramowitzandIreneAStegun(0-486-61272-4)ABSTRACTANDCONCRETECATEGORIESTHEJOYOFCATSJiriAdamekHorstHerrlichGeorgeEStrecker(0-486-46934-4)MATHEMATICSITSCONTENTMETHODSANDMEANINGADAleksandrovANKolmogorovandMALavrentrsquoev(0-486-40916-3)REALVARIABLESWITHBASICMETRICSPACETOPOLOGYRobertBAsh(0-486-47220-5)PROBLEMSOLVINGTHROUGHRECREATIONALMATHEMATICSBonnieAverbachandOrinChein(0-486-40917-1)VECTORCALCULUSPeterBaxandallandHansLiebeck(0-486-46620-5)

INTRODUCTIONTOVECTORSANDTENSORSSECONDEDITION-TWOVOLUMESBOUNDASONERayMBowenandC-CWang(0-486-46914-X)FOURIERANALYSISINSEVERALCOMPLEXVARIABLESLeonEhrenpreis(0-486-44975-0)THETHIRTEENBOOKSOFTHEELEMENTSVOL2Euclid(0-486-60089-0)

THETHIRTEENBOOKSOFTHEELEMENTSVOL1EuclidEditedbyThomasLHeath(0-486-60088-2)ANINTRODUCTIONTODIFFERENTIALEQUATIONSANDTHEIRAPPLICATIONSStanleyJFarlow(0-486-44595-X)DISCOVERINGMATHEMATICSTHEARTOFINVESTIGATIONAGardiner(0-486-45299-9)ORDINARYDIFFERENTIALEQUATIONSJackKHale(0-486-47211-6)

METHODSOFAPPLIEDMATHEMATICSFrancisBHildebrand(0-486-67002-3)

BASICALGEBRAISECONDEDITIONNathanJacobson(0-486-47189-6)

BASICALGEBRAIISECONDEDITIONNathanJacobson(0-486-47187-X)

GEOMETRYANDCONVEXITYASTUDYINMATHEMATICALMETHODSPaulJKellyandMaxLWeiss(0-486-46980-8)COMPANIONTOCONCRETEMATHEMATICSMATHEMATICAL

TECHNIQUESANDVARIOUSAPPLICATIONSZAMelzak(0-486-45781-8)MATHEMATICALPROGRAMMINGStevenVajda(0-486-47213-2)

FOUNDATIONSOFGEOMETRYCRWylieJr(0-486-47214-0)

SeeeveryDoverbookinprintatwwwdoverpublicationscom

BOOKSBYAALBERTKLAF

CalculusRefresherforTechnicalMenTrigonometryRefresherforTechnicalMen

ArithmeticRefresher

AAlbertKlaf

Copyrightcopy1964byMollieGKlafAllrightsreserved

ArithmeticRefresherwasfirstpublishedbyDoverPublicationsIncin1964underthetitleArithmeticRefresherforPracticalMen

LibraryofCongressCatalogCardNumber64-18856InternationalStandardBookNumber

9780486141930

ManufacturedintheUnitedStatesbyCourierCorporation21241622wwwdoverpublicationscom

FOREWORD

MyfatherwrotethisArithmeticRefresherforPracticalMenforthemassaudienceofprofessionalsandlaymenwhoarefrequentlyfacedwithnumericalproblemsThebookincludestheknowledgeandpracticalexperiencegatheredduringalifetimeofsearchingcuriosityHecompletedthemanuscriptayearbeforehispassingItisthetestamentofacareerdedicatedtopublicserviceandmathematicalenlightenment

IwishtoexpressmydeepappreciationtomyfatherrsquoscolleagueMrVictorFeigelmanBCEMCEforsolvingthesampleproblemsandcheckingthemanuscriptThanksarealsoduetoMrHaywardCirkerPresidentofDoverPublicationsIncwhowasmyfatherrsquosvaluedfriendaswellashispublisher

ThisbookwastohavebeenoneofaseriesthatbeganwithhisCalculusRefresherforTechnicalMenandprogressedtohisTrigonometryRefresherforTechnicalMenThesucceedingvolumeswillofcourseremainunwrittenButthebesthasbeensaidNowitmustbeusedbythosewhoseektoexperiencethejoyofmathematicsmyfathersodeeplyfelt

FRANKLINSKLAFMD

TableofContents

DoverBooksonMathematicsBOOKSBYAALBERTKLAFTitlePageCopyrightPageFOREWORD

INTRODUCTIONCHAPTERI-ADDITIONCHAPTERII-SUBTRACTIONCHAPTERIII-MULTIPLICATIONCHAPTERIV-DIVISIONCHAPTERV-FACTORSmdashMULTIPLESmdashCANCELLATIONCHAPTERVI-COMMONFRACTIONSCHAPTERVII-DECIMALFRACTIONSCHAPTERVIII-PERCENTAGECHAPTERIX-INTERESTCHAPTERX-RATIOmdashPROPORTIONmdashVARIATIONCHAPTERXI-AVERAGESCHAPTERXII-DENOMINATENUMBERSCHAPTERXIII-POWERmdashROOTSmdashRADICALSCHAPTERXIV-LOGARITHMSCHAPTERXV-POSITIVEANDNEGATIVENUMBERSCHAPTERXVI-PROGRESSIONSmdashSERIESCHAPTERXVII-GRAPHSmdashCHARTSCHAPTERXVIII-BUSINESSmdashFINANCECHAPTERXIX-VARIOUSTOPICSCHAPTERXX-INTRODUCTIONTOALGEBRAAPPENDIXA-ANSWERSTOPROBLEMSAPPENDIXBTABLESINDEXACATALOGOFSELECTEDDOVERBOOKSINALLFIELDSOF

INTEREST

INTRODUCTION

1WhatisarithmeticThescienceofnumberandtheartofcomputation

2WhatisournumericalsystemcalledandwhyisitsocalledItiscalledtheArabicsystembecauseitwasgiventousbytheArabswho

developeditfromtheHindusystem

3WhatisadigitAnywholenumberfrom1through9iscalledadigitThus1234567

89arecalleddigits

4WhatisacipherandwhatisitssymbolThewordldquocipherrdquocomesfromanArabicwordmeaningldquoemptyrdquoandmeans

ldquonodigitrdquoThesymbolforacipheris0

5WhatothercommonlyusedwordsmaybesubstitutedforthewordldquocipherrdquoldquoZerordquoandldquonoughtrdquomaybeusedforldquocipherrdquo

6WhatisthefoundationoftheArabicnumericalsystemThefoundationconsistsoftheninesymbolscalleddigitsmdash1234567

89mdashandonesymbolcalledacipherzeroornought

7WhatisadecimalpointandwhatisitssymbolAdecimalpointisapointthatisusedtoseparatethefractionalpartofa

numberfromawholenumberanditssymbolisadot[]

8Whatismeantbycomputationorcalculation

ComputationorcalculationistheprocessofsubjectingnumberstocertainoperationsThewordldquocalculationrdquocomesfromaLatinwordmeaningldquopebblerdquoasreckoningwasdonewithcountersorpebbles

9HowmanyfundamentaloperationsarethereinarithmeticTherearesixoperationsallgrowingoutofthefirst

Thesixoperationsaredividedintotwogroups(a)threedirectoperationsand(b)threeinverseoperationseachofwhichhastheeffectofundoingoneofgroup(a)

Group(a ) Direct Operat ion Group(b ) InverseOperat ion

1Addition 4Subtraction

2Multiplication 5Division

3Involution 6Evolution

10Whatarethesymbolsfor(a)ldquoequalstordquoorldquoequalsrdquo(b)addition(c)subtraction(d)multiplication(e)division(f)involution(g)evolutionand(h)ldquothereforerdquo(a)Theequalssign[=]meansldquoequalstordquoorldquoequalsrdquo

1+1=2oneplusoneequalstwo

(b)Theplussign[+]meansldquoplusrdquoldquoandrdquoorldquoaddedtordquo

2+2=4twoplustwoequalsfourortwoandtwoequalsfourortwoaddedtotwoequalsfour

(c)Theminussign[mdash]meansldquominusrdquoldquosubtractedfromrdquoorldquofromrdquo

5ndash3=2fiveminusthreeequalstwoorthreesubtractedfromfiveequalstwoorthreefromfiveequalstwo

(d)Themultiplicationsign[times]meansldquomultipliedbyrdquoorldquotimesrdquo

5times3=15fivemultipliedbythreeequalsfifteenorfivetimesthreeequalsfifteen

ldquoTimesrdquomayalsobeindicatedbyadotinthecenterofthelinebetweenthetwonumbers

5bull3=15fivetimesthreeequalsfifteen

(e)Thedivisionsign[divide]meansldquodividedbyrdquo

10divide2=5tendividedbytwoequalsfive

Thesigns or meanldquodividedintordquo

twodividedintotenequalsfive

twodividedintotenequalsfiveThisformisusedinlongdivision

expressedasafractionmeansldquotendividedbytwoequalsfiverdquo

(f)Asmallnumberplacedintheupperright-handcornerofanumberisusedtoindicatethenumberoftimesthenumberistobemultipliedbyitself

25=2times2times2times2times2=32

ReadTwotothefifthpowerequalsthirty-twoTheprocessoffindingapowerofanumberisinvolution

(g)Theradicalsign[radic]meansldquorootofrdquoAfigureisplacedabovetheradictoindicatetheroottakenItisomittedinthecaseofthesquareroot

Itistheinverseoperationofinvolutionandiscalledevolution

Thefifthrootofthirty-twoistwowhichisthenumberthatwhenmultipliedbyitselffivetimeswillgivethirty-two

(h)Thesign[there4]meansldquothereforerdquo

11WhatisthesignificanceofparenthesesenclosingnumbersThepresenceofparenthesesmeansthattheoperationswithintheparentheses

aretobeperformedbeforeanyoperationsoutsideAnumberprecedingparenthesesmeansthatthefinalfigurewithinparenthesesistobemultipliedby

thatnumber

EXAMPLE3(5+2)=21Theoperation(5+2)isperformedfirst=7Then3times7=21Theoperationof3timesisthenperformed

12WhatismeantbyaunitAnyonethingiscalledaunit

13WhatismeantbyanumberAunitorcollectionofunitsiscalledanumber

14WhatismeantbyanintegerwholenumberoranintegralnumberNumbersrepresentingwholeunitsarecalledintegerswholenumbersor

integralnumbers

EXAMPLES128751342659areintegersorwholenumbers

15WhatsymbolsareusedtoexpressnumbersDigitsorfiguresareusedtoexpressnumbers

Thesymbol0=zeroisusedtoexpressldquonodigitrdquo

16HowaredigitsusedtoexpressnumbersinourArabicsystemThevalueofthedigitisfixedbyitspositionstartingfromtherightandgoing

towardstheleft

ThefirstpositionisthatofldquounitsrdquoThenextpositionisthatofldquotensrdquoThethirdpositionisthatofldquohundredsrdquoThesearecalledthethreeldquoordersrdquoAgroupofthreeordersiscalledaperiod

17HowaretheordersandperiodsarrangedintheArabicsystem(Rarelyisthereuseforanynumberlargerthanldquotrillionsrdquo)

18HowdowereadanumberwrittenintheArabicsystemSeparatethenumbersbycommasintoldquoperiodsrdquoorgroupsofthreefigures

beginningattheright

Nowbeginattheleftandreadeachperiodasifitstoodaloneaddingthenameoftheldquoperiodrdquo

EXAMPLE7653460534646(above)

ReadSeventrillionsixhundredandfifty-threebillionfourhundredandsixtymillionfivehundredandthirty-fourthousandsixhundredandforty-six

Notethatthewordldquoandrdquomayinallcasesbeomitted

19WhatistherelationofaunitofanyperiodtothatofthenextlowerperiodTheunitofanyperiod=1000unitsofthenextlowerperiod

EXAMPLE

Onethousand=1000=1000unitsOnemillion=1000000=1000thousandsOnebillion=1000000000=1000millionsOnetrillion=1000000000000=1000billions

20HowwouldyouwriteanumberinfiguresBeginattheleftandwritethehundredstensandunitsofeachldquoperiodrdquo

placingzerosinallvacantplacesandacommabetweeneachtwoperiods

EXAMPLE400536080209

Fourhundredbillionfivehundredthirty-sixmillioneightythousandtwohundrednine

21HowdozerosbeforeorafteranumberaffectthenumberAzeroinfrontofanumberdoesnotaffectit

EXAMPLE0008060=eightthousandsixty

Azeroafteranumbermovesthenumberoneplacetotheleftormultipliesitby10

EXAMPLE8060Nowaddazeroafterthenumberor80600Theeightthousandsixtybecomeseightythousandsixhundred

Twozerosaddedattherightmovesthenumbertwoplacestotheleftormultipliesitby100

EXAMPLE80600Addtwozerosgetting8060000=eightmillionsixtythousandAndsoonwithaddedzeros

ForanothermethodofwritingverylargenumbersseeQuestion534

22Whatarethenamesoftheperiodsbeyondtrillionsuptoandincludingthetwelfthperiod5Trillions6Quadrillions7Quintillions8Sextillions9Septillions10Octillions11Nonillions12Decillions

23HowmaywethinkoftheordersofthesuccessiveperiodsasbeingbuiltupofbundlesoflowerunitsTaketheldquounitsrdquoperiodThelargestdigitthatcanappearintheunitsorderis

9Nowadd1to9anditbecomesabundleoften=10Thismeansdigit1intheldquotensrdquoorderandzerointheunitsorderNotethattheldquotensrdquopositionis10times

theunitsposition

Thelargestnumberthatcanappearintheldquotensrdquoandldquounitsrdquoordersis99Nowadd1to99anditbecomesabundleofonehundred=100Thismeansdigit1intheldquohundredsrdquoorderandzeroinboththetensandunitsorders100mayalsobethoughtofasmadeupof10bundlesofldquotensrdquoNotethattheldquohundredsrdquopositionistentimestheldquotensrdquoposition

NowtaketheldquothousandsrdquoperiodThelargestnumberthatcanappearintheldquounitsrdquoperiodis999Nowadd1to999anditbecomesabundleofonethousand=1000Thismeansdigit1intheldquounitsrdquoorderofthisperiodandzerosintheordersoftheunitsperiodTheldquothousandsrdquopositionistentimestheldquohundredsrdquoposition1000mayalsobethoughtofasmadeupof10bundlesofonehundredsor100bundlesoftens

Thelargestnumberthatcanappearintheldquounitsrdquoorderofthisperiodtogetherwiththeunitsperiodis9999Nowadd1to9999anditbecomesabundleoftenthousand=10000Thismeansdigit1intheldquotensrdquoorderofthisperiodandzerosinalltheotherplaces10000mayalsobethoughtofasmadeupof10bundlesofonethousands100bundlesofonehundredsor1000bundlesoftensTheldquotenthousandsrdquopositionistentimestheldquothousandsrdquoposition

Thelargestnumberthatcanappearinthetensandunitsordersofthisperiodtogetherwithentireunitsperiodis99999Nowadd1to99999anditbecomesabundleofonehundredthousand=100000Thismeansdigit1intheldquohundredsrdquoorderofthisperiodandzeroinalltheotherplaces100000mayalsobethoughtofasmadeupof10bundlesoftenthousands100bundlesofonethousands1000bundlesofonehundredsor10000bundlesoftensTheldquohundredthousandrdquopositionistentimestheldquotenthousandsrdquoposition

FollowasimilarprocedureintheldquomillionsrdquoperiodAdd1to999999gettingabundleofonemillion=1000000Digit1isintheunitsorderofthisperiod1000000maybegottenby10bundlesofonehundredthousands100bundlesoftenthousands1000bundlesofthousands10000bundlesofhundredsor100000bundlesoftensTheldquomillionsrdquopositionistentimestheldquohundredthousandrdquoposition

Add1to9999999gettingabundleoftenmillion=1000000010000000mayalsobegottenby10bundlesofonemillions100bundlesofonehundredthousands1000bundlesoftenthousands10000bundlesofthousands100000bundlesofhundredsor1000000bundlesoftensTheldquotenmillionsrdquopositionistentimestheldquomillionsrdquoposition

Add1to99999999gettingabundleofonehundredmillion=100000000whichmayalsobegottenby10bundlesoftenmillions100bundlesofmillions1000bundlesofonehundredthousands10000bundlesoftenthousands100000bundlesofonethousands1000000bundlesofhundreds10000000bundlesoftens

100000000=10times10000000

Thisprocedurecanbecontinuedtotheotherperiodswhichfollowthisone

NotetherelationofthebundlesAnybundleistentimesthesizeofthebundleonitsrightandonetenththatofabundleatitsimmediateleft

24WhenisadecimalpointusedItisusedtoexpressvalueslessthanone

EXAMPLES

02=twotenthsofoneunit= infractionform

002=twohundredthsofoneunit= infractionform

0002=twothousandthsofoneunit= infractionform

00002=twotenthousandthsofoneunit= infractionform

ForanothermethodofwritingdecimalsseeQuestion536

25Whatarethenamesofthedecimalorfractionalplaces

NotethevalueofthedecimalbecomessmallerandsmallerasyouadvancetotherightAlsothereisnounitsplaceafterthedecimalpointThisreducesthenumberofplacesby1ascomparedwithawholenumber

26HowisadecimalreadReadexactlyasifitwereawholenumberbutwiththeadditionofthe

fractionalnameofthelowestplaceTheabovenumberisreadasldquosixhundredeightymillionfiftyseventhousandninehundredtwenty-threebillionthsrdquoThelowestorsmallestplacehereisbillionths

27WhatistherelationofeachplaceinadecimaltotheplacethatprecedesitEachplaceisone-tenth( )oftheprecedingplaceItisthusaten(10)times

smallerfraction

EXAMPLE

ReadTwohundredforty-seventhousandeighthundredninety-sevenmillionths

28CanyoushowthatzerosaddedafterthelastdigitdonotaffectthevalueofthedecimalEXAMPLE

29HowdoesazeroplacedbeforeadigitaffectthevalueofthedecimalThevalueofadigitisdividedbytenasyoumovefromlefttorightSo

addingazerobeforethedigitmovesthedigitoneplacetotherightandmakesitsvalueonetenthofwhatitwas

EXAMPLE

Addingtwozerosmovesthedigittwoplacestotherightandmakesitsvalueonehundredthofwhatitwas

EXAMPLE

Eachadditionalzeroreducesitsformervaluebyonetenthagain

30HowisanumberreadthatconsistsofawholenumberandadecimalThepointseparatesthewholenumberfromthedecimalThedecimalpointis

readldquoandrdquo

EXAMPLE2451ReadTwenty-fourandfifty-onehundredthsItmayalsobereadTwenty-fourpointfifty-one

Toavoidanypossibilitythatthedecimalpointwillbeoverlookedwrite06insteadof6(=sixtenths)

31HowdowewritedollarsandcentsPlaceadecimalpointbetweenthedollarsandcents$1643=sixteendollars

forty-threecents

Numberstotheleftofthedecimalpointaredollarstotherightofitarecentsinthefirsttwoplaceswithanumberinthethirdplaceasmills$16437=sixteendollarsforty-threecentssevenmills

Note10mills=1cent=$001Thereforeforty-threecentssevenmills=fourhundredthirtysevenmills

Whenthenumberofcentsislessthan10writeazerointhetenthsplaceattherightofthedecimalpoint

$308=threedollarseightcents$310=threedollarstencents

32WhataretheessentialsymbolsintheRomansystemofnumerationInheritedfromtheEtruscanstheRomansystemofnotationusessevencapital

lettersofthealphabetandcombinationsoftheseletterstoexpressnumbers

Abaroveralettermultipliesitsvalueby1000

33Whataretherulesforthevaluesofthesymbolswhenusedincombinations(a)Eachrepetitionofaletterrepeatsitsvalue

EXAMPLES

II=2III=3XX=20XXX=30CCC=300MM=2000

(b)Aletterafteroneofgreatervalueisaddedtoit

EXAMPLES

(c)Aletterbeforeoneofgreatervalueissubtractedfromit

EXAMPLES

(d)Aletterbetweentwolettersofgreatervalueissubtractedfromtheletterwhichfollowsit

EXAMPLES

PROBLEMS1

1Howmanyunitsin379

2Howmanytensin304060

3Howmanytensandunitsin1937467296

4Howmanybundlesofhundredsin300500700900

5Howmanybundlesofhundredstensandunitsin76523448953697765885456798548958842891346738

6Whatis1000calledandhowmanybundlesofhundredsareinit

7Howmanybundlesofthousandshundredstensandunitsaretherein748680909935580325002925762392604087607978503374783959749294

8Whatis10000calledandhowmanybundlesofthousandsareinit

9Howmanybundlesoftenthousandsthousandshundredstensandunitsarein603084695137568453828946563895349569285798975203064595199358349259887229573

10Howmanybundlesofthousandsarein100000andwhatisthisnumbercalled

11Howmanybundlesofhundredthousandstenthousandsthousandshundredstensandunitsarein369243780979703148282297503005386470460007386364117008204951596382245520498287995193579697

12Whatis1000000calledandhowmanybundlesofthousandstenthousandsandtensareinit

13Howmanybundlesofmillionshundredthousandstenthousandsthousandshundredstensandunitsarein1753002752060082852394289594723795000946028017373111427550005830310047328500015590389214237295296086000829307118392862863401

14Whatis1000000000calledandhowmanybundlesofhundredmillionsandthousandsareinit

15Howmanybundlesofbillionshundredmillionstenmillionsmillions

hundredthousandstenthousandsthousandshundredstensandunitsaretherein27392496000140676200170024060104078410751073964325701900800005

16Howwouldyouexpressthefollowinginfiguresusingacommatoseparatetheperiods(a)Fivehundredeighty-four(b)Threehundredseventeen(c)Sixhundredninety-nine(d)Threehundredseven(e)Onethousandfourhundredeighty-three(f)Eightthousandsixty(g)Ninethousandfourhundred(h)Fourteenthousandsixhundredforty(i)Eighty-eightthousandsix(j)Sixty-sixthousandeighteen(k)Threehundredseventhousandtwohundredforty(l)Eightthousandeight(m)Fourthousandninety-nine(n)Seventythousandtwenty-three(o)Sevenhundredninety-fourthousandthree(p)Sixty-twothousandtwohundredthree(q)Twomilliontwohundredeighty-fivethousand(r)Thirty-eightmilliononehundredforty-eightthousand(s)Sevenmilliontwo(t)Sixty-onemillionfifty-eightthousandsix(u)Onehundredtwenty-twobillionseventythousandseven(v)Fivebillionsevenmillioneightthousandninehundrednine(w)Eighteenbilliononemilliontwohundredthreethousandsixteen(x)Tentrilliontwobilliononemillionsevenhundredsix(y)Onehundredmilliontwenty(z)Sixtymillionsixhundredthousandsixhundred

17Howarethefollowingexpressedasdecimals(a)Seventy-threethousandfivehundredeighty-sixhundred-thousandths(b)Eightthousandandeightthousandths(c)Fivetenthsthreetenthstwoandonetenth(d)Sevenandninethousandthstwelvemillionths(e)Twohundredthirty-fivethousandthsfourhundredninety-one

thousandthssixten-thousandthsthreehundredandthreehundredths(f)Fourandsevententhsnineandtwotenthseighty-sixhundredthsfivehundredandfivethousandths

(g)(h)Threehundredsixty-fourthousandfivehundredseventy-fivemillionths(i)Ninehundredeightmillionsixthousandthirty-fourbillionths

18Whatisthenameoftheplaceattherightoftenthsattherightofhundredthsattherightofthousandthsthefourthplacethefifththesixththeseventh

19Howarethefollowingread(a)16005(b)50607(c)00002(d)879375(e)35201(f)865392(g)23441(h)2003487(i)202074(j)20610057(k)30564(l)974356

20Howarethefollowingreadindollarstenthsandhundredthsofadollar(a)$457(b)$555(c)$666(d)$999

21Howis$356356read

22Howarethefollowingreadasdollarsdimesandcentsandasdollarsandcents(a)$652(b)$344(c)$555(d)$975

(e)$444(f)$888

23Howarethefollowingwrittenascentsusingthedollarsign(a)Sixty-sixhundredthsofadollar(b)Eightyhundredthsofadollar(c)Forty-sevenhundredthsofadollar(d)Tenhundredthsofadollar(e)Onedollarandtwentyhundredths(f)Sevendollarsandtwelvehundredths

24Howarethefollowingwrittenindecimalform

(a)

(b)

(c)

(d)

(e)(f)Fivehundredths(g)Fifty-sixten-thousandths(h)Eleventhousandandthirty-sixtenths(i)Fivehundredhundredths(j)Sixhundredforty-threeten-thousandths

25Howmanymillsaretherein(a)$0475(b)$5621(c)$0022(d)$1054(e)$10765(f)$02555(g)$010(h)$04444

26HowarethefollowingexpressedinArabicnotation(a)XI(b)VIII

(c)XX(d)XIV(e)XXX(f)XXXV(g)XL(h)LXXV(i)XVI(j)XCIV(k)LV

(l)DCCC(m)MCMXX(n)LXXXIII(o)(p)XLIX(q)MDCCCXCVI(r)XCV(s)MDLXXXIX(t)MCXLV(u)MCXL(v)CDIX(w)DCIX(x)MDLIV(y)MDLX(z)MDXLVII(arsquo)MMDCCXCII(brsquo)(crsquo)(drsquo)(ersquo)MMMDCCXIX(frsquo)(grsquo)

27HowwouldyouexpressthefollowinginRomannotation(a)12(b)18(c)19(d)43(e)33

(f)28(g)56(h)82(i)76(j)97(k)117()385(m)240(n)512(o)470(p)742(q)422(r)942(s)1426(t)1874(u)5872(v)24764(w)257846(x)1450729(y)4840005(z)10562942

CHAPTERI

ADDITION

34WhyisadditionmerelyashortwayofcountingIfwehavefourapplesinonegroupandfiveinanotherwemaycountfrom

thefirstobjectinonegrouptothelastobjectintheotherandobtaintheresultnineButseeingthat4+5=9underallconditionswemakeuseofthisfactwithoutstoppingtocounteachtimewemeetthisproblem

TheadditionoftwonumbersisthusseentobeaprocessofregroupingWedonotincreaseanythingwemerelyregroupthenumbers

35WhatisourstandardgrouporbundleOurnumbersystemisbasedongroupsorbundlesoften

EXAMPLE9+8=17Twogroupsof9and8areregroupedintoourstandardarrangementof17oronebundleof10and7unitsWhilewesayldquoseventeenrdquowemustthinkldquotenandsevenrdquoorldquo1tenand7unitsrdquo

36WhatisthusmeantbyadditionItistheprocessoffindingthenumberthatisequaltotwoormorenumbers

groupedtogether

37WhatismeantbysumItistheresultobtainedbyaddingnumbers

38Ofthetotalnumberof45additionsoftwodigitsatatimeforalltheninedigitswhichgivesinglenumbersasasumandwhichgivedoublenumbers(a)Thefollowing20pairsresultinone-numbersums

(b)Thefollowing25pairsgivedoublenumbers

39WhatistheruleforadditionWritethenumberssothatunitsstandunderunitstensundertenshundreds

underhundredsetcBeginattherightandaddtheunitscolumnPutdowntheunitsdigitofthesumandcarrytheldquotensrdquobundlestothenextcolumnrepresentingtheldquotensrdquobundlesDothesamewiththiscolumnPutdownthedigitrepresentingthenumberoftensandcarryanyldquohundredsrdquobundlestothehundredscolumnContinueinthesamemannerwithothercolumns

40WhatistheproperwayofaddingAddwithoutnamingnumbersmerelysums

EXAMPLE

41WhatisthesimplestbutslowestwayofaddingColumnbycolumnandonedigitatatimeAddfromthetopdownorfrom

thebottomupeachwayisacheckontheother

EXAMPLE

42WhatisavariationoftheaboveAddeachcolumnseparatelyWriteonesumundertheotherbutseteach

successivesumonespacetotheleftAsubsequentadditiongivesthetotalorsum

EXAMPLE(asabove)

43HowcangroupingofnumbershelpyouinadditionAddtwoormorenumbersatatimetotwoormoreothersinthecolumns

EXAMPLE

44HowisadditionaccomplishedbymultiplicationoftheaverageofagroupWhenyouhaveagroupofnumberswhosemiddlefigureistheaverageofthe

groupthen

sum=averagenumbertimesnumberoffiguresinthegroup

EXAMPLES(a)Of45and6number5=averageofthethree

there4Sum=5times3=15=(4+5+6)(b)Of89and109=average

there4Sum=times3=27=(8+9+10)(c)Of1213and1413=average

there4Sum=13times3=39=(12+13+14)

(d)Of6789and108=averagethere4Sum=8times5=40=(6+7+8+9+10)

(e)Of11121314and1513=averagethere4Sum=13times5=65

Notethatwheneveranoddnumberofequallyspacedfiguresappearsyoucanimmediatelyspotthecenteroneoraverageandpromptlygetthesumofallbymultiplyingtheaveragebythenumberoffiguresinthegroup

45Whatistheprocedureforaddingtwocolumnsatatime

37StartatbottomAdd96to80ofabovethenthe2getting24178Add178tothe20abovethenthe4getting202Add82202tothe

30abovethenthe7getting239=sum

Avariationwouldbetoaddtheunitsofthelineaboveitfirstandthenthetensas

46HowarethreecolumnsaddedatonetimeStartatbottomAddhundredsthentensthenunitsasyoucontinueup

EXAMPLES(a)

(b)

47WhatisaconvenientwayofaddingtwosmallquantitiesbymakingadecimalofoneofthemMakeadecimalofonebyaddingorsubtractingandreversethetreatmentfor

theother

EXAMPLE96+78

Add4to96getting100=decimalnumberSubtract4from78getting74there4Sum=174atonce

48HowmaydecimalizedadditionbecarriedouttoafullerdevelopmentReduceeachnumbertoadecimalAddthedecimalsAddorsubtractthe

increments

EXAMPLE

49Howmaysightreadingbeusedinaddition

Byuseofinstinctyougetanimmediateresult

EXAMPLES

(a)Add26to53

(b)Add67to86

Fixeyesbetweenthetwocolumnswherethedotsareandatonceseea7anda9ora13anda14tomake153Actually70isaddedto9and140to13buteachisdoneinstinctively

50WhatsimplemethodisusedtocheckthecorrectnessofadditionofacolumnofnumbersFirstbeginatthebottomandaddupThenbeginatthetopandadddown

WhenthecolumnsarelongitisoftenbettertowritedownthesumsratherthantocarrytheldquobundlesrdquofromcolumntocolumnPlacesumsinpropercolumns

EXAMPLE

51WhatismeantbyacheckfigureinadditionOnewhichwheneliminatedfromeachnumbertobeaddedandfromthesum

willgiveakeynumberthatmayindicatethecorrectnessoftheadditionThechecknumbers9and11aregenerallyused

52Whataretheinterestingfactsontheuseofthechecknumber9(1)Thefactthattheremainderleftafterdividinganynumberby9isthesame

astheremainderofthesumofthedigitsofthatnumberdividedby9

Ex(a)

Ex(b)

(2)Alsonotethatthesumofthedigitsalonewillgivethesamenumberasaremainderasthedivisionofthenumberby9Thusin(a)6+5+4=15and1+5=⑥In(b)2+6+7+7=22and2+2=④(3)Alsothefactthat9rsquoscanbediscardedwhenaddingthedigitsThusin(a)

6+5+4discard4+5rightawayandtheremainderisagain⑥In(b)2+6+7+7discard2+7butadd6+7=13and1+3=④

53Whatistheprocedureincheckingadditionbytheuseofthecheckfigure9oftencalledldquocastingoutninesrdquo(a)Addthedigitsineachnumberhorizontallyandgeteachremainder

(b)Addthedigitsoftheseremaindersandgetthekeyfigure

(c)Addthedigitshorizontallyoftheanswerandgetthesamekeyfigureiftheansweriscorrect

EXAMPLE

Inpracticeitissufficienttoaddthenumbersmentallytogettheremainders

Notethatall9rsquosanddigitsthataddupto9arediscardedrightawayEachdigitsodiscardedisshownwithadotattheupperrightcorner

54WhyisldquocastingoutninesrdquonotaperfecttestofaccuracyinadditionItispossibletoomitoraddninesorzeroswithoutdetectionAlsofiguresmay

betransposed27isquitedifferentinvaluefrom72althoughthesumofthedigitsisthesame

ThismethodisnotgenerallyrecommendedasapracticaltestinadditionworkbuthasitsgreatestvalueinmultiplicationanddivisionworkHoweveritissometimesusefulasaquickcheckofaddition

55Whataretheinterestingfactsontheuseofthechecknumber11(1)Theremainderleftafterdividinganynumberby11isthesameasthe

remainderleftaftersubtractingthesumofthedigitsintheevenplacesfromthesumofthedigitsintheoddplacesIfthesubtractioncannotbemadeadd11oramultipleofittotheodd-placessum

EXAMPLES

(a)

(b)

(2)ThesameremainderisalsoobtainedbystartingwiththeextremeleftdigitinthenumberandsubtractingitfromthedigittoitsrightWhennecessaryadd11tomakethesubtractionpossibleSubtracttheremainderfromthenextdigitAgainadd11ifnecessaryRepeattheprocessofsubtractionuntilallthedigitsofthenumberhavebeenused

56Whyisthecheckingofadditionworkbytheuseofthecheckfigure11(oftencalledldquocastingoutelevensrdquo)superiortothatofldquocastingoutninesrdquoldquoCastingoutelevensrdquocanindicateanerrorduetotranspositionofdigits

whichisnotpossiblewiththeldquoninesrdquomethod

EXAMPLESupposeournumberis8706

8from(11+7)=1010from(11+0)=1Ifrom6=⑤=Remainder=Checknumber

Nowsupposethetransposednumberis8076

8from(11+0)=3 3from7=44from6=②=Remainder=Checknumber

Thechecknumbersareseentobedifferentandwehaveuncoveredatranspositionofdigits

57Whatistheprocedureincheckingadditionbytheuseofthecheckfigure11(a)Castoutelevensfromeachrowandgeteachremainder

(b)Addtheremaindersandcastoutelevensfromthissumgettingthekeyfigure

(c)CastoutelevensfromtheanswerandgetkeyfigureCompare

EXAMPLE

PROBLEMS

1Countfrom3to99by3rsquos




5Startwith3andcountby2rsquos4rsquos6rsquos8rsquostojustbelow100




9Add269745and983

10Addusingldquocarryoversrdquo

11Add$525$1760$085$175$4565

12Findthesumof

(a ) (b ) (c)

$380865 $987367 $887406

37692 38898 51856

38623 573200 129897

48008 898719 54265

88842 782492 38600

75182 608604 4209

13Whatisthesumof102030bytheaveragemethod








21Addtwocolumnsatatime

22Addthreecolumnsatatime

23Addthefollowingbythedecimalizingmethod(a)94+75(b)86+69(c)92+48(d)89+52(e)468+982+429(f)346+899+212(g)589+913+165(h)862+791+386

24Addbysightreading(a)27+56(b)21+43(c)32+65(d)49+57(e)68+87(f)76+82

25Agasolinestationownerhad275gallonsleftafterselling632gallonsHowmanygallonsdidhehaveoriginally

26Onepipefromatankdischarges76gallonspersecondwhileanotherpipefromthesametankdischarges16gallonsperminutemorethanthefirstHowmanygallonswillbothpipesdischargeinaminute

27Anautomobiletravels386milesonthefirstdayand416milestheseconddayatwhichtimeitis237milesfromitspointofdestinationWhatisthedistancefromitsstartingpointtoitsdestination

28Asuburbanhousewasbuiltwiththefollowingexpensesmasonry$3565lumber$4850millwork$1485carpentry$3800plumbing$2758painting$679hardware$1508heating$1250andelectricity$687Whatdidthehousecostwhencompleted

29Ifafamilyoftwopersonsspends$135forrent$205forfood$85forclothing$35forfuel$7forlight$22forinsurance$6forcarfare$12forcharityandsaves$18whatistheincomeaftertaxesandotherpayrolldeductions

30Thetwenty-secondofFebruaryishowmanydaysafterNewYearrsquosHowmanydaysfromNewYearrsquostothefourthofJuly

31CheckthefollowingbyfirstaddingupandthenbyaddingdownPlacecheckmarksasproof

32Provethefollowingbyuseofthecheckfigure9


34Addhorizontallyandvertically

(a)

(b)

CHAPTERII

SUBTRACTION

58WhatissubtractionItisthereverseofadditionSinceweknowthatfiveapples+threeapples=

eightapplesitfollowsreverselythattakingfiveapplesawayfromeightapplesleavesthreeapples

Ortakingthreeapplesawayfromeightapplesleavesfiveapples

8minus5=3 8minus3=5

Aswithadditionsubtractionisthusseentobemerelyaregrouping

group(a)+group(b)=group(c)=8group(c)ndashgroup(a)=3 group(c)ndashgroup(b)=5

59WhymaysubtractionbesaidtobeaformofadditionEx(a)9ndash4=5

Maybethoughtofasldquo4andwhatmake9rdquo4and5make9

Ex(b)16minus9=7

9andwhatmake169and7make16

60Whatthreequestionswillleadtotheprocessofsubtraction(a)Howmuchremains

(b)Howmuchmoreisrequired

(c)Byhowmuchdotheydiffer

In(a)ifBerthas$10andpaysout$6howmanydollarsremainHerethe$6wasoriginallyapartofthe$10

In(b)Berthas$65andwouldliketobuya35-mmcamerathatcosts$89Howmuchmoredoesherequire

In(c)ifBerthas$10andCharleshas$6byhowmuchdotheydifferHerethe$10andthe$6aredistinctnumbers

61Whatarethetermsofasubtraction

IfthesubtrahendwasoriginallyapartoftheminuendthentheansweriscalledtheldquoremainderrdquoIftheminuendandsubtrahendaredistinctnumberstheansweriscalledtheldquodifferencerdquo

62WhyisitsaidthatwecanalwaysaddbutwecannotalwayssubtractSubtractionisnotalwayspossibleItisnotwhenthenumberofthingswhich

wewishtosubtractisgreaterthanthenumberofthingswehave

Ex(a)

Addition5apples+3apples=8applesSubtraction8applesminus3apples=5applesAddition5apples+7apples=12applesSubtraction5applesminus7apples=impossible

ThereexistnonegativeapplesAtbestwecanonlyexpresstherelationas2applesmissing

Ex(b)

7foot-candlesofilluminationminus5foot-candles=2foot-candles

7foot-candlesminus9foot-candlesisimpossiblebecausetherecannotbeanegativeilluminationof2foot-candlesThelimitiszeroilluminationordarkness

Ex(c)Fromanelectriccordof8feetwecancutoff3feetleaving5feetbutwecannotcutoff10feetleavingminus2feetofcord

63WhenisitpossibletosubtractwiththenumberexpressingthesubtrahendgreaterthanthenumberexpressingtheminuendByintroductionoftheconceptofldquodirectionrdquotothequantitiesexpressedby

thenumbersandcallingallnumbersinonedirectionpositivenumbersandnumbersinthereversedirection(fromthestartingpointzero)negativenumbers

Ex(a)

Nowifwestepoff5stepstotherightandthenstepoff7totheleftwelandatminus2

there45minus7=minus2

Ex(b)Ifweletzero=freezingtemperaturethen+5degis5degreesabovefreezingandifitfalls3degreesitwillbe2degreesabovefreezingIfitfalls7degreesitwillbe2degreesbelowfreezingor

Ex(c)Ifzeroislatitudethen+5deglatminus7deglat=minus2deglatThiswouldbeintheSouthernHemisphere

Ifwehave$5inthebankandifwehavecreditwemaybeabletodrawout$7inwhichcase$5minus$7=minus$2overdraftAgainifwehave$10inourpocketandbuysomethingthatcosts$25weareindebtfor$15$10minus$25=minus$15debt

Thenegativenumberisnotaphysicalbutamathematicalconceptionwhichmayormaynothaveaphysicalrepresentationdependingonhowitisapplied

64Whatisthesubtractiontablethatshouldbestudieduntiltheanswerscanbegivenquicklyandcorrectly

SubtractionTable

65Whatistheruleforsubtraction(a)Writethesubtrahendundertheminuendunitsunderunitstensundertens

etc

(b)Beginattherightandsubtracteachfigureofsubtrahendfromthecorrespondingfigureoftheminuendandwritetheremainderunderneath

(c)Ifanyfigureofthesubtrahendisgreaterthantheminuendincreasetheminuendby10(whichuses1unitofthenexthigherorder)andsubtractNowreducetheminuendofthenexthigherorderby1andcontinuetosubtractuntilallthedigitshavebeentakencareof

NotethatyoudonotactuallyaddortakeawayanythingfromthenumberYoumerelyregroupabundlebyunscramblingitandplacingitwiththelowerordertomakethesubtractionpossibleInEx(c)abovewecanseethatwewillneedonethousandsbundletounscrambleto10hundredsonehundredsbundletobecome10tensandonetensbundletobecome10unitsThenumbersthenbecome

66WhatisknownasthemethodofldquoequaladditionsrdquoinsubtractionThemethodisbasedonthefactthatthesamenumbermaybeaddedtoboth

minuendandsubtrahendwithoutchangingthevalueofthedifference

Ex(a)

Ex(b)

ThismethodisquickandsimpleAllyouneedtorememberistoadd1tothenextcolumninthesubtrahendeverytimeyouadd10totheminuendtomakesubtractionpossible

Ex(c)

67WhatisthemodeofthinkingofsubtractionthatiscalledtheAustrianmethodorthemethodofmakingchangeAgooddealofsubtractioninthebusinessworldisconcernedwithmaking

changeItconsistsinbuildingtothesubtrahenduntiltheminuendisreached

Ex(a)

Whensubtractionistobemadepossibleinanycolumnitbecomesamodificationoftheaboveldquoequaladditionrdquomethod

Ex(b)

68HowmaysubtractionbesimplifiedAddorsubtractaquantitytogetamultipleof10Itiseasiertosubtracta

multipleof10fromanotherquantitythantosubtractanyotherdoubledigitnumber

EXAMPLE

Notethattheansweristhesamewhenyouaddorsubtractthesamenumberfromboththeminuendandsubtrahendandthatitiseasiertosubtractwhenthesubtrahendismadeamultipleof10

69HowmaytheabovebeextendedDividethenumbersintocouplesandmakeeachcoupleamultipleof10

(whichisknownasadecimalnumber)

Ex(a)

Ifthesubtrahendinonecoupleislargerthantheminuendtherewillbe1tocarrywhichissubtractedfromthedifferencesofthecouplenextontheleft

Ex(b)

Insubtracting70from52borrowone(hundred)thensubtract1fromthedifferenceof(99ndash40)

Ex(c)

70Howcanthesubtractionoftwo-figurenumbersbedonebysimpleinspectionusingdecimalizationEx(a)

Ex(b)

89minus47=40+9minus7=4298minus36=60+8minus6=6295minus22=70+5minus2=73

71Howcaninvertedorleft-handsubtractionbedoneStartfromtheleftandsubtractnotingwhetherthereisonetocarryfromthe

columnattheright

Ex(a)

Ex(b)

72WhatismeantbythearithmeticalcomplementofanumberAbbreviatedacarithmeticalcomplementistheremainderfoundby

subtractingthenumberfromthenexthighestmultipleof10

EXAMPLE

acof2is10minus2=8acof57is100minus57=43acof358is1000minus358=642acof0358is1000minus0358=0642

73WhatisthesimplestwayofcalculatingtheacofanumberSubtractitsright-handdigitfrom10andeachoftheothersfrom9Thisdoes

awaywithcarryingof1rsquos

EXAMPLEacof68753=31247

Startatleft

6from9=38from9=17from9=25from9=43from10=7

74WhenandhowistheacusedinsubtractionWhenaquantityistobesubtractedfromthesumofseveralothersTo

subtractbymeansoftheacaddtheacofthesubtrahendandsubtractthemultipleof10usedingettingtheac

Ex(a)Subtract9431from9805byac

Nothingisgainedbyuseofacinsosimpleacase

Ex(b)Subtract1284fromthesumof97471283and1292

Ex(c)Frombankdepositsof$22680$34261and$18734deductawithdrawalof$56079togetthenetincrease

75Howdoweproceedtogivechangetoacustomerbytheuseoftheso-calledldquoAustrianmethodrdquoofsubtractionAddfromtheamountofthepurchaseuptothenexthighermoneyunitthen

tothenextandsoonuntilyoureachtheamountofthebilltenderedinpayment

EXAMPLEIfthebillgiveninpaymentis$5andthepurchaseis$238givecustomerthefollowingaschange2centstomake$24010centstomake$25050centstomake$300$2tomake$5

Totalchangeaddsupto$262

76WhatisthebestcheckinsubtractionThesumofremainderandsubtrahendmustequaltheminuendThismeans

wehavetakenawayacertainnumberwenowputitbackandreturntotheoriginalnumberThischeckshouldalwaysbemadeItisdonementally

EXAMPLES

77IsldquocastingoutninesrdquoapracticalcheckinsubtractionItisnotandtoomuchtimemustnotbespentonthismethod

Ex(a)

Itisseenthatthedifferencebetweentheremaindersoftheminuendandsubtrahend=remainderofanswer

Ex(b)

78MaycastingoutofelevensbeusedasacheckYesbutherealsotoomuchtimeshouldnotbedevotedtothismethod

Ex(a)

TaketheminuendStartatleft

TakethesubtrahendStartatleft

Ex(b)

PROBLEMS

Performthefollowingsubtractions

1

2

3

4

5

6

7Ifwesayacertaintreeisinzeropositionandwetake8stepstotherightofthetreewhichwecallthepositivedirectionandthenwestepoff12stepstotheleftwherewillweland

8Ifzeroisfreezingtemperaturewhatdoes+7degmeanWhatdoesminus8degmean

9Ifyourlatitudeiszeroandyoutravelnorthto+11deglatandthensouthwardfor15degwhatwouldbeyourlastposition

10Ifyouhad$85inthebankandyouissuedacheckfor$97whatwouldbeyouroverdraft

11Ifyouhadonly$63andyouwantedtobuya35-mmcamerathatcost$87howmuchwouldyoubeindebt

12Subtract

13Checktheanswerstoproblem12byadditionChecktheanswersbycastingoutninesChecktheanswersbycastingoutelevens

14Whatisthesubtrahendforeachofthefollowingsetsofvalues

15Checktheanswerstoproblem14byadditionandbycastingoutnines

16Usethesimplifiedmethodofsubtractionbymakingthesubtrahendamultipleoften

17Extendthesimplifiedmethodofsubtractiontotwocouplesmakingeachamultipleoftenoradecimalnumber

18Dothefollowingsubtractionsoftwo-figurenumbersbysimpleinspectionusingdecimalization

19Dothefollowingbyinvertedorleft-handsubtraction

20Whatisthearithmeticalcomplementof(a)7(b)69(c)472(d)1282(e)0472(f)79864(g)864348

21(a)Subtract8562from9983byacmethod(b)Subtract46827from87962byacmethod

22Subtract4976fromthesumof84321343and1565byacmethod

23Frombankdepositsof$34276$56259and$13459deductawithdrawalof$63248byacmethod

24Ifa$20billisgiveninpaymentandthepurchaseis$1289whatchangewillthecustomergetusingtheso-calledldquoAustrianrdquomethodofsubtraction

25Ifarailroadcarries2325879passengersoneyearand3874455passengersthefollowingyearwhatistheincrease

26IftheFederalincometaxcollectedoneyearis$67892762945and$71432652982thefollowingyearwhatistheincrease

27(a)Beginwith53andsubtractby2rsquos4rsquos6rsquos8rsquos(b)Beginwith89andsubtractby3rsquos5rsquos7rsquos9rsquos(c)Beginwith74andsubtractby5rsquos7rsquos3rsquos9rsquos

28Amanboughtafarmfor$17500Hekeptittwomonthsduringwhichtimehepaid$43950intaxesand$78275forrepairoffencesHethensolditfor$21500Whatwashisprofit

CHAPTERIII

MULTIPLICATION

79WhatismultiplicationItismerelyasimplifiedformofadditionSupposewehaveeightapplesina

rowandtherearefourrowsWecanaddthemas8+8+8+8=32orwecansaysimply4times8=32Alsoifwehavefourapplesinarowandthereareeightrowsthen

4+4+4+4+4+4+4+4=32or8times4=32

Youseethat4times8=8times4=32Ineachcasethesumis32Whenseveralequalnumbersaretobeaddeditismuchshortertoobtaintheresultbymultiplication

80Whatarethetermsofamultiplication(a)Thenumbertoberepeatediscalledthemultiplicand

(b)Thenumberoftimesthemultiplicandistoberepeatediscalledthemultiplier

(c)Theresultofthemultiplicationiscalledtheproduct

(d)Themultiplicandandthemultiplierarealsoknownasthefactorsoftheproduct

EXAMPLE

81Whatis(a)aconcretenumber(b)anabstractnumber(c)thetypeofnumberofthemultiplierinmultiplication(a)Anumberthatisappliedtoanyparticularobjectiscalledaconcrete

numberExamplesanappleanauto2hoursetc

(b)AnumberthatisnotappliedtoaparticularobjectisanabstractnumberExamples1562

(c)Inmultiplicationthemultiplierisalwaysanabstractnumber

82Whatarethemostusefulproductsthatshouldbecommittedtomemory

MultiplicationTable

83WhenseveralnumbersaremultiplieddoesitmatterinwhatorderthemultiplicationisperformedTheorderofmultiplicationdoesnotmatter

EXAMPLE2times6times4=2times(6times4)=(2times4)times6=48

The2maybemultipliedby6andthisresult(=12)maythenbemultipliedby4toget48orthe6and4mayfirstbemultipliedandthenthe2usedetc

84Whatistheruleinmultiplicationwhen(a)thetwosignsofthenumbersarebothplus[+](b)bothsignsareminus[ndash](c)thetwosignsareunlike(a)Twoplusesproduceaplusproduct

(b)Twominusesproduceaplusproduct

(c)Twounlikesignsproduceaminusproduct

(+4)times(+6)=+24(+4)times(minus6)=minus24(ndash4)times(minus6)=+24(ndash4)times(+6)=minus24

NoteItisnotnecessarytowritetheplusinfrontoftheproduct

85WhatistheeffectuponanumberwhenyoumoveitonetwothreeplacestotheleftintheperiodMovingafigureoneplacetothelefthasthesameeffectasmultiplyingitby

10Example76times10=760Sotomultiplyby10placeazeroattherightofthemultiplicandthusmovingeachdigitoneplacetotheleftandincreasingitsvalue10times

Tomultiplyby100placetwozerosattherightofthemultiplicandExample76times100=7600

Tomultiplyby1000placethreezerosattherightofthemultiplicandetcExample76times1000=76000

86WhatistheruleformultiplyingwheneithermultiplierormultiplicandendsinzerosMultiplythemultiplicandbythemultiplierwithoutregardtothezerosand

annexasmanyzerosattherightoftheproductasarefoundattherightofthemultiplierandmultiplicand

EXAMPLE

87HowisordinarysimplemultiplicationperformedWritethemultiplierunderthemultiplicandplacingtheunitsofthemultiplier

underunitsofmultiplicandandbeginattherighttomultiply

EXAMPLE

Notethattheworkcanbeshortenedbydoingtheldquocarryingrdquomentally

88WhatistheprocedurewhenthenumberstobemultipliedcontainmorethanonedigitEXAMPLE698times457Itwouldnotbeconvenienttosetdown698tobe

added457times

Multiplyingby457isthereforethesameasmultiplyingby7by50andby400andaddingtheresults

(a)Firstmultiply698by7

7times8=56 Write6carry57times9=63+5=68 Write8carry6

7times6=42+6=48 Write48

(b)Thenmultiplyby50Write0inunitscolumnandthenmultiply698by5

5times8=40 Writezerocarry45times9=45+4=49 Write9carry4


(c)Thenmultiply698by400Write00andmultiply698by4

4times8=32Write2carry34times9=36+3=39Write9carry3

4times6=24+3=27Write27

Nowaddthethreeresultstoget318986=productOfcourseyoumayomitwritingthezeroswhenyouremembertomovetheproductoneplacetotheleftwhenmultiplyingbythedigitinthetenscolumnandtwoplacestotheleftwhenmultiplyingbythedigitinthehundredscolumnetc

89HowcanthefactthateithernumbermaybeusedasthemultiplierservetoprovideacheckonourmultiplicationEXAMPLE(asabove)ReverseUse698asthemultiplier

90Howcanweextendthemultiplicationtablebeyond12times12bymakinguseofthesmallerproductsby2orby4EXAMPLES

(a)14times13=2times7times13=91times2=182Split14into7times2(b)16times13=2times8times13=104times2=208Split16into8times2(c)18times13=2times9x13=117times2=234Split18into9times2(d)16times16=4times4x16=4times64=256Split16into4times4

91Howcanmultiplicationbytwo-digitnumbersbesimplifiedConvertonetwo-digitnumberintotwoone-digitnumbers

Ex

(a)27times16=27times2times8=54times8=432(b)27times15=27times3times5=81times5=405

92Howcanthemultiplicationoftwo2-digitnumbershavingthesamefigureinthetensplacebesimplified(a)Multiplytheunits

(b)AddtheunitsandmultiplythesumbythetensdigitAnnexazero

(c)MultiplythetensAnnex2zeros

(d)Add(a)+(b)+(c)

EXAMPLES(1)

(2)

(3)

93HowcanmultiplicationbesimplifiedbymultiplyingonefactoranddividingtheotherfactorbythesamequantityEx(a)

Theproductisthesamebecause

Thiscouldalsobedoneas

94WhatcanbedonewhenmultiplicationmaysimplifyoneofthefactorsbutwhentheotherfactorisnotdivisiblebythesamenumberIfmultiplicationofonefactormakesthatfactorsimplerusetheresultasthe

multiplieranddividetheproductbythesamenumberusedtosimplifythemultiplier

Ex(a)45times29

Multiplyfactor45by2getting90Now90times29=2610

Dividethisby2getting

Ex(b)323times35

Notethissimplificationappliestonumbersendingin5upto55togiveprocedureswithintherangeofthemultiplicationtable

Ex(c)271times55

95Whenthetensdigitsarealikeandtheunitsdigitsaddupto10howismultiplicationsimplifiedWritetheproductoftheunitsdigitsIncreaseoneofthetensdigitsby1and

multiplybytheother

Ex(a)

Ex(b)

Ex(c)

96Whentheunitsdigitsarealikeandthetensdigitsaddupto10howismultiplicationsimplifiedWritetheproductoftheunitsdigitsAddunitsdigittoproductoftensdigits

Ex(a)

Ex(b)

Ex(c)

97Whenneitherofabovecombinationsisapplicablehowmayso-calledcrossmultiplicationbeappliedtoadvantageEx(a)

Ex(b)

Ex(c)

98Whentheunitsdigitsare5andthesumofthetensdigitsisevenhowismultiplicationsimplifiedTheproductwillendin25Multiplythetensdigitsandaddhalftheirsum

Ex(a)

Ex(b)

99Whentheunitsdigitsare5andthesumofthetensdigitsisoddhowismultiplicationsimplifiedTheproductwillendin75Multiplytensdigitsandaddhalftheirsum

discardingfraction

Ex(a)

ThismethodmaybeusedwhenthereareonlytwoandnotmorethanthreedigitsineithermultiplierormultiplicandWhendollarsandcentsareinvolvedthetwoenddigitsarecentsanddigitstotheleftaredollars

Ex(b)

Ex(c)

Ex(d)

100Whatismeantbyleft-handmultiplicationorwhatissometimescalledinvertedmultiplicationMultiplyleft-handfiguresfirstandthenthenextandaddtheproducts

Ex(a)

Ex(b)

101Whatismeantbyanaliquot(ălrsquoi-kwŏt)partofanumberItisaquantitywhichcanbeadivisorofanumberwithoutleavinga

remainderItisthereforeafactorofthenumber

Ex(a)5isanaliquotpart(orfactor)of20orof35When20or35isdividedby5thereisnoremainder5isafactorofeithernumber

Ex(b) and25gointo100863and4timesrespectivelyandarealiquotpartsof100orfactorsof100

Ex(c) cent10centand25centarealiquotpartsof$100sincetheyarecontained1210and4timesrespectivelyin$100

102WhatismeantbyafractionalequivalentofanaliquotpartBydefinition

Ex(a) (=aliquotpartof100)100isthebaseThen =fractionalequivalentofthealiquotpartof100( )

Ex(b) (=aliquotpartof100)Then =fractionalequivalentofaliquotpartof100

Itisseenthatthefractionalequivalenthasanumeratorof1andadenominator

whichisthenumberoftimesthatthealiquotpartiscontainedinthegivennumber

103WhenaresomenumbersusefulwhilenotaliquotpartsthemselvesTheyareusefulwhentheyareconvenientmultiplesofaliquotparts

Ex(a) isnotanaliquotpartof100sinceitdoesnotgointo100awholenumberoftimesbut isanaliquotpartof100and is Thefractionalequivalentof is of100

there4 is of100

Ex(b) is Thefractionalequivalentof is

there4 is of100

Ex(c)75is3times25Thefractionalequivalentof25is

there475is of100

104Whataresomeofthealiquotpartsof100andtheirfractionalequivalentsWeknowthatanaliquotpartof100isafactorof100

105Howmayaliquotpartsof100bewrittenasdecimalsAnaliquotpartof100meanssomanyhundredthsandmaybewrittenasa

decimalThebaseis100

EXAMPLE(asabove)

Cipherinfrontofaliquotpart

020405062506660833

Decimalpointinfrontofaliquotpart

125133316662025

106WhyarealiquotpartsusefulincalculationsinvolvingdollarsAliquotpartsof100have100partsastheirbasesAs$100=100centsthenofadollar= centsand ofadollar=20cents

EXAMPLEFindcostof72articleswhenthepriceofoneis16

Ifthepriceofanarticlewereadollarthetotalcostwouldbe$7200butsincethepriceisonly ofadollarthetotalcostis =$1200

107Howmayaliquotpartsof100beusedinmultiplication(a)Tomultiplyby50( of100)Multiplyby100byannexingtwozeros

Thendivideby2tomultiplyby50( of100)

EXAMPLE

(b)Tomultiplyby25( of100)Annextwozerostomultiplyby100Thendivideby4tomultiplyby25( of100)

EXAMPLE

(c)Tomultiplyby20( of100)Annextwozerostomultiplyby100Since20is of100divideby5

EXAMPLE

Inthiscaseitwouldgenerallybeeasiertomultiplydirectly

(d)Tomultiplyby75( of100)Annextwozerostomultiplyby100Sinceof100multiplyby

EXAMPLE

108WhatisthepracticaluseofaliquotpartsinmultiplicationAliquotpartsenableustodispensewithfractionsForourusealiquotpartsare

applicabletobasesofhundredsandotherdecimalnumbers

Ex(a)Whatisthecostof65articlesat$250eachThebasehereis10andis of10Thenaddonezeroanddivideby4

Ex(b)Howmuchwill49itemsat costMultiply49by3=$147andaddtoit

Ex(c)Whatisthecostof38articlesat of$100ButThen

Ex(d)Whatistheresultof37519times125

As125is of1000annexthreezerosanddivideby8Thismultipliesthenumberfirstby1000andthendividesby8tofind125asamultiplier

Alsosince125=(100+25)then

Ex(e)Whatisthecostofeachofthefollowing

109MaythenumberofarticlesandthepricebeinterchangedasameansofsimplifyingaprobleminaliquotpartsYesThus yardsat$315canbechangedto315yardsat

EXAMPLEWhatisthecostof16 yardsofclothat69centayardThiscanbechangedto69yardsat ayard

At$100peryard69yardswouldcost$69

But of$100there4 Ans

110Whatisthecostof1780lboffeedat$1500aton

At1centperlb($100per100lb)1780lbcosts$1780 of$100

there4 costof1780lbat$1500perton

111Howcanwesimplifythemultiplicationby24Multiplyby25byannexingtwozerosanddividingby4Subtracttheoriginal

numberfromtheresult

Ex(a)

Ex(b)Avariation261times124124=(100+24)

Then

112Howcanwesimplifythemultiplicationby26Multiplyby25byannexingtwozerosanddividingby4Addtheoriginal

numbertothis

Ex(a)

Ex(b)

113Howcanwemultiplyanumberby9usingsubtractionEXAMPLE

66492times9=59842866492(10minus1)=664920minus66492

114Howcanwemultiplyby11usingadditionEXAMPLE

Inoneline

Putdown2Addthenextfigure9tothe2Putdown1carry1Then4+1+9=14Putdown4carry1Then6+1+4=11Putdown1carry1Then7+1+6=14Putdown4carry1Then7+1=8

115Howcanwemultiplyby111byusingadditionEXAMPLE

Inoneline

76492times111 Putdown2

Add9+2=11 Putdown1carry1

Add4+9+2+carry1=16 Putdown6carry1

Add6+4+9+1carry=20 Putdown0carry2


Add7+6+1carry=14 Putdown4carry1

Add7+1carry=8 Putdown8

116Howcanwesimplifythemultiplicationby8andby7Tomultiplyby8annexazeroandsubtracttwicethenumber

EXAMPLE

Tomultiplyby7annexazeroandsubtract3timesthenumber

EXAMPLE

117Howdowemultiplyby999897orby999998997Annexthepropernumberofzerosandsubtracttherequirednumberoftimes

118WhatismeantbythecomplementofanumberThedifferencebetweenthatnumberandtheunitofanexthigherorder

Ex(a)Complementof7is3becausethedifferencebetween7and10is310isthenexthigherorderof7

Ex(b)Complementof58is42because100minus58is42100isthenexthigherorderof58

119Howiscomplementmultiplicationperformed(a)Findthecomplementofeachnumber

(b)Multiplythecomplementstogether

(c)Subtractoneofthecomplementsfromtheothernumberandmultiplythisby100

(d)Add(b)to(c)

Ex(a)

Multiply92x96 100minus92=8=complement

100minus96=4=complement

8times4=32=productofcomplementsNumber92minus4(=complementof96)=88

88times100=88008800+32=8832Ans

Ex(b)Multiply86times93Complementsare14and7

14times7=98=productofcomplements86minus7=7979times100=7900

7900+98=7998Ans

Ex(c)Multiply942times968Complementsare58and32

Itmaynotpaytousethismethodwiththreefigures

120Howcanwemultiplybyanumberbetween12and20usingonlyonelineintheproductMultiplyasusualbytheunitsfigureofthemultiplierCarryasusualbutalso

addthefigureontherightofthefiguremultipliedThislatteradditiontakescareofthetensfigureofthemultiplier

EXAMPLE

AlltheabovecanbedonementallyofcourseAsyouseebyordinarymultiplicationthemultiplicationofthetensfigure1ofthemultipliermovestheentiremultiplicandoneplacetotheleftandaccountsfortheadditionofthefiguretotherightoftheonebeingmultipliedintheone-lineprocess

121WhatismeantbycrossmultiplicationAmethodofmultiplyingbyanumberofmorethanonedigitwithoutputting

downthepartialproductsThepartialproductsarekeptinmindandonlyonelineresultsastheanswerThesecretistostartwiththeright-handdigitofthemultiplierandcontinuetoprogresstoeachdigitofthemultiplierandaseachisfinishedstartanothertotheleftGettheunitsfirstthenaddupthetenshundredsthousandsetcusingeachdigitofthemultiplierorthemultiplicandAddthecarry-overfigurePuteachproductinitsproperplace

122Whatistheresultof76times64usingcrossmultiplication


Thousands7times8+8carry=64Putdown64

124Howcanwecheckamultiplicationbyldquocastingoutninesrdquo(a)Gettheremainderbyaddingdigitsofmultiplicand

(b)Gettheremainderbyaddingdigitsofmultiplier

(c)Multiplyremainders(a)and(b)togetherandgetremainderofthisproduct

(d)Getremainderoftheanswer(orproduct)

Ifremainderof(c)and(d)arealikethemultiplicationisinallprobabilitycorrect

All9digitsorthosewhichaddupto9arediscardedrightaway

EXAMPLE

Remainderofmultiplicand(4)xremainderofmultiplier(3)=12

1+2=③=sameasremainderofanswerorproduct

ThisisnotanabsoluteproofbutonlyatestofthecorrectnessofthemultiplicationThereversingofmultiplierandmultiplicandrequiresmoretimebutitismoreaccuratebecauseiteliminatesthepossibilityoftransposedfiguresorofninesandzerosbeingaddedoromittederroneously

PROBLEMS

1Multiply54by10by100by1000



4Multiply631by60

5Multiply45by40by400by4000by400000

6Multiply4700by4by40by400by4000by40000

7Multiply6390by300

8Multiply

(a)870by3600(b)785340by4700(c)98750by400(d)87953by45000(e)48800by78000(f)780000by630(g)387470by4000

9Whatistheproductof

(a)4738multipliedby6(b)892by8(c)953by67(d)628by86(e)438by99(f)673by83(g)768by57(h)4174by647(i)587by756(j)9046by839(k)3490by874(l)5947by638(m)6084by519(n)7493by349(o)9486by305(p)9385by3005(q)3795by803(r)9476by8007(s)2583by7001(t)9434by8002(u)8754by408(v)7004by1371(w)8745by49(x)6354by684(y)2851by1212(z)8172by899

10Multiply

(a)$3885by375(b)$73140by457(c)$87234by741(d)$40010by856(e)$134035by704(f)$465020by708

11Amechanicearns$2885adayWhatwillhispaybeforafive-dayweekForamonthof22days

12If28yardsofcarpetarerequiredforafloorwhatwillbethecostat$925ayard

13OnOctober1Johngotatemporaryjobpaying$82aweekHowmuchdidheearnin23weeks

14Ifitcosts$4065forlaborand$3629formaterialtosprayanacreofvineyard5timeswhatwillbethecosttospray8acres5times

15Thereare21750cubicfeetinthefirst6inchesoftopsoilofanacreofgroundHowmuchwillthissoilweighat80lbpercubicfoot

16Amanbought1124acresoflandat$225anacreHespent$83700forimprovementsandthensold8acresat$450anacre270acresat$535anacre325acresat$380anacre360acresat$660anacreandtherestat$100anacreHowmuchdidhegainorlose

17Ifyoubought$15worthofbooksamonthfor28monthshowmuchwouldyouhavespent

18Joedroveacar400milesat40milesperhourfor20daysHowmanymilesdidhecover

19Whatis(a)14times17(b)16times17(c)18times17(d)16times19Makeuseofthesmallerproductsby2orby4

20Whatis(a)29times18(b)29times15(c)37times16(d)46times14Convertonetwo-digitnumberintotwoone-digitnumbers

21Multiply(a)85times87(b)48times49(c)58times53(d)37times32(e)65times67(ƒ)99times94(g)74times72(h)26times28(i)17times18bythemethodusedwhenthetensfiguresarealike

22Multiply(a)45times16(b) (c)32times18(d) (e)18times18(ƒ)15times16(g) bymultiplyingonefactoranddividingtheotherfactorbythesamequantity

23Multiply(a)35times27(b)237times35(c)117times55(d)42times15(e)89times45by

multiplyingthefactorendingin5tosimplifyitanddividingtheresultsbythesamenumber

24Multiply(a)52times58(b)63times67(c)79times71(d)48times42(e)85times85(ƒ)23times27(g)37times33bythemethodusedwhenunitsaddupto10andtensdigitsarealike

25Multiply(a)63times43(b)75times35(c)94times14(d)47times67(e)58times58(ƒ)84times24(g)26times86bythemethodusedwhenunitsdigitsarealikeandtensdigitsaddupto10

26Multiplybycrossmultiplicationmethodgettinganswerinoneline(a)63times54(b)82times23(c)72times48(d)52times43(e)48times69(ƒ)91times18

27Multiply(a)95times45(b)75times65(c)65times85(d)35times55(e)95times35(ƒ)75times55(g)35times35(h)85times75(i)145times65(j)$135times45(k)$156times75(l)$215times95bysimplifiedmethod

28Multiply(a)87times7(b)92times8(c)64times6(d)657times9(e)49times5(ƒ)432times7byleft-handmultiplication

29Whatpartof100is(a)50(b) (c) (d) (e) (f) (g) (h)(i)

30Whatpartof10is(a)125(b) (c) (d) (e)75(ƒ) (g) (h)

31Whatpartof1is(a)25(b)375(c)625(d)125


33Whatisthecostof84articleswhenthepriceofoneis

34Multiplythefollowingbythealiquot-partmethod

(a) (b) (c)25times5744(d)(e) (ƒ) (g) (h)75times48(i) (j) (k) (l)20times85(m)58times50(n)48times25(o)2840times75

35Whatisthecostof

(a)85articlesat$250eachusingaliquot-partmethod

(b)58articlesat (c)46articlesat(d)36lbat perIb(e)48lbat25cent(ƒ)56lbat(g)24lbat75cent(h) ydat$624peryd(i) ydat72cent

36Whatisthecostof1860lboffeedat$12atonMakeuseofaliquot-partmethod

37Findthecostof72lawnmowersat$125eachusingaliquotpart

38Whatisthecostof48radiosat$6250eachUsealiquot-partmethod

39Multiply(a)32times24(b)68times24(c)242times124(d)57times24usingsimplifiedmultiplicationby24

40Multiply(a)242times26(b)242times26(c)32times26(d)68times26(e)57times26usingsimplifiedmultiplicationby26

41Multiply(a)57384times9(b)58761times9(c)4328times9(d)98989times9(e)847632times9usingsubtractionmethod

42Multiply(a)87583times11(b)9898times11(c)57384times11(d)58761times11(e)4328times11(ƒ)847632times11usingadditionmethod

43Multiply(a)687times8(b)687times7(c)432times8(d)432times7(e)982times8(ƒ)982times7byannexingazeroandsubtractingeithertwiceorthreetimesthenumber

44Multiply(a)687times99(b)687times98(c)687times97(d)982times99(e)982times98(ƒ)982times97byaddingtwozerosandsubtractingtherequirednumberoftimesthenumber

45Multiply(a)84times98(b)94times96(c)86times93(d)79times95(e)82times88(ƒ)982times978byusingcomplementmultiplication

46Multiply(a)37512times16(b)8762times14(c)982times18(d)76582times12(e)8462times13(ƒ)6879times19usingonlyonelineintheproductasshownintextexamples

47Multiply(a)84times76(b)758times84(c)68times47(d)832times59(e)54times132(ƒ)38times78(g)176times42(h)872times74usingcrossmultiplicationandcheckresultsbyldquocastingoutninesrdquo

CHAPTERIV

DIVISION

125WhatismeantbydivisionDivisionistheinverseofmultiplicationAswehaveseenthat

multiplicationismerelyasimplifiedformofadditionwecanconcludethatitsinversedivisioninitssimplestformismerelyrepeatedsubtraction

Ex(a)Whenwemultiply8fourtimesweget8times4=32whichissimplifiedaddition8+8+8+8=32=productNowdividingtheproduct32by8weget4

32minus8=2424minus8=1616minus8=888minus8=0

Wehavesubtracted8successivelyfrom32infourstepstoget

Ex(b)Supposeyouhave972applesandyouwanttodividethemequallyamong324menHowmanyappleswilleachmanreceive

972minus324=648648minus324=324324minus324=0

Countthenumberofsubtractionswhichis3andyouget3applesforeachman

Ex(c)Howmany2rsquosin8Subtract2from8asmanytimesaspossiblenotingthenumberoftimes4astheanswer

126Inwhatotherwaysmaydivisionbethoughtof(a)Divisionproperaspeciesofmeasurementasfindinghowmanytimesone

numberiscontainedinanother

(b)PartitionwhichisdividinganumberintoequalpartsthenumberofsuchpartsbeinggivenThisisimportantwithconcretenumbersandisofnoimportancewithabstractnumbers

Ex(a)Howmanytimesis7containedin35

Ex(b)If3gallonsofmilkyield21ouncesofbutterhowmanyounceswill1gallonyield

Thinkof21ouncesasdividedinto3equalpartswhichwillresultin7ouncesineachpart

127Whatarethetermsofadivision

Dividend=ThenumbertobedividedorseparatedintoequalpartsNumberinfrontofdivisionsign

Divisor=ThenumberofequalpartsintowhichdividendistobeseparatedorthenumberbywhichdividendistobedividedNumberfollowingdivisionsign

Quotient=Resultobtainedbydivision

EXAMPLES

(a)42divide7=6orDividend

(b) or

(c) orDivisor(=7

128WhenthedividendisconcreteandthedivisorisabstractwhatisthequotientThequotientislikethedividend

EXAMPLEIf3gallonsofmilkyield21ouncesofbutterwefindthenumberofouncescontainedin1gallonofmilkbydividing21ouncesby3(notby3gallons)getting7ouncesThedivisorhere(3)isanabstractnumberandtheterm3gallonsservesonlytoindicatethenumberofgroupsintowhich21ouncesistobeseparated

129WhatistheresultwhenboththedividendanddivisorareconcreteThedividendanddivisormustbealikeandthequotientwillbeabstract

EXAMPLE

Sevenouncesgoesinto21ouncesthreetimes

130WhatismeantbyaremainderindivisionWhendivisionisnotexactthepartofthedividendremainingiscalledthe

remainder

EXAMPLE

17divide2=8with1asaremainder

Theremainderisplacedoverthedivisoras here

131WhymaywethinkofdivisionastheprocessoffindingonefactorwhentheproductandtheotherfactoraregivenEXAMPLEIn7times3=21wehavemultiplication

Factor(=7)timesFactor(=3)=Product(=21)

In =7wehavedivision

132HowcanwemakeuseofthefactthatdivisionistheoppositeofmultiplicationEXAMPLEWhatnumbermultipliedby324wouldgive972

Weknowthat324=300+20+4

972=900+70+2

133Ifwewantedtodivide3492meninto4groups

howwouldweproceed

(a)8times4=32or800complete 873(=800+70+3)

4rsquos=3200 4)3492

(b)Subtract3200from3492 -3200 (=4times800)

(c)7times4=28or70times4=280 292

leaves292menstilltobecounted

(d)Subtract280from292 -280 (=4times70)

(e)3times4=12

12leaves12menstilltobecounted

(ƒ)Addingthequotientsweget -12 (=4times3)

800+70+3=873

ThisprocesscanbeshortenedbyomittingthezerosasisdoneinmultiplicationBringdownonlythenumberornumberstobeusedinthenextpartoftheexampleBecarefulinplacingthenumbersdirectlyunderthecolumnsinwhichtheyfirstappeared

Whendividingwithonlyonedigitwemayshortenthestepstillfurtherbyldquothinkingrdquothesubtractionsandcarryingtheremainders

ldquoThinkrdquosubtract8times4=32from34carry2tothe9 making29

ldquoThinkrdquosubtract7times4=28from29carry1to2making12

ldquoThinkrdquosubtract3times4=12from12getting0whichiszeroremainder

134WhatismeantbyldquoshortdivisionrdquoandwhatistheprocessinsimpleformWhenthedivisorissosmallthattheworkcanbeperformedmentallythe

processiscalledshortdivision

EXAMPLEDivide9712by4Writeas

(a)BeginatleftFindhowmanytimesdivisor4iscontainedinthefirstfigureofthedividend

4iscontainedin9twotimeswitharemainder1

(b)Reducethe1tothenextlowerordermaking10whichwith7makes17

4iscontainedin17fourtimeswitharemainder1

(c)Reducethis1tothenextlowerordermaking10whichwith1makes11

4iscontainedin11twotimeswitharemainderof3

(d)Reducethis3tothenextlowerordermaking30whichwith2makes32

4iscontainedin32eighttimeswithnoremainder

135Howdowedivide3762by7usingshortdivision

(a)7isnotcontainedinthefirstfigureofthedividend3and3mustbereducedtothenextlowerordermaking30whichwith7makes37

(b)7iscontainedin37fivetimeswith2remainderReduce2tonextlowerordermaking20whichwith6makes26

(c)7iscontainedin26threetimeswith5remainderReduce5tonextlowerordermaking50whichwith2makes52

(d)7iscontainedin52seventimeswith3remainderwhichiswritten

136HowdoweproceedwithlongdivisionEXAMPLETodivide73158(=Dividend)by534(=Divisor)

(a)Sincedivisorhas3digitstakethefirst3digitsofthedividendandaskhowmanytimesdivisor534iscontainedin731(=first3digitsofdividend)(Usuallyaclueisgivenbytrialofthefirstfigureofdivisorwhichhereis5andfindinghowmanytimesitiscontainedinfirstfiguresofdividendhere7)Divide5into7or1

(b)Writepartialquotient1overthelastfigureof731Here1goesoverthe1of731

(c)Subtract1times534from731getting197andbringdownthe5whichisthenextdigitofthedividendThisresultsinthepartialdividend1975

(d)Dividefirstfigure5ofdivisorinto19(=thefirsttwofiguresofpartialdividend)Writepartialquotient3over5oftheoriginaldividend

(e)Subtract3times534=1602from1975getting373andbringdownthe8whichisthenextdigitofthedividendThisresultsinthepartialdividend3738

(ƒ)Divide5ofdivisorinto37ofpartialdividendWritepartialquotient7over8oforiginaldividend

(g)Subtract7times534=3738from3738ofpartialdividendgettingzeroremainderQuotientistherefore137exact

137WhatdowedowhenthelastsubtractionisnotzeroEXAMPLEDivide73170by534

Theremainder12isexpressedas12overthedivisoror hereThequotientis

Sometimesweplaceadecimalpointafterthelastdigitofthedividendaddzerosandcontinuetheprocessofdivisiontoexpresstheremainderasadecimal

138WhatistheprincipleofthetrialdivisorinlongdivisionEXAMPLEDivide236987by863

(a)Ordinarilytryfirstleft-handdigitofdivisorintothefirsttwodigitsofdividendas8ofdivisorinto23ofdividend

(b)Butwhentheseconddigitofdivisorisnumber5orgreater(6inthiscase)thenincreasethefirstdigitofdivisorby1andtryindividendHeretry9into23

getting2asquotient

(c)Inthenextpartialdividendtry9into64getting7asquotient

(d)Inthefollowingpartialdividendtry9into39getting4asquotient

(e)Remainderhereis Quotientis

139Whatistheruleforlongdivision(a)WritedivisoratleftofdividendwithacurvedlinebetweenthemTakethe

fewestnumberofdigitsatleftofdividendthatwillcontaindivisorandwritethispartialquotientontopovertheright-handdigitofthepartialdividend

(b)Multiplyentiredivisorbythispartialquotientandwritetheproductunderthepartialdividendused

(c)Subtractthisproductandtoremainderannex(bringdown)thenextfigureofdividendforthesecondpartialdividend

(d)Divideasbeforeandcontinueprocessuntilalldigitsofdividendhavebeenusedtomakepartialdividends

(e)Whenthereisaremainderwriteitwiththequotient

140WhatisapureproofofanydivisionMultiplydivisorbyquotientandtothisproductaddtheremainderifanyThe

resultshouldequalthedividend

EXAMPLE

141WhatistheprocedurefordivisionwithUnitedStatesmoneyDivideasinintegralnumberswritingthefirstdigitofthequotientoverthe

right-handdigitofthefirstpartialdividend(Placethedecimalpointinthequotientdirectlyoverthedecimalpointinthedividend)

EXAMPLEDivide$82911by87

142Whatisthequotientofthedivisionof$4536by$027Changethedividendanddivisortocentswhichgives4536centsdividedby

27centsThequotientis168whichisanabstractnumbershowingthenumberoftimes27centgoesinto4536cent

143HowcanfactoringofthedivisorbeusedtoreduceaproblemoflongdivisiontoaseriesofshortdivisionsEXAMPLEDivide27216by432Herethedivisor432canbefactoreddown

farenoughtogiveaseriesofshortdivisionsbythefactorswhichprocedureissubstitutedforthelongdivision

432=Divisor=12times12times3

144WhatistheprocedurefortheabovewhenthereisaremainderEXAMPLEDivide47897by18

Factordivisor18as2times3times3=18Divideby2thenthequotientofthisby3andthequotientofthisby3

Quotientis

Thefirstremainder1remainsunchanged

Theseconddivisionhasaremainder2Asthisdivisionisofonehalfthenumberby3youmultiplytheremainder2by2getting4andaddingthistothe

previousremaindergetting4+1=5

Thenextdivisionisofonesixthofthenumberby3Youthenmultiplythisremainderby6getting12andaddthistotheprevious5getting17whichisthefinalremainder

Itisseenthateachremainderexceptthefirstismultipliedbythefactorsofthedivisionsprecedingitsownandthesumoftheproductsisthetotalremainder

145Whatisthequotientof65349by126usingthefactoring-ofthe-divisormethodDivisor126=2times3times3times7

Quotient=

146Whatistheprocedurefordividingby101001000etcSetoffasmanyfiguresattherightofthedividendasthereareciphersinthe

divisorThefiguresthussetoffaretheremainderTheotherfiguresarethequotient

Ex(a)65divide10=6with5asremainderor (OnecipherindivisorSetoff1figureatrightofdividend)

Ex(b)579divide100=5with79asremainderor (TwociphersindivisorSetoff2figuresatrightofdividend)

Ex(c)

Dividing200by100weget2


Nowdividing5670by2weget2835(Ans)

Ex(d)

WhenthedivisorendsinoneormorecipherscuttheseoffandalsocutoffanequalnumberoffiguresfromtherightofthedividendThendividebythefiguresremaining

Ex(e)8743divide700=008743divide7=001249

147Howdoweapplytheexcess-of-ninesmethodtoprovethecorrectnessofadivision(a)Getexcessof9rsquosindivisor

(b)Getexcessof9rsquosinquotient

(c)Multiplythesetwoexcessesandgetexcessof9rsquosoftheproduct

(d)Addtothistheexcessof9rsquosinremainderGetexcessofsum

(e)Getexcessof9rsquosindividendandcompare

EXAMPLE

Aquotientmaybeincorrecteventhoughtheexcess-of-ninesmightcheckbutthishappensrarely

148WhatismeantbyanevennumberAnumberdivisibleby2iscalledanevennumberAnevennumbermayend

in2468orinazero

EXAMPLES42547668970areevennumbersEachdividedby2results

in21273834485

149Howcanweknowwhenanumberisdivisibleby3Whenthesumofitsdigitsisdivisibleby3thenumberitselfisdivisibleby3

Ex(a)Number=213Adddigits2+1+3=6Nowsum6isdivisibleby3Thereforenumber213isdivisibleby3Ans=71

Ex(b)Number=531Adddigits5+3+1=9Sum9isdivisibleby3Thereforenumber531isdivisibleby3Ans=177

150Ifwehaveanevennumberanditisdivisibleby3bywhatothernumberisitalsodivisibleThenumberisalsodivisibleby6becauseanevennumberisdivisibleby2

and2times3=6

EXAMPLEGivennumber=162whichisanevennumberAdddigits1+6+2=9whichisdivisibleby3

there4162isdivisibleby6or Ans

151Whenisanumberdivisibleby4Whenitslasttwodigitsaredivisibleby4

EXAMPLE7624Lasttwodigits24aredivisibleby4

152Whenisanumberdivisibleby5Whenitendsin5orzero

Ex(a)

Ex(b)

153Whatnumberoranymultiplesofitcanbedividedby711or13Number1001oranyofitsmultiplescanbedividedby711or13

Ex(a)

Ex(b)

154Whenisanumberdivisibleby8Whenthenumberendsinthreezerosorwhenthelastthreedigitsaredivisible

by8

Ex(a)

Because1000isdivisibleby8whateverprecedesthelastthreefiguresmerelyaddsthatmanythousandsanddoesnotaffectthedivisibilityby8

Ex(b) Nowadd1000getting1136Then

Againadd1000getting2136Then

Ex(c)

Nomatterhowmanyfiguresareplacedinfrontoftheoriginal136thenumberisdivisibleby8

Ex(d)29632Consider divisibleby8

there429632isdivisibleby8getting3704Ans

155Whenisanumberdivisibleby9Whenthesumofitsdigitsisdivisibleby9

Ex(a)Numberis8028Adddigits8+0+2+8=18

Ex(b)Number Adddigits3+8+9+3+4=27and

156Whenisanumberdivisibleby25Whenitendsintwozerosorintwodigitsformingamultipleof25

Ex(a)

Ex(b)

157Whenisanumberdivisibleby125Whenitendsinthreezerosorinthreedigitsformingamultipleof125

Ex(a)

Ex(b)

158Whatisthecriterionforanumberdivisibleby11(a)Whenthesumofeven-placeddigitsequalsthesumofodd-placeddigits

Ex(a)

(b)Whenthedifferencebetweenthesumoftheodd-andeven-placeddigitsisdivisibleby11thenumberitselfisdivisibleby11

Ex(b)

Ex(c)

159Howcanwetellinadvancewhattheremainderwillbewhenthedivisoris9AddthedigitsandthenaddthedigitsofthissumThislastistheremainder

Ex(a)867

Adddigits8+6+7=21(=Sum)Adddigitsofsum2+1=3(=Remainder)

Ex(b)973285

Adddigits9+7+3+2+8+5=34(=Sum)Adddigitsofsum3+4=7(=Remainder)

160Whatisashort-cutwayofdividingby5Multiplyby2andpointoffonedecimalplacetotheleft

Ex(a)23divide523times2=46Pointoffoneplacetoleft46Ans

Topointoffonedecimalplacemeansdividingby10

Ex(b) 832times2=1664Pointoffoneplace1664Ans

161Whatisasimplewayofdividingby25Multiplyby4andpointofftwoplacestotheleft

Ex(a)1394divide251394times4=5576Pointofftwoplaces5576

Topointofftwoplacestotheleftmeansdividingby100

Ex(b)

162Whatisasimplewaytodivideby125Multiplyby8andpointoffthreeplacestotheleft

EXAMPLE7856divide1257856times8=62848Setoffthreeplacestoleftgetting62848

Dividingby1000meanssettingoffthreeplacestotheleft

163Whatistheshort-cutwayofdividingbyanyaliquotpartof100Multiplybytheinvertedfractionrepresentedbythealiquotandpointofftwo

placestotheleft

Ex(a) Invert getting

875times3=2625Pointofftwoplacestoleftgetting2625Ans

Ex(b) Invert getting

Pointofftwoplacestoleftgetting90Ans

Ex(c) Invert getting


164Whatisasimplewayofdividingby99(a)Addthetworight-handdigitstotherestofthenumberPutthissumdown

undertheoriginalnumber

(b)Addthetworight-handdigitsofthistotherestofitsnumberandputthisdownundertheothertwo

(c)Keepupthisprocessuntil99oraquantitylessthan99isleftThisistheremainder

(d)Addupthedigitstotheleftexcludingthetworight-handdigitsofeachnumber

Ex(1)

Remainder Add1toquotient7890getting7891

Ex(2)

165Howcanwemakeanumberdivisibleby3(a)AddthedigitsDividethissumby3andgetremainder

(b)Subtractremainderfromoriginalnumber

EXAMPLE13477Adddigits1+3+4+7+7=22

Subtractremainder1from13477getting13476AnsNow

166Howcanwemakeanumberdivisibleby9(a)AddthedigitsDividethissumby9Getremainder



Subtractremainder4from13477getting13473

167HowdoweobtainanaverageofanumberofitemsDividethesumoftheitemsbythenumberofitemsadded

EXAMPLEFindtheaveragesalesmadebyasalesmanduringtheweekwhenhisdailysalesare

Monday $26860

Tuesday $32985

Wednesday $9745

Thursday $23990

Friday $29670

Numberofitems=5) $123250 (=Sum)

$24650 (=Average)

WeseethatthesalesforMondayTuesdayandFridaywereabovetheaveragewhileforWednesdayandThursdaytheywerebelowaverage

168WhatistheruleforfindingthevalueofoneofanythingAlwaysdividebythatofwhichyouwanttofindthevalueofone

Ex(a)If75bookscost$300whatisthecostof1book

Youwantthecostof1booksodividebythenumberofbooks

Ex(b)Ifadozenhatscost$72whatisthecostof1hat

Youwantthecostof1hatsodividebynumberofhats

Ex(c)Ifapoundofcoffeecosts80centhowmanyouncescanyougetfor10cent

Firstyouwantthenumberofouncesfor1centsodividebycents

For10cent

Ex(d)Ifajeepused16gallonsofgasolineindriving288mileshowmuchdoesitconsumeona486-miletrip

Youwantfirstthenumberofmilesfor1gallonsodividebygallons

then

Ex(e)Ifittakes8minutesforapipetofillatankhowmuchofthetankwillbefilledin1minute

Youwanttheamountfor1minutesodividebyminutes

PROBLEMS

1Howmanytimesis8containedin56

2If3gallonsofmilkyield18ouncesofbutterhowmanyounceswill1gallonyield

3Ifyouhave1048padsofwritingpaperandyouwanttodividethemequallyamong262employeeshowmanypadswilleachonereceive

4Findthequotientsofthefollowingandproveeachbymultiplyingthefactorstogether

(a)6divide2=because2times=6

(b)9divide3=because3times=9

(c)12divide4=because4times=12

(d)18divide9=because9times=18

(e)28divide7=because7times=28

(ƒ)42divide6=because6times=42

(g)48divide8=because8times=48

(h)66divide6=because6times=66

(i)72divide9=because9times=72

(j)84divide7=because7times=84

5If$1ischangedtofive-centpieceshowmanyarethere

6Ifamanearns$16whileaboyearns$6howmuchwilltheboyearnwhilethemanearns$96

7Ifamancanpicktwiceasmuchfruitasaboyand4boysand3menpick5acresoforchardinadaywhatamountofgrounddoeseachcover

8Ifamaneats380gramsofcarbohydrates130gramsofproteinand60gramsoffatseachdayhowmuchdoesheaveragepermeal

9Divide

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

(u) (v) (w) (x)

(y) (z)

10Findthequotientof

(a)1607divide19(b)6548divide89(c)3402divide81

(d)3485divide873(e)54963divide863(ƒ)861618divide843

(g)879384divide508(h)938764divide879(i)42896divide269

(j)98641divide679(k)3862847divide76298(l)

(m) (n) (o)

11Ifthereare266pagesinabookandyoucanread38pagesinanhourhowlongwillittakeyoutoreadit

12Findthequotientof

(a)$1836divide12(b)96750divide43(c)$96750divide$43

(d)$43890divide$21(e)$43890divide21cent

13Dividebyfactoringmethod


14Divide

(a)490divide10(b)487divide10(c)5300divide100(d)15874divide100

(e)385divide10(ƒ)8745divide100(g)490divide20(h)487divide30

(i)5300divide400(j)385divide20(k)8745divide700

(l)697divide1000(m)16720divide800

15Applyexcess-of-ninesmethodtoprovethecorrectnessofthedivisionsofproblem10

16

(a)Is7893divisibleby3(Usingsum-of-digitsmethod)

(b)Is3876divisibleby6(Usingshort-cutmethod)(c)Is3876divisibleby

4(Usinglast-two-digitsmethod)

(d)Is8695divisibleby5(Usingcriterion)

(e)Is14014divisibleby711or13(Usingcriterion)(ƒ)Is7462768divisibleby8(Usingdivisibility-of-last-3-digitsmethod)

(g)Is8658divisibleby9(Usingsum-of-digitsmethod)

(h)Are7800and9864175divisibleby25(Usecriterion)

(i)Are7860000and76375divisibleby125(Usecriterion)

(j)Are3657654and78947divisibleby11(Usecriterion)

17

(a)Whatwillbetheremainderof948divide9(withoutdividingfirst)(b)Canyoutellinadvancetheremainderof864893divide9

18

(a)Divide39by5atoncebyshort-cutmethod(b)Divide482by25byshort-cutmethod(c)Divide6743by125byshort-cutmethod

19Dividethefollowingbyuseofaliquotpartsof100

(a) (b)

(c)

(d) (e)

(ƒ)34560divide5(g) (h)


20Divide(a)872317divide99(b)867432divide99bysimplemethodshownintext

21Make(a)25694(b)85642divisibleby3bymethodshownintext


23Ifsixrankingcandidatesonanexaminationhadmarksof921873856807802and791respectivelywhatistheaveragemark

24Sixteenstudentsinaclassinarithmeticmadethefollowinggradesonatest849674938886817781949986716976and84Whatwastheaveragegradeoftheclass

25Anauthorreceivedroyaltiesfromhispublisherduringasix-yearperiodasfollows$89765$91759$89325$99775$114679and$123832Whatistheaverageyearlyroyalty

26Ifyouhaveanappleorchardof2000treesifyouuse4gallonsofsprayingmixtureforeachtreeandyoumix1lbofParisgreenat80centperlbwith150galofwaterwhatwouldbethecostoftheParisgreenfor2sprayingsWhatwouldbethecostpertree

CHAPTERV

FACTORSmdashMULTIPLESmdashCANCELLATION

169WhatisaprimenumberAnumberdivisibleonlyby1anditself

EXAMPLES123571113171923293137etcareprimenumbersEachisdivisibleonlyby1anditself

170WhatisacompositenumberOnethatisdivisiblebyothernumbersinadditionto1anditself

EXAMPLES46810121416183644etcarecompositenumbers

171WhatisafactorofanumberAnexactdivisorofthenumber

Ex(a)2isafactorof6because2isanexactdivisorof6

Ex(b)2346arefactorsof12becauseeachisanexactdivisorof12If3isonefactorof12then4istheotherfactor

172WhatismeantbyfactoringTheprocessofseparatinganumberintoitsfactors

173WhatisaprimefactorAfactorwhichisaprimenumber

Ex

(a)22and3areprimefactorsof12(b)222and3aretheprimefactorsof24

Ofcourse46812arealsofactorsof24butthesearenotprimefactors

174Whatdowecallanumberthathasthefactor2AnevennumberNumbersnotdivisibleby2arecalledoddnumbers

175WhatismeantbyacommondivisororfactorOnethatiscommontotwoormorenumbers

EXAMPLE

4isafactorcommonto12and363isafactorcommonto12and361262arefactorscommonto12and36

Numbersthathavenocommonfactorsaresaidtobeprimetoeachother

176Whatfactsregardingthedivisibilityofnumbersareofassistanceinfactoring(a)2isafactorofallevennumbers

(b)3isafactorwhenthesumofthedigitsisdivisibleby3

(c)4isafactorwhenthetwodigitsattherightarezerosoranumberdivisibleby4

(d)5isafactorwhentheunitsfigureis5orzero

(e)6isafactorofallevennumbersthataredivisibleby3

(f)8isafactorwhenthethreedigitsattherightarezerosoranumberdivisibleby8

(g)9isafactorwhenthesumofthedigitsisdivisibleby9

(h)11isafactorwhenthesumofthedigitsintheevenplacesequalsthesumofthedigitsintheoddplacesorwhenthedifferencebetweenthesetwosumsis11orsomemultipleof11

177HowdowefindtheprimefactorsofanumberDividebyaprimefactorandcontinuetodividebyaprimefactoruntilthelast

quotientisaprimenumber

Ex(a)Whataretheprimefactorsof720

Ex(b)Findtheprimefactorsof7644

178WhatismeantbythegreatestcommondivisororfactorabbreviatedGCDorgcdThelargestdivisororfactorcommontotwoormoregivennumbersisthe

GCD

Ex(a)6isthegreatestcommondivisorof24and30

Ex(b)8isthegreatestcommondivisorof1624and32becauseitisthelargestnumberthatwillexactlydivideeachofthenumbers

179WhatistheruleforfindingtheGCDoftwoormorenumbersSeparatethenumbersintotheirprimefactorsandgettheproductoftheprime

factorsthatarecommontoallthenumbers

Ex(a)

Factors2and3arecommontoboth24and30

there42times3=6=GCD

Ex(b)

Factors222arecommontoallthreenumbers

there42times2times2=8=GCD

180WhatisamoreconvenientmethodoffindingGCDArrangethenumbersasbelowanddividebysomenumberwhichwillexactly

divideeachofthemContinuedoingthisuntilnodivisorcanbefoundtodivideeachlastquotientMultiplyallthecommonfactors

Commonfactorsrarr2times2times2times3=24=GCD

181WhatismeantbyamultipleofanumberItistheproductofthatnumbermultipliedbyaninteger

Ex(a)24isamultipleofnumber12because12multipliedbyaninteger2=24

Ex(b)Whatnumbersaremultiplesof8

2times8=163times8=244times8=32etc

Thus162432etcaremultiplesof8

182WhatismeantbyacommonmultipleoftwoormorenumbersAnumberthatisamultipleofeach

Ex(a)16isacommonmultipleof4and8becauseeitherofthemmultipliedbyaninteger=16

Ex(b)18isacommonmultipleof236and9becauseanyofthesemultipliedbyaninteger=18

183Whatismeantbytheleastcommonmultiple(LCM)oftwoormorenumbersTheleastnumberthatisamultipleofeach

Ex(a)18isacommonmultipleof3and6but12istheleastcommonmultipleof3and6because12isthesmallestnumberwhichcontainseachwithoutaremainder

Ex(b)72isacommonmultipleof69and12but36istheLCMbecauseitisthesmallestnumberwhichcontainseachwithoutaremainder

184Whatisamethodoffindingtheleastcommonmultiple(LCM)of1828and36SeparateeachnumberintoitsprimefactorsMultiplythefactorsusingeach

factorthegreatestnumberoftimesitoccursinanyofthegivennumbersthatarefactored

2doesnotappearasafactormorethantwiceinanynumber


7appearsonce

there42times2times3times3times7=252=LCMthatwillcontain1828and36withoutaremainder

185WhatisanothermethodofgettingtheLCMof1828and36Dividethenumbersbyanyprimenumberthatwillexactlydividetwoormore

ofthemAnynumbernotsodivisibleisbroughtdownintactContinuethisprocessuntilnofurtherdivisioncanbemadeMultiplyalldivisorsandthequotientsremainingtogettheLCM

186WhatismeantbycancellationEliminationoffactorsinthedividendanddivisorbeforedividingThe

quotientisnotaffectedbyeliminationoffactorswhicharecommontobothdividendanddivisor

Ex(a)Divide4368by156byfactoringandcancelling

ThesameanswercanbeobtainedbylongdivisionItisnotnecessarytoseparate

thenumbersintotheirprimefactorsThecriteriafordivisibilityofnumbersmaybeusedasshowninquestion176

Ex(b)Compute bymeansofcancellation

Ex(c)Computebycancellation Ans

13isafactorof39and65threeandfivetimesrespectively

Then3iscontainedin105thirty-fivetimes

Theproductoftheremainingfactors5times35=175Ans

Ex(d)Computebycancellation

Findfactorscommontonumbersabovethelineandnumbersbelowthelineandcancelthem

PROBLEMS

1Nametwofactorsof18303681120

2Namethreefactorsof1832455066

3Nameafactorcommonto12and36

4Nameallthefactorsorexactdivisorsof3717

5Makealistofallprimenumbersbelow100

6Makealistofalloddnumbersbelow50

7Separateintoprimefactors45781012131416182124253034

8Separatetheprimecompositeevenandoddnumbersinthefollowing167101112141920212425273334

9Givetheprimefactorsof

(a)310(b)297(c)670(d)741(e)981(f)385(g)2650

(h)1215((i)321(j)1575(k)10935(l)420(m)497

(n)378(o)462(p)2430(q)25344(r)73260(s)599676

(t)273564(u)15625(v)10675(w)12625(x)976

(y)8050(z)3848

10FindtheGCD(greatestcommondivisor)of

(a)68112240(b)2184126147(c)212877

(d)457281(e)4477121(f)1498112(g)248096

(h)284236(i)457281(j)31522679012

(k)144576(l)820697(m)1251751792(n)60043318

(o)125423618163(p)1086905

11Givetwomultiplesof

(a)9and3(b)7and5(c)9and2(d)3and7(e)8and5

(f)6and3(g)8and2(h)92and8(i)36and9

(j)86and4

12FindtheLCM(leastcommonmultiple)of

(a)9and12(b)21and36(c)5and15(d)1215and18

(e)3642and48(f)3918and27(g)51525and35

(h)148135and15(i)324835and70(j)728896and124(k)112255and110

(l)212426and28(m)92142and63

(n)367548and24(o)71456and84(p)2472128and240

13Dividebycancellationmethodoffactorsandprovebylongdivision


(d)88368divide3682(e)32768divide2048

14Solvebycancellation

(a)3times27times48times81=6times9times54times210(b)81times16times10times12=9times27times2times5(c)8times12times18times32=4times6times9times16(d)42times36times77times22divide11times6times24times21(e)5times30times65times125=15times75times95

15Howmanylbofbutterat55centalbcanbeexchangedfor30dozeggsat66centadoz(Bycancellation)

16Howmanydaysof8hreachwouldoneneedtoworkat$230anhourtopayfor8tonsofcoalat$2760aton(Solvebycancellation)

17If14menearn$725760working27daysof8hourseachat$240anhourhowlongwillittake21menworking8hoursadayatthesameratetoearnthesameamount(Solvebycancellation)

18Ifyoudrove20000milesonnewtiresbeforereplacementandyoupaid$120forthe4newtireswhatwasthetirecostforeach100miles(Solvebycancellation)

CHAPTERVI

COMMONFRACTIONS

187WhatdoesafractionmeanTheLatinfrangeremeansldquotobreakrdquoTheLatinfractusmeansldquobrokenrdquoThus

afractionisabrokenunitorapartofaunitAlsoldquofractionrdquocomesfromthesameLatinrootasthewordldquofragmentrdquomeaningldquoapartrdquoActuallyafractionisanyquantitynumericallylessthanaunit

188WhatarethetermsofafractionEveryfractionhasanumeratorplacedaboveahorizontallineanda

denominatorplacedbelowthelineThedenominatoristhedivisorofthenumerator

EXAMPLE

189WhatisassumedinexpressingfractionaldivisionItisassumedthatallofthepartsintowhichanobjecthasbeendividedareof

exactlyequalsize

190WhatismeantwhenwesaythatathingisdividedequallyintotwopartsandhowisthefractionexpressedTheobjectissaidtobedividedintohalvesTheobjectisdividedintotwo

partsTheobjectorunittobedividedisplacedasthenumeratorofthefractionthenumberofdivisionsisthedenominator

Thus

191Whatismeantby

(a)

(b)

(c)

(d)

192WhatismeantbyaunitfractionWhenthenumeratorofafractionis1itiscalledaunitfractionas

193WhatisavulgarfractionandhowisitclassifiedAvulgarfractionisoneexpressedasadivision

ThedivisorclassifiesthefractionEx(a) isclassifiedasthirdsfromitsdivisor3

Ex(b) isclassifiedastwenty-fifthsfromitsdivisor25

194WhatarethepartsofavulgarfractionandhowisitwrittenThenumeratoristhedividendthedenominatoristhedivisorItiswrittenasa

numeratoraboveanddenominatorbelowashorthorizontalordiagonallineorbar

Ex(a) Numeratortellsusthatonly1ofitsclassisconsidered

Ex(b) Numeratortellsusthat11ofitsclassaretaken

195WhatothermeaninghasthebarinafractionThebarmeansldquodivisionrdquointhesamewayasthesign[divide]

Ex(a)

Ex(b) Bothexpressionsmeanthesamething

Ex(c)

196WhatarethethreewaysinwhichafractionmaybeinterpretedThefraction forexamplemaybethoughtofas(a)3unitsdividedinto2

equalparts

(b)1unitdividedinto2equalpartswith3ofthesepartstakenas3times

(c)Asanindicateddivisionnotyetperformed

EXAMPLESAssume1orunityisaline1inchlong

ThreeunitsdividedintotwoequalpartsEachpart

(b)

(c) canbethoughtofasadivisionnotyetperformed

197Whenweaddupallthefractionalpartsofaunitwhatdowegetasaresult

Wegetthewholeunit

Ex(a)

Ex(b)

Ex(c)

Oranyfractionalexpressionofanumberdividedbyitself=1=unityas

198WhatisasimplefractionOnewhosenumeratoranddenominatorarewholenumbers

EXAMPLE and aresimplefractions

199WhatisacompoundfractionItisafractionofafraction

EXAMPLE of and of arecompoundfractions

200WhatisacomplexfractionOneinwhicheitherthenumeratorordenominatororbotharenotwhole

numbers

Ex(a) Numeratorisnotawholenumber

Ex(b) Denominatorisnotawholenumber

Ex(c) Bothnumeratoranddenominatorarenotwholenumbers

Alltheabovearecomplexfractions

201Whatisaproperfraction

Oneinwhichthenumeratorislessthanthedenominator

EXAMPLE areproperfractionsEachhasavaluelessthanaunitNotethatthenumeratordoesnothavetobe1

202WhatisanimproperfractionOneinwhichthenumeratorequalsorexceedsthedenominatorThefraction

isthusequaltoorgreaterthan1unit

Ex(a)

Ex(b)

203WhatisamixednumberAwholenumberandafractiontakentogether

EXAMPLE aremixednumbers

204HowmayweshortentheprocessoffindingthevalueofanimproperfractionDividethenumeratorbythedenominatorWritethequotientasawhole

numberfollowedbyafractioninwhichtheremainderisexpressedasanumeratoroverthesamedenominator

Ex(a) Thirteengoesinto48threetimeswitharemainderof9 isamixednumber

Ex(b)

205HowdowechangeamixednumberintoanimproperfractionMultiplythewholenumberbythedenominatoraddthenumeratorandplace

thissumoverthedenominator

Ex(a)

Ex(b)

Ex(c)

Thereasoningis

Then =Thisiswhywemultiplythewholenumberbythedenominatorandaddthenumeratortogetthetotalnumberoffifthsinthiscase

206WhathappenstothevalueofafractionwhenwemultiplyordivideboththenumeratorandthedenominatorbythesamenumberThevalueofthefractionisunchanged

Ex(a)

Ex(b)

207WhenisafractionsaidtobereducedtoitslowesttermsWhenthetermsareprimetoeachother

Ex(a) isexpressedinitslowesttermsbecause5and6areprimetoeachother

Ex(b) isnotexpressedinitslowesttermsbecause2isafactorcommontobothnumeratoranddenominator

208HowdowereduceafractiontoitslowesttermsDividebothnumeratoranddenominatorbyacommondivisorandcontinueto

divideuntilallcommondivisorsareeliminatedThisisdonebycancellingthe

commonfactors

Ex(a)

Ex(b)

209HowcanwechangeafractiontohighertermsMultiplybothnumeratoranddenominatorbythesamenumber

Ex(a)Change totwenty-fourths

Multiplybothnumeratoranddenominatorby6

Ex(b)Change tohundredths


210Whatmustbedonetofractionsingivingtheanswertoaproblem(a)Reducefractionstolowestterms

EXAMPLE

(b)Reduceimproperfractionstomixednumbers

EXAMPLE

211Howcanweincreasethevalueofafraction(a)Bymultiplyingthenumeratorbyanumbergreaterthan1

EXAMPLE isincreasedto

bymultiplyingnumeratorby2forexample

(b)Bydividingthedenominatorbyanumbergreaterthan1


bydividingdenominatorby2forexample

Thevalueofthefractionhasbeendoubledineachcase

EXAMPLEIncreasethevalueof threetimes

212Howcanwedecreasethevalueofafraction(a)Bydividingthenumeratorbyanumbergreaterthan1

EXAMPLE isdecreasedto

bydividingnumeratorby2forexample

(b)Bymultiplyingthedenominatorbyanumbergreaterthan1


bymultiplyingthedenominatorby2forexample

Thevalueofthefractionisreducedone-halfineachcase

EXAMPLEDecrease toone-sixthofitsvalue

213HowdowechangeacompoundfractiontoasimplefractionPlacetheproductofthenumeratorsovertheproductofthedenominators

Ex(a)

Ex(b) of=simplefraction

214HowdowechangeacomplexfractiontoasimplefractionDividethenumeratorbythedenominator

Ex(a)

Ex(b)

215WhatisanothermethodofsimplifyingacomplexfractionMultiplybothnumeratoranddenominatorbyanumberthatdoesnotchange

thevalueofthefraction

EXAMPLE

216WhatistheconditionforaddingorsubtractingoffractionsThefractionsmustallbeofthesameclasswhichmeansthedenominators

mustallbethesame

Addthenumeratorsandplaceoverthecommondenominator

Ex(a)Add and

Ex(b)Ifthereareanywholenumbersaddthemalso

Add

Addwholenumbers1+3+12=16

Addfractions

Then

217WhatistheprocedurewhenthedenominatorsarenotthesameFindtheldquolowestcommondenominatorrdquowhichisthesmallestdenominator

intowhichallwilldivideevenlyThisisthesameastheLCMpreviouslystudied

Ex(a) +Thelowestcommondenominator(LCD)of23and6is6Allthedenominatorsdivideinto6evenly

Now

Ex(b)Add (LCD=20)

Ex(c)Add (LCD=20)MultiplyeachnumeratorbyasmanytimesasthedenominatorgoesintotheLCD

218Whatistheprocedureforsubtractionoffractions(a)Workwithonlytwotermsatatime

(b)Changeamixednumberfirsttoanimproperfractionwhenthemixednumberissmall

(c)FindtheLCD(sameasLCM)

(d)SubtractsmallernumeratorfromlargerPlaceresultoverLCD

(e)Reducetolowestterms

Ex(a)Subtract from (LCD=10)

Ex(b)Subtract from

219Howdowesubtractmixednumberswhentheyarelarge(a)Findthedifferencebetweenthetwofractionsandthenfindthedifference

betweenthewholenumbersBorrow1fromtheminuendtoincreaseitsfractionwhennecessary

Ex(a)

Ex(b)From take Before or canbetakenfrom youmustborrow1or fromtheminuendtomakethefraction Theminuendthenbecomes

220CanawholenumberalwaysbeexpressedinafractionalformYesEXAMPLE Denominatoris1

221InaddingorsubtractingtwofractionshowcanweusecrossmultiplicationtogetthesameresultaswiththeLCDmethod

Ex(a) Cross-multiplynumeratorswithoppositedenominatorstogetnumerator

Multiplydenominatorstogetdenominator

Ex(b)

Ex(c)

222WhatistheprocedureinmultiplyingoneproperfractionbyanotherPlacetheproductofthenumeratorsovertheproductofthedenominators

Ex(a)

Ex(b)

Shortentheworkbycancellationwhenpossible

Ex(c)

Ex(d)

223HowdowemultiplyaproperfractionbyawholenumberEithermultiplythenumeratorordividedenominatorbythewholenumber

Ex(a)

Ex(b)Multiply by11

Ex(c)

Theresultisthesamewhenthemultiplierandmultiplicandareinterchanged

inposition

224WhatistheprocedureformultiplyingonemixednumberbyanotherChangethemixednumberstoimproperfractionsandmultiplyintheusual

waybyplacingtheproductofthenumeratorsovertheproductofthedenominators

Ex(a)

Ex(b)Multiply

225Whatisthefour-stepmethodofmultiplyingonemixednumberbyanother(a)Multiplythefractioninthemultiplierbyeachpartofthemultiplicand

(b)Thenmultiplythewholenumberofthemultiplierbyeachpartofthemultiplicand

(c)Addthepropercolumns

EXAMPLEMultiply

226Howdowemultiplyamixednumberbyaproperfraction(a)Changethemixednumbertoanimproperfractionandmultiplyasusual

(b)Ormultiplythefractionstogetherthenmultiplythewholenumberbythefraction

Ex(a)

Ex(b)Multiply by

Ex(c)Multiply by

Orchangemixednumbertoanimproperfractionfirst Then

227WhatwordisfrequentlyusedinsteadofthemultiplicationsignorthewordldquomultiplyrdquoThewordldquoofrdquo

EXAMPLE

228WhatismeantbythereciprocalofanumberThereciprocalofanumberis1dividedbythenumber

Ex(a)Thereciprocalsof3810and25are and respectively

Since3810and25areequivalentto and respectivelyinfractionformweobtainthereciprocalofafractionbyinvertingthefraction

Ex(b)Thereciprocalsof and are and respectively

229Whentheproductoftwonumbersequals1whatiseachofthetwonumberscalledEachiscalledthereciprocaloftheother

Ex(a) Hence4isthereciprocalof and isthereciprocalof4

Ex(b) Hence isthereciprocalof and isthereciprocalofTogetthereciprocalofafractionweinvertthefraction

230HowcanweshowthattomultiplybythereciprocalofanumberisthesameastodividebythatnumberWehaveseenabovethat Weheremultiplyby toget1

Itisalsotruethat Herewedivideby toget1

But isthereciprocalof

Thereforemultiplyingby isthesameasdividingby

231Howmanytimesare(a) and containedin1(b) and containedin2

(a)

(b)

232IneachcasewhatcanwedowhenwewanttodivideawholenumberbyafractionorafractionbyawholenumberorafractionbyafractionMultiplybyitsreciprocal

EXAMPLEDivide by

Thismeansthat goesinto oneandfour-fifthstimes

233Specificallyhowdowedivideaproperfractionbyawholenumber

Divideitsnumeratorormultiplyitsdenominatorbythewholenumber

Ex(a)Divide by2

Multiplyingthedenominatorbythewholenumberisequivalenttomultiplyingbythereciprocalofthewholenumber

Ex(b)

234HowdowedivideawholenumberbyafractionDividethewholenumberbythenumeratorandmultiplybythedenominator

Ex(a)Divide24by

Ex(b)Divide17by or

Ineachcasethemethodisequivalenttomultiplyingbythereciprocalofthefraction

235HowdowedivideonemixednumberbyanotherChangethemixednumberstoimproperfractionsinvertthedivisorand

multiply(Invertingthedivisorgivesthereciprocalofthedivisor)EXAMPLEDivide by

236HowdowedivideamixednumberbyawholenumberChangemixednumbertoanimproperfractionanddividethenumeratoror

multiplydenominatorbythewholenumber

EXAMPLEDivide by3

Herealsothemethodisequivalenttomultiplyingbythereciprocalofthewholenumber

237WhatisanothermethodtousefortheabovecasewhenthedividendisalargenumberDivideasinwholenumbersandsimplifytheremainingcomplexfraction

EXAMPLEDivide by6

238WhataresomeothermethodsofdividingwholemixednumbersEx(a)Divide482by

Multiplyingbothnumeratoranddenominatorby5doesawaywiththemixednumberinthedivisorbutdoesnotchangethevalueofthefraction

Ex(b)Divide by

TochangetowholenumbersmultiplynumeratoranddenominatorbythecommonmultipleofthedenominatorsofthefractionsLCMhereis12

239WhatisthedifferencebetweenafractionapplicabletoanabstractnumberandoneapplicabletoaconcretenumberThefraction meansthatanabstractunitisdividedinto4equalpartsand3

partsareexpressed

Theexpressionldquo ofadozenrdquoisapplicableto12becausethatisthenumberofunitsinadozenandmaybeexpressedas9

Thefractionldquo ofagallonrdquomaybeexpressedas2quartsbecausethereare4quartsinagallon

240HowdowefindwhatpartthesecondoftwonumbersisofthefirstDividethesecondbythefirst

Ex(a)Whatpartof63is9

Ex(b)Whatpartof74is18

Ex(c)Whatpartof is

Ex(d)Whatpartof is7

241IfyouaregivenanumberthatisacertainfractionofawholehowwouldyoufindthewholeDividethegivennumberbythefraction

Ex(a)6is ofwhatnumber

Ex(b)72is ofwhatnumber

Ex(c)99is ofwhatnumber

Notethatineachcaseyoumultiplybythereciprocalofthefraction

Ex(d)If78is ofthelotwhatisthewholelot

Ex(e)Findthenumberofwhich40is

Ex(f) ofsomeradioequipmentisworth$350Whatisthevalueoftheentirestock

242HowdowetellwhichoneoftwofractionsisthegreaterReducethefractionstotheirlowesttermsbycancellation

GettheLCD(lowestcommondenominator)andchangeeachfractiontohavethisLCDComparenumerators

EXAMPLEWhichofthefollowingisgreater or

(LCD=72times19=1368)

Weseethat792isgreaterThus isgreaterthan middot

243Whatisachain(oracontinued)fractionOneinwhichthedenominatorhasafractionthedenominatorofwhichhasa

fractionthedenominatorofwhichhasafractionetc

EXAMPLE

244WhatchainfractionsareofinteresttousOnlythoseinwhichallnumeratorsare1orunitymdashtheso-calledintegerchain

fractions

245HowisaproperfractionconvertedintoachainfractionWeknowthatdividingbothnumeratoranddenominatorofafractionbythe

samequantitydoesnotchangethevalueofthefraction

DividebothnumeratoranddenominatorbythenumeratorThenumeratorbecomes1andthedenominatorbecomesawholenumberandafractionasaremainder

ConvertthefractionalremainderbydividingbothitstermsbythenumeratorAgainthenumeratorbecomes1andthedenominatorbecomesawholenumberandafractionasaremainder

Continuethisprocessuntilthefractionalremainderhas1asanumerator

EXAMPLEConvert toachainfraction

246HowcantheabovebesimplifiedEachtimedividethepreviousdivisorbytheremainderThequotientsbecome

thedenominatorsofthechainfractionwithunitsfornumeratorsThedenominators11182aretheintegralpartsofthequotients

247HowisachainfractionconvertedtoaproperfractionByinverseprocessstartfromtheendandgoupIntheabovestartwiththe

lastfractionaldenominator

Thenextfractionaldenominatoris

Next

Next

Finally

248OfwhatpracticalusearechainfractionsForonethingtheyenableustofindanotherfractionexpressedinsimpler

terms(smallernumbers)andofavaluenearorveryneartheonewithlargenumbers

EXAMPLEWhatfractionexpressedinsmallernumbersisnearinvalueto

Dividingbothtermsby31weget

expressedasachainfraction

Nowifwerejectthe thefraction willbelargerthan becausethedenominatorwasdecreased

Tocompare with gettheLCDofbothor

157times5=785=LCDThen and

Thus isseentobenearthevalueof

249Whatfractioninsmallertermsnearlyexpresses

Dividenumeratoranddenominatorby3937

isalittlelargerthan butitgivesusaprettygoodideaofitsvalue

250Howcanwegetacloserapproximation

whichissmallerthan003937

Togetstillnearertakethenextpartofthechainfraction

Startfromthebottom

Thisisthenearestfractionto003937unlesswereducetheentirechainfractionwhichwouldgiveus003937itself isonlylargerthan whichisquiteclose

Wethusseethatachainfractioncangiveusaseriesofsuccessiveapproximations

251WhatfeatureofachainfractionmakesitvaluabletousTheapproachtothetruevalueisextremelyrapidItgivesveryrapidly

convergingapproximations

EXAMPLEOfabovevaluesof

Weseethatthesecondapproximationbringsuswithin039percentofitstruevalueVeryrapidindeed

PROBLEMS

1Iftherearefourweeksinamonththreeweeksareequaltowhatpartofthreemonths

2Ifaunitisdividedintotenequalpartswhatisonepartcalled

3Readthefollowing Whatpartofthesefractionsshowsthenumberofpartsintowhichtheunitisdivided

4In whatshowshowmanypartsaretaken

5Whichareproperfractionsimproperfractionsandmixednumbersinthefollowing

(a) (b)

(c) (d)

(e) (f)

6Writeascommonfractionsormixednumbers(a)Twenty-ninetenths(b)Forty-nineelevenths(c)Eightfifteenths(d)Nineone-hundredths(e)Ninety-twoandthree-fourths(f)Onehundredandthirty-fivefifty-sixths(g)Eighty-sevenandninetenths(h)Sixhundredtenths(i)Twenty-threethirty-sevenths(j)Eighteenandsixtwenty-firsts(k)Thirty-oneandseventeennineteenths(l)Onehundredforty-fiveandonehundredthirty-threeonehundredthirty-fifths7

Changetowholeormixednumbers

(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)(k) (l) (m) (n) (o)8Changetoimproperfractions

(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)(k) (l) (m) (n) (o)(p)9

(a)Howmanyfourteenthsinoneunit(b)Howmanyfourteenthsintwounits(c)Howmanyfourteenthsinonehalfunit(d)Doeschanging toitslowerterm changeitsvalue

10Reducethefractionstolowestterms(a) (b) (c) (d) (e)(f) (g)

11Changetohigherterms

(a) to20ths(b) to64ths(c)to84ths(d) to96ths(e) to100ths(f) to24ths

12Findthemissingnumerators

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

13ReducetofractionshavinganLCD

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

14ChangetoimproperfractionsandreducetoLCD

(a) (b)

(c) (d)

15

(a)Increasethevalueof threetimes(b)Increasethevalueof twoandone-halftimes(c)Increasethevalueof fourandone-sixthtimes

16

(a)Decreasethevalueof to thevalue(b)Decreasethevalueof to thevalue(c)Decreasethevalueof to thevalue

17Changetoasimplefraction

(a) of (b) of (c) of(d) of (e) of (f) of


(a) (b) (c) (d) (e) (f)

19Add

(a) (b) (c)(d) (e) (f) (g)(h) (i) (j)(k) (l)

20Subtract

(a) from (b) from (c) from(d) from(e) from (f) from(g) from

21Multiply(a) by (b) by (c) by (d) by4

(e) by12(f)17by (g) by (d) by

(i) by (j) by (k) by (l) by

22Expressthereciprocalsof(a)491135(b)

23Howmanytimesare(a) containedin1(b) containedin2

24Divide(a) by2(b) by3(c)27by(d)19by(e) by(f) by4(g) by7(h)574by(i) by

25Whatpartof(a)72is9(b)86is16(c) is (d) is15(e) is (f) is72(g) is (h) is (i) is (j) is

26(a)8is ofwhatnumber(b)84is ofwhatnumber(c)144is ofwhatnumber

27(a)Findthenumberofwhich60is (b)Five-eighthsofashipmentisworth$430whatisthevalueoftheentireshipment

28Whichfractionhasagreatervalue or

29Express asachain(orcontinued)fraction

30Convert toachainfraction

31

32Whatfractioninsmallernumbersisnearinvalueto

33Whatfractioninsmallertermsnearlyexpressesπ=31416or (Usechain-fractionmethod)34Thewidthofadooropeningis ofitsheightWhatisthewidthwhentheheightis ft

35IfindthatIspent$88whichrepresents ofmytotalallowanceHowmuchdoIhaveleft

36Threecasesofmerchandiseweighing and IbwereshippedThecasesweighed and lbWhatisthetotalweightofthecasesgrossweightandthenetweightofthemerchandise

37Ifalbofbreadhad9sliceshowmanyouncesarethereperslice

38Howmanyreamsofpaperarelistedonthisinvoice andreams

39Ifinatestrunacartraveled26milesin30minuteshowmanymileswillittravelin hoursatthisrate

40Acrateofapplescontaining148appleswasboughtat anappleandsoldat ofthecostWhatwastheprofit

41Twopartnersboughtaparceloflandfor$3600eachpaying Theneach

sold ofhisinteresttoathirdpartyatcostWhatfractionalpartofthetotalinvestmentdoeseachpartynowownandhowmuchiseachworth

42Amanspends ofhissalaryforasuitofclothes foranovercoat forshoesand forahatWhatparthasheleft

43Iftheabovepersonhas$41lefthowmuchhadhetobeginwithandwhatdoeseachitemcost

44Thesidesofanirregularlyshapedyardhavethefollowingmeasurementsyd yd yd ydHowmanyyardsoffencingwillbeneededto

encloseit

45Ifthemineralmatteroftheorgansofthebodyisbones muscles lungs brain howmuchmoremineralmatteristhereinbonethanineachoftheotherorgansgiven

46Ifaboyof10yearsneedsdaily gramsofprotein gramoffatandgramsofcarbohydratesforeachpoundofweighthowmuchofeachwillaboyof10weighing69lbrequire

47Alotis feetwideby feetdeepHowmanyrods( fttoarod)ofwirewillbeneededtofencethelot

CHAPTERVII

DECIMALFRACTIONS

252WhatisdecimaldivisionDivisionofunitsintotenthshundredthsthousandthsetc

EXAMPLES

253WhatisadecimalfractionThepartofaunitobtainedbydecimaldivisionDecimalfractionsareoften

calleddecimalsItisafractionalvalueexpressedintenthshundredthsthousandthsetcThismeansthatthedenominatoris10orsomemultipleof10

254WhatdowecallthedecimalpointTheperiodplacedattheleftoftenthshundredthsetc

EXAMPLES

(threetenths) (sevenhundredths)(fivethousandths)

255Howmaydecimalfractionsbeexpressed(a)Bythepositionofthedecimalpoint

(b)Byadecimaldenominatorintheformofacommonfraction

Ex(a)0207008024017

Ex(b)

256Whatarethenamesofthedecimalplacesandhowaredecimalswritten

EXAMPLES

Toexpresstenthsoneplaceispointedoffas2

Toexpresshundredthstwoplacesarepointedoffas28

Toexpressthousandthsthreeplacesarepointedoffas287

Toexpresstenthousandthsfourplacesarepointedoffas2875

ReadaboveldquoFourandtwohundredeighty-seventhousandfivehundredeighty-threemillionthsrdquo

257HowisadecimalreadThedecimalpointisreadldquoandrdquoReadadecimalexactlyasifitwereawhole

numberandthenaddthefractionalnameofthelowestplace

EXAMPLE5631056923

ReadldquoFiveandsixhundredthirty-onemillionfifty-sixthousandninehundredtwenty-threebillionthsrdquoThelowestdecimalplacehereisbillionths

258WhatistherelationofthenumberoffiguresinadecimaltothenumberofzerosinitsdenominatorwhenexpressedasacommonfractionTheyarethesame

Ex(a)0345hasthreefigurestherefore hasthreezerosinthedenominator

Ex(b)001679hasfivefigurestherefore hasfivezerosinthedenominator

259IsthevalueofadecimalfractionchangedbyaddingoromittingzerosontherightNoEXAMPLE4=40=400Also

Addingzerostotherightdoesnotchangethevalue

260WhatistheeffectondecimalfractionsofmovingthedecimalpointtotheleftMovingthepointoneplacetotheleftdividesthedecimalby10twoplaces

dividesitby100threeplacesdividesitby1000etc

EXAMPLES

Thedecimalpointismovedtotheleftfordivisionby10rsquostomakethedecimalsmaller

261WhatistheeffectofmovingthedecimalpointtotherightMovingthepointoneplacetotherightmultipliesthedecimalby10two

placesby100threeplacesby1000etc

EXAMPLES

Thedecimalpointismovedtotherightformultiplicationby10rsquostomakethedecimallarger

262WhatmustbedonewhenthereisnotasufficientnumberoffiguresinthenumeratortoindicatethedenominatorofadecimalfractionZerosareplacedbetweenthedecimalpointandthefigureorfiguresinthe

numerator

Ex(a)Towriteninehundredthsasadecimalplaceazerobetweenthe9andthedecimalpointotherwisethefractionwouldbeninetenths

Placesufficientzerostotherightofthedecimalpointtomakeupasmanyfiguresinthenumeratorastherearezerosinthedenominatorwhenthefractionalvalueiswrittenasacommonfraction

Ex(b)Towrite notethatthedenominatorhasfivezerosthereforethenumeratormusthavefivefigurestotherightofthedecimalpointItalreadyhastwofiguressoaddthreezerostotherightofthedecimalpointor

263Howaredecimalsclassified(a)Asimpledecimalhasawholenumbertotherightofthedecimalpointas

048386356

(b)Acomplexdecimalhasawholenumberandacommonfractionwrittentotherightofthedecimalpointas

264DoweneedadecimalpointaftereverywholenumberNoThedecimalpointisunderstoodasattherightoftheunitsplace

EXAMPLE6=6=60=600

265HowdowedivideanynumberbyadecimalnumberShiftthedecimalpointoneplacetotheleftforeveryzerointhedivisor

EXAMPLES(a)132divide10=132OnezeroindivisorMove1placetoleft(b)132divide100=132TwozerosindivisorMove2placestoleft(c)132divide10=0132Move1placetoleft(d)132divide100=00132Move2placestoleft

266HowdowemultiplyanynumberbyadecimalnumberShiftthedecimalpointoneplacetotherightforeveryzerointhemultiplier

EXAMPLES

(a)132times10=1320Shift1placetoright(b)132times100=13200Shift2placestoright(c)132times1000=132000Shift3placestoright(d)132times10=132Shift1placetoright(e)132times100=132Shift2placestoright

(f)132times1000=132Shift3placestoright(g)132times10000=1320Shift4placestoright

267WhatisamixednumberindecimalformandhowdowemultiplyanddivideitbyadecimalAnumberthatconsistsofawholenumberandadecimalfractionas132465

Thesamerulesapplyasabove

EXAMPLES

(a)132465times10=132465Move1placetoright(b)132465times100=132465Move2placestoright(c)132465divide10=132465Move1placetoleft(d)132465divide100=132465Move2placestoleft

268HowcanwechangeacommonfractiontoadecimalAnnexzerostothenumeratoranddividebythedenominator

EXAMPLES

(a) or

(b) or

(c) or

(d) or

(e)

(f)

WhentheresultisacomplexdecimaltwoplacesareusuallyfarenoughtocarryoutthedecimalFormostpurposesthreeorfourplaceswillsuffice

269HowcanweextendacomplexdecimalAddzerostothenumeratorofthefractionanddividebythedenominator

Whenthedivisioncomesouteventhefractionistherebyremovedotherwisethedecimalmaybeextendedasmanyplacesasaredesired

Ex(a)Extendthecomplexdecimal

Addthreezerostothenumerator5anddividebydenominator8

Ans=9625Thedivisioncameouteven

Ex(b)Extend to6decimalplaces

Addfourzerostothe5anddivideby12

Ans=394166=sixdecimalplaces

270HowcanweconvertadecimalexpressiontoacommonfractionExpressthedecimalasanumeratoroveradenominatorandreducetolowest

termsThedenominatorisamultipleof10asindicatedbythedecimalpointThenumeratorisawholenumber

Ex(a)Change5toacommonfraction

Thedecimalpointindicates10asthedenominatorThus reducedtolowestterms

Ex(b)

Denominatoris1000Thus

reducedtolowestterms

Ex(c)Change5736toacommonfraction

TherearefourplacestotherightofthedecimalpointthereforetherearefourzerosinthedenominatorThus

271WhatistheprocedureforaddingwholenumbersandsimpledecimalsPlacethenumbersincolumnswiththedecimalpointsdirectlyunderone

anotherandaddintheusualwayThedecimalpointofthesumisdirectlyunderthepointsinthecolumn

EXAMPLEAdd2638745209537283and935

Addingzerosattherightofthedecimaldoesnotaffectthevalue

272WhatistheprocedureforaddingwholenumbersandcomplexdecimalsExtendthecomplexdecimalsthesamenumberofplacesandthenaddinthe

usualway

273WhatistheprocedureforsubtractingsimpledecimalsPlacethedecimalpointinthesubtrahenddirectlyunderthedecimalpointin

theminuendandsubtractasusualThedecimalpointoftheremainderisdirectlyunderthepointsaboveit

EXAMPLESubtract520953from7283

274WhatistheprocedureforsubtractingasimpledecimalandacomplexdecimalExtendtheshortercomplexdecimaluntilthefractionisremovedorthereare

thesamenumberofplacesintheminuendandsubtrahendandthensubtractintheusualway

EXAMPLEFrom subtract

275WhatistheprocedureformultiplyingsimpledecimalsMultiplyintheusualwayandpointoffintheproductasmanyplacesasthere

areplacesinboththemultiplierandmultiplicand

Ex(a)Multiply38by6

Ex(b)

Ex(c)

276WhatistheprocedureformultiplyingcomplexdecimalsExtendthedecimaltoremovethefractionwhenitcanbedoneorchangeto

improperfractions

EXAMPLE

277WhatistheprocedurefordividingonesimpledecimalbyanotherThetermsinadivisionare

(1)Thedivisormustbemadeawholenumberbymovingthedecimalpointtotheextremeright(ortheendofthenumber)Countthenumberofplacesyoumovedthepoint

(2)MovethedecimalpointinthedividendanequalnumberofplacesIfthedividendisawholenumberthenaddasmanyzerosinsteadandplacethepointattheend

(3)Placethedecimalpointinthequotientjustabovethepointinthedividend

Rememberthatadecimalpointisunderstoodaftereverywholenumber

Ex(a)Divide192by06

Sixone-hundredthsiscontainedin192thirty-twohundredtimes

ProofMultiply3200by06(2places)

3200times06=19200(2places)

Ex(b)Whatistheresultofdividing06118by14

Thedecimalpointinthequotientisalwaysdirectlyabovethedecimalpointinthedividend

Ex(c)Divide4030496by478

278WhatistheprocedurefordividingonecomplexdecimalbyanotherChangethecomplexdecimalstosimpledecimalsifpossibleandthendivide

otherwisemultiplybothnumbersbytheLCDofthedenominatorsofthefractionsbeforeyoudivide

EXAMPLEDivide by (LCD=6)

279HowisadecimalnumbershortenedforallpracticalpurposesIfarejectedordiscardeddecimalis5orover1isaddedtothenextfigureto

theleft

EXAMPLE44746143752canbeshortenedto44746144whichisconsideredtobecorrectto4decimalplaces(orfoursignificantfigures)Sincethefifthplacewhichis7isgreaterthan5then1isaddedtothenumbertotheleftofit3whichbecomes4

Nowin44746144thefourthplaceis4Thisislessthan5andisdroppedleaving4474614whichissaidtobecorrecttothreedecimalplaces

447461iscorrectto2decimalplaces44746iscorrectto1decimalplace

280WhatothermethodofdecimalapproximationhasbeeninternationallyapprovedThatofmakingthedecimaleven

Ex(a)48655isshortenedto4866

Thelast5isdroppedand1isaddedtothe5toitslefttomakethedecimaleven

Ex(b)48645isshortenedto4864

Since4isanevennumberyoumerelydropthe5Itisclaimedthatacloseraverageresultisobtainedwhenadecimalismadeeven

281WhatistheleastnumberofsignificantfiguresthatmustbekeptwhenthedecimalispurelyfractionalandcontainsanumberofzerostotherightofthedecimalpointAtleastonesignificantfiguremustbekept

EXAMPLE000072184maybeshortenedto00007

282Whatistheresultof03024times0196correctto2significantfigures

Onecantellatoncethat006iscorrectto2places(byadding1tothe5toget6because9issolarge)

283Whyisittheruletoworkaproblemtoonemoredecimalplacethanweneed

Ithelpsustodeterminewhetherthenextfigurewouldbegreaterorlessthan5andenablesustoknowwhetherornotthefigureweuseissufficientlyaccurate

284Whatcanwedotosimplifythingswhenwewanttogetananswercorrecttotwodecimalplacesinmultiplying4879by3765Thereisnoneedtogothroughthemultiplicationoftheentirenumbers

Ifweweretomultiply5times4(=20)wethusdropalldecimalsandweguessouranswertobesomewhatlessthan20Thisgivesusnodecimalplaces

Now49times38=1862Ifweretainonedecimalplaceinthemultiplierandmultiplicandwegetananswerwithtwodecimalplacesbutwearenotsureofthe62

Soourrulesaystoretainonemoreplacethanrequiredandweget488times377=183976or1840approximatelycorrectto2places

Thecompletemultiplicationwouldbe

4879times3765=18369435

Weseethatthislengthymultiplicationisnotjustified

285WhatisanotherwayofapproximatingthedesiredresultinvolvingdecimalsContractedmultiplicationSincethefigurestotheleftofthedecimalpointare

mostimportant

(1)Multiplyallofthemultiplicandbytheleft-handdigitofthemultiplier

(2)Droprightdigitofmultiplicandandmultiplyremainderbynextfigureofmultiplier

(3)Droptwodigitsofmultiplicandandmultiplyremainderbynextfigureofmultiplier

(4)Continuesuccessivelydroppingonemoredigitofmultiplicandeachtimeyoumultiplybyanotherfigureofthemultiplier

EXAMPLE

286WhatisarecurringdecimalWheninsomecasesofdecimaldivisionthecalculationcanbecarriedon

indefinitelywithrepeatingnumbersorsetsofnumberssuchadecimalisknownasarecurringdecimal

EXAMPLES

(a)(b)(c)

287Howarerecurringcirculatingorrepeatingdecimalsdenoted(a)ByadotovertherecurringfigureThus404means404444etcto

infinity

(b)BydotsplacedoverthefirstandlastfiguresoftherecurringgroupThus

288HowcanweconvertpurerecurringdecimalstofractionsUseninesinthedenominatormdashone9foreverydecimalplaceintherecurring

group

EXAMPLES(a)Recurringdecimal(b)Recurringdecimal (142857times7=999999)

Notethatapurerecurringdecimalisoneinwhichallthedigitsrecur

289HowcanweconvertmixedrecurringdecimalstofractionsInamixedrecurringdecimalthedecimalpointisfollowedbysomefigures

whichdonotrecur

(1)Subtractthenonrecurringfiguresfromallthefiguresandmaketheresultthenumerator

(2)Thedenominatorconsistsofasmanyninesastherearerecurringfiguresfollowedbyasmanyzerosasnonrecurringfigures

EXAMPLES(a)

(b)

(c)

(d)

(e)

290Whyinparticularshouldyouknowthedecimalequivalentsof and

ItisthensimpletofindotherfractionalequivalentsinthisseriesThus

291Howcanwesometimesproduceadecimalequivalentbymultiplyingbothnumeratoranddenominatorbyasuitablenumber

292HowdowefindthewholenumberwhenadecimalpartofitisgivenEx(a)56is8ofwhatnumber

Ex(b)If4ofanumberis64whatisthenumber

293HowisUnitedStatesmoneyrelatedtodecimalfractionsTheunitisthedollarexpressedbythesign$as$15=fifteendollarsDollars

maybedividedintotenthshundredthsandthousandths

294IfaBritishpound(pound)isworth$280andthereare20shillingstothepoundand12pencetotheshillinghowmuchis(a)1shillingworth(b)1pennyworthRememberIfyouwanttogetthevalueofoneunitofanyelementina

problemyoushoulddividebythatelement

(a)Youwanttofindthevalueof1shillingthendividebyshillings

Dividenumeratoranddenominatorby10orwhatisthesamethingmovethedecimalpoint1placetotheleftinnumeratoranddenominator

(b)

295AmanufacturersubmittedabidtotheUnitedStatesgovernmentformilitaryinsigniainthesumof$6839970at31cents millsperdozenHowmanydozenwouldbedelivered

PROBLEMS

1Writeindecimalform(a)Sixtenthsfourtenthstwoandonetenth(b)Sevenandninethousandthsnineandfifty-threethousandthsthreeten-thousandthselevenmillionths

(c)Onehundredfifty-fivethousandthsfourhundredninety-twothousandthssixten-thousandthsthreehundredandfourhundredths

(d)Sixandsevententhseightandtwotenthseighty-sixhundredthsfivehundredandsixthousandths

(e)Fourandthree-eighthshundredthsthirty-sixandfive-seventhsthousandthseightandtwo-thirdsofathousandtheightandfourandtwo-thirdsthousandths

2Writethefollowingfractionsasdecimalfractions

3Read12584062018

4Distinguishbetween0400and000004

5Whatisthedenominatorof45602763expressedinfractionform

6Expressascommonfractions025025002500

7Annexingaciphertoawholenumberincreasesitsvaluehowmanytimes

8Doesannexingaciphertoadecimalaffectitsvalue

9Selectthequantitiesthathavethesamevalueinthefollowing(a)0880088080080(b)04646004600046046004600(c)7387380738000073807380738

10Arrangethefollowinginascendingvalues

260260026260260

11Movethedecimalpointin4soastomakethedecimalsmallerby by

12Movethedecimaltomultiply004by10by100by1000

13Divide246by10by100




17Multiply246576by10by100


19Changethefollowingtodecimals(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)

20Extendthecomplexdecimals(a)(b)(c) to5places(d) to6places

21Changetocommonfractions(a)6(b)86(c)625(d)1875(e)0125(f)750(g)4765(h)

22Add(a)74866922536245and6286(b)486652366803643986and257(c)3749856309648394and824

23Add(a) and

(b) and(c) and(d) and

24Subtract(a)630842from8394(b)2884from49836(c)49486from23957(d)81564from128096(e)1489736from197134(f)3874from4

25(a)From subtract(b)From subtract(c)From subtract(d)From subtract(e)From subtract

26Multiply(a)49by7(b)054by8(c)845327by58986(d)1232by98736(e)184236by49

27Multiply(a) by(b) by(c) by(d) by(e)6836by

28Divide(a)283by07(b)07229by16(c)5040587by589(d)48735by6486

(e)64575by165(f)9686by136

29Divide(a) by(b) by(c) by(d) by(e) by(f)9957by

30Shorten57857254863tobecorrectto(a)4decimalplaces(b)3decimalplaces(c)2decimalplaces(d)1decimalplace

31Shorten(a)59767(b)59755

32Shorten0000083273totheleastnumberofsignificantfigures

33Findtheresultof04035times0287correctto2significantfigures

34Gettheresultof5987times4876correctto2decimalplacesbyshortenedmultiplication

35Gettheapproximateresultof5987x4876bycontractedmultiplication

36Convertthefollowingrecurringdecimalstofractions(a)(b)(c)(d)(e) (f)(g)(h)(i)

37Whatisthedecimalequivalentof(a) (b) (c) (d) (e) (f) (g) (h)

38(a)78is7ofwhatnumber(b)If6ofanumberis86whatisthenumber(c)81is9ofwhatnumber(d)99is75ofwhatnumber

39IftheBritishpound(pound)isworth$280andthereare20shillingstothepoundhowmucharethreeshillingsworthIfthereare12pencetoashillinghowmuchissixpenceworth

40Ifthetotalcostofashipmentis$7948865at millsperdozenitemswhatisthenumberofdozensintheshipment

41Ifafamilyfoundthatattheendoftheyearithadsaved$455andduringtheyearithadspent29ofitsincomeforfood17forrent25forclothingand21formiscellaneousitemswhatwastheamountofitsincome

42Inacollegetheregistrationwas33inpuresciencecourses26inliberalarts21insocialscienceandtheremainderinengineeringThenumberofstudentsinengineeringwas520WhatisthetotalregistrationofthecollegeHowmanystudentsineachcategory

43Amaninvests22ofhismoneyinbonds32incommonstocks36inrealestateandhehas$3400incashleftoverHowmuchishistotalequityHowmuchhasheineachcategory

44Specificationsforphosphorbronzerequire86copper065tin0007iron002lead0035phosphorusandtheremainderzincHowmanylbofeachelementarerequiredtomake1200lbofphosphorbronze

45Afarmersold8460poundsofapples(eachbushelweighing60lb)for$180abushelWiththeproceedshebought9000lboffertilizerWhatisthe

costofthefertilizerper100lb

46Thedistanceroundawheelis31416timesitsheightWhatisthedistanceroundawheel385feethighRounda32-inchhighwheel

47If100lbofmilkyield5563lbofbutterandagallonofmilkweighs87lbhowmuchbutterwill2gallonsofmilkyield

48Whatisthecostofarailroadticketat$045amileifthedistanceyouaretotravelis475miles

49If6370piecesofcutlerycost$75369tomanufacturewhatisthecostofeachincentsandmills

50Ifyoumade$260onaninvestmentof$4000whatfractionalpartoftheinvestmentdidyoumake

51If2lbofcoffeecost$165howmanylbcanyoubuyfor$2640

52Ifyouboughtsix$1000bondsfor andsoldthemfor (a)whatisthetotalamountpaidforthebonds(b)theamountreceivedforthem(c)theprofit(d)theprofitexpresseddecimallyinthousandths(Note meansoneach$100ofthebondor$96750foreachbond)

53TwoballteamsAandBeachhavingplayed46gameshavearespectivestandingof826and739IfAwinsonly4ofthenext10gamesandBwins6ofthenext10gameshowwilltheclubsstand

CHAPTERVIII

PERCENTAGE

296Whatismeantby(a)percent(b)percentage(a)Percentmeansldquobythehundredrdquothenumberofhundredthsofanumber

InLatinpercentummeansldquobythehundredrdquo

EXAMPLEIfoutof100students30failedinthefinalexaminationthen30percentfailedand70percentpassed

(b)PercentagemeansldquobyhundredthsrdquoandincludestheprocessofcomputingbyhundredthsIndealingwithpercentagewearethusworkingwithdecimalswhosedenominatoris100

EXAMPLE

297Whatisthesymbolusedtorepresentthedenominator100Thetermpercentisexpressedbythesign[]

EXAMPLES(a)(b)

(c)

(d)

(e)(f)(g)

Thepercentsign[]takestheplaceofthefractionlineandthedenominator100

298Inwhatwaysmayagivenpercentoragivennumberofhundredthsofanumberbeexpressed(a)Asawholenumber6(b)Asadecimal06(c)Asafraction

299Whendoweexpressquantitiesaspercentages

Whenwewishtocomparetwoquantitieswhicharenoteasilycommensurableitismoreconvenienttoexpressthemaspercentages

EXAMPLEItisobviousthat4ofaquantityisgreaterthan whileitisnotsoapparentthat268isagreaterproportionof6700than315of8400

300HowdowereduceanumberwrittenwithapercentsigntoadecimalDropthepercentsignandmovethedecimalpointtwoplacestotheleftThis

isequivalenttodividingby100whichisthemeaningofpercentDroppingthemeansdividingby100

EXAMPLES(a)35=35(move2placestolefttodivideby100)(b)135=135(move2placestolefttodivideby100)

301HowdoweconverttoadecimalwhenthepercentisexpressedasanumberandafractionCarryoutthefractioninordertoconvertittoadecimal

EXAMPLES

NoteYoumaycarryoutthefractiondirectlyandaddittothedigitnumbers

302HowcanweconvertawholenumberadecimalfractionafractionoramixednumbertoapercentIneachcasemultiplyby100toannexasign

EXAMPLES

NoteTomultiplyby100movedecimalpoint2placestotherightwheneverthatcanbedonedirectly

303Whatarethepercentequivalentsofverycommonfractions

304WhatpercentofthelargesquareistheshadedpartLargesquarecontains25smallsquares

Shadedpartcontains6smallsquares

Shadedpartis24oflargesquare

305WhatisthemostcommonmethodoffindingagivenpercentofanumberWritethepercentasadecimalandmultiply

Ex(a)Find6of$6700(6=06)Then

Ex(b)Find14of$9751(14=14)

306Whatisanothermethodoffindingagivenpercentofanumber

Find1ofthenumberfirstandthenmultiplybythegivenpercent

Ex(a)Find6of$6700

1of

(Move2placestolefttodivideby100)Then

6of$6700=6times$67=$402

Ex(b)Find4of$1860

1of$1860=$1860there44is4times$1860=$7440

Ex(c)Find of$7000

307WhatisthethirdmethodoffindingagivenpercentofanumberWritethegivenpercentasacommonfractionandmultiply

Thismethodisusefulwhenthegivenpercentistheequivalentofasimplecommonfraction

Ex(a)Find25of$51

Ex(b)Find of$8475

308Whattermsarecommonlyusedinpercentage(a)Thenumberofwhichsomanyhundredthsoracertainpercentistobe

takeniscalledthebase(=B)

(b)Thepercenttobetakenistherate(=R)

(c)Theresultoftheratetimesthebaseisthepercentage(=P)

P(percentage)=R(rate)timesB(base)orP=RtimesB

Ex(a)Findthepercentagewhentherateis4andthebaseis$1860

Ex(b)Find9of50

309WhatistheruleforfindingthepercentagewhenthebaseandratearegivenMultiplythebasebytherateexpressedeitherasadecimaloracommon

fraction

Ex(a)Intestingacertainore25ofitwasfoundtobeironHowmuchironwascontainedin552poundsofore

Ex(b)Suppose27wasironHowmuchironwastherein578poundsofore

27times578lb=27times578=15606lbiron(rate)(base)(percentage)

310WhatistheruleforfindingtheratewhenthepercentageandbasearegivenDividethepercentagebythebasetogettherateSince

Notethatrateisapercentandisafractionoradivision(=acomparisonbetweenpercentageandbase)

Ex(a)$114iswhatpercentof$3800

Dividethequantitybythatwithwhichitisbeingcompared

Ex(b)Aninvestorreceived$38250onaninvestmentof$8500Whatratepercentdidtheinvestmentpay

Youarecomparingthepercentagewiththebase

Ex(c)Amanearns$9000ayearHepays$1800ayearforrentWhatpercentofhissalaryishisrentComparethepercentageof$1800withthebase$9000

311WhatistheruleforfindingthebasewhentherateandthepercentagearegivenDividethepercentagebytherateexpressedeitherasacommonfractionoras

adecimalSince

NoteDividingbythepercentgivesyouthepercentagefor1percent(or1partinahundred)Thenmultiplyingby100givesyouthewholeamount

Ex(a)$435is20ofwhatamount

or

RememberIfyouwanttogetthevalueofoneunitofanyelementintheproblemyoushoulddividebythatelementDividebypercenttofindvalueof1percentTherefore

$2175times100=$2175=valueof100=base

Ex(b)$18720is16ofwhatamount

Ex(c)Whatistheamountofabillif2discountforcashcomesto$285

If$285is2then

or

312Whatismeantby(a)amount(b)differenceinpercentageproblems(a)Amount=base+percentage(b)Difference=basendashpercentage

313Howcanwefindthebasewhentherateandamountaregiven

Ex(a)Therentofanapartmentis$1848peryearandthisisanincreaseof10overthepreviousyearWhatwastherentthepreviousyear

Base=rentpreviousyearAmount=$1848rentthisyear10=rateincrease=10

or

100=base=Rentpreviousyear10=Advancethisyear110=$1848(=Rentthisyear)1=$1848divide110=$1680there4Base=100=100times$1680=$1680

Ex(b)AstorekeepersellsaTVsetfor$270andmakesaprofitof onthetransactionWhatdidtheTVsetcosthim

314Howdowefindthebasewhentherateanddifferencearegiven

Ex(a)Ifthewasteinminingandhandlingcoalamountsto4howmanytonswouldhavetobeminedtoload20carswith30tonseach

Base=tonstobeminedDifference=20times30=600tons

or

100=base=Tonstobemined4=Loss96=600tons1=600divide96=625tons100=625times100=625tonstobemined

Ex(b)Amansellshiscarfor$1500andloses25onthetransactionWhatdidhepayforit

315OnwhatdowealwaysbasethepercentofincreaseinsomequantityItisbasedontheoriginalquantityandnotontheincreasedquantity

Ex(a)Ifthepriceofanewspaperwasraisedfrom5centsto10centswhatwasthepercentofincreaseinpriceTheoriginalpriceis5centsTheincreaseis5cents

Thustherewasa100increaseinprice

Ex(b)Ifatthebeginningoftheyearyouhadabankbalanceof$4500andattheendoftheyearyouhad$5400bywhatpercenthadyourbalanceincreased

316OnwhatdowealwaysbasethepercentofdecreaseinsomequantityItisbasedontheoriginalquantityandnotonthedecreasedquantity

Ex(a)Anewautomobilewhichcost$2200wasworth$1800ayearlaterBywhatpercenthasitdecreasedinvalue

Ex(b)Ifabankdroppeditsinterestratefrom to whatwouldbethepercentofdecreaseintheinterestrate

317Howarepercentslessthan1percentorfractionalpartsof1percentwrittenandusedinbusinessandfinancialmatters

EXAMPLEIfthetaxonahouseisincreasedby whatistheamountofincreaseonahouseassessedat$15750

$15750times0025=$39375=$3938Ans

318HowistheexpressionofldquosomuchperhundredrdquocommonlyusedinbusinessItisusedineachofthefollowingexamples

Ex(a)Whatistheamountofthepremiumona$12000fireinsurancepolicyat55centahundreddollars

120times$55=$66Ans

Ex(b)Abrokerchargesyou$1250per100sharesHowmuchwillitcostyoutobuy500sharesofstock

5times$1250=$6250Ans

Ex(c)Abankruptfirmpaysyou43centonthedollarHowmuchdoyougetwhenyourclaimamountsto$46375

$46375times43=$19941Ans

319Howisthemillusedintaxmatters

EXAMPLEApropertyassessedat$12500istaxedat287millsperdollarHowmuchisthetax

320HowarepercentsaddedsubtractedmultipliedordividedTheyareconvertedtodecimalsfirstandcarriedoutinthesamemanneras

similaroperationsinvolvingdecimals

321IfanumberisincreasedbyacertainpercenttogetanamountwhatpercentmustbesubtractedfromthisamounttogettheoriginalnumberagainTogetbacktotheoriginalnumberadifferentpercentmustbesubtracted

fromtheamount

EXAMPLEIf6of85isaddedtoitweget

06times85+85=51+85=901=Amount

Nowwhatpercentof901mustbesubtractedfromittoget85again

Weseethat51isonly566of901whereas51is6of85theoriginalnumber

322IfBostonhasapopulationof2000000andPhiladelphiais50largerhowmuchsmallerisBostonthanPhiladelphia

(AlsoPhiladelphiais50largerthanBoston)

ThisagainemphasizestherulethatthepercentofincreaseordecreaseofsomequantityisalwaysbasedontheoriginalquantityForBostontheoriginalquantityis2000000andforPhiladelphiaitis3000000

323Ifamanspends30ofhisincomeforrentand10oftheremainderforclotheswhatishissalaryifthelandlordgets$1150morethantheclothier

30ofincome=Rent10ofremainder(100ndash30)=1times7=07=7=Clothes30ofincome=7ofincome+$1150or23ofincome=$1150

there4$1150divide23=$5000=Income

324Amansellshiscartohisfriendandtakesalossof20Hisfriendsellsthecarlatertoathirdpartyfor$1500losing25Howmuchdidtheoriginalownerpayforthecar$1500represents75ofhiscost

$2000represents80oforiginalownerrsquoscost

PROBLEMS

1Whatdoes27meanintermsofpercentage

2Whatpercentof4000is1800

31400iswhatof3600

4Reducetoadecimal

(a)5(b)(c)(d)6(e)75(f)(g)(h)115(i)(j)926(k)003(l)(m)225(n)6(o)250(p)73(q)03(r)(s)(t)(u)60(v)(w)(x)(y)

5Express asdecimalsofapercentandasdecimals

6Expressascommonfractionsinlowestterms(a)1212(b)2525(c)3636(d)7575(e) (f) (g)15015(h)375375(i) 14(j) 05

(k) (l)

7Changetoapercent(a)9(b)6(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)84(n)(o)65(p)(q)8(r)(s)07(t)0425(u)(v)

8Whatpercentofthecircleistheshadedpart

9Whatpercentofthelargesquareistheshadedpart

10Find(a)4of$4800(b)16of$8642(c)6of$8500(d)7of$1940(e) of$6000(f)25of$62(g) of$7625(h) of$1600(i) of1500(j)150of500(k) of7254(l) of6542

11Findtheresultbyfirstfinding1ofthegivennumberinthefollowing(a) of10000(b)4of1600(c) of4000(d) of10000(e) of6000(f)6of7000

12Amanowned960acresoflandHesold ofitHowmanyacresdidhesell

13Amanhad$24000incashHeinvested ofitinbondsand46instocksHowmuchdidheinvestineachandhowmuchmoneyhadheleft

14Intestingacertainore27ofitwasfoundtobeironHowmuchironwascontainedin645lbofore

15Thereare2760votersinacertaintownIf69ofthevotersgotothepollshowmanyvoteswillbecast

16Aninvestorreceived$46050onaninvestmentof$9200Whatratepercentdidtheinvestmentpay

17Amanearns$8000ayearHepays$1600ayearforrentWhatpercentofhissalaryishisrent

18$565is20ofwhatamount


20Whatistheamountofabillif2discountforcashcomesto$345

21Whatpercentof(a)138is56(b)495is65(c)9860is1260(d)125is05(e)03is0085(f) is (g)47830is6458(h)2736is5985(i)93is1546(j)66is24(k)107is765(l)1235is05486(m)289is1485

22Findthenumberofwhich(a)360is15(b)459is40(c)56is(d)420is125(e)52is(f)112is(g)306is(h)132is(i)89653is6

23Whatis4of ofanacreofland

24Ifamerchantrsquosscalesweigh14ozforapoundwhatpercentdoesthepurchaserlose

25Whatpercentis of6

266is5ofwhatnumber10ofwhatnumber


28$250is ofwhat ofwhat




32Therentofanapartmentis$1656andthisisanincreaseof12overthepreviousyearWhatwastherentthepreviousyear

33Amansellsarefrigeratorfor$340andmakesaprofitof onthetransactionWhatdidtherefrigeratorcosthim

34Ifthewasteinminingandhandlingcoalamountsto howmanytonswouldhavetobeminedtoload40carswith30tonseach

35Amansellshishousefor$12000andloses12onthetransactionWhatdidthehousecosthim

36Ifthepriceofamagazinewasraisedfrom15centto25centwhatwasthepercentincreaseinprice

37Ifatthebeginningoftheyearyourbankbalancewas$3800andattheendoftheyear$4600bywhatpercenthadyourbalanceincreased

38Anewcarwhichcost$3100wasworth$2700ayearlaterBywhatpercenthaditdecreasedinvalue

39Ifabankdroppeditsinterestratefrom to3whatwouldbethepercentdecreaseintheinterestrate

40Expressinfractionsofapercentandindecimals(a) of1(b) of1(c) of1(d) of1(e) of1(f) of1

41Ifthetaxonahouseisincreasedby whatistheamountofincreaseonahouseworth$14700

42Whatisthepremiumonan$18000fireinsurancepolicyat64centperhundreddollars

43Ifyouarecharged$1250per100sharestobuystockshowmuchwillitcostyoutobuy1200sharesofstock

44Abankruptfirmpaysyou67centonthedollarHowmuchdoyoureceive

whenyourclaimamountsto$58545

45Apropertyassessedat$14500istaxedat243millsperdollarHowmuchisthetax

46If8isaddedto$96toget$10308whatpercentof$10308mustbesubtractedfromittogetbackto$96

47IfuniversityAhasanenrollmentof12000studentsanduniversityBis35largerhowmuchsmallerisuniversityAthanB

48Ifamanspends25ofhisincomeforfoodand12oftheremainderforeducationwhatishissalaryifthelandlordgets$960morethantheschool

49Amansellshishouseandtakesalossof15Thepurchaserlatersellsthehousetoathirdpartyfor$14000losing20Howmuchdidtheoriginalownerpayforthehouse

50Thepriceofeggsdroppedfrom63centadozento56centadozenWhatwasthepercentdecreaseinprice

51Anarticlethatcost$12wassoldfor$16WhatpercentofthecostwasthedifferencebetweenthesellingpriceandthecostWhatpercentofthesellingpricewasthedifferencebetweenthesellingpriceandthecost

52Acollegehadanenrollmentof2600in195022morethanin1940Atthesamerateofincreasehowmanystudentswereenrolledin1960Whatwastheenrollmentin1940

53Whatis(a)64increasedby ofitself(b)45increasedby ofitself(c)054increasedby24ofitself

54Whatnumberincreasedby(a)10ofitselfis462(b) ofitselfis299(c)8ofitselfis3024

55Whatnumberdecreasedby(a) ofitselfis266(b) ofitselfis450(c)7ofitselfis2139

CHAPTERIX

INTEREST

325WhatismeantbyinterestInterestistheamountpaidfortheuseofborrowedmoneyortheamount

receivedfortheuseofmoneyloanedorinvestedInbookkeepingthesegoundertheitemsofinterestcostandinterestearned

326Whatarethethreefactorstoconsiderincalculatinginterest(a)Principal=thesumloanedorthecapitalinvested

(b)Time=durationoftheperiodOneyearisthecustomaryunitoftimeForapartofayearthesubdivisionusedisthemonthortheday

(c)Rate=ratepercent=numberofunitspaiduponeachhundredunitsofborrowedsumTheunitsareexpressedinthemoneyofthecountryconcernedasdollarspoundssterlingfrancsmarkskronerflorinsorpesos

EXAMPLEIf$6arepaidasinterestforeveryhundreddollarsloanedattheendofeachyearthentherate=6per100or6percentor6

Thustherate=theratiooftheinteresttotheprincipalforeachunitoftime

327Howdoweexpressarateofinterest(a)Asanintegraloramixednumberwithapercentsignafterit

EXAMPLE

5=fivepercent=anintegralwithasign

=sixandthree-quarterspercent=amixednumberwithasign

(b)Asadecimalthecorrectwaytowriteit

EXAMPLE

005=fivepercent=

00675=sixandthree-quarterspercent=

328WhatismeantbysimpleinterestInterestcalculatedontheoriginalprincipalforthetimetheprincipalisused

SimpleinterestisnothingmorethanpercentagewithatimeelementinvolvedTheoriginalprincipalremainsconstantandthequantityofinterestforeachunittimeintervalremainsunchanged

EXAMPLE

6intereston$100for1year=$6=simpleinterest06of$100=$66of$100=$6

Thussimpleinterest=apercentagewithatimeelement

329WhatismeantbycompoundinterestItisinterestcalculateduponboththeprincipalandtheinterestwhichhas

alreadyaccruedTheinterestiscompoundedquarterlysemiannuallyorannuallyaccordingtoagreementYoumerelycomputesimpleinterestonthenewprincipalatthevariousperiodsagreedupon

EXAMPLEFindtheinterestfor3yearsat6on$200withinterestcompoundedannually

Forfirstyearinterest=6of$200=06times$200=$12Newprincipal=$200+$12=$212

Forsecondyearinterest=6of$212=06times$212=$1272Newprincipal=$212+$1272=$22472

Forthirdyearinterest=6of$22472=$1348Newprincipal=$22472+$1348=$23820

Originalprincipal=$20000Compoundinterestfor3years=$3820

Notethatthesimpleinterestforthe3yearswouldbe

$200times06times3=$3600

330Whatistheformulaforfiguringsimpleinterest

Interest=principaltimesratetimestimeI=Ptimesrtimest=Prt

EXAMPLEWhatistheintereston$2000at6peryearforahalfyear

331WhatismeantbytheldquoamountrdquoandwhatisitssymbolThesumobtainedbyaddingtheinteresttotheprincipal=amount=S

orS=Principal+Interest=P+IorS=P+PrtsinceI=PrtorS=P(I+rt)sincePisacommonfactorofPandPrt

EXAMPLEIfyouborrowed$500atsimpleinterestfor3yearsat5howmuchwillthecreditorreceiveinall

S=amount=P(1+rt)=$500(1+05times3)

=$500(115)=$575Ans

Creditorwillreceive$575ofwhich$500istheprincipaland$75istheinterest

332Infiguringsimpleinterestforlessthanayearwhatistheruleforestablishing(a)theterminaldays(b)theduedate(a)IntheUnitedStatesweexcludethefirstdayandincludethelastday

EXAMPLEForabankloanmadeJanuary4andfallingdueJanuary27interestwouldbechargedfor23days

(b)Dateofmaturityofaloanisdeterminedbythewordingoftheagreementiftimeisstatedinmonthspaymentisdueonthesamedateofduemonthiftimeisstatedindaysthentheexactnumberofdaysiscountedtogetduedate

EXAMPLEIfinatransactiononJuly5adebtoragreestorepayaloaninfivemonthsthemoneyisdueDecember5Iftheagreementistorun150daystheduedatewouldbeDecember2

NoteGenerallyintheUnitedStatesloansfallingdueonSaturdaySundayoraholidayarepayableonthenextbusinessdayandthisextratimeiscounted

333Howarethemethodsforfiguringsimpleinterestcommonlyreferredto(a)Theordinarymethod(b)Theexactoraccuratemethod(c)Thebankersrsquomethod

Thedifferenceinthesemethodsisinthewaythetimeisfigured

334HowdowefindthetimebytheordinarymethodIntheordinarymethodayearisconsideredtohave12monthsof30days

eachor360days

Thetimeisfoundeasilybycompoundsubtraction

EXAMPLEFindthetimebetweenFebruary81959andMay151957

Year Month Day

1959 2 8

1957 5 15

_____ ____ _____

1 8 23

Borrow1month=30daysandaddittothe8daystomake38days

Subtract15daysfrom38daystoget23days

Borrow1year=12monthsandaddittothe1monthtomake13months

Subtract5monthsfrom13monthstoget8months

Now1957from1958leaves1year

Theresultis1year8monthsand23days

335Howdowefindthetimebytheexactmethod(a)Theactualnumberofdaysineachmonthiscounted

EXAMPLEFindtheexacttimefromMay81958toJanuary121959

May 23days

June 30days

July 31days

August 31days

September 30days

October 31days

November 30days

December 31days

January 12days

249days

(b)UseTable1inAppendixBEachdayoftheyearisindicatedasthetotalnumberofdaysfromJanuary1tothedayinquestioninclusiveFindthenumberoppositethelastdateandfromthissubtractthenumberoppositethefirstdatetogetthenumberofdaysbetweenthedates

EXAMPLEUseabovedatesMay8isthe128thdayDecember31isthe365thdayThen365ndash128=237daysin1958Nowadd12daysinJanuary1959to237daystoget249daysinall

336HowdowefiguretimebythebankersrsquomethodTimeisexpressedinmonthsanddaysorinexactdaysonlyThismethodis

usedtofindthetimeforshortperiods

EXAMPLEWhatisthetimefromJune4toOctober21

FromJune4toOctober4is4monthsFromOctober4toOctober21is17daysAns=4months17days

Or(fromTable1inAppendixB)

October21=294June4=155Ans=294ndash155=139days

The360-dayyearisusedwithexactdays

337Findtheintereston$3000at6fromNovember181958toApril61959(a)bytheordinarymethod(b)bytheexactmethod(c)bythebankersrsquomethod(a)

Year Month Day

1959 4 6

1958 11 18

4 18 =138days

Ayear=12months30dayseachor360days=ordinarymethod

$3000times06times =$69interest=Ordinarymethod

(b)Table1AppendixBNovember18is322nddayoftheyear365ndash322=43daysin1958

April6isthe96thdayoftheyear

Then43+96=139days(exact)

there4$3000times06times =$6855interest=Exactmethod

(c)$3000times06times =$6950interest=Bankersrsquomethod(Exactdaysand360-dayyearareused)

NoteExactmethodproducestheleastinterestofthethreeandthebankersrsquomethodproducesthemost(becausethedenominatorissmallerwhilethenumberofdaysisexact)

338WhatistheconstantrelationshipofexactinteresttoordinaryorbankersrsquointerestbasedonexactnumberofdaysLetN=exactnumberofdays

Then

and

Then

and

Wecanrememberthisbynotingthatexactisalwayslessthanordinaryinterestso

Thereforetogetexactwesubtract ofordinaryfromordinaryTogetordinaryweadd ofexacttoexact

339Whatisthe60-day6percentmethodofcalculatinginterest60daysare ofayear

Theniftheinterestrateis6percentayeartheinterestratefor60daysis

Thereforetofindtheinterestfor60daysat6percentonanyprincipalpointofftwoplacestotheleft

Ex(a)Theintereston$1360for60daysat6is$1360

Now6daysare

Theninterestfor6daysat

Thereforetofindtheinterestfor6daysat6onanyprincipalpointoffthreeplacestotheleft

Ex(b)Theintereston$1360for6daysat6is$136

Ex(c)Findtheintereston$570for66daysat6

340Abusinessmanborrowed$850for75daysat6Howmuchinterestdidhepay

341Howarethealiquotpartsof60usedwhenthetimeisgreaterorlessthan60daysinfindinginterestbythe60-day6methodEXAMPLEWhatarethealiquotpartsof60dayscontainedin(a)49days

(b)58days(c)77days

(a) 30days (b) 30days (c) 60days

15days 20days 15days

4days 6days 2days

49days 2days 77days

58days

58days

342Whatistheintereston$95370for124daysat6

343Whatistheintereston$59860for48daysat6Togetinterestfor30daysfirstget$5986interestfor60daysanddivideby

2

344Howcanwesometimessimplifythe60-day6process(a)Byexchangingtheamountoftheprincipalandthenumberofdays

EXAMPLEFindtheintereston$120for176daysat6Makeit$176for120daysbyexchangingonefortheother

Ans=$352intereston$120for176days

(b)Bydeductingfromtheinterestfor60daystheinterestforthedifferenceintimebetweenthetimegivenand60days

EXAMPLEFindtheintereston$170for50daysat6

345Howdowefindtheinterestatarateotherthan6Firstfindtheinterestat6thentoget

(a)3take oftheinterestat6

(b)4subtract oftheinterestat6

(c) subtract oftheinterestat6

(d)5subtract oftheinterestat6

(e)7add oftheinterestat6

(f) add oftheinterestat6

(g)8add oftheinterestat6

(h)9add oftheinterestat6

EXAMPLEFindtheintereston$790for145daysat andat

346HowcanwemakeuseoftheinterestformulainfindingoneofthefourfactorsmdashinterestprincipalrateandtimemdashwhentheotherthreearegivenWehaveseenthatinterestismerelyapercentageproblemwithatimefactor

or

I(interest)=Prt(principaltimesratetimestime)

Ex(a)Ona360-day-per-yearbasisatwhatratepercentwouldyouhavetoinvest$970for72daystoearn$970interest

Ex(b)Howmuchmoneywouldyouneedtoinvestat5for96daystoearn$1160interest

Ex(c)Howlongwillittaketoearn$1530interestonaninvestmentof$1080at6

347Whatisthe6-day6methodoffindinginterestandwhatisitsprincipalvalueTheinterestfor6daysat6canbefoundbymovingthedecimalpointthree

placestotheleftbecause6is of60

(a)Movedecimalpoint3placestotheleftfor6-dayinterest

(b)Dividethenumberofdaysby6togetthenumberof6-dayunits

(c)Multiplytheresultsoftheabove

Thismethodcanbeusedtochecktheresultofthe60-daymethod

Ex(a)Findtheintereston$300for27daysat6

$30=interestfor6daysat6(move3placestoleft)

there4$30times =$135=interestfor27daysat6

Ex(b)Whatistheintereston$52936for78daysat6


there4$52936times13=$688168=$688=interestfor78days

348WhatisthesignificanceofcompoundinterestInsimpleinteresttheprincipalremainsconstantduringthetermofaloan

Incompoundinteresttheprincipalisincreasedbytheadditionofinterestattheendofeachinterestperiodduringthetermofaloan

WhenevertheinterestisaddedtotheprincipalattheendofaperioditissaidtobeconvertedorcompoundedTheprincipalthenbecomeslargeratthebeginningofthesecondperiodthanitwasatthebeginningofthefirstperiodInturntheinterestdueattheendofthesecondperiodislargerthanthatdueattheendofthefirstperiodThisconditioncontinuesforeachsuccessiveperiodduringtheindebtedness

349Whatismeantby(a)compoundamount(b)compoundinterest(c)conversionperiod(d)frequencyofconversion(a)Compoundamount=principal+compoundinterest

(b)Compoundinterest=compoundamountmdashoriginalprincipal

(c)Conversionperiod=intervaloftimeattheendofwhichinterestiscompounded

(d)Frequencyofconversion=numberoftimesayearthattheinterestisconvertedintoprincipal

MostNewYorksavingsbanksconvertinterestfourtimesayearThustheconversionperiodis3monthsandthefrequencyis4

350Whatwill$450amounttointhreeyearsat4ifinterestiscompoundedannually

$45000=principalatbeginningoffirstyear$45000times04=$18=firstyearrsquosinterest

$45000+$18=$468=principalatbeginningofsecondyear$46800times04=$1872=secondyearrsquosinterest

$46800+$1872=$48672=principalatbeginningofthirdyear$48672times04=$1947=thirdyearrsquosinterest

$48672+$1947=$50619=principalorcompoundamountatendofthirdyear

351WhatisashortermethodoffiguringthecompoundamountTheamountatthebeginningofthesecondyearwasseentobeequaltothe

principalatthebeginningofthefirstyearplusoneyearrsquosinterestuponit(seeQuestion350)

$468=$450+$450times04

or

$468=$450(1+04)($450isacommonfactor)

and

$468=$450times104=amountatbeginningofsecondyear

Thustogettheamountforoneyearmultiplytheprincipalby(1+theannualinterestrate)

Theabovemultiplicationsareexpressedinonelineas

$50619=$450times104times104times104

or

$50619=$450times(104)3=amountatcompoundinterest

Thesmallfigure3attheupperright-handsideoftheparenthesisiscalledanexponentandmeansthatthequantityintheparenthesisistobeusedasafactorinmultiplicationthatnumberoftimesInthiscase3correspondstothenumberofyearsforwhichinterestwascomputedandmeansthat(104)istobemultiplied3timesSimilarly(1035)4meansaninterestrateof for4years

352Whatistheformulafortheamountatcompoundinterest

S=amount=$50619(inQuestion351)P=principal=$450(inQuestion351)

r=interestrateperyear=04(inQuestion351)t=numberofyears=3(inQuestion351)

Therefore

S=P(1+r)tS=$450(1+04)3=$450times(104)3=450times1124864

S=$50619

Thusthecompoundamountof$450in3yearswithinterestat4compoundedannuallyis

$50619Ans

353Inordertohave$6000attheendof3yearshowmuchmustyouinvestnowat5compoundedannually

Youmustinvest$518326nowtohave$6000attheendof3yearswhentheinterestiscompoundedannuallyat5

354WhatisusedinactualbusinessandfinancialpracticetosaveagreatdealoftimelaborandcomputationinfiguringcompoundinterestAtablewhichhasbeencomputedgivingtheamountof1(unity)atcompound

interestforvaryingperiodsoftimeandatdifferentratesofinterestThistableiscalledtheldquoCompoundAmountof1rdquo(seeTable2AppendixB)

S=(1+r)t=Formulaforthecompoundamountof1

wheninterestiscompoundedannuallyHereP=1(1+r)tisknownastheaccumulationfactorsincethecompoundamountindicatestheaccumulationofinterest

Accumulationfactortimesanyprincipal=compoundamounttowhichthatprincipalaccumulatesatcompoundinterestduringaspecifiedtime

Youfindinthetablethecompoundamountof1forthepropertime(numberofperiods)andrateandthenmultiplythisfigurebytheprincipalThesymbolforthetimeornumberofperiodsisusuallygivenasnThetablecanbeusedforanydenominationofcurrencysuchaspoundssterlingfrancsmarkslirapesosetcorforanyrequiredunit

Ex(a)Tofindwhat$1willamounttoinoneyearat5entercolumnheadnat1andrunhorizontallyuntilthecolumnheaded5isreachedwhereyouwillfind105

Ex(b)Tofindthecompoundamounton$1for4yearsat5entercolumnnat4andgohorizontallyuntilyoureachthecolumnheaded5whereyouwillfind$121551

Ex(c)Whichisgreater(1)asumofmoneyaccumulatingfor10yearsat2compoundinterestor(2)thesamesumaccumulatingfor5yearsat4compoundinterest

10yearsat2rarr$121899=compoundamountof15yearsat4rarr$121665=compoundamountof1there410yearsat2givesalargercompoundamount

355Whatwould$12000amounttoifinvestedfor7yearsat4compoundedannually

S=$12000times131593=$1579116Ans

(Compoundamountof$1forn=7yearsand4=131593fromtable)

356Whatamountofmoneyinvestedat5fornineyearswouldamountto$589505

(AccordingtoTable2AppendixBcompoundamountof$1for9yearsand5=$155133)

357Ifyoudeposited$1800inabankwhichpays4perannumhowlongwillittakeforthisdeposittogrowto$227758ifinterestiscompoundedannually

RefertoTable2andgodownundercolumnheaded4andyoufind126532isinahorizontallinerunningouttowhere

n=6=t=6yearsAns

Iftheresulthadbeenmoreorlessthan126532thenthetimewouldnothavebeenawholeyearandthetimewouldhavetobeinterpolatedbetweentwointegralyears

358WhatismeantbythenominalrateofinterestWheninterestiscompoundedorconvertedmorethanonceayearthestated

rateofinterestperyeariscalledthenominalrate

EXAMPLEIfasavingsbankpays ondepositscompoundedeveryquarteryearthenominalratewhichyoureceiveis Actuallyyougetalittlemorethan becauseeachbalanceisincreasedateach3-monthintervalbytheinterestaddedtoit

359WhatismeantbytheeffectiveannualrateofinterestRateofinterestactuallyearnedinayear

EXAMPLEHowmuchwill$700amounttoinoneyearifinterestiscompoundedquarterly

Thusarateof4compoundedquarterlyfor1yearwillproducethesameresultasarateof1compoundedannuallyfor4years

Weseethattheoriginal$100earned$406inoneyearThismeans

actuallyearnedduringtheyear

406isknownastheactualoreffectiveannualrate

Thusanominalrateof4compoundedquarterlyisequivalenttoaneffectiverateof406compoundedannuallybecausethesameamountofmoneyis

producedattheendofayear

360WhenarenominalandeffectiveratesequivalentWhentheyproducethesameamountofmoneyattheendofayear

Inabove

Dividebothsidesby$100togetthecompoundamountfor$1

Weseethattheeffectiverate0406isequivalenttothenominalrate04compounded4timesayear

361Whatistheformulashowingtherelationshipbetweenaneffectiverateiandanequivalentnominalraterpcompoundedptimesayear

Inabove

362Whatistheformulaforthecompoundamountof1ataraterp compoundedp timesperannumfort years

Theformulaforthecompoundof1wasshowninQuestion354tobeS=(1+r)twheninterestiscompoundedannually

Toobtainaformulaforthecompoundamountof1ataraterpcompoundedptimesperyear

ismerelysubstitutedfor(1+r)inabovebecauseiandrparetakenasequivalentratesThus

Theexponentpt=thetotalnumberofconversionperiodsduringtheindebtedness

EXAMPLEIf$800isleftondepositfor1yearatanominalrateofcompoundedsemiannuallywhatwillbetheamountattheendoftheyear

363Whatistheruleforuseofcompound-amount-of-1tableswhereinterestiscompoundedatanominalratemorethanonceayear(a)Findvalueofpt=totalnumberofconversionperiodsduringtimeof

indebtedness=nintables

(b)Findrpp=rateperperiod=percentinterestintables

(c)Lookinthecalculatedpercenttablesforthepercentforaquantityinlinehorizontallywiththencolumn(=pt)

EXAMPLEWhatistheamountof1at6compoundedquarterlyfor4years

Lookat gohorizontallyacrossfromn=16andget

126898555Ans

364Amaninvests$8000for12yearsat5compoundedquarterlyWhatamountwillhegetafter12years

Lookat interestforn=48horizontally

S=$8000times181535485=145228388

Thereforehewillreceive

$1452284Ans

PROBLEMS

1(a)Whatpercentof100is6(b)Whatpercentof$1is6cent

(c)If$6ischargedfortheuseof$100whatpercentofthesumloanedisthesumcharged

2Findtheintereston(a)$5for1yearat4at5at6(b)$300for2yearsat2at7at9(c)$400for3yearsat6for2years3monthsat7(d)$1200for1yearat3for3yearsat7for6monthsat8

3Ifyouborrowed$800atsimpleinterestfor4yearsat4howmuchwillthecreditorreceiveattheterminationofthecontractHowmuchwouldtheinterestamountto

4ForabankloanmadeonMarch6andfallingdueonMarch28interestwouldbechargedforhowmanydays

5(a)IfinatransactiononSeptember4adebtoragreestorepayinsixmonthswhenisthemoneydue

(b)Iftheagreementwastorun180dayswhenwouldtheduedatebe

6Findthetimebycompoundsubtractionbetween(a)June141958andAugust281958(b)September121957andJuly181958(c)December141955andMay121958(d)October181954andFebruary61959(e)July291955andMay141959

7FindtheexacttimebetweenthefollowingdatesusingTable1AppendixB(a)May101958andJanuary141959(b)October18andJanuary10(c)July16andNovember11(d)March5andNovember8(e)February161960andJuly71960(rememberthataleapyearhas366days)

8Findtheintereston$2500at5fromOctober171959toMay71960(a)bytheordinarymethod(b)bytheexactmethodand(c)bythebankerrsquosmethodWhichproducestheleastinterestwhichthemost

9Findtheexactintereston$1000fromJanuary12toApril18at3

10Findtheordinaryintereston$6200fromApril6toJuly12at3

11Obtaintheinterestat4on$12000forsixmonthsfromApril15

12Howmuchwill$5000beworth120daysafterApril211960ifinvestedat6ordinaryinterestandwhatistheduedate

13Findtheexactintereston$3800for135daysat

14HowwouldyoufindtheexactinterestgiventheordinaryinterestHowwouldyoufindtheordinaryinterestwhengiventheexactinterest

15Findtheexactinterestwhentheordinaryinterestis(a)$4783(b)$38640(c)$295(d)$1202(e)$29000(f)$375(g)$3479(h)$368(i)$4980

16Findtheordinaryinterestwhentheexactinterestis(a)$328(b)$5490(c)$65860(d)$8136(e)$622(f)$904(g)$22790(h)$446900(i)$6438

17Whatistheprincipalwhichat5for146dayswillyieldanexactinterest$120lessthantheordinaryinterest

18Findtheordinaryandexactintereston$6950fromMay10toAugust23at5

19Findtheinterestfor60daysat6on$1438



22Findtheintereston$680for66daysat6

23Whatarethealiquotpartsof60inthefollowing(a)27days(b)75days(c)39days(d)96days(e)40days(f)87days(g)129days(h)105days(i)145days(j)21days(k)126days(l)99days



26Findtheintereston$140for191daysat6(byinterchangingthedaysandprincipal)

27Findtheintereston$180for50daysat6(bydeductingfromtheinterestfor60days)

28Byproperdivisionofdaysfindtheinterestbythe60-day6methodof(a)$697000for156days(b)$386for84days(c)$61775for48days(d)$5900for222days(e)$8749for23days

29FindtheinterestfromApril1toJuly9bythe60-day6methodon$5850

30Byproperdivisionofdaysfindtheinterestbytheappropriatemethodon

(a)$487for142daysat45(b)$653for180daysat(c)$9825for192daysat(d)$3760for164daysat8(e)$217975for105daysat5(f)$470for85daysat(g)$2130for120daysat4(h)$423for129daysat9(i)$3570for75daysat3

31Ona360-day-per-yearbasisatwhatratepercentwouldyouhavetoinvest$860for78daystoearn$840interest

32Howmuchmoneywouldyouneedtoinvestat4for82daystoearn$1290interest

33Howlongwillittaketoearn$1645interestonaninvestmentof$1160at6

34Whatprincipalwillproduce(a)$1870interestat6for72days(b)$835interestat6for126days(c)$14interestat6for96days(d)$1574interestat6for75days

35Inwhattimewill(a)$700produce$14at6(b)$960produce$2235at6(c)$1400produce$11at6(d)$2200produce$84at6

36Atwhatratewill(a)$1400produce$2830in126days(b)$760produce$1160in96days(c)$1680produce$21in75days(d)$3200produce$1820in36days

37Findtheinterestbythe6-day6methodon(a)$300for24days

(b))$150for27days(c)$63842for78days(d)$400for36days(e)$25for66days(f)$500for51days

38Whatwill$550amounttoin3yearsat4ifinterestiscompoundedannually


40Findthecompoundamounton$1for5yearsat4usingTable2AppendixB

41Whichisgreater(1)asumofmoneyaccumulatingfor8yearsat2compoundinterestor(2)thesamesumaccumulatingfor4yearsat4compoundinterest(usetable)

42Whatwould$10000amounttoifinvestedfor6yearsat compoundedannually

43Whatamountofmoneyinvestedat5for8yearswouldamountto$384140


45If$1000isleftondepositfor1yearatanominalrateof4compoundedsemiannuallywhatamountwilltherebebytheendoftheyear

46Whatistheamountof$1at6compoundedquarterlyfor6years(usetable)

47Ifamaninvests$10000for10yearsat5compoundedquarterlywhatamountwillhegetafter10years

48Findthecompoundintereston$2000for8yearsat5compounded(a)annually(b)semiannuallyand(c)quarterly

49Findtheamountof$5placedannuallyfor10yearsat5compoundinterest(usetable)

50Ifinterestat5iscompoundedsemiannuallyfor3yearsitamountstothesameasinterestat compoundedannuallyforhowmanyyears

51Atrustfundof$20000earnsinterestat3ayearcompoundedsemiannuallyWhatwillthefundamounttoin10yearsHowmuchwilltheinterestbeinthattime

CHAPTERX

RATIOmdashPROPORTIONmdashVARIATION

365Whatarethetwowaysofcomparinglikequantities(a)Subtractingthesmallerfromthelargermdashthedifferencemethod

EXAMPLEIfyouare35yearsoldandyoursonis5yearsoldyouare30yearsolderthanyourson(35minus5=30)

(b)Dividingonebytheothermdashtheratiomethod

EXAMPLEYouare7timesasoldasyourson( )

366WhatismeantbyaratioAcomparisonoftwolikequantitiesbydividingonebytheotherAsaratiois

arelationshipoftwoquantitieswemustbespecificandindicatetheorderoftheirrelationship

Ex(a)IfmachineAproduces300unitsperhourwhilemachineBproduces450unitsperhouritisincorrecttosaythattheproductionratioofthesemachinesis WemustsaytheproductionratioofmachineAtothatofmachineBis middot

Ex(b)InQuestion365youmustsaythattheratioofyouragetoyoursonrsquosageis 7andnotthattheratiooftheagesis Youmayalsosaythattheratioofyousonrsquosagetoyoursis

367WhattwotermsaregiveninallratiocalculationsThefirsttermgivenisthenumeratorandiscalledtheantecedentThesecond

termgivenisthedenominatorandiscalledtheconsequent

Ex(a)Whatistherelationbetween4and12

Here4isthefirstterm=antecedentand12isthesecondterm=consequent

Ex(b)Ifonehousecosts$54000andanothercosts$18000theratiobetweenthefirstandsecondhouseis

orratiois3to1Onecoststhreetimestheother

368WhatsymbolisusedtoindicateratioColon[]=ldquotordquo

EXAMPLES

$54000$18000=31412=13(to)(to)(to)(to)

Thecolonisactuallyanabbreviationfor[divide]withthehorizontallineomitted

369Howmayratiosbeexpressed(a)ByasinglewholenumberTheratioof35yearsto5yearsis7(35divide5=7)

(b)AsafractionalnumberTheratioof1ouncetoapoundis

(c)AsadecimalfractionTheratioofonesideofatriangle4incheslongtoasecondside5incheslongis or08

(d)Infractionalformandtreatedlikeafractionmaybereadastheratioof4to5

(e)Withtwodotsseparatingtheterms45meanstheratioof4to5

Notethatwhenaratioisexpressedbyasingleintegralfractionalordecimalnumberthenumber1isthesecondtermoftheratiobutisnotwrittendownTheratioof35to5istheratioof7to1orsimply7

370CantherebearatioofunlikethingsNoThetermsmustbeoflikethingsTherecanbenoratiobetweendollars

andbeansorbetweenhousesandyachtsUnlessthingscanbechangedtosomethingthatmakesthemaliketherecanbenoratioTherecanbearatiobetweenthecostofahouseandthecostofayachtasexpressedindollarsAlsothecomparisonmustnotonlybebetweenquantitiesofthesamekindbutbetweenquantitiesexpressedinthesameunitsWecannotcomparepoundsandinchesfortheyarenotquantitiesofthesamekindandwecannotcomparealengthinincheswithalengthinyardswithoutfirstmakingtheunitsalikethatiswemusteitherreduceyardstoinchesorconvertinchestoyards

371IsaratiodependentupontheunitsofmeasureNoTheratioitselfisalwaysabstractandthetermsmaybewrittenasabstract

numbers

EXAMPLEIftwoboardsare10feetand12feetlongrespectivelytheratioofthefirsttothesecondboardis56whetherweexpresstheirlengthsasinchesfeetoryardsTheunitscanceloutandtheratiois56

372DoesmultiplyingordividingbothtermsofaratiobythesamenumberchangeitsvalueNoEx(a)

Ex(b)

373WhatistherelationbetweenratioandpercentSincearatioisalwaysafractionwemaythinkofapercentasaratioRatios

arefrequentlyexpressedaspercents

EXAMPLEWhenwesay$100is20of$500wemeanthattheratioof

$100to$500is

Problemsinvolvingpercentcanhoweverbesolveddirectlywithoutreferringtoratio

374HowisaratiosimplifiedAratioisalwaysreducedtoitssimplestformPerformtheindicateddivision

andreducetheresultingfractiontoitslowesttermsExpressthefractionasaratio

Ex(a)Ratio simplified

Ex(b)Simplifytheratio

375WhatcanbedoneinordertocomparereadilytwoormoreratiosReducetheratiostosuchformsthatthefirsttermsoftheratiostobe

comparedshallbethesameusually1

Ex(a)Reduce927toaratiohaving1foritsfirstterm

Dividebothtermsby9getting13

Ex(b)Reduce1639toaratiohaving1foritsfirstterm

Dividebothtermsby16getting and

Ex(c)Reduce7849toaratiohaving1foritsfirstterm

Ex(d)Reduce toaratiohaving1foritsfirstterm

Dividebothtermsby

376WhatwouldyoudowhenrequiredtoworkoutacomplicatedratiocontainingfractionspercentsordecimalsSimplifytheratiofirst

(a)Ifthedenominatorsofbothfractionsarealiketheyareintheratiooftheirnumerators

EXAMPLEFindtheratiobetween and

(b)Ifthedenominatorsarenotalikemakethemalikeordividethefirstfractionbythesecondfraction

EXAMPLES(1)Findtheratiobetween and ( )

(2)Findtheratiobetween and

377Howdowedividesomegivennumberinagivenratio(a)Addthetermsoftheratioandmakeitthedenominatorwiththegiven

numberasthenumerator

(b)Multiplythequotientbyeachtermoftheratio

Ex(1)Given65Divide65intheratio23

As65=26+39therefore65isdividedintotwoterms26and39intheratioof23

Ex(2)Ashipmentof1200TVsetsistocontaincolorsetsintheratioof35Howmanyofeachkindarethere

there435=450colorsets750blackandwhitesetsAns

Ex(3)1600booksaretobeallottedtothreeclassesintheratioof479Howmanybookswilleachclassreceive

4+7+9=20=denominator

80times4=320=quotienttimesfirsttermofratio=bookstoclass180times7=560=quotienttimessecondtermofratio=bookstoclass280times9=720=quotienttimesthirdtermofratio=bookstoclass3Total=1600books

there4320560720=479Ans

378Howcanwedivide65intheratio

ReducefractionstoacommondenominatorFirstterm= and =secondtermAddthenumeratorsofthese3+2=5

Divide65by5anduse3and2asnumerators

Firstterm= and =secondterm

there4 Ans

379HowdowesolvearatioprobleminwhichtheratioisnotgivenFirstweassignaratiovalueof1tothegivenbasicquantityWethen

computetheratiovaluesofalltheotherquantitiesbasingourcalculationsonthegivenfactsthusarrivingataratio

ThenweproceedasinQuestion377aswhenratioisgiven

EXAMPLEAcompanybought3trucksThefirstcost timesasmuchasthesecondThethirdcost timesasmuchasthesecondThecompanypaid$30000forthe3trucksHowmuchdiditpayforeach

Addthetermsoftheratio (=denominator=5partsonepartofwhichisthebasictruck)

380Ifthewingspanofaplaneis76ft6inwhatwillthewingspanofamodelhavetobewhentheratioofthelengthofanypartofthemodeltothelengthofthecorrespondingpartoftheactualplaneis172Thelengthofmodelisthus ofthecorrespondinglengthoftheactualplane

or

381Ifabankruptfirmcanpay60centonthedollarandifitsassetsamountto$28000whatareitsliabilitiesPaying60centonthedollarmeansthatitsratioofassetstoliabilities=60

382WhatsellingpriceshouldbeplacedonaTVsetifthecostis$250andthedealeroperatesonamarginof30ofcostAmarginof30ofcost=ratioofmarginonsettoitscost

Thusmarginhere ofcost

Ormargin

there4Sellingprice=$250+$75=$325Ans

383Ifyouallow12ofyourincomeforclothingand21forrent(a)whatistheratioofthecostofrenttothecostofclothing(b)howmuchdoyouspendforrentpermonthwhenyourincomeis$8400peryear(a)

(b) forrent

384Ifatownestimatesthatithastoraise$300000intaxesandtheassessedvaluationofitsrealpropertyis$9000000whatisthetaxrateTaxrate=ratioofamounttoberaisedtoassessedvaluation

385Acertainconcretemixtureistobemadeupof1partcement3partssandand5partsstoneWhatis(a)ratioofsandtostone(b)theratioofcementtosand(c)percentofsandintheconcretemixture(a)Sandstone=35(b)Cementsand=13(c)1+3+5=9partsintheentiremixture

there4Sandmixture=39= Ans

386Ifthebedroomofahouseisshownontheprinttobe intimes inandifthescaleoftheblueprintis in=1ftwhataretheactualdimensionsoftheroom

387WhatismeantbyanldquoinverseratiordquoItismerelyaratiowithreversedterms

EXAMPLEWhatistheinverseratioof408

Reversetheratiogetting840= =ldquoinverseratiordquoof408

388Whatwouldbeyourshareinanautomobilethatcostyouandyourbrother$880if ofyourshareisequalto ofyourbrotherrsquosTheratiosinthiscasewouldbe or Sinceindivisionoffractions

onefractionisreversedouranswer isthereverseofthetrueratioThereforethetrueratioisthereverseofthisor Theratio isknownasanldquoinverserdquoratio

Nowaddthetermsoftheratio6+5=1Then

389Whataresomegeneralrulesforratiocalculation(a)Togetaratiodividethefirsttermbythesecondterm

EXAMPLEWhatistheratioof1yardto1inch

1yard=36inchesthere4 Ans

(b)Togetthefirsttermmultiplythegivensecondtermbythegivenratio

EXAMPLE3=9

there43times9=27=FirsttermAns(Check273=9)(c)Togetthesecondtermdividethefirsttermbytheratio

EXAMPLE36=12

there4 =3=SecondtermAns(Check36divide3=12)

390HowdowecompoundratiosChangetheexpressionstofractionformThentreatthetwofractionsasa

probleminmultiplication

EXAMPLEWhatisthecompoundratioof84and2436

Theproductoftwoormoresimpleratiosisacompoundratio

391Howdowesolveinamannersimilartothatofa

ratioproblemaprobleminwhichthesamenumberofarticlesareboughteachatadifferentpriceAddthevariouspricesanddividethissumintothetotalprice

EXAMPLEIfyoubuythesamenumberoforangesat6cent8centand10centandyouspend$288howmanyateachpricedidyoubuy

Tobuyoneofeachwouldcost6+8+10=24cent

392HowdowesolveinamannersimilartothatofaratioproblemaprobleminwhichadifferentnumberofarticlesareboughtatdifferentpricesProceedinthesamemannerasinaratioproblemwhentheratioisnotgiven

(a)Findthebasicquantitywithwhichalltheothersarecompared

(b)Assignvalue1toitandcomputevalueofotherquantitiesaccordingtogivenfactsorrelations

(c)Multiplythepricesbytheirrespectivevalues

(d)AddtheseproductsanddividethissumintothetotalcosttogetthebasicquantityMultiplythisbasicquantitybytheratiovaluetogettheotherquantities

EXAMPLEIfyourfirmbuys4timesasmanytrucktiresat$37eachaspassenger-cartiresat$18andtwiceasmanystation-wagontiresat$24eachhowmanyofeachdiditbuyifitspent$2354

Thebasicquantityisldquopassenger-cartiresrdquoAssignavalue1tothisbase

Value=4

fortrucktiresasthereare4timesasmanytrucktires

Value=2

forstation-wagontiresastherearetwicethenumberoftheseascomparedwiththebase1

Sincewecannotcomparearatioofunlikethingstheratiocannotbeexpressedintiresbutincostoftires

Thuspassenger-cartirescost=$18each=base1

Trucktirescost$37eachtimes4value=$148

Station-wagontirescost$24eachtimes2value=$48

Thereforetheratiois$148$18$48

$148+$18+$48=$214(=costpergroupof4+1+2=7tires)

Foreachtypethefirmspent

393WhatismeantbyaproportionAstatementthattworatiosareequal

EXAMPLE

48=12ratio=ratioratio=ratio

394Howareproportionswritten

[]=ldquoasrdquo

68346isto8as3isto4or68=346isto8equals3isto4or =fractionalform

395WhatarethetermsofaproportionldquoExtremesrdquo=firstandlastterms

ldquoMeansrdquo=thetwomiddleterms

396WhatisthetestastowhetherthetermsareinproportionTheproductoftheextremes=theproductofthemeans

3times12=4times9=36=Testforaproportion(extremes)(means)

397FromtheabovehowdowefindeithermeanthatisnotgivenMultiplytheextremesanddividebythegivenmean

398FromtheabovehowdowefindamissingextremeMultiplythemeansanddividebythegivenextreme

399Youbuy8tonsofcoalfor$208Whatwill12tonscost

400A9-foot-hightreecastsashadowof feetWhatistheheightofaradiotowerthatcastsashadowof203feet

9-fttree165-ftshadowheightoftower203ftshadow

401WhenarequantitiessaidtobeindirectproportionWhenthefirstistothesecondasthesecondistothethird

EXAMPLE3612=adirectproportion

402WhatismeantbyameanproportionalWhenthesecondtermisequaltothethirdeachisameanproportionaltothe

othertwo

Ex(a)36612

6isameanproportionalto3and12

Ex(b)

5isameanproportionalbetween2and

Ex(c)3xx12

xisameanproportionalbetween3and12orx2=36Productofmeans=productofextremes

there4x=6=themeanproportionalbetween3and12Thisisalsoknownasthegeometricmean

403HowdoesstatingaproblemasasimpleproportionsimplifythefindingofanunknownterminaproblemEXAMPLEIf36gallonsofgasolinecost$864howmuchwill60gallons

cost

3x=5times$864

Productofmeans=Productofextremes

there4 Ans

Byelementaryarithmeticwecanfindthecostofonegallon

Thismethodcanbelengthy

404Analloyconsistsof4partsoftinand6partsofcopperHowmanypoundsofcopperwouldbeneededwith120poundsoftintomaintainthegivenratio


405WhatismeantbyaninverseproportionQuantitiesaresaidtovaryinverselywhenonequantityincreasesastheother

decreasesMostofsuchproblemsdealwithldquospeed-and-timerdquoorldquowork-and-timerdquo

Ex(a)Asspeedincreasestimetakendecreases

Ex(b)Thegreaterthenumberofmenemployedonajobthelesstimeittakesforcompletion

Ex(c)Thedistancebetweentwoairfieldsis1000milesIftheaveragespeed

ofaplaneis100mphthetripwilltake10hoursIftheaveragespeedis200mphitwilltake5hours

Oneistheinverseoftheother

406Drivingtoyourofficeat45mphyoumakeitin55minutesAtwhatspeedwouldyouhavetotraveltogettherein50minutes

Notethatthespeed45mphandthetime55minutesmustbesosetuptoprovideforcrossmultiplicationinthefractionalformtogiveldquospeed-timerdquo(45times55)

407HowisaninverseproportionsetupEXAMPLEIf24mendoajobin15dayshowmanymenwillberequiredto

doitin5days

Setupproportioninfractionalformtoutilizecrossmultiplicationsothat24menand15daysaremultipliedtogiveldquoman-daysrdquoThiswillgivethesetupforaninverseratio

Anyoneofthefollowingwilldothat

Furthersimplificationcanbeobtainedbyreducingthefractioninwhich5occursgetting

408If130yardsofacopperwireoffer18ohmresistancewhatwillbetheresistanceof260yardsofcopperwireof timesthecross-sectionalareaThegreaterthecross-sectionalareaofawirethelesstheresistance

Firsttheincreasedlengthwillincreasetheresistance

Secondthelargerareawilldecreasetheresistanceintheratio

409WhatisacompoundproportionOneinwhicheitherorbothratiosarecompound

Wesometimeshavetodealwithunitsthathavetobemultiplied

EXAMPLEAprivatenursinghometookcareof16citywelfarepatientsfor5monthsandanothergroupof20patientsfor7months

(a)Whatistheratioofthemaintenancechargeforthetwogroups

(b)Ifthechargeforthesmallergroupwas$16000whatwouldthechargeforthelargergroupbe

(c)Ifthechargeforthelargergroupwas$35000whatwouldthesmallerbe

(a)Theratiobetweenthegroupswouldbe

(b)Chargeforsmallergroupisthus ofthelargerandthechargeforlargergroupis ofthesmaller

Ifsmallerchargeis$16000

(c)Ifchargeforlargergroupis$35000

410Whatistheruleforsolvingacompoundproportion(a)Placetheunknownquantityasthefourthtermoftheproportion

(b)Placeasthethirdtermthegivenquantityexpressingthesamekindofthingastheunknownquantity

(c)Arrangeeachoftheotherratiosaccordingtoitsrelationtotheratioalreadystated

(d)Gettheproductofallthemeansanddivideitbytheproductofalltheextremesexcepttheunknownonetofindtheanswer

411If20menworking6hoursperdaycandigatrench80feetlongin30dayshowmanymenworking10hoursadaycandigatrench120feetlongin12days(a)Placex=unknownquantityasfourthterm(=men)

(b)Place20=menasthirdtermThen (whichisthethirdtofourthtermratio)

(c)Nextratio isaninverseratioandmustbesetupsothat30daysand20mencanbecross-multipliedtogiveldquoman-daysrdquo

Nextratio isadirectratio

Nextratio isaninverseratioandissosetupthat6hoursperdaytimes20mengivesldquoman-hoursperdayrdquo

Thus

412Whyisitpossibletosetupthesecondmemberoftheproportionasasingleratio(a)Intheabove20mendigatrenchin30daysThenin12days

(b)Nowif50mendigan80-fttrenchin12daysthenfora120-fttrench

(c)If75mendiga120-fttrenchin12daysworking6hoursperdaythenworking10hoursperday

Thismethodofproceduremaybeshortenedbymultiplyingthecompletedproportions(a)(b)and(c)togethertermbytermtogetanewproportionwhichisexpressedasaratio

WeseethattheanswersobtainedfromthefirsttwoproportionscancelleavingthesecondmemberasimpleratioTheratiomaynowbeexpressedasaproportion

andsolvedasfollows

Asthefirsttwoanswerscancelitwasunnecessarytoobtainthemtoarriveatthefinalanswer

413If2mencut8cordsofwoodin4dayshowlongwillittake12mentocut36cords

414Iftheeggslaidby30hensin15weeksareworth$108whatwillbethevalueoftheeggslaidby60hensin10weeks

415Whataresomeofthepropertiesofproportionthatcanbeobtainedbyelementaryalgebraicchangesintheformoftheequationwhichexpressestheproportion(a)If =whereabcanddarenumbersinproportiontheproductofthe

means=theproductofextremes

ad=bcbymultiplyingdiagonallyldquocornertocornerrdquo

EXAMPLEIf =346and8areinproportionand

3times8=4times6

(b)If then

ThenumbersareinproportionbyinversionYoumerelyinvertbothsidesoftheproportion

EXAMPLEIf =then 3

(c)If then

ThenumbersareinproportionbyalternationThefirstistothethirdasthesecondistothefourth

EXAMPLEIf then

(d)If then

ThetermsareinproportionbycompositionYouaddthesecondtothefirstandthefourthtothethird

EXAMPLEIf then or

(e)If then

ThetermsareinproportionbydivisionYousubtractthesecondfromthefirstandthefourthfromthethird

EXAMPLEIf then or

(f)If then

Thetermsareinproportionbycompositionanddivision

EXAMPLEIf then or

416Whatproportionsof3milkand5milkmustbemixedtoget milkIfyouhaveaunitvolumeof5butterfatmilkyoucanreduceits

concentrationbyaddingxpartsofaunitof3milk

Thesumoftheconcentrationsoverthecombinedvolume=therequiredconcentrationThen

or

Thismeansthatforeveryunitvolumeof3milkyoumusthave3unitvolumesof5milk

417HowisproportionappliedtotheprincipleoftheleverTheleverisarigidstructureoftenastraightbarwhichturnsfreelyonafixed

pointorfulcrumandwhichisusedtotransmitpressureormotionfromasourceofpower(orforce)toaweight(orresistance)

Whentheleverisinequilibriumthepowerandtheweight(orresistance)areininverseratiototheirrespectivedistancesfromthefulcrum

Whenthesetupissuchthatthereiscrossmultiplicationbetweenthe

correspondingfactorsyouhaveaninverseratioorproportion

EXAMPLEUsinga14-footplankwheremustyouputthesupportundertheplankssothattwochildrenweighing45and55poundsrespectivelycanplayseesaw

Supporttobeplaced77ftfromsmallerchild

418WhatistherelationbetweenratioandproportionandthelanguageofvariationRatioandproportionmayattimesbeconvenientlystatedinthelanguageof

variation

EXAMPLEIfyoudividethecircumferenceCofanycirclebyitsdiameterdyouwillget

(a) Thisisastatementofaratio (b)Thisratiohowevermaybewrittenasavariationintheform

C=πd=variationform

ThismeansthatcircumferenceCvariesasdiameterd

IfdishalvedthenCishalved(πisconstant)

IfdisdoubledthenCisdoubled(πisconstant)

419WhatmaybesaidabouteachofthestatementsofratioandproportionEachimpliesanequationinvolvingaconstant

Ex(a)HookersquoslawstatesthattheelongationEofaspringbalancevariesdirectlyastheweightWisapplied

Ex(b)Boylersquoslawstatesthatthevolumevofagasataconstanttemperaturevariesinverselyasthepressurep(ldquoinverselyasrdquomeansldquoreciprocalofrdquo)

(suppliedtotakecareofdifferentgasesandvarioustemperatures)

AsingleexperimentwilldeterminekIfforacertaingasatacertaintemperatureavolumeof250ccresultsfromapressureof20lbpersqinthen

andBoylersquoslawwouldforthiscasebev=5000p

420WhatisimpliedinadirectvariationandhowisadirectvariationexpressedThestatementldquoyvariesdirectlyasxrdquo(orabbreviatedasldquoyvariesasxrdquoorldquoyα

xrdquo)(αmeansldquovariesasrdquo)impliesthatthereisaconstantksuchthat

istrue(symbolαisreplacedby[=]andaconstantk)

Thedirectvariationisexpressedasy=kx

kinappliedworkisfoundnumericallybyanexperimentandisinsertedtogetaparticularequationforlateruse

EXAMPLEWeknowthatthesurfaceSofaspherevariesdirectlyasthesquareofitsradiusr

Sαr2

Thisimpliestheequation

andthedirectvariationisexpressedas3=kr2Bytheoryandmeasurementwecandeterminethatk=4πandtheequationbecomes

S=4πr2

whichistheusualformulaforthesurfaceofasphere

421WhatisimpliedinaninversevariationandhowisaninversevariationexpressedThestatementldquoyvariesinverselyasxrdquooryα1ximpliesthatthereisa

constantksuchthaty=kxistrue(symbolαisreplacedby[=]andaconstantk)

Theinversevariationisexpressedasyx=k

EXAMPLEInQuestion419whatisthevolumeofthegasforapressureof25lbpersqin

422WhatismeantbyajointvariationandhowisitexpressedAjointvariationmaybeanycombinationofoneormoreofeachofthedirect

andinversetypes

Ifzvariesasxandinverselyasyorzαxythenwemaywrite

byreplacingthesymbolαwith[=]andaconstantkandthisimpliesthatthereisaconstantksuchthatzyx=kistrueThisisanexpressionofajointvariation

EXAMPLEThusifweknowthatwhenz=6x=4y=2wecanfindthevalueofzwhenx=5andy=3

From

then

423Whatistheelectricalresistanceof1000feetofcopperwire inchindiameterusingk =103Theresistanceofanyroundconductorvariesjointlyasthelengthand

inverselyasthesquareofthediameter

where

R=resistanceinohmsL=lengthinfeetd=diameterinmilsk=constantdeterminedbysubstitutingL=1d=1andgettingk=RThusk=resistanceof1ftofwirewhichis1mildiaHencek=circularmil-ftconstantormil-ftresistance

PROBLEMS

1Expressthefollowingcommonfractionsintheformofratios

2Expressthefollowingratiosasfractions(a)710(b)1070(c)59(d)1312(e)112(f)121

3IfmachineAproduces350unitsperhourwhilemachineBproduces630unitsperhourwhatistheproductionratioofmachineAtothatofmachineB

4Ifyouare40yearsoldandyoursonis8yearsoldwhatistheratioofyoursonrsquosagetoyours

5Ifonehousecosts$12000andanothercosts$22000whatistheratiobetweenthesecondandthefirsthouse

6Writetheratioof(a)1footto1inch(b)1inchto1foot(c)1centto1dollar(d)1dollarto1cent

7Ifthelengthofarectangleis110ftanditswidthis80ftwhatistheratioofitslengthtoitswidthandtheratioofitswidthtoitslength

8Iftwoboardsare8ftand10ftlongrespectivelywhatistheratioofthefirsttothesecond

9Ifonesideofatriangleis3ftandanother5ftwhatistheratioofthefirsttothesecondexpressedasadecimalfraction

10Whenwesay$200is25of$800whatdoesthatmeaninratioterms

11Simplifyeachofthefollowingratios

(a)1525(b)2415(c)824(d)2724

12Simplify(a)610(b)3624(c)(d)(e)(f)728(g)(h)1846

13Reduceeachofthefollowingtoaratiohaving1foritsfirstterm(a)39(b)612(c)721(d)660(e)1972(f)981(g)1123(h)96600(i)14(j)74(k)6958(l)5412(m)(n)(o)(p)(q)

14Whatistheratiobetween hoursand45minutes

15Whatistheratioof$650to$4

16If6bushelsofwheatcost$9and8bushelsofcorncost$8findtheratioofthevalueof10bushelsofwheattothevalueof10bushelsofcorn

17Ifaphotographis12inby8inanditisenlargedsothatthelargersidebecomes24ininwhatratioistheareaincreased

18Findtheratiosbetween(a) and(b) and(c) and(d) and(e) and

19Divide35intotwopartswhoseratiois23

20560childrenarriveatacampandaredividedbetweentwolodgingsintheratio35Howmanyareassignedtoeachlodging

21Ashipmentof200radiosTVsetsandrecordplayersisreceivedintheratio578respectivelyHowmanyofeacharethere

221200booksaretobeallottedtothreeclassesintheratio6910Howmanybookswilleachclassreceive

23Divide85intheratio

24AcitydepartmentboughtthreebusinessmachinesThefirstcosttwiceasmuchasthesecondThethirdcostthreetimesasmuchasthesecondItpaid$4800forthethreemachinesHowmuchdiditpayforeach

25Whatistheinverseratioof255

26JohnandBillbought$105worthofmerchandiseIf ofJohnrsquosshareisequalto ofBillrsquoswhatwasthecostofthemerchandiseeachbought

27Ifthewingspanofaplaneis85ft6inwhatisthewingspanofamodeliftheratioofthelengthofanypartofthemodeltothelengthofthecorrespondingpartoftheactualplaneis172

28Ifabankruptfirmcanpay55centonthedollarandifitsassetsamountto$24000whatareitsliabilities

29Whatsellingpriceshouldbeplacedonarefrigeratorifthecostis$325andthedealeroperatesonamarginof35ofcost

30Ifyouallow22ofyourincomeforfoodand18forrent(a)whatistheratioofthecostoffoodtotherent(b)howmuchdoyouspendforfoodpermonthwhenyourincomeis$7200peryear

31Ifatownestimatesthatithastoraise$406250intaxesandtheassessedvaluationofitsrealpropertyis$12500000whatisitstaxrate

32Acertainconcretemixtureistobemadeof1partcement partssandand partsstoneWhatis(a)theratioofsandtostone(b)theratioofcementtosandand(c)percentofsandintheconcretemixture

33Ifthelivingroomofahouseisshownontheblueprinttobe inby inandthescaleoftheprintis in=1ftwhataretheactualdimensionsoftheroom


35Whatistheratioof1yardto1foot

36(a)4=16(b)5=4(c)

37(a)24=6(b)49=7(c)

38Whatisthecompoundratioof128and415

39Ifyoubuythesamenumberofcigarsat10cent15centand20centandyouspend$360howmanyateachpricedidyoubuy

40IfyouboughtfivetimesasmanygradeAarticlesat$28asgradeBarticlesat$16andthreetimesasmanygradeCarticlesat$22asgradeBarticleshowmanyofeachgradedidyoubuyifyouspent$3552

41Completethefollowingproportions(a)243(b)448(c)6912(d)1224(e)5156(f)9248(g)61020(h)367264

42Findthemissingtermsinthefollowingproportions

(a)(b)

(c) (d)

(e)

(f)

(g)

(h)

(i)

43If10bushelsofapplescost$25whatwill15bushelscost

44If25lbofsugarcost$350whatwill75lbcost

45Ifatacertainmomentapost32fthighcastsashadow48ftlonghowlongistheshadowofatreewhichis48fthigh

46MeasuretheheightofapostandthelengthofitsshadowAlsoatthesametimemeasurethelengthoftheshadowofanytallobjectandcalculatetheheightofthetallobject

47Acertainbrandofwhitepaintcontains21partsoftitaniumdioxideand37partsofwhiteleadbyweightIfyouhave600lboftheoxidehowmanypoundsofwhiteleadwouldyouneedtomakeabatchofpaint

48Equalsumsofmoneyareinvestedat and Iftheincomeat is$819whatistheincomeat

49Calculatethemeanproportional

(a)

(b)

(c)

50If42galofgasolinecost$1260howmuchwill85galcost

51Analloyconsistsof partstinand partscopperHowmanypoundsofcopperwouldbeneededwith150poundsoftintomaintainthegivenratio

52Ifittakesyou45minutestodrivetoworkat40mphatwhatspeedwouldyouhavetotraveltomakeitin38minutes

53If28mendoajobin18dayshowmanymenwillberequiredtodoitin12days

54If110yardsofcopperwireoffer12ohmresistancewhatwillbetheresistanceof600yardsofcopperwireof timesthecross-sectionalarea

55Ahotelputsup8guestsfor12daysandanothergroupof12guestsfor21daysWhatistheratioofthetwohotelbillsIfthesmallerbillwas$1344whatwouldthelargerbillbeIfthelargerbillwas$4032whatwouldthesmallerbe

56If16menworking6hoursperdaydigacanal120feetlongin40dayshowmanymenworking8hoursadaycandigacanal160feetlongin10days

57If4mencut16cordsofwoodin9dayshowlongwillittake10mentocut30cordsofwood


59Whatproportionof milkand milkmustbemixedtoget4milk

60Usinga12-footplankwherewouldyouputthesupportundertheplanksothattwochildrenweighing40and50poundsrespectivelycanplayseesaw

61Whatistheelectricalresistanceof800ftofcopperwire indiausingk=1025

62TheweightofabodyabovethesurfaceoftheearthvariesinverselyasthesquareofitsdistancefromthecenteroftheearthIfamanweighs160poundsatsealevelwhatwillheweighatthetopofamountain3mileshighAssume4000miles=radiusoftheearth

63ThedistancethatabodyfallsfromrestvariesasthesquareofthetimeIfabodyfalls16ftthefirstsecondhowfarwillitfallinthefirst5seconds

64Writethefollowingasequations(a)xvariesasy3

(b)zvariesinverselyasx2anddirectlyasy(c)xvariesinverselyasy2

65Ifyvariesinverselyasxandy=6whenx=3findxwheny=3

66ThevelocityVofafreelyfallingbodyfromarestingpositionis

proportionaltothetimetIfithasavelocityof322ftsecattheendofthefirstsecondwhatisthevelocityattheendofthefifthsecond

67ThepressureofaconfinedgasatconstanttemperaturevariesinverselyasthevolumeIfagashasapressureof60poundspersquareinchwhenconfinedinavolumeof120cuinwhatisthepressurewhenthevolumeisreducedto80cuin

68Ifittakes2cuydofconcretetomake40posts howmanycubicyardswillittaketomake700posts4intimes4intimes5ft

69Ifittakes17menworking7hradaytobuildabridgein22dayshowmanymenworking10hoursadaywillittaketobuildthebridgein4days

70Amapisdrawntoascaleof1500Whatisthedistancebetweentwoplacesthatare inchesapartonthismap

CHAPTERXI

AVERAGES

424WhatismeantbyanaverageinstatisticsAnaverageisasignificantrepresentativevalueforanentiremassofdataIt

standsfortheessentialmeaningofthedetailedfacts

Individualmeasurementsusuallyhavemeaningonlywhentheyarerelatedtootherindividualmeasurementsusuallytosometypicalvaluewhichrepresentsanumberofsuchmeasurementsmdashforexampleaveragecostoflivingaveragewageaverageweightforageandheightandaveragebirthrate

425Whataretheusesofaveragesinstatistics(a)Theygiveusaconciseideaofalargegroup

EXAMPLEWedonotgetaclearmentalimagewhenwearegiventheheightofeverytreeinaforestbuttheaverageheightofthetreesissomethingdefiniteandunderstandable

(b)Theygiveusabasisforcomparisonofdifferentgroupsbysimplerepresentativefacts

EXAMPLETwoforestscanmorereadilybecomparedbymeansofaveragesofsomekind

(c)Theygiveusanideaofacompletegroupbyusingonlysimpledata

EXAMPLEItisnotnecessarytomeasuretheheightofeachpersonofaracetogetthetypicalheightofthatraceAnaverageobtainedfromalimitednumbersayafewthousandsampleswouldgenerallybesufficienttogiveafigureclosetotheexactaverage

(d)Theyprovideuswithanumericalconceptoftherelationshipbetweendifferentgroups

EXAMPLEWemaysaythatthepeopleofoneracearetallerthanthoseof

anotherbuttogetanydefiniteratioofheightsweneedaverages

426WhymayanaveragebeamorereliablefiguretorepresentagroupthanasamplefigureselectedfromthegroupItrepresentsmanyindividualmeasurementsItlevelsoutalldifferencesby

disregardingthevariationsamongtheitemsoftheseriesgivingsignificancetotheentireseriesScientistsfrequentlyperformafinemeasurementanumberoftimesandthenaveragetheresultbecausebysodoingtheyhopeerrorswillcanceloutIfsomemeasurementsaretoolargeandotherstoosmallmistakeseachwaywillaboutbalanceThustheaveragedescribestheseriesofvaryingindividualvaluesandispresumedtobethebestpossiblerepresentationoftheseries

427CanaveragesbecomparedwhentheyarederivedfromdatarepresentingwidelydifferentconditionsandgroupsNoThedatamustbehomogeneous

ThearithmeticalaverageofaseriesofwagedatawherewagesofbothmenandwomenareincludedisnottypicalofeithermenrsquosorwomenrsquoswagesAusefulaveragemustbetypicalofactualconditionsnotmerelyaresultofamathematicalcalculation

428WhatismeantbyadeviationfromtheaverageOnceavaluerepresentativeofanentiregroupisestablishedthesingleitem

canbecomparedwithitThedifferenceiscalledthedeviationfromtheaverage

429WhatisthesignificanceofasmalltotalamountofdeviationsThesmallerthetotalamountofthedeviationsthegreateristhehomogeneity

ofthedatathecloserthegroupingabouttheaveragethesmallerthevariabilityamongtheindividualitemsThiscanservetodecidewhetherornottheaverageistypical

430Whatarethetwoclassesofaveragesingeneral(a)Averagesofordinarynumbersrepresentingtimemoneyandgeneral

things

(b)Averagesofratiosrepresentingspeedandotherratios

431HowdowefindthearithmeticaverageormeanvalueofanumberofsimilarquantitiesAddthequantitiesanddividethissumbythenumberofthequantities

Ex(a)If10menearn$80$96$102$78$92$65$59$110$150and$87respectivelyperweekwhataretheaverageearningsofthe10men

Ex(b)Ifacartravels180milesin4hourswhatisitsaveragespeed

Ex(c)Whatistheaverageof42865379203593

432WhenisanaverageanexcellentwayofshowingthemiddleormosttypicalfigureWhenthefiguresarefairlyclosetogether

Ex(a)Whatistheaveragemarkofagroupof5studentswhentheirrespectivegradesare75788081and77

Ex(b)Whatistheaveragemarkwhenthegradesare758810050and77

Example(a)describesfairlywelltheperformanceofthestudentsExample(b)doesnotreallydescribetheperformanceofthegroupeventhoughtheaverageisthesame78

433Ifatraintakesthefollowingtimesbetweenstopsmdash48minutes55minutes1hour8minutesand42minutesmdashwhatistheaveragetimebetweenstops

48+55+68+42=213minutes

434Acartravels10milesupasteepgradeat30mphandthen90milesonalevelroadat50mphWhatisitsaveragespeedSpeedisaratiooftwothingsdistanceandtime30mphand50mphare

ratiosandwecannotgettheaverageofthetworatiosbydividingtheirsumby2Theaveragespeedisnot

Toaverageratioswemustdividethesumofonekindofthingbythesumoftheotherkindofthing

Herethesumofthemilestraveledis10+90=100miles

435TwoplanesleaveatthesametimefromSeattleWashingtonforElPasoTexasmdashadistanceof1381milesOneplaneAfliesat400mphandreturnsat400mphTheotherplaneBfliesat600mphfromSeattleandreturnsat200mphbecauseofdefectiveenginesIfeachplaneremains12hoursinElPasowhichcomesbackfirstWecannotsaythattheybothgetbacktogetherWhileitistruethatthe

averageofthenumbers400and400isthesameasof600and200thespeedsthemselvesareratiosandwemustineachcasedividethetotaldistancebythetotaltime

Totaldistance=1381times2=2762milesPlaneAtakes goingand345hrreturning

there4

PlaneBtakes goingand returning

there4ThusPlaneAflyingat400mphreturnsfirst

436Ifyoupaidanincometaxof22on$3400oneyearand28on$4600thefollowingyearhowmuch

didyoupayaltogetherSince22and28areratiosyoumustnotfigurethat

istheaverageonthetotalincomeof$8000Insteadyoufigure

437Howwouldyoufindthetotalgiventheaveragewithordinarynumbers(notratios)Multiplytheaveragebythenumberofitemsinvolved

EXAMPLEIftheaverageweightofapersonisassumedtobe150poundswhatwouldthecarryingcapacityofapassengerelevatorbewhenonly12peoplearepermittedtoride

150lbtimes12=1800lb=Capacity

438Anappliancedealersells15TVsetsthatcost$180persetatanaverageprofitof30and20otherTVsetsthatcosthim$260persetatanaverageprofitof35Whatisthetotalprofitassumingthepercentagesarebasedonthecostprice

Averageprofiton15TVsets=30times$180=$54Totalprofiton15TVsets=15times$54=$810Averageprofiton20TVsets=35times$260=$91Totalprofiton20TVsets=20times$91=$1820there4Totalprofiton35sets=$810+$1820=$2630Ans

439WhatismeantbyaweightedaverageOneobtainedbyfirstmultiplyingeachitembyitsappropriatefactorbefore

addingandthendividingbythenumberofitems

EXAMPLEInaCivilServiceexaminationtheweightsforthemeasurementsareOral1Arithmetic2Practical4Citizenship1English2WhatistheaveragemarkofacandidatewhosemarksareOral85Arithmetic92Practical79Citizenship80English76

Theratioofweights=12412whichaddsupto10

Theweightedmarksare

440HowcanwefindthevalueofonequantitythatisnotgivenwhentheweightsandthefinalaverageareknownEXAMPLEIntheaboveifwearegivenaminimumpassingaverageof

70whatmustacandidategetforthePracticalmarkinordertopass

12412=10=sumofweights10times70=700=totalweightedmarkinordertopass

Theaveragemustbe700pointsinordertopass

Hehasalreadyscored501points

Remainder=199

ButthePracticalhasaweightof4

mustbescoredonthePracticaltogetaminimum70average

Usuallyaminimumissetforeachpartofthetest

441Thereare8manufacturingplantshaving453699341621383562741and214employeesrespectivelyIftheemployeesinplants12and3worked38hoursperweekinplants45and640hoursperweekandinplants7and842hoursperweekhowcouldwe(a)getatruecomparisonoftheirproductivityexpressedinman-hours(b)determinetheaveragenumberofhourseachmanworkedinthegivenweek(a)Multiplythenumberofemployeesineachplantbythenumberofhours

eachisrequiredtoworkDividebythenumberofplantstogettheaveragenumberofman-hoursworkedperweekineachplant

(b)Togettheaveragenumberofhourseachemployeeworkedinthegivenweekdividethetotalnumberofman-hoursbythetotalnumberofemployees

Thereare4014employeeswhoworked157484man-hours

442Howcanwesimplifytheprocessofgettinganaverageofseveralnumbersthatdifferfromoneanotherbyacomparativelysmallamount(a)Determinementallytheapproximateaverage

(b)Geteachdeviationaboveorbelowthisfigure

(c)Subtractthesumofthedeviationsbelowthisamountfromthesumofthedeviationsabovetheamount

(d)Findtheaveragedeviationandaddittotheoriginalapproximatevalue

EXAMPLEWhatistheaveragedailysalesfigureifthedailysalesrecordis

Weseeatoncethattheaverageisapproximately$300aday

Deviations+$6852(above)minus$2374(below)=$4478

443ForscattereddatawhattwootherwaysarethereoffindingtheldquomiddlerdquothatstandformorethananaverageThemedianandthemodearetwowaysofsometimesgettingamore

representativepictureoftheldquomiddlerdquo

444WhatismeantbythemedianThemedianisthemiddlescoreinaseriesofscoresaftertheyhavebeen

arrangedinorderfromlowesttohighestThemedianscoreissuchthatthereareasmanyscoresaboveitastherearescoresbelowit

445HowisthemedianlocatedWhenthereisanoddnumberofscoresthemedianvalueisthatofthemiddle

caseWhenthereisanevennumberofscoresthemedianvalueislocatedbetweenthetwomiddleitemsIfthetwomiddlevaluesareidenticaltheneithermaybechosenasthemedianvalue

Ex(a)Whatisthemedianof8151231822313and9

Arrangetheseintheorderoftheirmagnitudegetting238912131518and23(9values=oddnumber)Themedianis12becauseitisthefifthormiddlevalueTherearefournumbersinthisserieshigherthanthemedianandtherearefournumberslowerthanthemedian

Themeanaverageis

Ex(b)Whatisthemedianof12384918523811and30

Arrangetheseinorderofmagnitudegetting45891112182330and38(10values=anevennumber)Thetwomiddlenumbersare11and12Themedianishalfwaybetweenthemat115

Themeanaverageis

446If25salesmeninanorganizationreporttheiraverageweeklyincomesas$260$200$95$200$220$160$160$800$240$240$235$350$150$260$200$275$450$275$175$200$500$225$250$650and$200whatistheaverageweeklyincomeofthegroupandisthisaveragerepresentativeofthegroup

Thisaveragedoesnotgiveatruepictureofwhatthesalesmengetbecausethe$800andthe$650incomesthrowitoff

447WhatisthemedianoftheaboveanddoesthismediangiveareasonableideaofthegroupincomeArrangetheincomesinorderofmagnitude95150160160175200200

200200200220225235240240250260260275275350450500650and800

Themedianvalueisthethirteenthvalueor$235Asmanysalesmanhaveincomesmorethan$235ashavelessthan$235Thisgivesusareasonableideaofhowmuchthisgroupearnsascomparedwithagroupwhosemedianis$500aweekActuallyhoweveronlyonepersonearns$235andthereforethiscannotbeconsideredasthemosttypicalfigure

448WhatismeantbythemodeItisthemostfrequentsizeofitemthepositionofgreatestdensityWhenwe

speakoftheaveragemantheaverageincomeweusuallymeanthemodalmanorthemodalincomeWemightsaythemodaltipatarestaurantis15themodalworkingmanrsquoshousehasfiveroomsmdashineachinstancethatisthemostusualoccurrencethecommonthingThefigurehavingthehighestfrequencyisthemodeThemodemeansthesinglemosttypicalfigure

449WhatisthemodeoftheweeklyincomesofQuestion446Makeafrequencytableshowinghowmanysalesmenreceiveeachweekly

amount

Moresalesmanhaveincomesof$200aweekthananyotheramountThisfigurehavingthehighestfrequencyisthemodeforthistable

450HowcanwewidentheconceptthatthemodeisthemosttypicalfigureandgetabettermeasureofthegroupGroupthefrequenciesofQuestion449

$700 and over (1)

$400 to $699 (3)

$300 to $399 (1)

$250 to $299 (5)

$200 to $249 (10)

$150 to $199 (4)

less than $150 (1)

Thelargestgroupreceivesfrom$200to$249andthatisthemodeforthistable

451WhatarethebestmeasuresoftypicalearningsofthegroupofsalesmenWehaveseenthatthemeanoraverageis$27880

Themedianis$235

Themodeis$200forthefrequencytable

Themodeis$200minus249forthegroupedfrequencytable

ThusherethemedianandmodearethebestmeasuresofwhattypicallythisgroupreceivesperweekTheygiveusabetterideaofindividualincomesthandoestheaverage

452Whataretheadvantagesofthearithmeticmeanoraverage(a)Itislocatedbyasimpleprocessofadditionanddivision

(b)Extremedeviationsaregivenweightwhichisdesirableincertaincases

(c)Itisaffectedbyeveryiteminthegroup

453Whatarethedisadvantagesofthearithmeticmeanoraverage(a)AverageisaffectedbytheexceptionalandtheunusualOneortwolarge

contributionsinachurchcollectionconcealtheusualortypicalcontributionAfewverylargeincomesproduceanaverageincomefarabovearepresentativeofthemajority

(b)Theaverageemphasizestheextremevariationswhichinmostcasesisundesirable

(c)ItmayfallwherenodataactuallyexistWemayfindthattheaveragenumberofpersonsperfamilyis512althoughsuchanumberisevidentlyimpossible

(d)Itcannotbelocatedonafrequencygraphwhensuchisalreadyinexistence

454Whataretheadvantagesofthemedian(a)Itiseasytodetermineandisexactlydefined

(b)ItisonlyslightlyaffectedbyitemshavingextremedeviationfromthenormalA$1000checkinthechurchcollectiondoesnotaffectthemodeatallandaffectsthemedianonlyasmuchasanyothersingleitemlargerthanthemedianwoulddothatistheitemreceivesthesameweightasanyotherinstanceandnomoreThusitisusefulwheneverextremeitemsareoflittleimportance

(c)Themedianisparticularlyusefulingroupstowhichameasurecannotbeappliedgroupsofnonmathematicaltype

(d)Itslocationcanneverdependuponasmallnumberofitemsasissometimesthecasewiththemode

(e)IfthenumberofextremeitemsisknowntheirvaluesarenotneededingettingthemedianThemedianisapositionaverageMerelythenumberofitemsnottheirsizeinfluencesthepositionofthemedian

(f)Onthewholeitisoneofthemostvaluabletypesforpracticaluseandforsuchstudiesaswagesanddistributionofwealthitisoftensuperiortoeitherthemodeorthemean

455Whatarethedisadvantagesofthemedian(a)Itisnotsoreadilydeterminedbyasimplemathematicalprocess

(b)Wecannotobtainatotalbymultiplyingthemedianbythenumberofitems

(c)Itisnotusefulwhereitisdesirabletogivelargeweighttoextremevariations

(d)Itisinsensitivewhichmeansthatwecanreplacecertainmeasurementsorvaluesofagivengroupbyothervalueswithoutaffectingthemedian

EXAMPLEInthevalues246⑧101214themedianis8theaverageis

Nowwemayreplacethethreevalueswhicharelargerthan8andthisreplacementwillhavenoeffectuponthemedianThusthevaluesare246⑧172134Themedianisstill8Butthemeanbecomes

(e)Unlikemodebutlikearithmeticmeanitisfrequentlylocatedatapointinthearrayatwhichactualitemsarefew

(f)Wheretherearemanyitemsofthesamesizeasthemedianthenumberofitemslargerthanthemedianmaybeverydifferentfromthenumberofitemssmallerthanthemedianandthevalueofthemedianasanaverageislargely

destroyed

456Whataretheadvantagesofthemode(a)Itisusefulincasesinwhichonedesirestoeliminateextremevariations

whichdonoteffectit

(b)Oneneedknowonlythatextremeitemsarefewinnumbernottheirsize

(c)Modemaybedeterminedwithconsiderableaccuracyfromwell-selectedsampledata

(d)ItisthebestwaytorepresentthegroupItmeansmoretosaythatthemodalwageofworkingmeninalocalityis$16perdaythantosaythattheaveragewageis$1632whichnooneactuallyreceives

457Whatarethedisadvantagesofthemode(a)Inmanycasesnosinglewell-definedtypeactuallyexistsThereisnosuch

thingasamodalsizecityWearelikelytofindseveraldistinctmodescorrespondingtothevariousgradesoflabor

(b)Modeisdifficulttodetermineaccuratelybyanymethod

(c)Itisnotusefulwhenyouwanttogiveanyweighttoextremevariations

(d)Modetimesthenumberofitemsdoesnotequalthecorrecttotalasinarithmeticmean

(e)UnlessgroupingisusedmodemaybedeterminedbyacomparativelysmallnumberoflikeitemsinalargegroupofvaryingsizeIfonly4peopleowned$3000eachinacommunityhavingagreatvariationinwealththiswouldbethemodalvaluewhilethewealthofallothersvaried

PROBLEMS

1Thewagesofamanforsixweeksare$92$87$9950$91$9750and$89Whatistheaveragewageforthesesixweeks

2AschoolsystemhadthefollowingattendancesinoneweekMonday248585Tuesday248326Wednesday247963Thursday248658andFriday248597Whatistheaveragedailyattendance

3Ifacartravels235milesin5hourswhatistheaveragespeed

4Whatistheaveragemarkofagroupof8studentswhentheirrespectivegradesare83869092878281and84

5Whatistheaveragemarkwhenthegradesare869810060849177and89

6WhichaveragedescribestheperformanceofthegroupbettertheoneinProblem4ortheoneinProblem5

7Ifatraintakesthefollowingtimesbetweenstops37minutes44minutes1hour2minutesand31minuteswhatistheaveragetimebetweenstops

8Acartravels8milesupasteepgradeat32mphandthen80milesonalevelroadat52mphWhatisitsaveragespeed

9Ifyoupaidanincometaxof20on$3200oneyearand26on$4400thefollowingyearhowmuchdidyoupayaltogether

10Ifatotalofonly14personsarepermittedtorideinanelevatorandtheaverageweightofapersonisassumedtobe150lbwhatisthecarryingcapacityofthiselevator

11Ifyousell40radiosthatcost$35persetatanaverageprofitof and70setsthatcost$58persetatanaverageprofitof40whatisthetotalprofitassumingthepercentagesarebasedonthecostprice

12IftheweightsinanexaminationareArithmetic2English3Practical3Oral1Citizenship1whatistheaveragemarkofacandidatewhosemarksareArithmetic94English89Practical75Oral80Citizenship80

13Iftheminimumpassingaverageis75whatmustacandidategetfortheEnglishmarkinordertopassinProblem12

14Whatistheaveragedailysalesfigureifthedailysalesrecordis$43589$30764$39638$42907and$43679usingthesimplifiedmethodbyfirstdeterminingmentallytheapproximateaverage

15(a)Whatisthemedianof9161341932414and10(b)Whatisthemedianof13395019624912and31

16If10salesmenreporttheiraverageweeklyincomesas$370$310$105$310$560$385$760$300$260and$385whatistheaverageweeklyincomeofthegroupandisthisaveragerepresentativeofthegroup

17WhatisthemedianincomeofthegroupofProblem16

18WhatisthemodeoftheweeklyincomesofProblem16

19WhatisthemodewhenthefrequenciesaregroupedinProblem16

20Intheseries35791822and35whatisthemedian

21IsthemedianofProblem20affectedif1822and35arereplacedby1112and15

22Agroupof50personscontributedtoachurchcollectioninthefollowingamounts

$500(1)$50(12)$300(2)$25(22)$100(3)$15(6)$75(4)

(a)Howmuchdidthegroupcontribute(b)Whatwastheaveragecontribution(c)Whatwasthemediancontribution(d)Whatwasthemode(e)Whichtypegivesthetruestpictureofthecontributionsofthegroup

23Ifthemediangradeofaclassinabiologytestis81whatcanbesaidaboutthegradesinthattest

24Aplanecovers290milesinthefirsthourofitsflight504milesinthenext hoursofflightand376milesinthefinal hoursofflightWhatistheaveragespeedfortheentirejourney

CHAPTERXII

DENOMINATENUMBERS

458WhatisadenominatenumberItisaconcretenumberwhoseunitofvalueormeasurehasbeenfixedbylaw

orcustomItisusedtospecifytheunitsofmeasurementsWhenstandardunitsareusedwithastatedquantitytheyarecommonlyreferredtoasdenominatenumbers

EXAMPLE3feet4yards8poundsaredenominatenumbers

459WhatismeantbyreductionofdenominatenumbersItistheprocessofchanginganumberexpressedinonedenominationtoan

equivalentexpressedinanotherdenomination

EXAMPLES

3feetchangedtoinchesequals36inches3quartschangedtopintsequals6pints

460Whatismeantby(a)reductiondescending(b)reductionascending(a)Changinganumberfromahighertoalowerdenominator=reduction

descending

EXAMPLE2yards=6feet=72inches

(b)Changinganumberfromalowertoahigherdenomination=reductionascending

EXAMPLES200cents=2dollars36inches=3feet

Notethatinreductiontheexpressionischangedwithoutchangingthevalue

461Whatarethestandardlinearmeasures

12inches(in)=1foot(ft)320rods=1mile(mi)3feet=1yard(yd)1760yards=1mile51yards=1rod(rd)5280feet=1mile

161feet=1rod

Note(a)Marinemeasuresareexpressedinfathoms(=6feet)longcablelengths(=120fathoms)shortcablelengths(=100fathoms)knots(=115miles)andleagues(=3knots)

Note(b)TheunitsintheabovetablerepresentlengthonlyTheyareusedtomeasuredistanceslengthswidthsorthicknessesofobjectsTheunitoflengthisthestandardyard

Note(c)

Symbolforinches=[Prime]placedatupperright5Prime=5inSymbolforfeet=[prime]placedatupperright5prime=5ft

462Whatistheresultofthereductionofthefollowing(a)5ft5intoinches5times12rdquo+5rdquo=65in(b)5yd3fttofeet5times3prime+3prime=18ft(c)5rdtoyards5times51yd271yd(d)108intofeet(e)4mitorods4times320rd=1280rd(f)1rdtoinches in(g)66fttoyards yd(h)72intoyards yd(i)66fttorods rd(j)2rdtofeet ft(k) ydtorods rd

(l)15840fttomiles158405280=3mi

463WhatistheprocedureforreductiontolowerdenominationswhenthelengthisexpressedinseveraldenominationsReduceeachunittothenextlowerdenominationinregularorder

EXAMPLEWhatisthereductiontoinchesof6rd5yd2ft6in

464WhatistheprocedureforreductiontohigherdenominationsReduceeachunittothenexthigherdenominationinregularorder

EXAMPLEWhatisthereductiontorodsyardsfeetandinchesof1503inches

465Whataretheunitsusedinmeasuringtheareasofsurfaces(squaremeasure)

1sqmi=102400sqrd=3097600sqyd=27878400sqft

Notethat12intimes12in=144sqin=1sqft

3fttimes3ft=9sqft=1sqydetc

NoteAsquare10primetimes10prime=100sqftiscommonlyusedinroofing

466Whatistheresultofthereductionofthefollowing(a)442sqintosqft442divide144=3sqft(b)45sqfttosqyd45divide9=5sqyd(c)4sqydtosqft4times9=36sqft(d)640sqrdtoacres640divide160=4A(e)432sqmitotwp432divide36=12twp(ƒ)10sqmitoacres10times640=6400A(g)10twptoacres10times36times640=230400A(h)120A240sqrdtosqyd

(i)24sqyd14sqfttosqin

(j)2sectionstosqrd2times640times160=204800sqrd(k)24320sqrdtoacres24320divide160=152A(l)152460sqfttoacres A

467Whatarethemeasurementsforsolids(cubicmeasure)Cubicmeasureisusedtomeasurethecontentsorcapacityofbinstanksand

thelikeaswellassolids(volume)

NoteAcordofwoodis8ftlongtimes4ftwidetimes4fthigh=128cuftAperch(usedtomeasurestonemasonry)is ftlongtimes ftwidetimes1fthigh=cuft

468Whataretheunitsapplicabletoliquidmeasure4gills(gi)=1pint(pt) gallons=1barrel(brl)

2pints=1quart(qt)63gallons=1hogshead(hgs)

4quarts=1gallon(gal) gallons=1cubicfoot

231cubicinches=1gallon(US)

277274cuin=1gallon(imperialgallonofEngland)

Agallonofwater(Englishgallon)weighs10pounds

Agallonofwater(USgallon)weighsabout pounds

Acubicfootofwaterweighs pounds

Liquidmeasureisusedinmeasuringliquidsexceptmedicine

NoteAfluidounceisequalto ofapintor ofagill

469Whataretheunitsapplicabletodrymeasure

Drymeasureisusedinmeasuringgrainsseedsproduceandthelike

470HowmanykindsofweightareinuseintheUnitedStatesFourkinds

(a)Avoirdupoisweightisusedinweighingheavycoarseproductssuchasgrainhaycoalironandthelike

(b)TroyweightisusedinweighingpreciousmetalsmdashmineralsgoldsilveranddiamondsItisalsousedbythegovernmentinweighingcoinsatthemint

(c)Apothecariesrsquoweightisusedinweighingdrugsandchemicals

(d)MetricordecimalsystemofweightsisusedextensivelyintheUnitedStatesinscientificwork

471Whatconstitutestheavoirdupoistableofweights

ThelongtonisusedbytheUSCustomHouseindeterminingdutyonmerchandisetaxedbythetonItisalsousedwhencoalandironaresoldatwholesaleattheminesUnlessotherwisespecifiedatonistakentobe2000pounds

472Whatconstitutesthetroytableofweights

Thecaratusedinweighingpreciousstonesisequivalentto3168grainstroyor2055milligramsThetermkaratisusedtodenotethefinenessofgoldandmeans byweightofgoldForexample24karatsfinemeanspuregold18karatsmeans puregoldbyweight

473Whatconstitutestheapothecariesrsquotableofweights

Althoughavoirdupoisweightisusedinbuyingandsellingdrugsandchemicalswholesaledruggistsandphysiciansuseapothecariesrsquoweightincompoundingmedicines

Apothecariesrsquofluidmeasure

Apothecariesrsquofluidmeasureisusedbydruggistsinpreparingmedicines

474Whataresomecomparisonsofweights

Pound Ounce

Troy 5760grains 480grains

Apothecariesrsquo 5760grains 480grains

Avoirdupois 7000grains 437 grains

ThegrainisthesameinallthreesystemsThetroyandapothecariesrsquopoundandouncearerespectivelyalike

475WhataretheunitsformeasurementoftimeThemeasuresarebasedonthemovementsoftheearthandotherbodiesofthe

solarsystemOnerevolutionoftheearthonitsaxisisdesignatedadayandonecompleterevolutionoftheeartharoundthesunisoneyearThemonthisderivedfromtherevolutionofthemoonaroundtheearth

60seconds(sec) = 1minute(min)

60minutes = 1hour(hr)

24hours = 1day(da)

7days = 1week(wk)

30days = 1month(mo) (SeeNote(b)below)

52weeks = 1year(yr)

12months = 1commonyear(yr)

365days = 1commonyear

366days = 1leapyear(1yr)

10years = 1decade

20years = 1score

100years = 1century(C)

Onerevolutionoftheeartharoundthesunrequires365days5hours48minutesand497secondsSincethefractionisalmostfrac14ofadayoneentiredayisaddedeveryfourthyeartomakealeapyearBecausethisdoesnotexactlytakecareofthefractioneverycentennialyearwhichisnotdivisibleby400isregardedasacommonyear

Note(a)Allmonthshave31daysexceptAprilJuneSeptemberandNovemberwhichhave30daysandFebruarywhichhas28daysinthecommonyearand29daysintheleapyear

Note(b)Itiscustomaryinbusinesstoregardayearas12monthsof30dayseachoras360daysThispracticeisforconvenienceonlyinmakinginterestcalculationsasexplainedearlier

476Whatarethemeasuresofcounting

20units = 1score

12units = 1dozen

12dozen = 1gross(gro)

12gross = 1greatgross(grgro)

477Whataretheunitsforpapermeasure

24sheets=1quire(qr)20quires=1ream(rm)2reams=1bundle(bdl)5bundles=1bale(bl)

Publishersandprintersestimateonabasisof1000sheetsandallow500sheetstoareamalthoughthereareusually480sheetsinaream

478Whataresomemeasuresofvalue

479WhatisthemetricsystemofweightsandmeasuresItisadecimalsysteminwhichthefundamentalunitisthemetertheunitof

lengthFromthistheunitsofcapacity(liter)andofweight(gram)werederivedDecimalsubdivisionsormultiplesofthesecomprisealltheotherunits

Onemeter(=3937in)wastakentobeonetenmillionthofthedistancefromtheequatortothepoleMoreaccuratemeasurementslaterprovedthistobeonlyapproximatelycorrect

Sixnumericalprefixescombinewithmetergramandlitertoformthemetrictables

TheLatinprefixesare

milli-=onethousandth=001= centi-=onehundredth=01= deci-=onetenth=1=

TheGreekprefixesare

deca-=ten=10hecto-=onehundred=100kilo-=onethousand=1000

480Whatisthelinearmeasuretableinthemetricsystem

MovethedecimalpointtotherighttochangefromahighertoalowerdenominationandtotheLefttochangefromalowertoahigherdenomination

Ex(a)Express826metersasdecimetersHighertolowermovepointtorightgetting826decimeters

Ex(b)Express83234centimeterstometersLowertohighermovepointtoleftgetting83234meters

Ex(c)Express15283metersintheproperdenominations

15kilometers2hectometers8decameters3meters

481WhatistheareameasuretableinthemetricsystemTheunitmeasureforsmallsurfacesisthesquaremeterOnehundredunitsof

anydenominationarerequiredtomakeoneunitofthenexthigherdenomination

100sqmillimeters(sqmm) = 1sqcentimeter(sqcm)

= 0001sqmeter

100sqcentimeters = 1sqdecimeter(sqdm)

= 01sqmeter

100sqdecimeters = 1sqmeter(sqm)

= 1sqmeter=1centare

= 1sqmeter=1centare

100sqmeters = 1sqdecameter(sqDm)

= 100sqmeters=1are

100sqdecameters = 1sqhectometer(sqhm)

= 10000sqmeters=1hectare

100sqhectometers = 1sqkilometer(sqkm)

= 1000000sqmeters

Movedecimalpointtotherighttochangefromahighertoalowerdenomination

Ex(a)Express826sqmetersassqdecimetersHighertolowermovepointtorightgetting8260sqdecimeters

Movepointtothelefttochangefromalowertoahigherdenomination

Ex(b)Express83234sqcentimetersassqmetersLowertohighermovepointtoleftgetting083234sqmeters

482WhatisthevolumeorcubicmeasuretableinthemetricsystemThecubicmeteristhepracticalunitofmeasuresofvolumeWhenusedin

measuringwoodthecubicmeteriscalledastereOnethousandunitsofanydenominationarerequiredtomakeoneunitofthenexthigherdenomination

1000cumillimeters(cumm) = 1cucentimeter(cucm)

= 000001cumeter

1000cucentimeters = 1cudecimeter(cudm)

= 001cumeter(=1liter)

1000cudecimeters = 1cumeter(cum)

1000cumeters = 1cudecameter(cuDm)

= 1000cumeters

1000cudecameters = 1cuhectometer(cuhm)

= 1000000cumeters

1000cuhectometers = 1cukilometer(cukm)

= 1000000000cumeters

Ex(a)Express826cumetersascubicdecimetersHighertolowermovepointtorightgetting82600cudecimeters

Ex(b)Express83234cucentimetersascumetersLowertohighermovepointtoleftgetting083234cumeters

483WhatisthetableformeasuresofliquidanddrycapacityinthemetricsystemTheliteracubethesideofwhichisonedecimeter(= meter)istheunitof

capacityforbothliquidanddrymeasures

10milliliters(ml) = 1centiliter = 01liter

10centiliters(cl) = 1deciliter = 1liter

10deciliters(dl) = 1liter = 1liter

10liters(l) = 1decaliter = 10liters

10decaliters(Dl) = 1hectoliter = 100liters

10hectoliters(hl) = 1kiloliter(kl) = 1000liters

484Whatisthetableformeasuresofweightinthe

metricsystemTheunitofweightisthegramwhichistheweightofacubeofdistilledwater

havinganedge meterinlengthOnepound=4535924grams

10milligrams(mg) = 1centigram(cg) = 01gram

10centigrams = 1decigram(dg) = 1gram

10decigrams = 1gram(g) = 1gram

10grams = 1decagram(Dg) = 10grams

10decagrams = 1hectogram(hg) = 100grams

10hectograms = 1kilogram(kg) = 1000grams

10kilograms = 1myriagram(Mg) = 10000grams

10myriagrams = 1quintal(Q) = 100000grams

10quintals = 1metricton(MT) = 1000000grams

485Whataretheunitsforcircularmeasure

60seconds(Prime) = 1minute(prime)

60minutes = 1degree(deg)

360degrees = 1circle(cir)

Anangleof90degrees(90deg) = arightangle

ofacircle(90deg) = aquadrant

ofacircle(60deg) = asextant

ofacircle(30deg) = asign

486Inreducing4bu3pk5qt2pttopintswhatistheprocedureReduceeachdenominationtopintsbymultiplyingbytheappropriateunits

andfindthetotal

487Whatistheresultofreducing galtolowerdenominations

488Whatistheresultofreducing10qt2pttothefractionofabushel

489Whatistheresultofreducing fttothefractionofarod

=thefractionofayd=thefractionofardAns

490Whatistheresultofreducing2pk6qt pttoadecimalofabushel

2pt pt Divide by2pt(=1qt)

8qt

(or25)qt+6qt=625qt

625qt Divide625qtby8qt(=1pk)

4pk

78125pk+2pk=278125pk

278125 Divide278125pkby4pk(=1bu)

6953125buAns

491Whatistheresultofreducing27lbapothecariesrsquotolowerdenominations

492Whatistheresultofreducing62gilltoadecimalofagallon

62gilldivide4gill(=1pt)=155pt155ptdivide2pt(=1qt)=0775qt

there40775qtdivide4qt(=1gal)=019375galAns

493WhatistheprocedureforadditionofdenominatenumbersArrangesothatlikeunitsareunderlikeunits(poundsunderpoundsounces

underouncesetc)Beginwiththelowestdenominationandworktotheleft

EXAMPLEAdd

Sumofthegris38grwhichdivideby24gr(=1pwt)=1pwt+14grremaining

Sumofpwt=33+1carry=34pwtwhichdivideby20pwt(=1oz)=1oz+14pwtremaining

Sumofoz=17+1carry=18ozwhichdivideby12oz(=1lb)=1lb+6ozremaining

Sumoflb=21+1carry=22lb

there422lb6oz14pwt14grAns

494WhatistheprocedureforsubtractionofdenominatenumbersPlacelikeunitsunderlikeunitsStartwiththelowestdenominationBorrow

fromhigherdenominationwhennecessary

EXAMPLE

Borrow1min=60secfrom35minleaving34min

Add60secto24sec=84secandsubtract32secleaving52sec

Borrow1hr=60minfrom7hrleaving6hr

Add60minto34min=94minandsubtract50minleaving44min

Subtract4hrfrom6hrleaving2hr

Borrow1mo=30daysfrom7moleaving6mo

Add30daysto14days=44daysandsubtract22daysleaving22days

Borrow1yr=12mofrom3yrleaving2yr

Add12moto6mo=18moandsubtract8moleaving10mo

Subtract1yrfrom2yrleaving1yr

there41yr10mo22days2hr44min52secAns

495Whatistheresultofmultiplying26sqrd10sqyd5sqft34sqinby8Multiplyeachdenominationby8andplaceresultsinposition

272sqin=Isqft+128sqinremaining

40sqft+1sqft=41sqft=4sqyd+5sqftremaining

80sqyd+4sqyd=84sqyd=2sqrd+23 sqydremaining

208sqrd+2sqrd=210sqrd

Change sqydto4sqft72sqinandadjusttheresultbyadding

72+128sqin=200sqin=1sqft+56sqinremaining

4+5+1sqft=10sqft=1sqyd+1sqftremaining

23sqyd+1sqyd=24sqyd

there4210sqrd24sqyd1sqft56sqinAns

496Whatistheresultofdividing18A142sqrd24sqydby7Indivisionstartattheleftwiththehighestdenominationanddivideeachin

turn

497Howmanypoundsofavoirdupoisare25poundstroyweight

Thereare5760grinthepoundtroyand7000grinthepoundavoirdupois

498Howcanwereduce6km4hm3m5dm9mmtometersInsertadecimalpointafterthemeasurerequiredfirstmakingsuretoinserta

zerowheneveranyunitisomitted

6km4hm0Dm3m5dm0cm9mm

HeremetersarecalledforInsertadecimalpointaftermetersgetting

6403509metersAns

499Howcanwereduce5327698dmtokm

Herethe6representswholedmthe7representswholemthe2representswholeDmthe3representswholehmthe5representswholekm

Askmarecalledforputthedecimalpointafterthe5getting5327698km

Anotherwayofdoingitistonotethatfromthetableofunits10times10times10times10or10000dm=1km

Thereforedivide5327698dmby10000ormovethedecimalpoint4placestotheleftgetting

5327698kmAns

500Whatistheresultofadding48m284cmand5Dm2dmwiththeanswerexpressedinmetersWritedowneachquantityinmeterskeepingthepointsunderneatheachother

501Howmanycentimetersremainwhenfromapipe283mlong167cmiscutoff

502Whatisthetotalweightinkgof3450cartonswheneachcartonweighs3600g3600g=36kg

there436times3450=12420kgAns

PROBLEMS

1Express(a)3ft3inininches(b)3yd3ftinfeet(c)6rdinyards(d)112ininfeet(e)5miinrods(ƒ)2rdininches(g)88ftinyards(h)96ininyards(i)92ftinrods(j)3rdinfeet(k)34860ftinmiles(l)6miinfeet

2Reduce5rd4yd4ft7intoinches

3Whatisthereductionof1608inchestorodsyardsfeetandinches

4Whatistheresultofthereductionof(a)562sqintosqft(b)36sqfttosqyd(c)6sqydtosqft(d)860sqrdtoacres(e)362sqmitotwp(ƒ)12sqmitoacres(g)8twptoacres(h)80A120sqrdtosqyd(i)12sqyd10sqfttosqin(j)3sectionstosqrd(k)12460sqrdtoacres(l)174240sqfttoacres

5Howmanycubicinchesarethereinabarofmetal4 incheslong3incheswideand1 inchesthick

6At$225acubicyardwhatwouldbethecostofexcavatingabasement25feet9inchesby34feet6inchesby9feetdeep

7Atankis40ft6inhighand5ft9insquareHowmuchwillthistankfullofwaterweighassumingwaterweighs625lbpercubicfoot

8Express(a)4gallonsinpints(b)96pintsinbushels(c)3pintsingills(d)6quartsinpints(e)2bushelsinpints(ƒ)12pecksinbushels(g)3pecksinpints(h)8quartsingills(i)2bushelsinpecks(j)692cuiningallons(k)4bushelsinquarts(l)12gillsinpints(m)12pintsinquarts

(n)24quartsinpecks(o)32pintsinpecks(p)22 gallonsincubicfeet

9Abinholds832bushelsofgrainWhatisitscapacityinbarrels

10Express(a)4000poundsintons(b)4poundsinounces(troy)(c)3pennyweightsingrains(d)5lbinounces(avoirdupois)(e)60pennyweightsinounces(ƒ)48ouncesinpounds(g)60hundredweightintons(h)3caratsingrains

11Whatisthevalueofagoldnuggetwhichweighs6ounces4pennyweights12grainsat$35anounce

12Howmanypoundsaretherein103680grains

13Express22longtonsinpounds

14Express(a)2weeksinhours(b)4hoursinminutes(c)3daysinhours(d)6hoursinminutes(e)3commonyearsindays(ƒ)4 minutesinseconds(g)72hoursindays(h)300secondsinminutes(i)7daysinminutes(j)4000yearsincenturies(k)414720secondsindays(l)1dayinseconds

15Howmanydayswilltheyear2000have

16Express18hours38minutes20secondsinseconds

17Howmanydaysinthesecondsixmonthsofacommonyear

18HowmanyhoursarethereinthemonthofSeptember

19Acratecontains504eggsmdashhowmanydozen

20Asinglecardcontains24hooksandeyesHowmanygrossarethereon48cards

21Howmanyyearsarethereinthreedecades

22Howmanyyearsarethereinthreescoreyearsandten

23Howmanysheetsaretherein(a)12quires(b)3reams(c)2bundles(d)3bales

24Howmanymillsin5 cents

25Express(a)735metersasdecimeters(b)74126centimetersasmeters

26Express18362metersintheproperdenominations

27Express(a)48261sqcentimetersassqmeters(b)748sqmetersassqdecimeters

28Express(a)746cumetersascubicdecimeters(b)94364cucentimetersascumeters

29Express8poundsand10ouncesindecigrams

30Express3kilogramsinounces

31Howmanygrainsaretherein4 hectograms

32Express3poundsand6ouncesinmilligrams

33Express2poundsand4ouncesincentigrams

34Howmanygrainsaretherein45grams

35Express4centigramsand3 milligramsingrains

36Howmanyminutesin10deg12deg28deg

37Howmanysecondsin4prime6prime2deg

38Reduce(a)50deg40prime30rdquotoseconds(b)43200rdquotodegrees

39Reduce5bu4pk3qt2pttopints

40Reduce galtolowerdenominations

41Reduce12qt2pttothefractionofabushel

42Reduce fttothefractionofarod

43Reduce3pk4qt pttoadecimalofabushel

44Reduce38lbapothecariesrsquotolowerdenominations

45Reduce58gilltoadecimalofagallon

46Add

5lb4oz8pwt10gr4lb7oz6pwt8gr14lb8oz16pwt18gr

47From4yr8mo12da8hr30min22secsubtract2yr10mo24da3hr45min30sec

48Multiply24sqrd8sqyd4sqft28sqinby9

49Divide20A138sqrd22sqydby6

50Howmanypoundsavoirdupoisare22poundstroyweight

51Reduce7km5hm4m6dm10mmtometers

52Reduce65438979dmtokm

53Addandexpressresultinmeters56m376cmand7Dm4dm

54Howmanycentimetersremainwhenfromapipe369mlong268cmarecutoff

55Whatisthetotalweightinkgof4860cartonswheneachcartonweighs2400g

CHAPTERXIII

POWERmdashROOTSmdashRADICALS

503HowcanweshowthatthesquareofanumberistheproductofanumberwithitselfInthefiguretherearethreeunitsoneachsideofthesquareThereare9

squareunitsinasquareof3unitsonasideTherefore9issaidtobethesquareof3Similarlyasquarewith5unitsoneachsidehasatotalof25squareunits5times5=25

504Howcanweshowthatthecubeofanumberistheproductofthenumbertaken3timesasafactorInthefiguretherearethreeunitsoneachedgeThereare27cubicunitsina

cubewith3unitsonanedgeTherefore27issaidtobethecubeof3Similarlyacubewith5unitsonanedgehasatotalof125cubicunits5times5times5=125

505WhatismeantbyraisinganumbertoapowerAnumbermultipliedbyitselfissaidtoberaisedtoapower

EXAMPLES

Otherhigherpowersaredenotedbynumbersindicatingthenumberoftimesthefactorisused

3times3times3times3times3=fifthpowerof3=243

506Whatismeantby(a) anexponent(b) abase(a)Theexponentisasmallfigurewrittentotheupperrightofanumbertobe

raisedtoapowerandindicatesthepowertaken(orhowmanytimesthenumberismultipliedbyitself)

EXAMPLES

32meansthesquareof3(3times3=9)exponentis243meansthecubeof4(4times4times4=64)exponentis378meanstheeighthpowerof7(7times7times7times7times7times7times7times7=5764801)exponentis8

(b)Thefactortoberaisedtoapoweriscalledthebase

EXAMPLEIn787isthebaseand8istheexponent

507HowdoweraiseanalgebraicsymboltoapowerBytheuseofanexponentwhichdenotesthenumberoftimesthesymbolis

used

Ex(a)xsquared=xbullx=x2whichmeansthattwoequalquantitiestimeshavebeenmultipliedtogether

Ex(b)xcubed=xbullxbullx=x3whichmeansthatthreeequalquantitiesxhavebeenmultipliedtogether

Ex(c)(3x)squaredmeans3squaredmultipliedbyxsquaredor3bull3bullxbullx=32x2=(3x)2=9x2whichisreadldquo9(xsquared)rdquo

Ex(d)3xraisedtothefourthpower=3bull3bull3bull3bullxbullxbullxbullx=34x4=81x4

508WhatistheoperationofraisingquantitiesortermstogivenpowerscalledTheprocessiscalledinvolution

509HowcanweshowthatthesquareofthesumofanytwonumbersisthesquareofthefirstplusthesquareofthesecondplustwicetheproductofthetwonumbersThesquareofanumberisthenumbermultipliedbyitselfThesquareof26is

26x26=676

Wemaywritethismultiplicationas

Sinceanynumbergreaterthan10maybeconsideredasthesumoftwonumbersthesquareofthesumofanytwonumbers=thesquareofthefirst+squareofthesecond+twicetheproductofthetwonumbers

510HowcantheabovebeshowngraphicallyCutthelinesofthesidesinto20unitsand6unitstorepresent2tens+6units

Thewholesquareof26consistsofthelargesquare=202+2timestherectangle20times6+smallsquare62or

400+240+36=676

If20=aand6=bwegettheformula

(a+b)2=a2+2ab+b2

511HowdowefindthenumberofsquareunitsinthesurfaceofanyplanefigureorflatsurfaceMultiplytheunitoflengthbytheunitofwidthofthesamedenomination

(inchesbyinchesfeetbyfeetetc)

Herethereare8unitsoflengthand6unitsofwidththerefore8times6=48squareunitsThismaybethoughtofas6rowsof8squareunitsperrow

512HowdowecalculateahigherpowerofacommonfractionRaisethenumeratortothepowerrequired

Raisethedenominatortothepowerrequired

Expressthepowersasafraction

EXAMPLEFindthefourthpowerof

513Whataretherulesaffectingthepowersofdecimalfractions(a)Thesquareofadecimalfractionmusthaveatleasttwodecimalplaces

EXAMPLES

22=0452=25

(b)Theremustbeanevennumberofdecimalplaces

EXAMPLES

514WhyisadecimalfractionraisedtoapowerofasmallervaluethantheoriginalfractionAdecimalfractionwhenconvertedtoanumeratorandadenominatorhasa

verylargedecimaldenominatorInraisingthefractionthesmallernumeratorisdividedbyalargerandlargerdenominatorasthepowertowhichthefractionisraisedincreases

EXAMPLE

515WhatistheprocedurewhentwopowersofthesamebaseornumberaretobemultipliedAddtheexponents

EXAMPLE

25times23=25+3=28(2bullsdot2bull2bull2bull2)times(222)=28

Now25=32and23=8

Therefore32times8=256=25times23=25+3=28Thisshowsthatwecanmultiply32times8bymeansofexponents

516WhatistheprocedurewhentwopowersofthesamebaseornumberaretobedividedSubtracttheexponentofthedivisor(ordenominator)fromtheexponentofthe

dividend(ornumerator)

Ex(a)Divide32by8

Ex(b)Divide243by9

Thisshowsthatdivisioncanbeperformedbymeansofexponents

517WhatlimitstheaboveprocessesTheyareonlygoodfordivisionandmultiplicationofexactpowersof23or

exactpowersofanyothernumbersorbasesforwhichyouhavebuiltuptables

518WhatistheprocedurewhenthepowerofanumberisitselftoberaisedtoapowerMultiplytheexponents

EXAMPLEFindthethirdpowerof42

(42)3=42bull42bull42=42times3=46=4096

Multiplyexponent2byexponent3togetexponent6

519Howcanweshowthatanynumberorbasetothezeropowerequals1Anyquantityorbaseraisedtothefirstpowerisrepresentedbythequantityor

baseitselfThus2raisedtothefirstpoweris21=2xtothefirstpoweriswrittenx1=x

Ex

(a)2divide2=1Bust there42deg=1(b)5divide5=1But there45deg=1(c)10divide10=1But there410deg=1

Thesameprocedurecanbefollowedforanybaseornumber

there41=Anynumbertothezeropower

520Howcanweshowthatthesignofanexponentmaybechangedbychangingthepositionofthenumberfromonesideofthedenominatorlinetotheother

Whenafactordoesnotappearitsexponentiszeroandthevalue1canbesubstitutedforit

EXAMPLEIn3times5=157isnotusedasafactorwhichmeansitsexponentiszeroorthefactorisusedzerotimesThismaybewritten

3times5times7deg=3times5times1=3times5

Now

Butsubtractingexponents

Thesignoftheexponentmaybechangedbychangingthepositionofthenumberfromonesideofthedenominatorlinetotheother

Thusanegativeexponentmeansdivisionof1bythenumberwiththesamepositiveexponent

EXAMPLES

521Whyisadecimalfractionraisedtoanegative

powerofgreatervaluethantheoriginaldecimalfractionInnegativepowerstheverylargedenominatorbecomesthenumeratorwhich

increasesthevalueofthefraction

Ex(a)

(b)

522WhyarethenegativepowersofwholenumberssmallerthantheoriginalnumbersAnegativepowermakesafractionofawholenumberandreducesitsvalue

Ex(a)

(b)

Highernegativepowersmaketheresultssmallerandsmaller

523Howcanwesimplifytheraisingofanumbertoapowerthatcanbefactored(a)Factorthepower

(b)Raisethenumbertothepowerofoneofthefactors

(c)Raisethisresulttothepowerofthenextfactorandsoonuntilallthefactorsareusedup

Ex(a)Raise3totheeighthpower

Factorexponent8into2times2times2(3factors)

Raise32=9Then92=81Then812a=6561=38

Ex(b)Raise5tothetwelfthpower

Factorexponent12into2times2times3

Raise52=25Then252=625Then6253=244140625=512

524Whatisthebasisforashortmethodofsquaringanumberfrom1to100Weknowfromalgebrathat(amdashb)(a+b)=a2ndashb2Theproductofthesum

anddifferenceoftwonumbersisthesameasthedifferenceoftheirsquares

EXAMPLEIfwewanttosquare29wesetup

(29+1)(29mdash1)=(292ndash1)

or

30times28=840=(292ndash1)292=840+1=841

525Whatthenistheprocedureforashortmethodofsquaringanumberfrom1to100(a)Addorsubtractanumbertomakeoneofthemultipliersadecimal

number

(b)Subtractthesamenumberfromtheoriginal

(c)Multiplytheaboveandaddthesquareofthenumberaddedorsubtracted

Ex(a) Addandsubtract2getting

(b)

526HowdoestheprocedureofQuestion509comparewiththeaboveasashortmethodofsquaringanumberfrom1to100EXAMPLE

Fornumbersbetween1to100theprocedureofQuestion525wouldappeartobesomewhatsimpler

527HowcanweapplytheprocedureofQuestion509tomixednumbersas etc

(a+b)2=a2+2ab+b2=a(a+2b)+b2

Inthiscaseaismadeanintegralnumberandbismadethefraction

Addtwicethefractiontotheintegralnumberandmultiplythisbytheintegral

numberThenaddthesquareofthefraction

Ex(a)

(b)

(c)

528HowmayaliquotpartsbeappliedtotheabovemethodConvertthenumbertoamixednumbertheoretically

EXAMPLETosquare825convertto )2theoreticallyandapplyaboverule

Now =0625andsincetheoriginalnumberhasnodecimaltheansweris680625

529Howisthesquaringofanumberthatisdivisiblebyfactor23or5madesimplerDividebythefactorsquarethequotientandmultiplybythefactorsquared

Ex(a)Tosquare36divide36by3getting12asthequotientSquare12getting144whichmultiplyby32getting1296

(b)Square35 =772=4949times52=1225

(c) =772=4949times22=196(d) =992=8181times22=324

530WhatistheprocedureforgettingthesquareofthemeanbetweentwonumbersMultiplythetwonumbersandaddthesquareofhalftheirdifference

Ex(a)Whatisthesquareofthemeanof12and16or14

(b)Whatisthesquareofthemeanof30and40or35

(c)Whatisthesquareofthemeanof24and25or

531Whatisaneasywayofsquaringanumberendingin Multiplytheintegralbythenexthigherintegralandadd (Thisissimilarto

Example(c)ofQuestion530)

Ex(a)

(b)

532Whatistheprocedurewhenthenumberendsin5insteadofThe5istakenasrepresentingthe oftheabove

Ex(a) issimilarto

(75)2=70times80+25=5625(b)(125)2=120times130+25=15600+25=15625

Theproofoftheabovewhenthenumberendsin is

533Whatistheprocedureforsquaringanumberconsistingof9rsquosPlace1astheright-handfigure

Thenzerosonelessthanthenumberof9rsquos

Thenfigure8

Then9rsquosonelessthanthenumberof9rsquos

Ex(a)

(b)

534Whatdoestheexponentofanypowerof10indicateItindicatesthenumberofzerosafterthe1inrepresentingtheresult

Eachpoweraddsonemorezerosuccessively

Thereversealsoholdsthatisiftheresultis10000000youcountthezerostogetthenumberoffactorsof10ortheexponentof10whichinthiscaseis107

Howcanlargenumbersbeexpressedconvenientlyintermsofpowersof10

Ex(a)3900=39times100=39times102

(b)4000000=4times1000000=4times106

(c)36300000=363times10000000=363times107

535Doestheaboveapplytonegativeexponentsofbase10Yes

536Howcanweexpressdecimalsaspowersof10

Negativepowerof10=adecimal

Howcandecimalsbeexpressedconvenientlyintermsofnegativepowersof10

Ex(a)003=3times001=3times10ndash2

(b)00021=21times00001=21times10ndash4

(c)00000462=462times000001=462times10ndash5

537Whatisdonewiththeexponentsinmultiplyingpowersof10Theexponentsareaddedalgebraically

Ex(a)

(b)

(c)

(d)

(e)

538Whatisdonewiththepowersof10indivisionSubtracttheexponentofthedenominatorfromtheexponentofthenumerator

Thesamethingisobtainedbychangingthesignoftheexponentofthedenominator

Ex(a)

(b)

(c)

539WhatismeantbyarootofanumberorpowerIfagivennumberortermcanbeproducedbymultiplyingtogethertwoor

moreequalnumbersortermstheneachoftheequalnumbersortermsissaidtobearootofthatproduct

Ex(a)9=3times3then3isarootof9(b)125=5times5times5then5isthecuberootof125

(c)81=3times3times3times3then3isthefourthrootof81(dx3=xmiddottimesmiddotxthenxisarootofx3

Therootofanumberisalwaysoneoftheequalfactorsofthatnumber

540WhatismeantbyevolutionItistheinverseprocessofinvolutionInevolutiontheproblemistodetermine

oneofagivennumberofequalfactorswhentheirproductaloneisgivenThefactorssofoundarecalledsquarerootcuberootfourthrootetcdependinguponthenumberoffactorsinvolved

541WhatisthesymbolofevolutionThesymbolisradicwhichisanabbreviationrforrootfollowedbyalineThis

symbolisknownastheradicalsignandindicatesthatarootistobetakenoftheexpressionbeforewhichitstandsAsmallnumbercalledanindexiswrittenovertheradicalsignandindicatestheroottobetakenexceptforasquarerootwhenitisusuallyomittedThequantityorexpressionwithintheradicalsignisknownastheradicandIn 81istheradicandand4istheindex

Ex(a)radic9indicatesthatthesquarerootistobeextracted(b) indicatesthatthecuberootistobeextracted(c) indicatesthatthefourthrootistobeextracted

542Whatismeantby(a)aperfectpower(b)animperfectpower(a)Anumberisaperfectpowerwhenitsrootcanbeextractedwithout

leavingaremainder

(b)Anumberisanimperfectpowerwhenitsrootcannotbeextractedexactly

Ex(a)81isaperfectpowerbecause

(b)87isanimperfectpowerbecauseitsrootcannotbeextractedexactly

543WhatisthesimplestmethodofextractingarootDividethenumberbyitslowestprimefactorandcontinuetheprocess

EXAMPLEFindthecuberootof216 2)216

Therearethreefactors2andthreefactors3 2)108

or =2times3Then 2)54

2times3=6=thecuberootof216 3)27

3)9

3

544WhatistheruleforextractingtherequiredrootofaquantityDividetheexponentofthequantitybytheindexoftherootandthenperform

indicatedoperationswhenpossible

Ex(a) =2times3=6(b) =3b3

(c) =a2b(d)

(e)(f)

545WhatistheruleforfractionalexponentsThenumeratorindicatesthepowertowhichthebaseistoberaisedandthe

denominatortherootwhichistobeextractedofthatpower

Ex(a) (Question544f)

Weseethatinthefractionalexponent ofthebase7thedenominator3istheindexoftherootandthenumerator2istheexponentofthebaseorquantity

Ex(b)(c)(d)

(e)(f)

(g)

546WhenareradicalssimilarWhentheyhavethesameindicesandthesameradicands

EXAMPLE and aresimilarradicals

547WhenmayafactoroftheradicandberemovedfromundertheradicalsignWhenthefactorisanexactpoweroftheindicatedorder

Ex(a)(b)

548HowmayafactorinthecoefficientofaradicalbeintroducedundertheradicalsignByraisingthefactortothepoweroftheindex

Ex(a)(b)

549HowmayafractionwitharadicalinthedenominatorbereducedtoafractionwitharationaldenominatorMultiplynumeratoranddenominatorbythesameradicalexpressionwhich

wouldmakethedenominatorrational

Ex(a)

(b)

550HowmayaradicalwithafractionalradicandbereducedtoafractionwhosedenominatorhasnoradicalMultiplythenumeratoranddenominatorbythesamenumberwhichwill

makethedenominatorarationalnumber

EXAMPLE

551HowmayaradicalbechangedtooneofahigherorderwithanindexthatisamultipleoftheoriginalindexMultiplythenumeratoranddenominatorofthefractionalexponentofthebase

bythesamenumber

EXAMPLE

552WhenmayaradicalbereducedtoaradicalofalowerorderWhentheexponentoftheradicandisafactoroftheindexoftheradical

EXAMPLE

553WhenisaradicalexpressionsaidtobeinsimplestformWhen

(a)theindexisassmallaspossible

(b)theradicandhasnofractions

(c)thedenominatoroftheexpressionhasnoradical

(d)everyfactoroftheradicandhasanexponentlessthantheindex

554Whatistheresultof reducedtoitssimplestform

Thisisthesimplestformastheindex4isassmallaspossibletheradicandhasnofractionthereisnoradicalinthedenominatoroftheexpressionandtheradicandy3hasnofactorwhichisafourthpowerofy

555Whatistheresultofreducing(a) (b) tothesimplestform(a)

(b)

556Howmanyfiguresdoesittaketoexpressthesquarerootofanumberof(a)1or2figures(b)3or4figures(c)5or6figures(a)Whenanumberhas1or2figuresthesquareroothas1figure

(b)Whenanumberhas3or4figuresthesquareroothas2figures

(c)Whenanumberhas5or6figuresthesquareroothas3figures

Ifawholenumberbedividedintogroupsof2figureseachbeginningattheunitsplacethenumberofgroupswillequalthenumberoffiguresintheroot

557(a)Whatistherelationofthenumberofdecimalplacesinthesquareofadecimaltothatofthedecimalitselfand(b)whatistherelationofthenumberofdecimalplacesinthesquarerootofadecimaltothatofthedecimalitself(a)Thesquareofadecimalhastwiceasmanydecimalplacesasdoesthe

decimalitself

EXAMPLES

Ineachcase2placesinthedecimalproduce4placesinthesquare

(b)Thesquarerootofadecimalhashalfasmanydecimalplacesasdoesthedecimalitself

EXAMPLES

Ineachcase4placesinthedecimalproduce2placesinthesquareroot

TogetthesquarerootofadecimaltheremustbeanevennumberoffiguresAnnexazeroifneedbeIfadecimalnumberbedividedintogroupsof2figureseachbeginningatthedecimalpointthenumberofgroupswillequalthenumberoffiguresintheroot

558Whatisthesquarerootof676Dividethenumberintogroupsoftwofiguresstartingfromtheunitsfigure

andgoingtotheleftgetting676Thereare2groupsandtherootwillhave2figuresoneoftensandoneofunits

FromQuestions509and510weknowthebasicformulaforthesquareofthesumoftwonumbersis

(a+b)2=a2+2ab+b2=a2+(2a+b)b

Ifa=2tensandb=6unitswegetbysubstitution

(2tens+6)2=(2tens)2+(2times2tens+6)6=202+(2times20+6)6=400+(240+36)=400+276=676

Wemaystartwith676andworkbacktogetthesquareroot

(b)Subtract400from676getting276=remainderofthenumberWehavenowaccountedforthea2partoftheformula

(c)Toaccountfortheremainder(2a+b)bgetatrialvalueofbbydividing4

(=2a)into27oftheremainderandgetting6(=b)Put6intheroot

(d)Addthe6tothe4tensgetting46(=2a+b)andmultiplyby6getting276(=46times6)=(2a+b)b

(e)Subtractthis276fromtheremainder276gettingzero

radic676=26

Toproveasquarerootmultiplythesquarerootbyitself

559Whatistherulefortheextractionofasquareroot(a)Separatethenumberintogroupsof2figuresgoingtotheleftfromthe

decimalpointforthewholepartofthenumberandtotherightforthedecimalpart

(b)DeterminethegreatestsquareinthefarthestleftgroupGetitsrootandputthisintheroot

(c)Subtractthesquareofthisrootfromthisleftgroupandbringdownthenextgrouptotheremainder

(d)Dividetheremainderbytwicetherootalreadyfoundconsideredastensasatrialdivisorgettingthenextfigureoftheroot

(e)Tothetrialdivisoraddthenewfigureoftherootthenmultiplybythelastfigurefoundandsubtractthisproductfromthelastremainder

(f)Bringdownthenextgrouptotheremainderandcontinueasbefore

Ifthenumberisnotaperfectsquareorifyouwantmoredecimalplacesintherootaddzerostothenumberandcontinuetheprocess

560Whatisthesquarerootof70225

(a)Therootwillhave2wholefiguresand1decimalfigure

(b)Thegreatestsquarein7(or700)is4(or400)whoserootis2(=2tens)=aPut2intheroot

(c)Subtract4from7getting3andbringdownthenextgroupgetting302=remainder

(d)Dividetwicetherootalreadyfoundor2times2=4asatrialdivisorinto30oftheremaindergetting6(7wouldbetoolarge)thenextfigureoftherootPut6intheroot

(e)Add6tothetrialdivisor4(astens)getting46andmultiplyby6(thelastfigurefound)getting276Subtract276fromthelastremainder302getting26

(f)Bringdownthenextgroup25getting2625=remainder

(g)Dividetwicetherootalreadyfoundor2times26=52asatrialdivisorinto262oftheremaindergetting5thenextfigureoftherootPut5intheroot

(h)Add5tothetrialdivisor52(astens)getting525andmultiplyby5(thelastrootfigurefound)getting2625Subtract2625fromthelastremainder2625gettingzero

there4

561Whatisthesquarerootof7043716(a)Therootwillhave2wholefiguresand2decimalfigures

(b)Greatestsquarein7is4whoserootis2Put2inroot

(c)Subtract4from7getting3Bringdownnextgroupgetting304=

remainder

(d)Dividetwicerootalreadyfoundor2times2=4asatrialdivisorinto30ofremaindergetting6thenextfigureofrootPut6intheroot

(e)Add6totrialdivisor4(astens)getting46andmultiplyby6(thelastfigurefound)getting276Subtract276fromlastremainder304getting28


(g)Dividetwicerootalreadyfoundor2times26=52asatrialdivisorinto283ofremaindergetting5thenextfigureofrootPut5intheroot

(h)Add5tothetrialdivisor52(astens)getting525andmultiplyby5(thelastfigurefound)getting2625Subtract2625fromlastremainder2837getting212

(i)Bringdownthenextgroup16getting21216=remainder

(j)Dividetwicerootalreadyfoundor2times265=530asatrialdivisorinto2121ofremaindergetting4thenextfigureofrootPut4intheroot

(k)Add4tothetrialdivisor530(astens)getting5304andmultiplyby4(thelastfigurefound)getting21216Subtract21216fromlastremainder21216gettingzero

NoteIneachstepyouconsiderthepartoftherootalreadyfoundastensinrelationtothenextfigure

562Whatisthesquarerootof94864(a)Therootwillhave3wholefigures

(b)Thegreatestsquarein9is9whoserootis3Put3inroot

(c)Subtract9from9gettingzeroBringdownthenextgroupgetting48=remainder

(d)Dividetwicerootalreadyfoundor2times3=6asatrialdivisorinto4ofremainderwhichresultsinzeroPutazerointherootandinthedivisorandbringdownthenextgroupgetting4864=remainder

(e)Dividethenewtrialdivisor60into486oftheremaindergetting8thenextfigureofrootPut8intheroot

(f)Add8tothetrialdivisor60(astens)getting608andmultiplyby8(thelastfigurefound)getting4864Subtract4864fromlastremainder4864gettingzero


Thefinalsubtractioncannotbemadeas105284isalittlelargerthantheremainder105242butisclosetoitsothattherootis

there426322Ans(approx)

564Howdowegettherootofafraction

Extracttherootofboththenumeratoranddenominatorseparately

Ex(a)

(b)

565Whatistherulefortheextractionofthecuberoot(a)Separatethenumberintogroupsof3figureseachtoleftofdecimalpoint

forwholenumbersandtorightfordecimalportion

(b)Findgreatestcubecontainedinfarthestleft-handgroupPutitscuberootintheroot

(c)Subtractthiscubefromthefirstgroupandbringdownthenextgrouptogettheremainder

(d)Divideremainderby3timesthesquareoftherootalreadyfoundconsideredastensasatrialdivisortogetthenextfigureoftherootPutthisfigureintheroot

(e)Totrialdivisoradd3timestheproductofthetwopartsoftherootplusthesquareofthesecondpartoftheroottomakethecompletedivisor

(f)MultiplythecompletedivisorbythesecondfigureoftherootSubtractandbringdownthenextgroup

(g)Continueinthismanneruntilallgroupshavebeenused

566Whatisthecuberootof245314376

(a)Separateintogroups

(b)Thecubeof6isthelargestcubecontainedinthefirstgroup Put6intheroot

(c)Subtract =216from245getting29Bringdownthenextgroupgetting29314=remainder_

(d)Therootalreadyfoundconsideredastensis60and602=36003times3600=10800=firsttrialdivisorThisiscontainedin29314twicePut2asthenextfigureoftheroot

(e)Thetwopartsoftherootalreadyfoundare60and260times2=120and3times120=360Thesquareofthelastfigurefoundis4Adding360+4tothetrialdivisorwegetthecompletedivisor=11164

(f)Multiplycompletedivisorbythesecondfigureoftherequiredroot11164times2=22328andsubtractfromtheremaindergetting6986Bringdownthenextgroupgetting6986376=remainder

(g)Therootalreadyfoundis62orconsideredastens620 =384400and3times384400=1153200=secondtrialdivisorThistrialdivisoriscontained6timesintheremainderPut6asthenextfigureoftheroot

(h)Thetwopartsoftherootalreadyfoundare620and6620times6=3720and3times3720=11160Thesquareofthelastnumberoftherootis =36Adding11160and36to1153200weget1164396=secondcompletedivisor

(i)Multiplycompletedivisorbythethirdfigureoftheroot

1164396times6=6986376andsubtractfromremainder6986376gettingzero

there4 =626Ans

NoteThereareasmanydecimalplacesinacuberootofadecimalasthereareperiodsof3figureseachinthedecimalIfthenumberisnotaperfectcubeannexzerosandcontinuetheprocesstoasmanyplacesasyoudesire

Thecuberootofafractionisfoundbytakingthecuberootofitsnumeratorandofitsdenominatororbyreducingthefractiontoadecimalandthenextractingtheroot

567Insummarywhataretheprinciplesapplyingtoexponents(a)Multiplicationammiddotan=am+n(b)Divisionamdividean=amndashn

(c)Raisingtoapower(am)n=amn(d)Extractingaroot(e)Negativeexponentandashm=1am

(f)Fractionalexponent (g)Zeroexponenta0=1y0

=1 =1

PROBLEMS

1Find(a)52

(b)82

(c)202

(d)14

(e)H2

(f)19

(g)103

(h)34

(i)252

(j)173

(k)833

(l)1253(m)(n)(o)(p)(q)xsdotxsdotxsdotx(r)(4x)squared(s)(2b)cubed(t)(125)3

2Findthesquareofthefollowingbytheformula(a+b)2=a2+2ab+b2(a)64(b)89(c)36(d)72(e)93(f)783(g)209

3Howmanysquarefeetarethereinalot40primetimes100prime

4Howmanyacresarethereinafield140rdsquare

5Howmanysquareyardsarethereinthefloorofaroom24feetlongand18feetwide

6Whatisthesquareof(a)3(b)6(c)14(d)134(e)07

7Whatisthevalueof(a)26times22(b)35times34(c)(d)35divide34(e)axdivideay(f)aItimesay(g)(43)2(h)(52)3(i)70(j)a0

(k)(l)4times6times80(m)53divide58(n)2-3(o)4-4

8Raise4tothe8thpowerbyfactoring-the-powermethod


10Reducethefollowingtoequivalentexpressionsfreefromzeroandnegativeexponents(a)3-3times2deg(b)a0a-1

(c)(d)3times4-1

(e)(05)-2

(ƒ)(a-m)-n

11Squarethefollowingbytheshortmethodasindicatedbytheformula(a‒b)(a+b)=a2‒b2(a)28(b)67(c)76(d)89

12Apply(a+b)2=a(a+2b)+b2tosquaring(a)(b)(c)

13Square975byaliquotpartmethodanda(a+2b)+b2

14Squarethefollowingbyfirstdividingby23or5(a)16(b)45(c)24(d)24

15Whatisthesquareofthemeanbetweenthetwonumbersin(a)14and18(b)40and50(c)25and26

16Squarethefollowingbythesimplemethod(a)(b)(c)

(d)65(e)225

17Whatisthesquareof(a)9999(b)99(c)999999

18Whatisthevalueof(a)107(b)109(c)10-6(d)10-4(e)105times102(f)104times103(g)105times10-2(h)1014times10-6(i)104divide106(j)103divide10-6

19Whatisthevalueof(a) (b) (c) (d)

(e)

(f) (g)(h) (i)

20Expressinradicalform(a)(b)(c)(d)(e)

(f)

21Removeafactoroftheradicandfromundertheradicalsign(a)(b)(c)

22Introducethecoefficientoftheradicalundertheradicalsign(a)(b)(c)

23Makethedenominatorrational

(a)

(b)

(c)

24Reducetoafractionwhosedenominatorhasnoradical(a)(b)

(c)

25Changetoahigherorderwithanindexthatisamultipleoftheoriginalindex(a)(b)(c)

26Reducetoaradicaloflowerorder

(a)(b)(c)

27Reducetosimplestform(a)(b)(c)

28Asquareroomcontains784sqftWhatisthelengthofoneside

29Ifthereare6084sqrdintheareaofasquareparkwhatisthelengthofoneside

30Ifthereare2916sqininasquaretabletopwhatisthelengthinfeetofoneside

31Findthesquarerootof39864tothreedecimalplaces


33Findthesquarerootof(a)(b)(e)(d)(e)(f)0178(g)(h)9(i)(j)(k)00065

34Extractthecuberootof242970624

35Whatisthevalueof

CHAPTERXIV

LOGARITHMS

568Whatismeantby(a)logarithm(abbreviatedldquologrdquo)(b)exponent(c)baseAlogarithmisanexponent

Aquantityraisedtoanexponentequalsanumber(powerofthequantity)

Thewordldquologarithmrdquomaybesubstitutedforldquoexponentrdquo

Then(quantity)logarithm=anumber

Nowthequantitytoberaisedtoapoweriscalledthebase

Thusbaselogarithm=anumber

Ex(a)(base)82(log)=64(number)

Hereexponent2isthelogofthenumber64Orthelogof64tothebase8is2

Ex(b)43=64


Weseethatthesamenumbermayhaveadifferentlogdependinguponthebaseused

Notecarefullythatwhenweraiseabaseoraquantitytoacertainpowerweapplyanexponenttothebaseandthenumberobtainedasaresultofthisprocessiscalledthepowerofthebase

569Whatarethetwoformsofexpressingthe

relationshipbetweenthebasethepowerandtheexponent(a)82=64=exponentialform

(b)log864=2=logarithmicform

logarithm=exponent

NoteInthelogarithmicformthequestionarisesldquoTowhatexponentmustthebase8beraisedtoproduce64rdquoAlwaysaskyourselfthisquestionwhenyouseethisformHoweveranyvaluemaybechosenasthebaseofasystemoflogarithms(orexponents)exceptthebase1

570Whattwosystemsoflogarithmsareingeneraluse(a)TheNapierianorthenaturalsystemHerethebaseisisin=epsilonwhich

denotestheirrationalnumber27182+(Anirrationalnumberisonewhichcannotbeexpressedasthequotientoftwowholenumbers)Itisusedprincipallyintheoreticalmathematicsengineeringandadvancedstatistics

(b)TheBriggsorthecommonsystemHerethebaseis10whichismostapplicabletoourdecimalnumbersystem

571Towhatexponent(logarithm)mustthebase10beraisedtoproduceanumberbetween1and10Wecanreadilygetthelogsofthefollowingnumbers

Inlogarithmicformthesearewrittenas

Weseethatthelog(exponent)ofanumberbetween1and10isadecimalfraction

Fromnowonweshallomitwritingthebase10whichwillbeunderstoodthuslog10=1willmeanlog1010=1

572Towhatexponent(log)mustthebase10beraisedtoproduceanumberbetween10and100


Weseethatthelog(exponent)ofanumberbetween10and100is1+afraction

NotethatthedigitsequenceofthenumberswhoselogsarerequiredisthesameasforQuestion571andthedecimalpartofthelogisthesameineachcaseTheonlydifferenceisinthepositionofthedecimalpointinthenumberwhichproducesacorrespondingwholenumbervalueofthelog



Foranumberbetween100and1000thelogis2+afractionThefractionalpartsofthelogsarethesameasbeforeforthesamesequenceofdigitsThewholepartofthelogisaffectedonlybythepositionofthedecimalpointinthenumber

574Howdoesthisconditionapplytohigherpowersof10foranynumberyoumaywanttoproduceEx(a)

Ex(b)

575Whyisthelogofanumberbetween1and1expressedasmdash1plusthesamepositivedecimalfractionasforQuestion571withthesamesequenceofdigitsinthenumber

ThepositivefractionalpartofthelogisthesameasinQuestion571forthesamesequenceofthedigitsofthenumberineachcase

576HowdoesthisapplytofindingthelogofstillsmallerdecimalfractionsEXAMPLES(a)04642=01times4642(log01=mdash2andlog4642=6667)

there4log04642=-2+6667(b)004642=001times4642(log001=-3andlog4642=6667)

there4log004642=-3+6667etc

577Whymaynumbersbetween1and10beconsideredasbasicnumbersforasystemoflogshaving10asabase306216438769and937482arecalledbasicnumbers

Logarithmsofallnumbershaving10forabasecanbeobtainedfromthelogsofthebasicnumbers

4642isabasicnumber

log4642=6667(Question571)

4642=4642times101there4log4642=166674642=4642times102there4log4642=266674642=4642times103there4log4642=3666746420=4642times104there4log46420=46667etc4642=4642times10ndash1there4log4642=-1+666704642=4642times10ndash2there4log04642=ndash2+6667004642=4642times10ndash3there4log004642=ndash3+6667

578WhatismeantbythecharacteristicofalogarithmThelogarithmofabasicnumberisadecimalfractionForothernumbersa

positiveornegativeintegermustbeaddedtothefractiontogetthelogarithmofthenumberThisintegralpartorintegeriscalledthecharacteristicofthelogarithm

EXAMPLEInlog4642=166671isthecharacteristic

579WhatismeantbythemantissaofalogarithmThedecimalpartofthelogarithmisthemantissa

EXAMPLEInlog004642=ndash3+6667ndash3isthecharacteristic6667isthemantissa

Themantissadependsonlyonthesequenceofthedigitsofthenumberandnotonthepositionofthedecimalpoint

580WhatistheruleforfindingthecharacteristicofthelogarithmofanumberCountthenumberofdigitsintheintegralpartofthenumberThe

characteristicisonelessthanthatnumberThisfollowsfromthefactthatabasicnumberhasoneintegraldigitanditslogarithmhasnocharacteristic

Ex(a)

Number Integral digi ts Characteris t ic

1to9 1 1mdash1=0

1to9 1 1mdash1=0

10to99 2 2mdash1=1

100to999 3 3mdash1=2etc

Ex(b)Thecharacteristicofthelogof8653794is4whichisonelessthanthenumberofintegraldigits

581WhatistheruleforfindingthecharacteristicofapurelydecimalnumberCountthenumberofplacesthedecimalpointmustbemovedtomakethe

numberbasicThenegativecharacteristicisthatnumber

Ex(a)Whatisthenegativecharacteristicofthelogof000865Movedecimalpoint4placestoget865whichisabasicnumberThenndash4isthecharacteristic

log000865=ndash4+9370

Ex(b)Whatisthenegativecharacteristicofthelogof00427Movedecimalpoint3placestoget427whichisabasicnumberThenmdash3isthecharacteristic

log00427=ndash3+6304

582WhyisanegativecharacteristickeptdistinctfromthemantissaofalogarithmIncomputationitisadvantageoustohavethemantissapositiveineverycase

andtokeepitequaltothemantissaofthelogofthebasicnumberThelogofapurelydecimalnumberthenconsistsofanegativeintegerplusapositivedecimal

583Howarenegativecharacteristicsgenerallyexpressed

(a)WithaminussignoverthecharacteristicThisindicatesthatitaloneisnegative

Ex(a)log000865=49370(b)log00427=36304

(b)Byaddingandsubtracting10

Ex(a)log000865=4+10+9730ndash10=69370ndash10(b)log00427=3+10+6304ndash10=76304ndash10

584MayanegativecharacteristicbeexpressedinotherwaysItmaysometimesbefoundusefultoaddandsubtractanumberotherthan10

EXAMPLElog00427=36304maybewrittenas

3+8+6304mdash8=56304mdash8

or

+30+6304ndash30=276304ndash30

Anycombinationmaybeusedaslongasthenetresultistheoriginal3

Howevertheform9middotmiddotmiddotmiddotmdash10ismostconvenientforoperationsofadditionandsubtractionoflogsandtheseoperationsarequitecommon

585WhatisatableofcommonlogarithmsAtableoflogsisatableofmantissasItisatableoftheexponentsof10

correspondingtobasicnumbersItanswersthequestionldquoWhatisthepowerof10requiredtogiveacertainbasicnumberrdquoFindingtheexponentisfindingthelogThedifferencesbetweensuccessivelogsarenotthesamebecausetheyformanexponentialscaleofpowersof10SeeTable3AppendixB

Thesamesequenceofnumbersgivesthesamelogindependentofthepositionofthedecimalpoint

586HowdowelookupaloginatableLookattheleftofthetabletogetthesequenceofdigitsinthenumberasfar

asitwillgoandthengotothetopforthenextdigitinthesequenceWhenthenumberhasmorethanthreesignificantfiguresaddtothelogreadingtheproportionalpartofthenumberbetweenthetwoadjacentlogsinthetableForlessthanthreesignificantfiguresaddzeros

Ex(a)Findthelogof42Lookup420figuresEnter42atleftand0columnontopandget62325forthemantissa2Thenadd1asthecharacteristic

there4log42=162325Ex(b)Forthesequenceoffigures420themantissaisthesamebutthe

characteristicisonelessthanthenumberofdigits

there4log420=262325Ex(c)

log42=62325log42=162325or962325ndash10log042=262325or862325ndash10log0042=362325or762325ndash10

587WhatismeantbyaproportionalpartofalogTheproportionalpartofthedifferencebetweentwoadjacentlogsrepresented

bytherequiredlogisknownastheproportionalpartofthelog

Ex(a)Findthelogof6816

Enter68atleftandmoverightuntilyoureachcolumn1atthetopoftableRead83315

Thenextadjacentlogisof682Read83378

Differenceis83378ndash83315=63

Now6ofthisdifferenceis6times63=378or38tonearestdigit

Then83315+0003883353

Characteristicof6816is2

there4log6816=283353AnsEx(b)Findlogof76452

588WhatismeantbyanantilogarithmAnantilogarithmisthenumbercorrespondingtoagivenlogarithmWhenthe

exponentisgivenandthenumberisrequiredtheprocessiscalledfindingtheantilogarithmItisthereverseoffindingthelogarithm

EXAMPLEIntheabove288339isthelogand76452istheantilog

589Howdoweobtainanantilogornumberfromatableoflogs(a)Findthenumbercorrespondingtothetwomantissasbetweenwhichthe

desiredmantissaislocated

(b)GettheirdifferenceFindthedifferencebetweenthelowermantissaandthedesiredone

(c)Findtheproportionalpartandaddthistothenumber

EXAMPLEFindtheantilogof861768ndash10=261768

590UponwhatlawsdocomputationswithlogsdependUponthelawsofexponentsTheessentiallawsofexponentsare

(a)Tomultiplyaddtheexponentsalgebraically

105times times = =104=10000

(b)Todividesubtracttheexponentsalgebraically

10-5divide10ndash8=10ndash5ndash(ndash8)=103=1000

(c)Toraisetoapowermultiplytheexponents

(10ndash3)ndash2=10(ndash3xndash2)=106=1000000

(d)Toextractarootdividetheexponents

=1093=103=1000

591WhatistheprocedureformultiplyingtwoormorequantitiesbylogsAnumbercanbeexpressedinexponentialformtoanybaseortobase10

EXAMPLE160=1022041236=102372928=1014472

NowbythelawsofexponentstomultiplyweaddtheexponentsButexponentsarelogsSotomultiplyaddthelogsThus

160times236times28=1022041times1023729times1014172=102middot2041+23729+14472=1060242there4log(160times236times28)=60242(characteristic=6mantissa=0242)andantilog=1057000=product

Theproceduremaybestatedinlogarithmicformas

592WhatistheprocedureforgettingthequotientoftwonumbersbylogsBythelawsofexponentstodividesubtracttheexponentsThusthelogofa

quotientisthelogofthenumeratorminusthelogofthedenominator

EXAMPLE135834=10213301896=1095230

Thus

Theantilogis151605

Thisproceduremaybeexpressedinlogarithmicformas

593WhatistheprocedureforraisinganumbertoapowerbylogsBythelawofexponentstoraisetoapowermultiplytheexponents

EXAMPLE374=1015729

Thismeansthatexponent15729isthelogof374Now(374)3=(1015729)3

=1015728x3

there4log(374)3=3times15729Thismeansmultiplythelogofthenumberbythepower

Inlogarithmicformthisisstatedas

log374=15729log(374)3=3times15729=47187(characteristic=4

mantissa=7187)there4antilog=52320Ans

594WhatistheprocedureforgettingtherootofanumberbylogsBythelawofexponentstoextractarootdividetheexponents

EXAMPLE

Thismeansdividethelogofthenumberbytheroot

Inlogarithmicformthisisexpressedas

595Howcanweexpressthelogof75intermsofthelogof5andthelogof3

75=52times3

Thenlog75=log(52times3)=log52+log3=2log5+log3

596Howcanweexpress asanalgebraicsumoflogs

597Howcanwereducelog7+3log5tothelogofasinglenumber

598Whatisthelogof1toanybaseWeknowthat

599WhatisthelogofthebaseitselfinanysystemWeknowthata=a1

there4loga=11=theaexponent=logofatobaseaAns

600Whatisthelogof0inanysystemwhosebaseisgreaterthan1Weknowthat

ndashinfin=theexponent=logof0toanybasegreaterthan1Ans

Thuslog0isnegative=numericallygreaterthananyassignednumberhowevergreat

601HowcanwefindthelogofanumbertoanewbasewhenthelogsofnumberstoaparticularbasearegivenDividethelogofthenumbertotheparticularbasebythelogofthenewbase

referredtotheparticularbase

EXAMPLEWehaveatableoflogs(exponents)tobase10andwewanttogetthelogof4725toanewbaseisin=2718

602Howarenaturalandcommonlogsrelatedasseenfromtheabove(a)Togetthenaturallogofanumbermultiplyitscommonlogby23026

EXAMPLE

log100=23026timeslog10100=23026times2=46052

(b)Togetthecommonlogofanumbermultiplythenaturallogby4343

EXAMPLE

log10100=4343logε100=4343times46052=2

603WhatismeantbythecologarithmofanumberThecologarithmofanumberisthelogarithmofthereciprocalofthenumber

EXAMPLEIfaisagivennumberthen

cologa=log

Butlog =log1ndashloga

there4cologa=0ndashlogaThismaybewrittenas

cologa=(10ndash10)ndashloga

604Whatistheruleforobtainingthecologofanumbertobase10Subtractthelogarithmofthenumberfrom(10ndash10)

EXAMPLEIfthelogofanumberis715625ndash10thenthecologis

605WhenarecologsusedtoadvantageInfindingthelogofafractionorquotient

InsteadofsubtractingthelogofthedenominatoraddthecologofthedenominatortothelogofthenumeratorInaseriesofmultiplicationanddivisionusecologsforthedenominatorsorthetermsbywhichyouhavetodivideThisenablesyoutocombinethelogvalueinoneoperationofaddition

EXAMPLEWhatisthevalueofof

606Whatistheresultof005864times2726times8465

607Whatistheresultof(262)4

608Whatistheresultof

609Whatisthevalueof(1834)ndash3log(1834)ndash3=-3timeslog1834=ndash3times12634=ndash(37902)

HeretheentirenumberincludingthedecimalpartisnegativeToobtainapositivemantissaforuseinthetableoflogschangetheformofthislogbyaddingandsubtracting10

Thisproblemmaybesolvedbyusingthecologmethodbecause

Then

and

610Whatisthevalueof(2718)-14


ThelogofanegativenumberisnotdefinedinrealnumbersHoweverthisproblemmaybesolvedbyconsideringthebaseasapositivenumberandprefixingaminussigntotheresult

Prefixminussigntoresult

there4ndash09683Ans

NoteSinceevenpowerscanneverbenegative(seeQuestion628)itisimpossibletoexpressanevenrootofanegativequantitybytheldquorealrdquosystemofnumbersInhighermathematicssuchevenrootsarecalledldquoimaginaryrdquonumbers



Usingthecologprocedure

Herethecologprocedureissimpler


615Whatistheresultof(3846)-16

616Whatistheresultof(42) 71 x(76)- 62 x(432ndash69)Perform(432ndash69)firstgetting363

Changethenegativenumberndash5461toapositivemantissabyaddingandsubtracting10

Nowaddallthefactors

617Whatistheresultof log =log875=99420ndash10

Nowsincelog875istobeusedasanumberandnotasalogevaluateitbygettingthedifferencebetween99420andndash10

Disregardthenegativesignof058duringcalculationandprefixittotheresult


Ingetting ofcologof006439firstmultiplyby5andthendivideby9toeliminateanyerrorthatwouldresultfrominexactdivisionanerrorthatwouldbemultiplied5times

619Whatistheresultof(58)y =567Takethelogsofbothsides

Carryouttheindicatedsubtractioninthedenominator

Then

620HowaccurateareresultsofnumericalcomputationsbylogsResultsobtainedbylogarithmiccomputationsareapproximate

AlogofanumbercannotingeneralbefoundexactlybutonlyapproximatelytofourfiveoranydesirednumberofdecimalplacesThereforetheresultsofnumericalcomputationsbymeansoflogsarenotinanycasecorrectbeyondthefourfiveorothernumberofdecimalplacesinthelogsusedtomakethecomputations

PROBLEMS1Givethelogandwritethelogformof(a)53=125(b)106=1000000

(c) =(d)9radic2=2235(e)34=81(f)2-2=

2Writethelogformof(a)43=64(b) =(c)10d=600(d)pt=n(e)(01)4=00000001(f)2ndash4=

3Expressinexponentialform(a)log4256=4(b)logxa=b(c)logb1=0(d)log10000001=-6(e)log1010000=4(f)log61296=4

4Ifthelogstothebase4are01234ndash1-2 whatarethenumbers

5Ifthebaseis5whatarethelogsofthefollowingnumbers1525125625


7Findthevalueoftimesineachofthefollowing(a)log10x=3(b)log16times=(c)x=log 243(d)logx64=(e)log5times=ndash5(f)logx10000=4(g)2log25x=-3

(h)x=log1001000(i)logx49=2

8Arethefollowingtruestatements(a)log1010000ndashlog101000+log10100+log1010+log101=4(b)log1000001+log100001ndashlog10001ndashlog1001=-4(c)3log33+4log3 +log31=-11

(d)3log5radic008+3log10 =ndash3

9Whatisthecharacteristicofthelogsofeachofthefollowingnumbers(a)9854(b)9854(c)9854(d)9854times106(e)985(f)000098(g)98541(h)985000000(i)0098541(j)985413(k)462915()31416(m)2718times10ndash14

(n)00054times10ndash4(o)3755000(p)4343

10Ifthemantissaofthelogofanumberis4064whereshouldthedecimalpointbeforeachofthefollowingcharacteristics(a)2(b)ndash11(c)0(d)ndash3(e)5(f)ndash4(g)1(h)3(i)6

(j)(3ndash1)(k)(11ndash10)()(10ndash10)(m)(2ndash3)(n)(8ndash10)(o)(27ndash30)(p)(34ndash38)

11Findthelogofeachofthefollowingnumbers(a)59433(b)9714(c)0642(d)008793(e)3793(f)1379(g)0306(h)00006794(i)5674times10-5

(j)00638times104

12Findtheantilogsofthefollowinglogs(a)9954(b)34789(c)19572(d)30358(e)43762(f)78617ndash10(g)186742ndash20(h)24169ndash5(i)31606(j)12168ndash07(k)5464ndash(l)ndash3649

13Express196intermsofthelogof7andthelogof4

14Express asanalgebraicsumoflogs

15Expresslog9+3log6asalogofasinglenumber

16Expresseachofthefollowingasthesumordifferenceoflogs(a)83times92times28(b)

(c)

(d)

17Expressinexpandedform

(a)log

(b)log

18Findthevalueofeachofthefollowing(a)log(01)3+log(b)log +log(c)log +log(d)log +log(001)2

(e)log(001)5mdashlog(100)2+log(f)log5 +log

19Contracteachofthefollowingexpressions(a)4log6+ log5mdash7log8(b) log25mdash log10ndash log5+log9(c) [6log2+6log5ndash log6‒ log7](d)3log2+log3ndash log4

20Evaluateeachofthefollowinggiventhatlog2=3010andlog3=4771(a)log8(b)log6(c)log12(d)log27(e)log15(f)log432

21Findtheresultofeachofthefollowing(a)log29

(b)log6112(c)log511(d)log89(e)log6122(f)log5(g)log41(h)log410(i)log43(j)log76(k)logs01(l)logs100

22Findthenaturallogofeachofthefollowingnumbers(a)8721(b)782(c)6928(d)0432(e)1872(f)000496

23Findthecommonlogifthenaturallogsareasgivenbyeachofthefollowing(a)782(b)8472(c)0083(d)9248(e)00062(f)378

24Evaluatethefollowingusinglogs(a)006943times3422times8243(b)(358)4

(c)(d)(2112)minus3(e)(2718)ndash12(f)

(g)(06493)minus(h)(5937)minus13

(i)(36)69times(53)minus58times(238ndash43)

(j)

(k)(l)(42)x=649

(m)(n)

CHAPTERXV

POSITIVEANDNEGATIVENUMBERS

621WhatismeantbyldquosignedrdquonumbersNumbersprecededbyaplussignoraminussignarecalledsignednumbers

Suchnumbersshowtheamountanddirectionofchangeandmaythusdenotequalityaswellasquantity

Ex(a)If+32degrepresents32degabovezerothen-32degrepresents32degbelowzero

Ex(b)If+8milesrepresents8milestotheeastthenminus8represents8milestothewest

Ex(c)If+$5representsacreditof$5thenmdash$5representsadebitof$5

Ex(d)If+100representsadistanceabovesea-levelthenminus100representsadistancebelowsea-level

622WhatismeantbyldquopositiverdquoandldquonegativerdquonumbersNumbersprecededbyaplus[+]signorbynosignatallarecalledpositive

numbersas32+5+711+

Numbersprecededbyaminus[-]signarecallednegativenumbersasminus7minus14minus minus28 minus23

623WhatismeantbytheabsolutevalueofanumberTheabsolutevalueisthevalueofthenumberwithoutthesign

EXAMPLES

Theabsolutevalueof+32is32Theabsolutevalueofmdash8is8

624Howcantherelationsbetweentheplusnumberstheminusnumbersandzerobeshownbythenumberscale

TheordinarynumbersofarithmeticarepositivenumbersandaregreaterthanzeroTheseareshowntotherightofzeroNegativenumbersaretotheleftofzeroCorrespondingto+4wehavemdash4whichisasmuchbelowzeroas+4isgreaterthanzero

EXAMPLEmdash6islessthanmdash5ormdash2or0or+1or+6Numbersincreaseasyougototherightanddecreaseasyougototheleft

625WhatarethetwomeaningsofplusandminussignsTheplussign[+]maydirectustoaddoritmayindicatethequalityofthe

numberasapositivenumber

Theminussign[-]maydirectustosubtractoritmayindicateanegativenumberoppositeinqualityorsensetoapositivenumber

Todistinguishthesignofoperationfromthesignofquality(positiveornegative)thequalitysignisenclosedinparentheses

EXAMPLES(a)18+(+3)(b)18ndash(+3)(c)18+(mdash3)(d)18mdash(mdash3)

Forthesakeofbrevity(a)and(b)maybewrittenas18+3and18mdash3sinceaplussignisnotnecessaryinfrontofapositivenumber

626WhatistheprocedureforadditionofpositiveandnegativenumbersIfthenumbershavethesamesignsaddthenumbersandprefixthecommon

(orsame)sign

Ifthenumbershaveunlikesignsfindthedifferenceandusethesignofthelargernumber

EXAMPLES(a)(+7)+(+5)=7+5=12(like[+]signs)+result(b)(mdash7)+(mdash5)=7+5=mdash12(like[mdash]signs)ndashresult

(c)(+7)+(mdash5)=7mdash5=+2(unlikesigns)

(d)(mdash7)+(+5)=7ndash5=mdash2(unlikesigns)

627WhatistheprocedureforsubtractionofpositiveandnegativenumbersChangethesignofthenumberbeingsubtractedandaddasinaddition

(Question626)

EXAMPLES(a)(minus5)minus(minus7)Changethesignof(minus7)andaddto(minus5)ormdash5+(+7)=2(b)(minus5)minus(+7)Changethesignof(+7)andaddto(minus5)orminus5+(mdash7)=mdash12

(c)(+5)minus(minus7)Changethesignof(minus7)andaddto(+5)or5+(+7)=12(d)(+5)mdash(+7)Changethesignof(+7)andaddto(+5)or

5+(mdash7)=mdash2

628WhatistheprocedureformultiplicationofpositiveandnegativenumbersTheproductispositivewhenthetwonumbershavethesamesignwhether

bothare(+)orbothare(mdash)

Theproductisnegativewhenthetwonumbershaveoppositesigns

EXAMPLES(a)(+12)times(+8)=+96=96samesign(b)(minus12)times(mdash8)=+96=96samesign(c)(mdash12)times(+8)=mdash96oppositesigns(d)(+12)times(mdash8)=mdash96oppositesigns

629Whatistheprocedurefordivisionofpositiveand

negativenumbersThequotientispositivewhenthedividendandthedivisorhavethesamesign

Thequotientisnegativewhenthedividendandthedivisorhaveoppositesigns

EXAMPLES(a)(+96)divide(+8)=+12=12samesign(b)(mdash96)divide(mdash8)=+12=12samesign(c)(+96)divide(mdash8)=mdash12oppositesigns(d)(mdash96)divide(+8)=mdash12oppositesigns

PROBLEMS1Howwouldyourepresentthefollowing(a)20mileseastand25mileswest(b)200feetabovesea-leveland200feetbelow(c)15degabovezeroand15degbelowzero(d)Againof$25andalossof$25

2Answerthefollowing(a)Ismdash12greaterorlessthanmdash8(b)Whichislarger+3ormdash6(c)Whichislargermdash50or+1

3Whatistheabsolutevalueof(a)+12(b)mdash6(c)+ (d)mdash16(e)350

4Whatistheresultof(a)(+3)+(+14)(b)(mdash16)ndash(mdash72)(c)(mdash20304)ndash(mdash123)(d)(mdash18604)+16(e)+14(f)mdash13minus8minus5

(g)mdash12(h)mdash6+7+11

5Whatistheresultof(a)(mdash122)times(mdash12)(b)(mdash7)times(mdash9)times(mdash6)(c) (d)(mdash6)times(mdash1 )times(1 (e)(mdash14)times(mdash6)(f)(mdash14)times(+6)

6Whatistheresultof(a)108divide12(b)(mdash108)divide(mdash12)(c)(mdash368)divide(mdash46)(d)1330divide38(e) (f)

CHAPTERXVI

PROGRESSIONSmdashSERIES

630WhatisaseriesAsuccessionoftermssorelatedthateachmaybederivedfromoneormore

oftheprecedingtermsinaccordancewithsomefixedruleororder

631WhatisanarithmeticprogressionAseriesofnumberseachofwhichisincreasedordecreasedbythesame

numberinadefiniteorder

Ex(a)24681012etc

Eachnumberisincreasedby2inanascendingorder

Ex(b)24201612840mdash4mdash8etc

Eachnumberisdecreasedby4inadescendingorder

632WhatisageometricprogressionOneinwhicheachtermisdividedormultipliedbythesamenumbertogetthe

nexttermThisconstantmultiplierordivideriscalledtheratio

Ex(a)28321285122048etc

Eachtermismultipliedby4togetthenexttermThisiscalledanascendingseriesorprogression

Ex(b)20485121283282

Eachtermisdividedby4togetthenextterminadescendingseries

633WhatisaharmonicprogressionAseriesoftermswhosereciprocalsformanarithmeticprogression

EXAMPLE1 isaharmonicprogressionbecausethereciprocalsoftheterms13579etcformanarithmeticprogression

634WhatisknownasamiscellaneousseriesAnypatternorcombinationofpatternsmayconstituteamiscellaneousseries

Ex(a)358101315182023

Togettheterms2then3then2then3areadded

Ex(b)2mdash24-46mdash68mdash8etc

ThenumbersarepairedoffinintervalsThenextpairwouldbe10-10

(a)and(b)areexamplesofmiscellaneousarithmeticseries

Ex(c)2223242526isavariedgeometricseries

Ex(d)2222428216isavariedgeometricseries

In(d)eachtermisthesquareoftheprecedingterm

635Whatistheprocedureforsolvinganascendingarithmeticprogression(a)Subtractthefirsttermfromthesecondtermtogetthecommondifference

(b)Addthedifferencetothelasttermtofindthetermthatfollows

EXAMPLE13579

(3ndash1)=2=differencethere42+9=11=nextterm

636Whatistheprocedureforsolvingadescendingarithmeticprogression(a)Subtractthesecondtermfromthefirsttermtogetthecommondifference

(b)Subtractthisdifferencefromthelasttermtogetthetermthatfollows

EXAMPLE25211713

(25mdash21)=4=differencethere4(13mdash4)=9=nextterm

637Howcanweobtainageneralformulaforsolvinganarithmeticprogression

Leta=thefirsttermd=thecommondifferencen=thenumberofterms(given)l=thelastterm(tobefound)

Theprogressioncanthenbestatedas

Notethatthecoefficientormultiplierofdinanytermis1lessthanthenumberofthetermThismeansthatthemultiplierofdforthenthorlasttermis(nmdash1)

there4l=lastterm=a+(nmdash1)d

Ex(a)Tofindthelastterm(thetwenty-seventhterm)oftheprogression1411852mdash1mdash4to27terms

Herea=14d=11mdash14=mdash3andn=27Thenl=a+(nmdash1)d=14+(27mdash1)times(mdash3)=14+[26times(mdash3)]=14mdash78=mdash64=twenty-seventhtermAns

Ex(b)Findtheseventeenthtermof58111417

Hered=8ndash5=3a=5andn=17Then

l=a+(nminus1)d=5+(17minus1)times3=5+16times3=5+48=53Ans

638Howcanwefindanexpressionforthesumofthetermsofanarithmeticprogression

Leta=thefirstterml=thelasttermn=thenumberoftermsS=thesumofthetermsd=thedifferencebetweenterms(common)

Then

S=a+(a+d)+(a+2d)++(ndashd)+l

Nowwritingthetermsinthereverseorderweget

S=l+(lminusd)+(lndash2d)++(a+d)+a

Addtheseequationstermbytermandget

there4S=(a+l)whichistheexpressionrequiredAddthefirsttermtothelasttermandmultiplythisbythenumberoftermsdividedby2

Alsowehavefoundpreviouslythatl=a+(nmdash1)dThus

whichisanotherformfortheexpressionrequired

639Whatisthesumofthefirsttwenty-seventermsof

1411852mdash1mdash4 Herea=14d=14mdash11=mdash3andn=27Then

AsacheckweknowfromExample(a)ofQuestion637thatl=mdash64Then

640WhenanythreeofthefiveelementsofanarithmeticprogressionaregivenhowaretheothertwofoundGivenanythreeoftheelementsadnlandStofindtheremainingtwo

elementssubstitutein

Ex(a)

Givena= n=10andS= Finddandl

Then

Now

Ex(b)

Givend=mdash4l=mdash48andS=mdash288Findaandn

Now

(1)

Factoringweget(nmdash9)(nmdash16)=0andn=9orn=16

Substitutingin(1)

a=4times9mdash52=mdash16forn=9a=4times16mdash52=12forn=16

Therearetwoprogressionsasananswer

Ifa=mdash16andn=9theprogressionis

ndash16mdash20ndash24ndash28ndash32ndash36ndash40ndash44ndash48

Ifa=12andn=16theprogressionis

12840mdash4mdash8mdash12mdash16mdash20

ndash24ndash28ndash32ndash36ndash40ndash44ndash48

Ineachcasethesumisndash288

641HowcanweinsertanynumberofarithmeticmeansbetweentwogiventermsUsel=a+(nmdash1)dtofindthecommondifferencedandthenformthe

series

EXAMPLEInsertfivearithmeticmeansbetween4andmdash6Thismeansthatwearetofindanarithmeticprogressionofseventermswiththefirsttermof4andthelasttermofmdash6Then

mdash6=4+(7mdash1)d=4+6d6d=ndash10ord= =

Thustheseriesis

4 minus1 minus6

642HowcanweshowthatthearithmeticmeanbetweentwoquantitiesisequaltoonehalftheirsumIfx=thearithmeticmeanbetweentermsaandbthenbythenatureofthe

progression

xmdasha=bmdashx

or

2x=a+b

and

x= =halftheirsum

EXAMPLEWhatisthearithmeticmeanbetween and

643Howcanwefindanexpressionforthelasttermlofageometricprogressionwhengiventhefirstterma theratior andthenumberoftermsnTheprogressionisaarar2ar3

NotethattheexponentofrinanytermisllessthanthenumberofthetermThismeansthatinthenthterm(last)theexponentofris(nmdash1)

there4l=arnminus1

EXAMPLEFindthelasttermfortheprogression

41 to7terms

Herea=4r= andn=7Then

Togettheratiodividethesecondtermbythefirstoranytermbythenextprecedingterm

644HowcanwefindanexpressionforthesumSofageometricprogressionwhengiventhefirstterma thelastterml andtheratior

(1)

Nowmultiplyeachtermbyrgetting

(2)

Subtract(1)from(2)getting

rSmdashS=arnmdasha

Alltheothertermscancelout

S(rmdash1)=arnmdasha

Then

But

l=arnndash1orrl=arn

there4S= =expressiondesired

EXAMPLEFindthesumoftheseriesofQuestion643

Therel= r= anda=4

645HowcanwefindtwoofthefiveelementsofageometricprogressionwhenanythreearegivenSubstitutein

EXAMPLEGivena=mdash2n=5andl=mdash32

FindrandS

646(a)Whatdowecallthelimittowhichthesumofthetermsofadecreasinggeometricprogressionapproacheswhenthenumberoftermsisindefinitelyincreased(b)Howcanwefindanexpressionforthislimit(a)Thislimitiscalledthesumoftheseriestoinfinity

(b)Wehavealreadyfoundthat

Thiscanbewrittenas

NowwhenwecontinueadecreasinggeometricprogressionthelasttermmaybemadenumericallylessthananyassignednumberhoweversmallThuswhenthenumberoftermsisindefinitelyincreasedlandthereforerlapproachesthelimit0

Thenthefraction approachesthelimit

EXAMPLEFindthesumoftheseries3minus toinfinity

Herea=3and

647HowcanwefindthevalueofarepeatingdecimalbytheuseofthesumofaseriestoinfinityEXAMPLEFindthevalueof

Now

Thetermsafterthefirstconstituteadecreasinggeometricprogressioninwhich

Then

Thevalueofthegivendecimalis

648WhatistheprocedureforinsertinganynumberofgeometricmeansbetweentwogiventermsUsel=arnminus1

EXAMPLEInsertfourgeometricmeansbetween3and729

Thismeansthatwemustfindageometricprogressionofsixtermswith3asafirsttermand729asalastterm

Herea=3andl=729=arnminus1

729=3r(6minus1)=3r5r5=243r=3there4392781243729istheprogression

649HowcanweshowthatthegeometricmeanbetweentwoquantitiesisequaltothesquarerootoftheirproductPutxbetweenaandbtermsasaxbThenbythenatureofthe

progression

Hence

EXAMPLEFindthegeometricmeanbetween1 and2

650WhatistheprocedureforsolvingaharmonicprogressionTakethereciprocalsofthetermsandapplytheproceduresandformulaeof

arithmeticprogression

Thereishowevernogeneralmethodforfindingthesumofthetermsofaharmonicprogression

EXAMPLEFindthelasttermoftheprogression3 totwelveterms

Takethereciprocalstogetanarithmeticprogression

Herea= d=1andn=12Now

Takereciprocalof toget =lasttermofthegivenharmonicprogression

651Howcanweinsertsixharmonicmeansbetween2andThismeanswehavetoinsertsixarithmeticmeansbetween and

Here andn=8Then

Thenthearithmeticprogressionis

Therequiredharmonicprogressionis

652HowcanwefindanexpressionfortheharmonicmeanbetweentwotermsLetx=harmonicmeanbetweentermsaandbThen

=arithmeticmeanbetween and

and

EXAMPLEWhatistheharmonicmeanbetween3and6

Theharmonicseriesisthen

346

Thearithmeticseriesis

653HowisthesumofanarithmeticseriesappliedincertaininstallmentpurchaseproblemsEXAMPLEAhutchcabinetisadvertisedfor$1000cashorontime

paymentsof$20perweekplus of1oneachweeklyunpaidbalancefor50weeksWhatwouldbethetotalamountpaidontheweeklyinstallmentbasis

Thefirstunpaidbalanceis$1000and0015times$1000=$150

(15times01=0015)

Thesecondunpaidbalanceis$980and0015times$980=$147

Thethirdunpaidbalanceis$960and0015times$960=$144

Thustheseriesofthecarryingchargesbecomes$150$147$144

Herea=$150=firsttermd=$150minus$147=minus$03=commondifferenceandn=50=numberofterms

Then

$1000+$3825=$103825=totalamountpaid

PROBLEMS

1Whattermcomesnextin(a)6912151821(b)1018161(c)45891213

(d)11119977(e)8127931 (f)2818325072(g)12481632(h)403430282218

2Findthelasttermandthesumofthetermsof(a) to12terms(b)3915to8terms(c) to14terms(d)minus7minus12minus17to10terms(e) to14terms

3Givend=4=71andn=15findaandS

4Givena=minus7n=12andl=56finddandS

5Insertsixarithmeticmeansbetween3and8

6Insertfivearithmeticmeansbetweenminus3and1

7Findthearithmeticmeanbetween and

8Findthesumofalltheintegersbeginningwith1andendingwith100

9Findthesumofalltheevenintegersbeginningwith2andendingwith1000

10Findthelasttermandthesumofthetermsoftheprogression31 toseventerms

11Findthelasttermandthesumofthetermsoftheprogressionndash26ndash18totenterms

12Givena=ndash3n=4andl=ndash45findrandS

13Findthesumoftheseries toinfinity

14Findthesumtoinfinityof16ndash41

15Findthevalueoftherepeatingdecimal85151

16Findthevalueof296296

17Insertfivegeometricmeansbetween2and

18Insertfivegeometricmeansbetween2and128

19Findthegeometricmeanbetween9and25

20Findthelasttermoftheprogression totwentyterms

21Insertfiveharmonicmeansbetween2andndash3

22Whatistheharmonicmeanbetween4and8

23ATVsetissoldfor$675cashorfor$150cashand$5250amonthplus1ofeachmonthlyunpaidbalancefor10monthsWhatwoulditcosttobuyitonthetimepaymentbasis

CHAPTERXVII

GRAPHSmdashCHARTS

654WhataregraphsWhenyouhavestatisticalfactsinsciencesociologybusinesseconomicsor

anyotherrelationshipsyoucanpresentthemgraphicallytoadvantageinavarietyofformsThepictorialrelationshipsthatarethusshownintrueproportionsarecalledgraphsTheymayrepresenttherelationbetweentwounitsofmeasureasquantitywithtimeorcostwithquantitypartswithreferencetothewholeandwithreferencetoeachotheretc

655Whataretheadvantagesofgraphs(a)Ataglancetheymayshowinformationthatwouldusuallyrequiremuch

verbaldescription

(b)Theymaystimulatethemindinamoredirectdescriptiveanddramaticmannerthanstatisticsexpressedinnumbers

(c)Theymayenableustounderstandthefactsbetterandhelpustolearnnewfactsmoreeasily

(d)Theymaysaveustimeandworkinmakingcomputationsandenableustodrawconclusionsinacomparativeway

ForexampleinaeronauticstheymayshowexperimentalortestdataandcalibrationofinstrumentsInbusinesstheymayshowchangesofcostwithtimeInsociologytheymayshowgrowthofpopulationwithtimeuseofwaterresourceswithpopulationetc

656Whatarethedisadvantagesofgraphs(a)Theyarenecessarilylessaccuratethanthefiguresonwhichtheyare

basedHoweverinmanycasesthisisofnogreatimportance

(b)Theycansometimesmisleadusintowronginterpretationswhenweare

notcarefulWemustthusexaminecloselythereliabilityofthesourceandthemethodofpresentation

657Whatquestionsshouldweaskaboutgraphs(a)Whatideaisthegraphtryingtoconvey

(b)Whatquantitiesarebeingcomparedmdashtimemoneypeoplespeedetc

(c)Whatmeasurementsareusedmdashfeetdollarspercentyearsweights

(d)ExactlyhowmuchinformationdoesthegraphsupplyWheredoesourinterpretationbegin

(e)IstheinformationreliableHowwerethedataobtainedIsthegraphplantedorhonestlypresentedbyareliableorganization

658Whattypesofgraphsarecommonlyused(a)Bargraphs(horizontalandvertical)(b)Blockgraphs(c)Rectanglegraphs(divided-barcharts)(d)Circleorpiegraphs(e)Broken-linegraphs(f)Curvedgraphs(smooth-lineorcurve)(g)Frequencydistributiongraphs(staircasediagrams)(h)Statisticalmaps(i)Pictographs

659WhatarehorizontalbargraphsandwhenaretheyusedTheyaregraphsthatshowacomparisonofdata

Theyareusedwhenthedataarecomparablebutseparate(discrete)aswhenyoucompareheights(sameaspect)ofdifferentpeople(separatedata)

Theymaybeusedtocompareamountsofdifferentkindsofthingsorofthesamethingattwoormoredifferenttimesorplaces

TheymayshowtheproductionorconsumptionofanitemforseveralperiodsortheamountsofseveralitemsduringasingleperiodTheyaresimpleandconvenient

660Howisabargraphconstructed

ItisconstructedonordinarygraphpaperThegraphhasatitledescriptionofeachbarahorizontalscaleandwhennecessaryaverticalscaleBarsaremadeofthesamewidthandareplacedequallyfarapart

Ex(a)DrawahorizontalbargraphtoshowthecomparativesalesofabusinessconcernforthemonthsofJanuaryandFebruarywhenthesalesforJanuarywere$208600andforFebruary$276500

Ex(b)Showwithabargraphtherangeofincomesoftheemployeesofacertaincompanywhenthestatisticsareasfollows

Incomerange Numberof employees

$4000-$4999 12400

$5000-$5999 10200

$6000-$6999 8100

$7000-$7999 3040

$8000-$8999 2200

$9000-$9999 1160

$10000andover 208

Ex(c)ShowwithabargraphthestoppingorbrakingdistanceofacarinrelationtospeedofvehicletravelingonaharddrysurfaceDistanceismeasuredfromtheinstantthebrakesareapplied

661WhatareverticalbargraphsandwhenaretheyusedWhenbarsaredrawnfrombottomtotopthedrawingisaverticalbargraph

Thespacingbetweenconsecutivebarsshouldbeuniformandshouldbearrangedinorderofsizeoraccordingtosequenceoftime

VerticalbargraphsarecommonlyusedtorepresentquantitiesoramountsatvarioustimesandarethenknownashistoricalbargraphsThehorizontalscaleisalwaysusedtorepresentthetimeandtheverticalscaletorepresentquantitiesoramountsatvarioustimesTheheightsofanytwoadjacentbarscomparetheincreaseordecreasefromonetimetoanother

EXAMPLEShowwithaverticalbargraphthecomparisonofafirmrsquossalesfor7yearswhenthestatisticsare

Year Sales

1954 $38260000

1955 $47840000

1956 $43190000

1957 $45000000

1958 $39080000

1959 $47040000

1960 $51000000

662Whattypesofchartsorgraphsareusedtoshowtherelationofthepartstothewholeofanitemandwhichtypeispreferred(a)The100barchart(b)Thedividedbarchart(orrectanglegraph)(c)Thecirclegraphorpiechart

TheseareusuallyexpressedintermsofpercentsbutnotnecessarilysoItisoftendesirablethatboththeactualfiguresandthepercentsbestateddirectlyonthechartorgraph

EXAMPLEStatisticsshowthatoutof100accidents65areduetofalls25duetoburnsbruisesandblowsand10duetoallothercausesShowthisinformationwitha100barchartdividedbarchartandcirclegraph

ThedividedbarchartistobepreferredMentalcomparisonofsectorshaving

differentcentralanglesisnotsosimpletomakeviewandinterpret

663WhenisacirclegraphorpiechartusedandhowisitdrawnItisusedtoshowtherelationofpartstothewholeofsomethingItisused

frequentlyinnewspapersandmagazinesYougetthedecimalfractionthateachpartrepresentswithrespecttothewholeandyoumultiplyeachfractionby360degtogetthecentralangleWithaprotractoryoulayoutthecentralanglesofound

EXAMPLEInacertainschooltheenrollmentisasfollows

Freshmen = 520

Sophomores = 410

Juniors = 380

Seniors = 290

Totalenrollment = 1600

Expresstherelationshipbetweentheenrollmentineachclassandthetotalenrollmentusingacirclegraph

Iftheenrollmentisgivenorfiguredinpercents

Then

Butas360=100times36wehave

Nowmultiplyeachsideby100andget

Soineachcasemultiplytheby36

middot Sophomores = 25625times36 = 9225deg

Juniors = 2375times36 = 855deg

Seniors = 18125times36 = 6225deg

Drawthecirclechartusingaprotractortolayoffeachangleindegrees

664Howisthesameinformationshownintheformofalongbarchart

Dividethe100lengthintothefractionalpartsrepresentedbytheThischartmaybepreferredtothecirclechartforeasiercomparisonsoflengthsratherthanthelesseasilycomprehendedsectorsofacircle

665WhatisablockgraphItisarectangularblockwhoselengthindicatesthequantitytobecompared

EXAMPLECompareusingablockgraphaschoolbudgetfortheyear1950of$286000000withthatfor1960of$465000000

666Whatisabroken-linegraphorlinediagramandwhenisitusedWhenyouselectsuitablescalesplotpointsinaccordancewiththegivendata

andjointhepointsbystraightlinesegmentsyougetabroken-linegraphorlinediagram

ThevaluesbetweenplottedpointsmayormaynothavesignificancedependinguponthenatureofthequantitiesrepresentedandtheimplicationisthatsuccessivevalueschangeuniformlyandcontinuouslyForexampleonagraphofaveragemonthlybankbalancesthein-betweenvalueshavenomeaning

AlinediagramisusedwhenthereisalongseriesofrelativelycontinuousitemsItisespeciallyadaptedtorepresentatimeseries

Ex(a)Showwithalinegraphtheprobablemillionsofdollarsinautosalesforeachmonthof1960intheUnitedStates

Ex(b)Showafeverchartasalinediagram

HererateofchangeisindicatedbecausebetweenthetimesthetemperatureistakenthepatientrsquostemperatureisslowlygoingupordownWhenthelineislevelornearlysothechangeisslowandwhenthelinegoesupordownsteeplythechangeisrapid

Linechartsareusefulinshowingrateofchangeevenwithnoncontinuousdata

667Whatisacurvedgraph(smooth-linegraph)andwhenisitusedItisverysimilartoabroken-linegraphWhentheldquoin-betweenrdquovaluesvary

continuouslyanduniformly(ornearlyso)fromoneobservedormeasuredvaluetothenextasmooth-curvelineisdrawnbetweenthepointseitherfreehandorwithaFrenchcurve

TwoormoregraphsmaybeshownoneundertheotherandtheseareknownascomparativecurvegraphsIncomparingtherelativeamountsofcollectionsandsalesduringeachmonthofayearinabusinesstheuppercurvemaybesalesandthelowercollections

EXAMPLEShowwithacurvegraphthemonthlynormaltemperaturesinNewYorkCity

668WhatarepictographsandwhenaretheyusedTheyaregraphsthatuseimagesorpicturestorepresentnumbersThey

portraykindsandquantitiesofthingsataglancewithaminimumofexplanationTheyarenotcommonlyusedexceptforlargedistribution

EXAMPLEShowwithapictographthecomparativeappleproductioninthecommercialcountiesoftheUnitedStatesfortheyears1930and1956

669Whatarefrequencydistributiongraphs(frequencypolygonssometimescalledldquostaircaserdquodiagrams)Whenanumberofmeasurementsorphenomenaaregroupedintoconvenient

intervalsthedistributionofthesefrequenciescanbeshownbyatimegraphorhistographcalledafrequencydistributiongraph

Thisshowsataglancetherangeofmeasurements(weights)mostpredominantthecompleterangebetweentheextrememeasurementstheprevalenceofextremelylargeandsmallmeasurementssymmetricaldistributiononeithersideofacentraltendencyormode

EXAMPLEShowwithafrequencydistributiongraphthefrequencydistributionoftheweightsofaclassofwomen5feet4inchesinheightand21to25yearsofage

Weight Numberof women

91to100lb 12

101to110lb 124

111to120lb 268

121to130lb 107

131to140lb 26

141to150lb 8

141to150lb 8

151to160lb 4

670WhatismeantbyanindexnumberandhowisitobtainedAnindexnumberisacalculatedoranassumednumberusedasabasefor

comparisonwithothervalues

InsteadofcomparingtheactualcostoflivingofatypicalfamilyforeachyearoveranumberofyearswecanaveragetheincomefortheperiodandusetheaveragefigureasanindexTheaveragefigure(orindex)isthenconsideredtobe100andthefigureforeachsingleyearcanbeexpressedasapercentofthatindex

EXAMPLEIftheaveragecostoflivingforafamilyfortheyears1955to1960is$6000mdashwhichwecalltheindexmdashandifwefindthatthecostis$8000for1961then

Thismeansthatthecostoflivingin1961is oftheaveragefor1955-1960

=anindexfigurebasedonthe1955-1960figureastheindex

671WhataretheadvantagesofindexnumbersChangesareshownmorevividlywithindexnumbers

Todiscoveratrenditismucheasiertocomparenumbersintermsof100thantocomparethenumbersthemselves

EXAMPLE52ascomparedwith100iseasiertounderstandthan346comparedwith665

Usingindexnumberswecanmorereadilycomparepresentconditionswithconditionsinthepastorwithamorenormalperiod

Wecanuseeitherasingleyearoranaverageofaperiodofyearsasanindex

672WhatismeantbyinterpolationInterpolationisthereadingbetweentwopointsorvaluesonagraphofa

missingpointthatisdesired

Ex(a)Ifonebookcosts$325andfourbookscost$1300itisreasonabletointerpolatethattwobookscost$650andsevenbookscost$2275

Ex(b)Ifin195832860peoplewereinjuredbyfallsfromstepladdersandin196038400peopleweresoinjuredarewejustifiedinsayingthatin1959theyearinbetweenthenumberofpeoplesoinjuredmustbe35630midwaybetween32860and38400Nowecannotsayso

673WhatismeantbyextrapolationToextrapolateistodrawaconclusion(topredict)thataprocesswillgoonin

thesamedirectionasitseemedtobegoingwhenthedatagaveoutandthegraphended

EXAMPLEIfthenumberofjuvenilecrimesin1958weregivenas282346andas341692in1959wecannotextrapolate(predict)thefigurefor1960Too

manyfactorsmayentertochangethepicture

674WhenareinterpolationandextrapolationadvisableOnlywhenthedataaremovingaccordingtoapredictablepathor

mathematicallaw

675Whenwouldwehandledatainpercentform

Whenthedataaretoolargepercentsbringthemdowntoasmallermorecomparablebasis

676WhatiseasiertocomparetwoareasorthelengthsoftwolinesThelengthoftwolines

677WhenandhowarestatisticalmapsusedTheyareusedtoshowgeographicdistributionTheycombinefigureswith

geographicalareasSometimesvariouscolorsshadingsorcross-hatchingareusedtoindicatedata

EXAMPLEToshowgraphicallythedistributionoftelephonesinthestatesoftheUnitedStatestabulatethephonesforeachstateandchooseascaleinwhichonedotrepresentsacertainnumberofphonesThenumberofphonesinanystateisthenindicatedbythedensityofthedotsinthatstate

678WhatismeantbyCartesiancoordinatesAsystemofcoordinatesinaplanethatdefinesthepositionofapointwith

referencetotwomutuallyperpendicularlinescalledtheaxesofcoordinates

PointOiscalledtheoriginLinesXXprimeandYYprimearecalledtheaxesofcoordinates

679WhatismeantbytheaxisofabscissasUsuallythehorizontallineXXprimeiscalledtheaxisofabscissasorxaxis

680WhatismeantbytheaxisofordinatesThelineperpendiculartothexaxisiscalledtheaxisofordinatesorthey

axisYYprimeistheaxisofordinates

681InwhatorderarethefourquadrantsformedbytheaxesofcoordinatesdesignatedThefourquadrantsthatareformedbytheaxesofcoordinatesarenumbered

fromrighttoleftorcounterclockwiseasshowninthefigure

682WhatdirectionsareconsideredpositiveandwhatdirectionsnegativeDistancesmeasuredtotherightoftheyaxisarepositive(+)

Distancesmeasuredtotheleftoftheyaxisarenegative(ndash)

Distancesmeasuredabovethetimesaxisarepositive(+)

Distancesmeasuredbelowthexaxisarenegative(ndash)

683HowarepointslocatedinCartesiancoordinatesEachpointislocatedbybothitsabscissaandordinateTheabscissaisgiven

first

EXAMPLEThecoordinatesofpointP1areabscissax=2andordinatey=6

PointP2coordinatesare(-45)

PointP3coordinatesare(-5-4)

PointP4coordinatesare(7-3)

Theseshowapointineachquadrant

NotethatineachcasetheabscissaandtheordinatearetakenfromtheaxistothepointP

684HowdoweplotastraightlinerelationshipWhenevertwoquantitiesaredirectlyproportionalthegraphoftheir

relationshipisastraightline

EXAMPLES

1cubicfootofwaterweighs625lb2cubicfeetofwaterweigh125lb4cubicfeetofwaterweigh250lb6cubicfeetofwaterweigh375lb10cubicfeetofwaterweigh625lb

685HowdoweplotthegraphofaquadraticformulaItisacurvedlinegraph

S=16t2=aquadraticformula(parabola)s=distanceinfeet(abodyfalls)t=timeinseconds(timeoffall)

PROBLEMS

1DrawahorizontalbargraphtoshowthecomparativesalesofanautoagencyforthemonthsofJanuaryandMaywhenthesalesforJanuarywere$396000andforMay$874000

2Showwithahorizontalbargraphtheincomeoftheemployeesofafirmwhenthestatisticsare

Income Numberof employees

$4000-$4999 8400

$5000-$5999 3200

$6000-$6999 2100

$7000-$7999 1800

$8000-$8999 760

$9000-$9999 139

$10000andover 68

3Showwithaverticalbargraphthecomparisonofincomefortheyears1950to1960whenthestatisticsare

1950mdash$54000000 1956mdash$46000000

1951mdash$52000000 1957mdash$45000000

1952mdash$51000000 1958mdash$39000000

1953mdash$47000000 1959mdash$47000000

1954mdash$37000000 1960mdash$52000000

1955mdash$48000000

4Showwitha100barchartdividedbarchartandcirclegraphwhereeachdollarwentinthefollowing

Materialsandservicespurchased $620000000 5340

Wagesandsalaries $421350000 3625

Pensionssocialsecuritytaxesinsuranceetc

$26500000 228

Depreciationandpatentamortization $21100000 182

Interestonlongtermdebt $6200000 53

Taxesonincomeandproperty $35400000 305

Preferredandcommonstockdividend $18300000 157

Reinvestmentinthebusiness $12800000 110

Total= $1161650000 10000

5Expresstherelationshipbetweentheenrollmentineachclassandthetotalenrollmentusingacirclegraph

Freshmen 650

Sophomores 530

Juniors 480

Seniors 390

6Showtheinformationof(5)intheformofalongbarchart

7Compareusingablockgraphthebudgetofatownfortheyear1959of$135500withthatfor1960of$194000

8Showwithalinegraphtheaverageconstructioncostpernewdwellingunitofone-familystructuresfortheyears1950to1956

Year Cost

1950 $8675

1951 $9300

1952 $9475

1953 $9950

1954 $10625

1955 $11350

1956 $12225

9Showafeverchartasalinediagram

8am99degF12noon998degF4pm1018degF8pm1027degF12midnight1001degF4am100degF

10ShowwithacurvegraphthelengthofdayforNewYorkCityforeachmonth

LengthofdayatNewYorkCityforthefirstofeachmonthgivenas

January92hr

February106hr

March112hr

April122hr

May136hr

June146hr

July154hr

August142hr

September136hr

October114hr

November102hr

December94hr

11ShowwithapictographthecomparativepeachproductionintheUnitedStatesfor1955and1956

1955mdash51852thousandbushels1956mdash68973thousandbushels

12Showwithafrequencydistributiongraphthedistributionoftheheightsofaclassofmenweighing140lband20to24yearsold

13IftheretailpriceindexofdairyproductsintheUnitedStatesfor1947-1949is100andtheindexfigurefor1956is1087whatwouldbethecostofaquartofmilkin1956ifthecostin1947was20cent

14Howcanwemorereadilycompare285with679

15Ifonegallonofpaintcost$875andfourgallonscost$33howmuchwillsevengallonscost

16Locatethepoints(43)(ndash28)(ndash7ndash3)(4ndash8)(04)and(ndash40)inCartesiancoordinates

17PlottherelationshipP=625hwhereP=pressureinlbpersqftandh=heightinfeet

18Plottherelationshipv= (thevelocityacquiredbyabodyfallingadistancehfeetthroughspace)whereg=322=constant

CHAPTERXVIII

BUSINESSmdashFINANCE

686Whatarethetwotypesofcost(a)Netorprimecost=costofgoodsalone

(b)Grosscost=netcost+buyingexpensesashandlingorfreightstoragecarryingchargesinsurancecommissionsandadditionalchargesconnectedwiththecostofdeliveredgoods

687Intowhattwogroupsisprofitdivided(a)Grossprofit(marginofprofit)=sellingpricendashgrosscost

(b)Netprofit=grossprofitndashtotalcostofdoingbusiness

688WhatconstitutescostofdoingbusinessCostofdoingbusiness(overheadoroperatingexpenses)includesadvertising

taxessellingexpensesemployeesrsquosalarieslightheatdeliveryexpensesdepreciationandotherexpensesexceptthosethatconstitutethegrosscostofgoods

689Whatismeantby(a)grosssales(b)netsales(c)grosspurchases(d)returnpurchases(e)netpurchases(f)depreciation(a)Grosssales=totalofsalesoveraperiodoftimeatinvoiceprices

(b)Netsales=amountofsalesafterdeductingreturnsandallowances

(c)Grosspurchases=totalamountofgoodsboughtfortradingpurposes

(d)Returnpurchases=totalamountofgoodssentbacktofirms

(e)Netpurchases=grosspurchasesndashreturnpurchases

(f)Depreciation=decreaseinvalueofpropertybecauseofuseorchanges

resultingindisuserecordedasacertainpercentofthecostvalueofthepropertyusuallyattheendofeachbusinessyear

690Whatare(a)tradediscounts(b)cashdiscounts(a)Tradediscounts=deductionsfromlistpricemadetothetrade

(b)Cashdiscounts=deductionsfrominvoicepricewhenpaymentismadewithinaspecifiedtimeas10days30daysetc210means2discountifbillispaidwithin10days410n60means4discountwithin10daysandfullamount60daysfromdateofinvoice

691Whatis(a)asalescommission(b)abuyingcommission(a)Salescommission=apercentageofasellingtransactionchargedbya

salesmanagentbrokerorjobberforservicesinsellinggoods

(b)Buyingcommission=apercentageofabuyingtransactionforservicesofbuyinggoods

692Whenisthere(a)aprofit(b)aloss(a)Thereisaprofitwhensellingpriceisgreaterthancostofgoods+all

expenses(operatingshippingsellingbuyingetc)

(b)Thereisalosswhensellingpriceislessthanthatofgoods+theotherexpenses

Whensellingprice=buyingprice+otherexpensesthereisnoprofitorloss

Profitsandlossesareusuallycomputedonthegrosscostoronthenetsales

693Infiguringprofitorlosswhatis(a)thebase(b)therate(c)thepercentage(a)Base=grosscost

(b)Rate=percentofgainorloss

(c)Percentage=actualgainorloss

694HowdowefindthesellingpricewhenthenetcostandtherateofprofitaregivenMultiplythecostbythepercentofprofitandaddthistothenetcost

EXAMPLEWhatisthesellingpriceifgoodscost$20andyouwanttomake

aprofitof60ofthecost

Sellingprice=costtimesprofit+netcostmiddot($20times6)+$20=12+20=$32=sellingpriceAns

695HowdowefindthesellingpricewhenthereisalossandyouaregiventhenetcostandtherateoflossMultiplythecostbythepercentoflossandsubtractthisfromthecost

EXAMPLEWhatisthesellingpriceifthecostis$20andthelossis60ofthecost

Sellingprice=netcostndash(costtimesloss)middot$20ndash($20times6)=$20ndash$12=$8=sellingpriceAns

696HowdowefindthepercentofprofitgiventhecostandsellingpriceSubtractthecostfromthesellingpricetogettheprofit

Dividetheprofitbythecostandmultiplyby100togetthepercentofprofit

EXAMPLEWhatisthepercentofprofitifthesellingpriceis$120andthecostis$80

697HowdowefindthepercentoflossgiventhecostandthesellingpriceSubtractthesellingpricefromthecosttogettheloss

Dividethelossbythecostandmultiplyby100togetthepercentofloss

EXAMPLEWhatisthepercentoflossifthesellingpriceis$80andthecostis$120

698HowdowefigureadiscountoracommissionMultiplythecostorthesellingpriceoftheitembythepercentofthetrade

discount

Ex(a)Ifthetradediscountis10andthecostoftheitemis$2then

=tradediscountAns

Ex(b)Ifthetradediscountis40andthesellingpriceis$2then

4times$2=$8=80cent=tradediscountAns

Notethedifferencebetweentheformsinwhichthediscountisgivenpercentagesanddecimals

699HowdowefindthecashdiscountwhentheamountofthebillandtherateofdiscountaregivenMultiplytherateofdiscountbytheamountofthebilltogetthediscount

EXAMPLEIfthetermsare410n60andthebillis$1240whatarethecashdiscountandthenetamount

700Whatismeantbybankdiscount

Bankdiscountisinterestchargedbyabankforadvancingmoneyonnotesandtimedrafts

TheownerofthenoteendorsesittothebankwhichholdsittomaturityassecurityThenthebankcollectsthefaceamountfromthemakerorfromtheonewhosignedthenoteShouldthemakernotpaytheneitherthepartywhohadthenotediscountedortheendorserhastopayit

701HowissimplebankdiscountfiguredThesamewaythatsimpleinterestisfigured

Interestisfiguredfortheactualnumberofdaysbetweenthediscountdateandtheduedate

EXAMPLEFindthebankdiscountat6andthenetproceedsofa92-daynotefor$3000whenthedateofthenoteisAugust11960andtheduedateisNovember11960

702Howdowefigurethenetpriceofanitemwhenthereisaseriesofdiscountsas405and2(meaning405and2)(a)Multiplythecostoftheitembythefirstdiscountandsubtractthisfrom

thecostgettingresult(I)

(b)MultiplyresultIbytheseconddiscountandsubtractthisfromresultIgettingresultII

(c)Multiplyresult(II)bythethirddiscountandsubtractthisfromresult(II)gettingthenetpriceofitem

EXAMPLEGivencost$300anddiscounts405and2findthenetprice(a)$300times40=$300times4=$120$300ndash$120=$180=resultI(b)$180times5=$180times05=$9$180ndash$9=$171=resultII(c)$171times2=$171times02=$342

middot$171ndash$342=$16758=netpriceofitemAns

NoteThediscountsmaybetwoorthreeinnumberortheymaybeacombinationoftradeandcashdiscounts

Inanycasedeductthefirstdiscountintheseriesfromthetotalamountandfollowthisbydeductingthenextdiscountfromtheremainderetc

703Howmaytheaboveprocessbeshortenedbyobtainingasingleequivalentoftheremainder afterdeductingallthediscountsTake100asthebaseregardlessofthecostofthegoods

EXAMPLEIfthegrosscost(orlistprice)=$300andthediscountsare405and2findthenetcost

If100=basethen100ndash40=60=remainder

Now5of60=05times60=3

Therefore60ndash3=57=remainder

Then2of57=02times57=114and57ndash114=5586=5586=singleequivalentremainder

middot$300times5586=$16758=netcostAns

704WhatistheprocedureforgettingasinglediscountwhichisequaltotwodiscountsbymentalcalculationSubtract oftheirproductfromtheirsum

EXAMPLEFindasinglediscountequalto30and4

Theirsumis30+4=34

oftheirproductis

Thedifferenceis

middot34ndash12=328=singleequivalentdiscountAns

705Usingthismethodhowcanwegetasinglediscountwhichisequaltoaseriesofdiscounts(a)Findasinglediscountequaltothefirsttwo(b)Combinetheresultofthefirsttwowiththethird(c)Combinethelastresultwiththefourthetc


(a)Combine40with10

(b)Combinetheresult46with5

706Ifafter8and4discountsaredeductedthenetcostofaninvoiceofgoodsis$168436whatisthelistprice

707Iftheamountofdiscountis$39842andthediscountsare40and2whatisthenetcostofthegoods

Now

$39842=412=discount

Then

708Ifthetermsona$2680invoiceofgoodsare410n60howmuchdoyougainifyouborrowmoneyfromabankat6for60daysandpaycashforthemerchandise

04times$2680=$10720=discount$2680ndash$10720=$257280=netcost

$2680at6for60days=$2680=interestonloan

middot$10720ndash$2680=$8040=gainAns

709Ifthegrosscostofanarticleis$672andthearticleissoldataprofitof30onthesellingpricehowmuchisthenetprofitif21ischargedtothecostofdoingbusiness

100ndash30=70=70middot$672(grosscost)=70ofthesellingprice

and

30ndash21=9=percentofnetprofitmiddot$960(sellingprice)times09=$864=amountofnetprofitAns

710Whatistheprocedureforgettingthesellingpricegiventhenetcostpercentageofprofitandcostofselling(a)FindthenetprofitNetcosttimesprofit=netprofit(b)Addnetprofittonetcost(c)Thisiswhatofthesellingprice(d)Findthesellingpricebydividingbythis

EXAMPLEWhatisthesellingpricewhenthenetcostofanarticleis$1260anditistobesoldtomakeanetprofitof15ofthecostandwhenthecostofdoingbusinessis20ofthesellingprice(a)Netcosttimesprofit=$1260times15=$189=netprofit(b)Netprofit+netcost=$189+$1260=$1249

(c)$1249is80(=100ndash20)ofthesellingprice

(d)middot =$1561=sellingpriceAns

711Howcanwefindtherelationofnetprofittosellingpriceinpercentage(a)FindnetcostNetcost=listbuyingpricendashdiscounts(b)Findsellingprice

Sellingprice=listsellingpricendashdiscounts(c)FindgrossprofitGrossprofit=sellingpricendashnetcost(d)Findofgrossprofitonsales

(e)Findofnetprofitonsalesofnetprofitonsales=ofgrossprofitonsales

ndashcostofdoingbusiness

EXAMPLEAnappliancedealerbuysacolorTVsetfor$460less30and5Hesellsitfor$490less15Ifthecostofdoingbusinessis14ofthesaleswhatofthesellingpriceishisnetprofit

(a)Netcost

$460times(100ndash335)=$460times665=$30590=netcost

(b)Sellingprice=$490ndash(490times15)=$490ndash$7350

=$41650=sellingprice

(c)Grossprofit=$41650ndash$30590=$11060

(d)ofgrossprofitonsales=

(e)ofnetprofitonsales=2655ndash14=1255Ans

712Howcanwefindtherelationofnetprofittogross

costortonetcostexpressedasapercentage(a)Findnetcost=listbuyingpricendashdiscounts

(b)Findsellingprice=listsellingpricendashdiscount

(c)Findgrossprofit=sellingpricendashnetcost

(d)Getcostofdoingbusiness=timessellingprice

(e)Findnetprofitonnetcost=grossprofitndashcostofdoingbusiness

(f)ofprofit= =100

EXAMPLEIfbasketballscost$12adozenless30and5andaresoldfor$7eachless15andthecostofdoingbusinessis20ofthesaleswhatistheofprofitonnetcost

(a)Netcost30+5ndash =35ndash15=335

=singleequivalentdiscount


(b)Sellingprice

$84(=12times7)ndash($84times15)=$84ndash$1260=$7140


(d)Costofdoingbusiness=20times$7140=$1428

(e)Netprofitonnetcost=$2352ndash$1428=$924

(f)of

713Ifshirtsareboughtfor$560less14and8andaresoldfor$740less10andthebuyingexpensesare4ofthenetcostandsellingexpensesare5ofnetsaleswhatofthegrosscostisthenet

profit

(a)Netcost14+8ndash


(b)Sellingprice=$740ndash$740times10=$740ndash$74=$666

(c)Costofbuying=$443times04=$18

(d)Grosscost=$443+$18=$461

(e)Netprofit

$666(sellingprice)ndash$666times15(costofdoingbusiness)

=$666ndash$100=$566

there4$566ndash$461=$105=netprofit

(f)ofprofitongrosscost= times100=2278Ans

714Ifweknowtheamountofprofitthepercentofprofitonthegrosscostandthepercentofbuyingcosthowdowegetthenetcostandthecostofbuying(a)FindthegrosscostDivideamountofprofitbytheofprofitongross

cost

(b)FindgrosscostAddofbuyingcostto100(thenetcost)

(c)FindnetcostDividegrosscostbygrosscost

(d)FindcostofbuyingGrosscostndashnetcost

EXAMPLEIf30=ofprofitongrosscostofanarticleandtheprofitis$1293and7=costofbuyingwhatarethenetcostofthearticleandthecostofbuying

(a)Grosscost$1293=30=profit

(b)Grosscost=7(costofbuying)+100(netcost)=107

(c)Netcost= =$4028Ans(d)Costofbuying=$107

$4310(grosscost)ndash$4028(netcost)=$282=costofbuying

715Ifweknowthenetcostpercentofbuyingexpensesandtheamountofprofithowdowefindthepercentofprofitandthesellingprice(a)Getthecostofbuying(ofbuyingexpensestimesnetcost)(b)Getthegrosscost(netcost+buyingexpenses)(c)Findprofitongrosscost(profitgrosscosttimes100)(d)Findsellingprice(profit+grosscost)

EXAMPLEThenetcostofanarticleis$56Thebuyingexpensesare5ofnetcostWhatistheofprofitonthegrosscostifthearticleissoldataprofitof$1860andwhatisthesellingprice(a)Costofbuying=05times$56=$280(b)Grosscost=$56+$280=$5880

(c) (d)Sellingprice=$5880+$1860=$7740Ans

716Ifyoubuyanarticleinvoicedat$3460less3discountandsellitat30profitwhatisthesellingprice

Discount=$3460times03=$104Netcost=$3460minus$104=$3356Profit=30times$3356=$1007

Sellingprice=$3356+$1007=$4363Ans

717IfadealerbuysaTVsetfor$360pays$12freightandcartageandsellsitataprofitof whatisthesellingprice

Grosscost=$360+$12=$372


718Ifamerchantpays$1860foranarticleandsellsitataprofitof25ofthesellingpricewhatisthesellingpriceSellingprice=100

Cost=100minus25=75=$1860

719Ifthegrosscostofanarticleis$865anditissoldataprofitof25onthesellingpricewhatisthenetprofitifthecostofdoingbusinessis12Astheprofitis25onthesellingpricethenthegrosscost$865=75of

thesellingpriceandsellingprice=$86575=$1153

Nowofnetprofit=25minus12=13there4amountofnetprofit=$1153times13=$150Ans

720Ifamerchantsellsapplesat$550abushelat

commissionandhiscommissionamountsto$14850whileotherchargesare35centabushelhowmanybushelsdoeshesellandhowmucharethenetproceeds

Othercharges=600times$35=$210Totalcharges=$14850+$210=$35850Netproceeds=$3300minus$35850=$294150Ans

721ThecostofaTVsettoanappliancedealeris$360less40and2Whatshouldhemarkthesetifhewantstomakeaprofitof25onthenetcostandallowthecustomera15discountonthemarkedprice

$360times(100minus412)=$360times588=$21168=hisnetcost$21168times25=$5292=his25profitonnetcost$21168+$5292=$26460=netsellingprice

100(markedprice)ndash15(customersrsquodiscount)

=85=sellingprice

orsellingpriceis85ofthemarkedprice

722WhatismeantbytheldquofutureworthrdquoorvalueofasumofmoneyWehaveseenthatmoneyatinterestincreasesoraccumulatesastimepasses

Futurevalue=amountinquestion(principal)timesinterestaccumulationfactor

FuturevalueS=P(1+rt)atsimpleinterestP=principaland(1+rt)=interestaccumulationfactorr=rateofinterestt=timeAtcompoundinterestS=P(1+r)t(1+r)t=accumulationfactor

723WhatismeantbytheldquopresentworthrdquoorvalueofasumofmoneyItistheprincipalwhichifputatinterestatagivenrateforagiventimewill

equalsomeassumedordesiredamountinthefuture

Asumofmoneyisworthlesstodaythaninthefuturebecauseyoucaninvestthemoneytodayandallowittoaccumulate

Forsimpleinterest

Forcompoundinterest

724WhatismeantbythetruediscountTruediscount=thedifferencebetweenfutureworthandpresentworthofa

debt=interestonpresentworthofadebtforthetimeithastorunbeforematurity

725Whatarethepresentworthandthetruediscountofadebtfor$1800duein8monthsifmoneyisworth6interest

$1for8monthsat6=$104or

=$104=accumulationfactor

Then

The$1800debtwhichisduein8monthsisworth$173077now

And$1800minus$173077=$6923=thetruediscount

Toprovethiswehave

$173077for8monthsat6

and

$173077+$6923=$1800=theamountatmaturity

726IfAowesB$1000whichisnotdueuntil3yearsfromnowandAofferstopayBtodaywhatsumshouldApaynowatcompoundinterestassumingthemoneytobeworth4

ThismeansthatAshouldpay$88900now

Alsoitfollowsthat

presentvaluetimesaccumulationfactor=futureworth

or

$88900times112486=$1000

Accumulationfactorscanbeobtainedfromappropriatetables

727Whatismeantbythepresentvalueof1andhowisitusedThepresentvalueof1=thereciprocaloftheaccumulationfactor

Itismucheasiertomultiplythantodividewithnumbersofmanyplacesandthatiswhythepresentvalueof1isuseful

EXAMPLEFindthepresentvalueof$1000duein3yearsat4compoundinterest

Theaccumulationfactoris112486

Soinsteadoffinding

multiply$1000bythereciprocaloftheaccumulationfactor(orthepresentvalueof1)

there4$1000times88900=$88900

Reciprocalsofaccumulationfactorsaregivendirectlybyatableofpresentvaluesof1SeeTable4AppendixBforasectionofsuchatable

728Inwhattwowaysmayconsumerfinancebeconsidered(a)CashLoanandfinanceagenciesgivecashandallowtheborrowera

certaintimetorepaytheprincipalandinterest

(b)InstallmentcreditBusinessmenofferinstallmentcreditandpermitpurchasestobepaidforininstallmentsatspecifiedregularintervals

729WhatismeantbyinstallmentbuyingorbuyinggoodsldquoontimerdquoPartofthepurchasepriceispaidonpossessionandthebalanceinfractional

paymentsatstatedintervalsuntiltheentiresumispaid

ThemerchantisconsideredtoextendcredittotheconsumerThepurchaserisconsideredtoborrowmoneyindirectly

EXAMPLEIfaTVsetispricedat$150cashandtheadvertisedpaymentplanis$25downand$3aweekfor45weekshowmuchmoredoesitcostontheinstallmentplan

$25 = downpayment

$135 = 45weeksat$3

$160 = totalcostoninstallmentplan

$150 = cashprice

$10 = carryingchargeAns

730Ifyoubuyawashingmachinefor$280aregivena$50trade-inallowanceforyouroldmachineandagreetopaythebalancein10monthlyinstallmentsplusafinalinstallmentof$35howmuchwouldyousavebybuyingforcash

$280minus$50 = $230=balanceforcash

$230 = 10times$23=10equalmonthlypayments

= finalpayment

$35 = finalpayment

$265 = totalinstallmentpayments

$230 = cash

$35 = savedbybuyingforcash

Youpaytheequivalentof$280+$35=$315forthemachineinsteadof$280

731Ifyouborrow$2400fromabankandpayitbackinmonthlypaymentsof$3805over6yearshowmuchdoyoupaythebankfortheloan

6times12 = 72monthlypaymentstomake

72times$3805 = $273960 = totalpayment

$240000 = amountborrowed

$33960 = amountpaidforloan

732WhyisbuyinggoodsoncreditthesameasborrowingmoneyYouactuallykeepforatimethemoneythatbelongstothemerchantandon

thisyoumustpayinterest

Theadditionalmoneyyoupayontheinstallmentplanrepresentsadefiniteinterestrate

733WhydoescreditorinstallmentbuyingcostmoreItismoreexpensivetothemerchantHehastowaitforwhatyouowehim

YouusethegoodswhileyouarestillpayingforthemThemerchanthastokeeparecordofwhatyouowehimHetakesextrarisksbecauseshouldyounotbeabletofinishpaymenthecanrecoverthegoodsbutcannotsellthemasnewagain

734WhydosomemerchantspreferthecreditplantocashdespiteallthisTheygetmoremoneyforgoodsevenwithalltherisksincethecustomer

paysacomparativelyhighrateofldquointerestrdquocarryingchargeorfinancingchargeoncreditpurchases

Theycanalsosellmoretothoseunabletoaffordcashbuying

735Whataresomeoftherangesofinterestchargedinconsumerfinance(a)Personalfinancecompanies to permonthonunpaidbalances

(b)Contractinterestrate6minus12peryear

Notethatachargeof permonth=anannualeffectiverateof345Achargeof3permonth=anannualeffectiverateof426

(c)Creditunions12peryearor1permonth

(d)Industrialbanks12minus34peryear

Notethattoavoidanillegalrateofinterestinstallment-buyingcontractsgenerallydonotmentioninterestbutrefertoafinancingchargeorcarryingchargewhichincludesinterestbookkeepingcostandotherexpensesinvolvedininstallmentbuying

736Whatisthe6methodofferedbysomecreditcompaniesandhowdowefindthemonthlypaymentOne-halfpercentisaddedtotheunpaidbalanceforeachmonthuptoalimit

of12monthsYoudividethisresultbythenumberofpaymentstofindthemonthlypayment

EXAMPLEIfyoubuyarefrigeratorfor$480andmakeadownpaymentof$150thenpaythebalanceof$330in1yearwhatwouldbeyourmonthlypayment

Notethatthis6planisnotthesameas6interestaswillbeshownlater

737Ifyouasamerchantdecidetochargeanadditional14onthegoodsyousellldquoontimerdquowhatwouldbethepriceona10-equal-paymentplanandtheamountofeachpaymentonaclockradiothatsellsfor$8860cash

$8860times$114=$10100=priceoninstallmentplan

738WhatisthekeyinfiguringtheannualrateofinterestchargeyoupaywhenyoubuyontheinstallmentplanorwhenyouborrowmoneyfromafinancecompanytoberepaidinmonthlyinstallmentsYoumustaddupthenumberofmonthsspecifiedintheplandivideitby12

toconverttoyearsandsubstitutethisinI=Prt(I=interestamountP=principalt=timeinyearsr=annualinterestrate)

EXAMPLEIftheinterestorcarryingchargeis$8andthereare6monthlypaymentsof$10onapurchasewhatistheinterestrate

Thesetermsmeanyouactuallyowethemerchant(oraloancompanyifitisaloan)$60cashwhichyoupaybackinmonthlyinstallments

Youthushavekeptorborrowedthe

First paymentof$10for 1month

Second paymentof$10for 2months

Third paymentof$10for 3months

Fourth paymentof$10for 4months


Fifth paymentof$10for 5months

Sixth paymentof$10for 6months

Oryoukept$10foratotalof21months= years=t

Thesumofthemonthsfrom1to6canbeobtaineddirectlyfromthesumofaseries

P=principal=$10here

I=interestorcarryingcharge=$8

739HowmuchinterestandfinancingchargedoyoupaywhenyoubuyaTVsetfor$280ifyouareallowed$50foryouroldsetastrade-inallowanceandyouagreetopaythebalancein10monthlyinstallmentsof$23plusafinalinstallmentof$35

$280minus$50=$230=balance=10times$23inpayments

I=$35=finalinstallment=interestandfinancingchargeYoukeeporborrowthe

First $23paymentfor 1month

Second $23paymentfor 2months

Third $23paymentfor 3months

Fourth $23paymentfor 4months

Fifth $23paymentfor 5months

Sixth $23paymentfor 6months

Seventh $23paymentfor 7months

Eighth $23paymentfor 8months

Ninth $23paymentfor 9months

Tenth $23paymentfor 10months

Youkeep $23foratotalof 55months

Sumofmonths

740WhatprecautionmustyoutakeingettingthesumofthenumberofmonthsyoukeeporborrowtheinstallmentpaymentWhenthetotalofpaymentsresultsinasumgreaterthanthecashpriceofthe

goodsfindthepaymentnumbernearesttothecashpriceThengetthepartofthatpaymentthathasgonetowardtheactualcostofthegoodsandbyproportionfindthepartofthetimethispaymenthasbeenkeptbyyou

EXAMPLEIfyoubuyalivingroomsuitefor$870andpay$150downandthebalancein10monthlyinstallmentsof$84whatistherateoffinancingcharge

$870minus$150=$720=cashbalanceyouowe

10times$84=$840=amountpaidin10installments

$840minus$720=$120=amountoffinancingorcarryingcharge

Youkeeporborrowthe









Ninth $84paymentfor months

Attheendoftheeighthpaymentyouhavepaidback8times$84=$672

Thecashbalanceyouoweis$720

$720minus$672=$48whichgoestowardmeetingtheactualcashbalance

Sinceduringtheninthmonthonly$48goestowardtheactualcostofthesuiteyoumustconsidertheninthpaymentashavingbeenkeptonly

Thusthe$84paymentiskeptonly months

Sumofmonthsfrom1to8is

741HowcanwesolvefortherateofinterestbygettingthetotalamountoftheinstallmentmoneyyoukeeporborrowforonemonthintheexampleofQuestion740Youkeeporborrow

$84for1month

$168for1month Thisisthesameasborrowing$84for2months







Nowyoudonotkeeptheentireamount$756(=9times$84)ofthenextinstallmentbecauseyouneedonly$720minus$672=$48toreachthecashbalanceof$120youowe

Thenbyproportion

Thusyoufinallykeep$432for1month

Getthesumofamountsfrom$84to$672

n=8termsintheprogression

Tothisaddthelastamount=$432Totalamountofmoneykeptfor1month=$3456=P

Question740inwhichyougetthetotalnumberofmonthsyoukeepthe$84paymentissomewhatsimpler

742Ifyouborrow$300fromafinancecompanytopayasurgicalbillandyouarecharged3permonthinterestontheunpaidbalanceoftheloanwhileyouarerequiredtorepaytheloanin12monthlyinstallmentsof$25eachhowmuchdoyoupaybackforthe$300loanandwhatistheannualinterestrateusingtheinstallmentplanmethod

Totalmonths

Totalinterest

P=principal=$25

Totalamountpaidonloan

Weseethat3amonth=36ayear

743Ifyouborrow$300fromacreditunionwheretheinterestchargeis1amonthontheunpaidbalanceandyoupaybacktheloanin12monthlypaymentsof$25plusinterestchargehowmuchdoyoupaybackandwhatistheannualinterestrateHowdoesthiscomparewithasecuredbankloanof$300for1yearat6

Totalmonths

Totalinterest


Weseethat1amonth=12ayearbutthe$300isnotkeptonefullyearbutisbeingpaidbackeverymonth

Asecondloanfromabankwouldbe

$300times06=$1800=interestpaid

Hereyoukeepthe$300theentireyear

Thisisalmostascheapasacreditunionloanwhereyoupaybackeverymonth

744Ifyougetaloanof$2500at5interestperyearandyouagreetopayitbackin20yearsat$1650permonthhowmuchisthetotalamountofrepaymentandhowmuchdoesitcostyou20times12=240months=numberofpayments

240times$1650=$3960=totalrepaymentonloan

there4$3960minus$2500=$1460=costtoyouAns

745Howdoestheabovecostcomparewithabankloanof$2500for20yearsat5

$2500times05=$125peryear20times$125=$2500=costofloan

YoupaylesswhenyoupaybackthemoneyeachmonthAns

746Ifyougetaloanof$7000at5ayearontheunpaidbalancefromamortgagecompanytofinanceyourhomeandyouagreetopayitbackin8yearsat$8862permonthwhatisthetotalrepaymentontheloanandhowmuchdoesitcostyouNoteThe$8862permonthisobtainedbymultiplying$7000byanannuity

factor01265992obtainedfromatablebasedonanannuityformulausedbythemortgagecompany

8times12=96months=numberofpayments96times$8862=$850752=totalrepayment$850752minus$7000=$150752=costtoyouforloan

Repaymentschedule

747WhatisacommonlyusedmethodofdeterminingtheannualrateofinterestwhenyoubuyorborrowontheinstallmentplanThisisamethodbasedontheassumptionthateachinstallmentpayment

containsprincipalandinterestintheratioofthestartingunpaidbalancetothecarryingcharge

EXAMPLEIfaloanisfor$180tobepaidin10monthsat$20amonthandthereisacarryingchargeof$20thenthe$180principalis ofthetotaldebtof$200andtheinterestis of$200or$20Thus

Herealltheinstallmentsareequalandtheproceduregivesareasonableapproximationtoatrueinterestrateasyouwillsee

Thisisknownastheequalinstallmentconstant-ratiomethodofdeterminingannualinterestrateininstallmentplans

748Whatistheformulafortheequalinstallmentconstant-ratiomethodoffindingannualinterestrateininstallmentplans

r=annualinterestrate(asadecimalfraction)m=paymentperiodsperyear

I=totalinterestorcarryingchargeindollars

P=unpaidbalanceatbeginningofcreditperiodorcashpricelessanydownpayment

n=numberofpaymentscalledforexcludingdownpayment

EXAMPLEWhatisthepercentinterestperyearonaloanof$180plus$20carryingchargetobepaidin10equalmonthlyinstallments

m=12(paymentsaremonthly)I=$20=carryingchargeP=$180=balancedue(nodownpayment)n=10=numberofinstallments

749Howistheconstant-ratioformulaobtainedWeknowthatI=Prt=simpleinterestformulaFromthisweget

Nowfindtheaveragelengthoftimetheinstallmentsareinthehandsoftheborrower

Ifm=numberofpaymentperiodsinayearthen

1paymentiswiththeborrower thyear


1paymentiswiththeborrower thyearetc

upto thyearm

Thesumofthetimeprogressionorseriesisobtainedfrom

whereS=sumn=numberofterms(payments)a=firstterm=1mandl=lastterm=nmThen

Nowdividethisbyntogettheaveragetimethepaymentsareheldorborrowedor

750IfaTVsetispricedat$150cashandtheadvertisedpaymentplanis$25downand$3aweekfor45weekswhatistheinterestrateHere

m=52sincepaymentsareweekly

I=$25+45times$3minus$150=$160minus$150=$10=totalinterest=carryingcharge

P=$150minus$25downpayment=$125=unpaidbalance

751Aclockradioisofferedfor$45cashorontimepaymentsfor10morewithadownpaymentof$950andthebalancein13weeklypaymentsWhatistheannualrateofinterestHere

m=52sincepaymentsareweeklyI=10of$45=1times$45=$450=carryingcharge

P=$45minus$950downpayment=$3550n=13payments

752Ahi-fisetcanbeboughtfor$380cashwithadiscountof$19orin12equalmonthlyinstallmentsbypaying$130andaddinga$30carryingchargeWhatistheannualrateofinterestHere

m=12sincepaymentsaremonthly

I=($380+$30)minus($380minus$19)=$410minus$361=$49=totalcarryingcharge

P=$361cashminus$130downpayment=$231n=12payments

753Ifyouborrow$150fromaloancompanyfor10monthsandrepayitin10equalinstallmentsof$1734

whatrateofinterestdoyoupayHere

m=12P=$150I=10times$1734minus$150=$2340=carryingchargen=10

754HowcanwegettheannualratepaidinQuestion753byfindingtheamountofmoneytheborrowerhadtheuseoffor1monthTheborrowerhadtheuseof$15for1month$30foranothermonth$45for

1monthetcto$150for1month

Sumoftheseriesfrom$15to$150

Theborrowerhadtheuseof$825for1month

755HowcanwegettheannualratepaidinQuestion753byfindingthetotaltimetheborrowerhadtheamountoftheinstallmentavailableforuse

Theborrowerhad

$15availablefor1month$15availablefor2months$15availablefor3months$15availablefor4monthsetcto$15availablefor10months

Sumoftheseriesfrom1to10months

Theborrowerhad$15availableforuseforatotalof55monthsor years=t

756Ifyouborrow$300fromabankfor15monthsandpayback$2157permonthwhatannualrateareyoupayingasfiguredbythethreemethodsshown(a)Constant-ratiomethod

Each$2157consistsof$20paymentonprincipaland$157carryingcharge

Herem=12I=15times$157=$2355P=$300andn=15

(b)Total-amount-used-for-1-monthmethod

Sumofseriesof$20for1monthto$300for1month

(c)Totaltimeamountofinstallmentwasavailableforusemethod

Sumofseriesof1monthto15monthsthe$20waskept

757Ifyoubuyontimeasetofdishesthatcosts$86

cashand$12isaddedforcarryingchargesonapaymentplanof$14downand$14amonthfor6monthswhatistherateofinterestyoupay

P=principal=$86minus$14down=$72=theamountofmoneyofwhichtheborroweractuallyhastheuse

I=$12=totalcarryingchargem=12n=6

758Whatistheinterestonthetimeplanifaclothesdryersellsfor$189cashor$20downand$21permonthfor10monthsHere

759Whatistheconstant-ratioformulaforfindingtheinterestratewhenallpaymentsareequalexceptthelastoneThelastpaymentmaybedifferentfromtheregularonetotakecareofany

remainingbalance

wherel=lastpaymentindollars

EXAMPLEWhatistheinterestrateperyearonthetimeplanofasetofcookingutensilsthatisadvertisedat$28cashor$5downand$5perweekfor5weekswithalastpaymentof$2inthesixthweek

$5down+5times$5+$2 = $32

Cashcost = $28

Carryingcharge = $4=I

m=52sincepaymentsareweeklyP=principal=$28cashminus$5down=$23l=$2=lastpaymentn=6payments

760Whatistheinterestrateperyearifaclockcosts$25cashor$5downand$5permonthfor4monthswitha$375paymentthefifthmonth

Here

m=12sincepaymentsaremonthlyI=($5+4times$5+$375)minus$25=$375P=$25cashminus$5down=$20l=$375

761WhatismeantbypartialpaymentsTheyarepaymentsonanobligationoranoteinwhichapartofthe

indebtednessispaideachtime

EXAMPLEApromissorynotefor$5000givenfor6monthsshouldnormallybepaidinfullwhendueHoweversubstantialpaymentsmaybemadeonitandthedateandtheamountshouldbeenteredonthebackofthenote

762Whattworulesareusedtosolvepartialpaymentproblemsanduponwhatdoesthemethoduseddepend(a)TheUnitedStatesrule

ThisrulewasfirstusedbytheUnitedStatesgovernmentwhenpaymentsandinterestwereinvolvedManystatesadoptedthemethodwhenitwasapprovedbytheSupremeCourtoftheUnitedStatessothatcompoundinterestwouldnotbecharged

Itisusedwhenpartialpaymentsaremadeonaninterestbearingnoteofoveroneyearmaturity

(b)Themerchantsrsquorule

Themethoduseddependsuponagreementorthelawinthestateinwhichthemakerofthenotelives

763HowdobanksacceptingpartialpaymentsofnotessubmittedfordiscountcollectcompoundinterestandyetavoidtheSupremeCourtrulingTheyhavetheoldnotecanceledandanewonedrawnfortheamountstill

unpaidInthiswaytheyareabletocollectcompoundinterestbecausetheycollecttheinterestinadvance

764ForhowlongdonotesandaccountsonwhichnopaymentshavebeenmaderemaininfullforceUndertheStatuteofLimitationsthetimeis6yearsfromtheduedateDuring

thistimethecreditormaytakecourtactiontorecover

765MustmortgagesmadeforadefinitetimebepaidonmaturityYesbutveryoftentheyarepermittedtocontinueindefinitelyaslongasthe

interestpaymentsaremadewhendueGenerallybanksholdingmortgagesacceptpartialpaymentsonanyinterestdate

766Whatistheprocedureforsolvingpartialpaymentproblemsbythemerchantsrsquorule(a)Gettheinterestonthefaceofthenotefromitsdatetothedateitispaidin

full

(b)Gettheinterestoneachpaymentfromitsdatetothedateofpaymentinfull

(c)Subtractthesumofthepaymentsplustheirinterestfromthefaceofthenoteplusitsinterest

EXAMPLEAnotefor$1000datedApril161961hasthefollowingpaymentsendorsedonthebackJuly141961$250September301961$200November241961$100IfthemakerwishestopayinfullonDecember311961whatistheamountdueatthattimewhentheinterestis6

Thepaymentperiodsarefoundbycompoundsubtractionunlessmorereadilydeterminedotherwise

Amountdueonnote=$1000+$4250=$104250

767WhatistheprocedureforsolvingpartialpaymentproblemsbytheUnitedStatesrule(a)Gettheinterestontheoriginalprincipalfromdateofnotetodateoffirst

payment

(b)SubtractfirstpaymentfromsumofprincipalandinterestifthefirstpaymentisgreaterthantheinterestthendueTheresultbecomesthenewprincipalonwhichinterestisfigureduntilthesecondpaymentismade

(c)Thepartialpaymentforanyperiodshouldbegreaterthantheinterestforthatperiodotherwiseyoumustaddthispaymenttothenextpaymentorpaymentsuntiltheirsumisequaltoorgreaterthantheinterestforthecombinedperiods

(d)Thesameprocedureiscontinueduntilthetimewhentheamountdueon

thenoteisdesired

EXAMPLEFindthebalancedueonDecember311961onthenoteofQuestion766for$1000datedApril161961wherethepartialpaymentsendorsedonthebackofthenoteareJuly14$250September30$200andNovember24$100andinterestis6

Faceofnote $100000

Addintereston$1000(April16toJuly14=2mo28days) +$1467

AmountdueonJuly14 $101467

SubtractpaymentofJuly14 minus$25000

NewprincipalonJuly14 $76467

Addintereston$76467(July14toSeptember30=2mo16days)

+$969

AmountdueonSeptember30 $77436

SubtractpaymentofSeptember30 minus$20000

NewprincipalonSeptember30 $57436

Addintereston$57436(September30toNovember24=1mo24days) +$517

AmountdueonNovember24 $57953

SubtractpaymentofNovember24 minus$10000

NewprincipalonNovember24 $47953

Addintereston$47953(November24toDecember31=1mo7days) +$296

BalancedueonDecember311961byUnitedStatesrule $48249

Weseethat

768BytheUnitedStatesrulehowmuchisrequiredtosettleonAugust11961ademandnotefor$10000datedFebruary11960withinterestat6andwiththefollowingpaymentsendorseduponitApril101960$2000August41960$100February11961$4000June11961$1000

Faceofnote $1000000

Addintereston$10000(February1toApril101960=2mo9days) +$11500

AmountdueonApril101960 $1011500

SubtractpaymentofApril101960 minus$200000

NewprincipalApril101960 $811500

Addintereston$8115(April10toAugust41960=3mo24days) $15419

Weseethatthepaymentof$100onAugust41960islessthantheinterest$15419ofAugust4

WemustthenfindandaddtheinterestfortwointerestperiodsandsubtractthesumofthetwopaymentsfromthisamountdueonFebruary11961



Addintereston$8115(April101960toFebruary11961=9mo21days) +$39359

AmountdueonFebruary11961 $850859

Subtracttwopaymentsof$100and$4000 minus$410000

NewprincipalFebruary11961 $440859

Addintereston$440859(February11961toJune11961=4mo) +$8818

AmountdueonJune11961 $449677

SubtractpaymentofJune11961 minus$100000

NewprincipalonJune11961 $349677

Addintereston$349677(June11961toAugust11961=3mo) +$5246

BalancedueonAugust11961 $354923

769Whatarethetwogeneralkindsoftaxes(a)Directtaxesleviedonpersonalincomeprofitsvalueofpropertyor

business

(b)IndirecttaxesleviedonimportedgoodstobaccosalestaxongoodswartaxetcTheseultimatelyarepassedontotheconsumerinthepricesofthethingshebuys

770Whatis(a)apolltax(b)apropertytax(c)anincometax(d)asurtax(a)Polltax=taxasarequirementforvotingincertaincommunities

(b)Propertytax=taxleviedonproperty

(c)Incometax=taxleviedonincome

(d)Surtax=anadditionaltaxaddedtoregulartaxrate

771Whatis(a)alicence(b)anassessment(a)Apermittodosomethingyoudesireortoenjoysomespecificprivilege

(b)Assessment=taxleviedbyappointedorelectedassessorsagainstanindividualoracompanyonrealpropertyoruseofsomeproperty

772InwhatformareassessmentsusuallystatedIntermsofpercentintermsofmillsper$100somuchper$100orso

muchper$1000

773Whatarethethreeitemsthatareusuallyinvolvedintaxation(a)Base=amounttobetaxed=assessedvaluation(b)Rate=taxrate(c)Taxamount=taxexpressedindollars

Ex(a)Whatisthetaxonapropertyvaluedat$8000(base)at (rate)

$8000times0225=$180=taxamountAns

Ex(b)Whatisthetaxona$9000propertywhentherateis30millsper$100

774Whatisthetaxonapropertyassessedfor$7500iftherateis$2885per$100andthecollectorsrsquofeeis2$2885per$100=2885=02885

775Howdowefindthetaxratewhengiventhebase(assessedvaluation)andthetaxamountDividethetaxamountbythebase

Ex(a)Whatisthetaxrateona$4000propertywhenthetaxis$80

Ex(b)Iftheassessedvaluationoftaxablepropertyinatownis$2383015andthetaxtoberaisedis$68750whatshouldbethetaxrateexpressedasapercenttherateper$100valuationandtherateper$1000valuation

$68750divide$2383015=02885=28852885=$2885per$1002885=$2885per$1000

776HowdowefindtheassessedvaluationwhengiventhetaxrateandthetaxDividethetaxbythetaxrate

Ex(a)Whatisthebase(assessedvaluation)whenthetaxis$300andtherateis3

Ex(b)Whatisthevalueoftheassessablepropertyofatownifthetaxrollis

$68750andthetaxrateis$2885per$100

$2885per$100=2885=02885

there4$68750divide02855=$2383015Ans

777Howdowecalculate(a)surtax(b)totaltax(a)Multiplythebasebythesurtaxrate(b)Multiplythebasebytheregulartaxrate

Add(a)and(b)

Ex(a)Whatisthetotaltaxon$16000iftheregulartaxis5andthesurtaxis3

$16000times05 =$800

$16000times03 =$480

there4 $1280=totaltaxAns

Thesurtaxmaynotstartatthesamepointastheregulartax

Ex(b)Whatisthetotalincometaxon$8000iftheregulartaxis5andthesurtaxis2afterthefirst$3000ofincome

$8000times05 = $400 = regulartax

$8000minus$3000 = $5000

$5000times02 = $100 = surtax

there4$400+$100 = $500 = totaltaxAns

PROBLEMS

1Whatisthesellingpriceifgoodscostyou$30andyouwanttomakeaprofitof40ofthecost

2Whatisthesellingpriceifthecostis$30andthelossis40ofthecost

3Whatisthepercentofprofitifthesellingpriceis$180andthecostis$130

4Whatisthepercentoflossifthesellingpriceis$130andthecostis$180

5Ifthetradediscountis20andthecostoftheitemis$15whatistheamountofthediscount

6Ifthetradediscountis30andthesellingpriceis$15whatistheamountofthediscount

7Ifthetermsare610n60andthebillis$1800whatarethecashdiscountandthenetamount

8Findthebankdiscountat6andthenetproceedsofa92-daynotefor$1000whenthedateofthenoteisJuly11961andtheduedateisOctober11961

9Givencost$500anddiscounts406and3findthenetpriceofthegoods

10Ifthegrosscost(orlistprice)is$425andthediscountsare406and2findthenetcostbyfirstgettingasingleequivalentremainderconsidering100asthebase

11Bymentalcalculationfindasinglediscountequalto35and5

12Findasinglediscountequalto405and3


14Iftheamountofdiscountis$28515andthediscountsare and3whatisthenetcostofthegoods

15Ifthetermsona$1800invoiceofgoodsare410n60howmuchwouldyougainifyouborrowmoneyatabankat6for60daysandpaycashforthegoods

16Ifthegrosscostofanarticleis$12anditissoldataprofitof35howmuchisthenetprofitif18ischargedtothecostofdoingbusiness

17Ifthenetcostofanarticleis$1840whatisthesellingpriceifitistobe

soldtomakeanetprofitof20ofthecostandthecostofdoingbusinessis18ofthesellingprice

18AdealerbuysaTVsetfor$380less40and2Hesellsitfor$425less10Ifthecostofdoingbusinessis18ofthesaleswhatpercentofthesellingpriceishisnetprofit

19Ifshirtscost$66adozenless40and2andaresoldfor$625eachless10andthecostofdoingbusinessis18ofthesaleswhatisthepercentofprofitonnetcost

20Iftrousersareboughtfor$840less20and5andaresoldfor$1020less10andthebuyingexpensesare3ofthenetcostandsellingexpensesare16ofnetsaleswhatpercentofthegrosscostisthenetprofit

21If35=ofprofitongrosscostofanarticleandtheprofitis$1640and6=costofbuyingwhatarethenetcostofthearticleandthecostofbuying

22Thenetcostofanarticleis$60Thebuyingexpensesare4ofnetcostWhatisthepercentofprofitonthegrosscostifthearticleissoldataprofitof$1430andwhatisthesellingprice


24Ifadealerbuysarefrigeratorfor$380pays$15freightandcartageandsellsitataprofitof30whatisthesellingprice

25Ifamerchantpays$2670foranarticleandsellsitataprofitof28ofthesellingpricewhatisthesellingprice

26Ifthegrosscostofanarticleis$1235anditissoldataprofitof30onthesellingpricewhatisthenetprofitifthecostofdoingbusinessis15

27Whatistheincometaxon$7500iftheregulartaxis4andthesurtaxis2afterthefirst$2500ofincome

28Thecostofawasher-dryertoanappliancedealeris$340less40and2Whatshouldhemarkthesetifhewantstomakeaprofitof28onthenetcostandallowthecustomera12discountonthemarkedprice


30IfAowesB$2400whichisnotdueuntil2yearsfromnowandAoffers

topayBtodaywhatsumshouldApaynowatcompoundinterestassumingmoneytobeworth6

31Findthepresentvalueof$2400duein3yearsat4compoundinterest

32IfaTVsetispricedat$195cashandtheadvertisedpaymentplanis$35downand$450aweekfor40weekshowmuchmoredoesitcostontheinstallmentplan

33Ifyoubuyawashingmachinefor$240witha$35trade-inallowanceonyouroldoneandagreetopaythebalancein10monthlyinstallmentsplusafinalinstallmentof$30howmuchwouldyousavebybuyingforcash

34Ifyouborrow$1800fromabankandpayitbackinmonthlypaymentsof$4229over4yearshowmuchwouldyoupaythebankfortheloan

35Onthebasisofthe6methodofferedbysomecreditcompaniesifyoubuyarefrigeratorfor$450makeadownpaymentof$150andthenpaythebalanceof$300in1yearwhatwouldbeyourmonthlypayment

36Ifamerchantwishestochargeanadditional16onthegoodshesellsontimewhatwouldbethepriceona10-equal-paymentplanandtheamountofeachpaymentonaradiothatsellsfor$98cash

37Iftheinterestorcarryingchargeis$12andthereare8monthlypaymentsof$12eachwhatistheinterestrateperyearbytheldquosumofthetimerdquomethod

38HowmuchinterestandfinancingchargedoyoupaywhenyoubuyaTVsetfor$250witha$40trade-inallowanceonyouroldsetandyouagreetopaythebalancein10monthlyinstallmentsof$21plusafinalinstallmentof$30usingtheldquosumofthetimerdquomethod

39Ifyoubuysomefurniturefor$760andpay$140downandthebalancein10monthlyinstallmentsof$73eachwhatistherateoffinancingchargebytheldquosumofthetimerdquomethod

40SolveProblem39bytheldquototalinstallmentmoneykeptforonemonthrdquomethod

41Ifyouborrow$200fromafinancecompanywitha3permonthchargeontheunpaidbalanceoftheloanandyouarerequiredtorepaytheloanin10monthlyinstallmentsof$20eachhowmuchdoyoupaybackforthe$200loanincludinginterestandwhatistheannualinterestrateusingtheldquosumofthetimerdquomethod

42Ifyouborrow$200fromacreditunionandarecharged1amonthontheunpaidbalanceandyoupaybacktheloanin10monthlyinstallmentsof$20plusinterestchargehowmuchdoyoupaybackandwhatistheannualinterestratebytheldquosumofthetimerdquomethod

43Ifyougetaloanof$2000at5interestperyearandyouagreetopayitbackin20yearsat$1250permonthhowmuchisthetotalamountofrepaymentandhowmuchdoesitcostyou

44HowdoesthecostinProblem43comparewithabankloanof$2000for20yearsat5

45Ifyougetaloanof$6000at5ayearontheunpaidbalancefromamortgagecompanytofinanceyourhomeandyouagreetopayitbackin12yearsat$5549permonthwhatisthetotalrepaymentontheloanandhowmuchdoesitcostyou

46Whatisthepercentinterestperyearonaloanof$200plus$25carryingchargetobepaidin10equalmonthlyinstallmentsusingtheldquoconstant-ratiordquomethod

47IfaTVsetispricedat$200cashandadvertisedonapaymentplanof$30downand$5aweekfor37weekswhatistheinterestrateusingtheldquoconstant-ratiordquomethod

48Aradioisofferedfor$65cashorontimepaymentsfor10morewithadownpaymentof$12andthebalancein12weeklypaymentsWhatistheannualrateofinterestusingtheldquoconstant-ratiordquomethod

49Aldquohi-firdquosetcanbeboughtfor$640cashwithadiscountof$20orin12equalmonthlyinstallmentsbyfirstpaying$150andaddinga$32carryingchargeWhatistheannualrateofinterestusingtheldquoconstant-ratiordquomethod

50Ifyouborrow$250fromaloancompanyfor10monthsandrepayitin10equalinstallmentsof$2880whatrateofinterestdoyoupaySolvethisbythe(a)ldquoconstant-ratiordquomethod(b)ldquosumofthetimerdquomethod(c)ldquototalinstallmentmoneyrdquomethod

51Ifyouborrow$500fromabankfor16monthsandpayback$3365permonthwhatannualrateareyoupayingasfiguredbythethreemethodsstudied

52Ifyoubuyontimeatypewriterthatcosts$98cashand$14isaddedforcarryingchargesonapaymentplanof$14downand$12amonthfor7months

whatistherateofinterestyoupayusingtheldquoconstant-ratiordquomethod

53Whatistheinterestonthetimeplanifaclothesdryersellsfor$215cashor$25downand$2280permonthfor10monthsUsetheldquoconstant-ratiordquomethod

54Whatistheinterestrateperyearonatimeplanonasetofcookingutensilsthatisadvertisedat$34cashor$5downand$6aweekfor5weekswithalastpaymentof$3inthesixthweekusingthespecialldquoconstant-ratiordquomethod

55Whatistheinterestrateperyearifaclockcosts$30cashor$6downand$6permonthfor4monthswitha$250paymentthefifthmonthUsethespecialldquoconstant-ratiordquomethod

56Anotefor$2000datedMay151961hasthefollowingpaymentsendorsedonthebackAugust121961$500October281961$400November291961$200IfthemakerdesirestopayinfullonDecember311961whatistheamountdueatthattimebythemerchantsrsquorulewithinterestat6

57FindthebalancedueonDecember311961onthenoteofProblem56usingtheUnitedStatesrule

58BytheUnitedStatesrulehowmuchisrequiredtosettleonSeptember11961ademandnotefor$8000datedMarch11960withinterestat6andwiththefollowingpaymentsendorseduponitMay121960$1600September31960$80March41961$3200July51961$800

59Whatisthetaxonapropertyvaluedat$10000at rate

60Whatisthetaxona$12000propertywhentherateisgivenas35millsper$100

61Whatisthetotaltaxonapropertyassessedfor$9500iftherateis$2963per$100andthecollectorrsquosfeeis2

62Whatisthetaxrateona$6000propertywhenthetaxis$120

63Iftheassessedvaluationoftaxablepropertyinatownis$3875680andthetaxtoberaisedis$89430whatwouldbethetaxrateexpressedasapercenttherateper$100valuationandtherateper$1000valuation

64Whatistheassessedvaluationofapropertywhenthetaxamountis$340andtherateis29

65Whatisthevalueoftheassessablepropertyofatownifthetaxrollis$89430andthetaxrateis$2910per$100

66Whatisthetotaltaxon$12000iftheregulartaxis5andthesurtaxis3

CHAPTERXIX

VARIOUSTOPICS

AWorkingratesofspeed

778Whatfactorsareinvolvedinanyproblemrelatingtomenworking(a)Thenumberofmenthatareworking(b)Theamountofworktobedone(c)Thetimeinvolved

779Howcanwefindthetimeitwilltakeonemantodotheamountofworkdonebyanumberofmenwhoworkatequalratesofspeed

Multiplythenumberofmenbythegiventime

EXAMPLESevenmenworkingatequalratesofspeedtake10daystofinishajobHowlongwillittakeonemantodothejob

7mentimes10days=70man-daysthere41mantakes70daysAns

780Howcanwefindthetimeitwilltakeanumberofmen(workingatequalratesofspeed)todoajobwhenweknowthetimeittakesonemantodoit

Dividethegiventimebythenumberofmen

EXAMPLEOnemanworks8daystofinishajobHowlongwillittakefourmentodothesamejob(allworkingatequalratesofspeed)

781Howcanwefindthetimeitwilltakeanumberofmentodoajobwhengiventhetimeforadifferentnumberofmen(workingatequalratesofspeed)todothejob

Multiplythegivennumberofmenbythegiventimetogettheman-daysequaltothetimeittakesonemantodotheworkThendividethisbytherequirednumberofmen

EXAMPLEHowlongwillittake5mentodoajobthatisdoneby8menin50days

8mentimes50days=400man-days=timeforoneman

782Iftheratesofspeedofthemenareunequalhowcanwefindthetimeitwilltakeoneofthementodoajobwhengiventhetimeandtheratioofthespeedswithwhichanumberofmendothejob

(a)Assumetheslowestmanasabaseof1andsetuparatiotogettheldquoequalrdquonumberofman-daysbasedontheworkoftheslowestman

(b)Multiplythegiventimebytheldquoequalrdquonumberofman-daystogetthetime

oftheslowestmantodothejobhimself

(c)Dividethisproductbythenumberofldquoequalrdquoman-daysrequired

EXAMPLEIfthreemendoajobin10daysandtwoofthemenaretwiceasfastasthethirdhowlongwillittakeoneofthefastermentodothejob

Theslowman=1=baseRatiois212

Therefore2+1+2=5=numberofldquoequalrdquoman-daysbasedontheworkoftheslowestman

Nowgiventime10daystimes5(ldquoequalrdquoman-days)=50days=timeforslowestmantodothejobhimself

Sinceoneofthefastermenistwiceasfast

783Howdowefindtheamountofworkamanwilldoinpartofthetimewhenweknowthetimeittakeshimtodotheentirejob

Expressthetimesasafraction

EXAMPLEIfittakesaman9daystodoajobhowmuchoftheworkwillhedoin3days

Expressasafraction

there4Hewilldoin3days oftheworkthathewoulddoin9daysAns

784Knowingthetimenecessarytocompleteafractionofajobhowcanyoufindthetimenecessarytodotheentirejob

Dividethegiventimebythefraction

EXAMPLEIf ofthejobisdonein6dayshowlongwillittaketocompletethejob

785Howcanwefindthetimeitwilltakeanumberofmenworkingtogethertodoajobwhenweknowtheirrespectiveratesofwork

Findthepartofthejobeachwoulddoin1day

Addthesefractionstogetthecombinedpartofthejobdonein1day

Divide1bythiscombinedfraction

EXAMPLEIfittakesA3daystopaintahouseB4daysforthesamejobandC8dayshowlongwillittakethemtodothejobworkingtogether

In1dayAwilldo ofthejobIn1dayBwilldo ofthejobIn1dayCwilldo ofthejob

Then oftheworkwillbedonein1dayallworkingtogether

786Knowingthetimeittakesanumberofmentocompleteajobandtheindividualratesofworkexceptonehowcanwefindthetimeitwouldtakethemanwiththeunknownratetodothejobbyhimself

(a)Fromthegiventimegetthefractionoftheworkdonein1daywhenallworktogether

(b)Getthefractionoftheworkdonebyeachwhoserateisknownandaddthesefractions

(c)Subtractsumof(b)from(a)togetthefractionorpartofthejobdonein1daybythemanwiththeunknownrate

(d)Divide1byfractionresultingin(c)togetthetimeitwouldtakehimtodothejobbyhimself

EXAMPLEAcandoajobin6daysandBin8daysABandCworkingtogethercandoitin3daysHowlongwillittakeCtodothejobbyhimself

In1dayAcando ofthejobIn1dayBcando ofthejobIn1day In1day ofthejobIn1day ofthejobforCalone

BMixturesmdashSolutions

787Whatistheprocedureforsolvinganordinarymixtureproblem

(a)Considertheelementofthemixturethatdoesnotchange(theconstantingredient)andfinditsamountintheoriginalmixture

(b)Findthepercentthisamountisofthefinalmixture

(c)Fromthisgettheamountofthefinalmixture

(d)Subtracttheoriginalmixturefromthefinalmixturetogetthequantityoramounttobeadded

EXAMPLEHowmuchalcoholwouldyouaddtoa20alcoholmixtureof180gallonsofalcoholandammoniatomakea25alcoholmixture

(a)Ammoniaistheconstantingredientwhichis80oftheoriginalmixtureor

8times180gal=144galammonia

(b)144galammonia=75ofthefinalmixture

(c) ofthefinalmixtureTherefore

(d)(Final)192galminus180gal(original)=12galofalcoholtobeaddedtomakea25alcoholmixture

788Whengiventwodifferentgradesofanarticleinamixturehowcanwefindtheamountofeach

EXAMPLEHowmanypoundsofgroatsthatsellfor16centperlbshouldbemixedwithgroatsthatsellfor24centperlbtogetatotalmixtureof100lbtosellfor18centperlb

(a)Findthevalueofthetotalmixtureatthegivenprice

100lbtimes18cent=$1800

(b)Findthevalueofthetotalmixtureatthelowerprice


(c)Subtractthelowerfromthehighervalue

$1800minus$1600=$200

(d)Subtractthepriceoftheloweritemfromthepriceofthehigheritem

24centminus16cent=8cent

(e)Nowthedifferencebetweenthevalues$200dividedbythedifferencebetweentheprices8centis

or25lbmdashthenumberofpoundsofthehigher-gradeingredient

(f)100lbminus25lb=75lb=amountofthelowergradeinthemixture

there4Youneed75lbofthe16centgroatsand25lbofthe24centgroatstomakea100lbmixtureofthe18centgroatsAns

789Ifweknowthepercentageconcentrationsofseveralsimpleingredientsofamixturehowcanwefindthepercentagestrengthofthemixture

EXAMPLEWhatisthepercentagestrengthofalcoholinamixtureof6galof12alcohol8galof14alcoholand12galof35alcohol

Ifwehave1gallonof12alcohol12ofthegallonispurealcoholand88ofthegalloniswater

Nowifweadd11gallonsofwatertomakeatotalvolumeof12gallonsofsolutiontheconcentrationorproportionofalcoholisreducedto

Thus1galof12alcohol=12galof1alcoholand

Thus26galofmixturecontainsasmuchpurealcoholas604galof1alcohol

790Howmanyquartsofwatermustbeaddedto5quartsofa35solutionofhydrochloricacidtoreduce

ittoa25solution

Asabove

1qtofa35solutionofhydrochloricacid=35qtofa1solutionofhydrochloricacid

Then5qtofa35solution=5times35=175qtofa1solutionofhydrochloricacid

Andxqtof25solution=175qtofa1solutionor

there47qtndash5qt=2qttobeaddedtomakeita25solutionAns

791HowistheabovesolvedbytheprocedureofQuestion787

Thehydrochloricaciddoesnotchange(istheconstantingredient)andis35oftheoriginalmixture

35times5qt=175qthydrochloricacidNow175qt=25ofthefinalmixture175

Then =7qt=thefinalmixture

there47qtndash5qt=2qtwatertobeaddedtomakea25solutionofhydrochloricacidandwaterAns

792Howmuchalcoholmustweaddto3quartsofa25solutionofalcoholandwatertomakea40solution


Wateristheconstantingredientwhichis75oftheoriginalmixtureor

75times3=225qtofwater

Now225qtofwater=6ofthefinalmixture

Therefore =375qt=finalmixture

And375ndash300=75qtofalcoholtobeaddedtomakea40alcoholsolution

794HowcanweusetheprocedureofQuestion789todeterminetheamountofeachofseveralsimple

ingredientswhosepercentageofconcentrationisknowntoproduceamixtureofadesiredconcentration

EXAMPLEInwhatproportionshould45and85alcoholmixturebemixedtogiveanalcoholmixtureof68strengthPercentagesarebyvolumes

xvolumesof45alcohol=xtimes45=45timesvolumesof1alcoholyvolumesof85alcohol=ytimes85=85yvolumesof1alcohol

Totalx+yvolumes=45x+85yvolumesof1alcohol

Or(x+y)volumesofmixturecontainasmuchpurealcoholas45x+85yvolumesof1alcoholThereforethestrengthofthemixtureisasmanypercentasthenumberof(x+y)volumescontainedin45x+85yor

Then

and

there4Mix17volumesof45alcoholwith23volumesof85alcoholtogeta68alcoholAns

795Howmaytheabovebeshowndiagrammatically

Placethedesired(new)percentageconcentrationattheintersectionoftwodiagonallinesPlacethepercentageconcentrationstobemixedattheleft-handcornersMerelytakethedifferencebetweenthecenterfigureandeachleft-handfigureandplaceitatthecorrespondingendofthediagonalThisgivesatonce

thepartorvolumetobemixedofthegivensolutionconcentration

ThismethodistheresultofthecalculationinQuestion794andgivesthesameanswer

796Howmaytheabovemethodbeappliedtomixturesofdifferentquantitiesofliquidsofknownspecificgravities

EXAMPLEHowmanygallonsofwatershouldbemixedwith12gallonsofglycerineofspecificgravity124togetadesired107specificgravity

Thus17volumesofwatermustbemixedwitheach07volumesofglycerineofspgr124toproduceamixtureof107spgror

Thenbyproportion

17water7glycerine=xgalwater12galglycerine

NoteTheabovecalculationsapplyonlywhenthemixedliquidsdonotcontractinvolumewhenmixedWhenalcoholandwateraremixedinequalvolumesthereisashrinkageofover55involumeThesolutionofsugarinwateralsoresultsinacontractionofvolume

797Whattypesofpercentagesolutionsoccurinpractice

(a)Weightinweightdesignatedwlw

Thismeansthatadefiniteweightofasubstanceistobedissolvedtoproduce100weightsofsolution

(b)Weightinvolumedesignatedwlv

Thismeansthatadefiniteweightofsubstanceistobedissolvedinenoughsolventtoproduce100volumesofsolution

(c)Volumeinvolumedesignatedvlv

Thismeansthatadefinitevolumeofliquidistobemixedwithenoughsolventtoproduce100volumesofsolutionIntheUnitedStatesvlvconcentrationisdesignatedforliquidsandwlvforsolidsdissolvedinliquids

798Howmayweconvert(a)fluidounces(UnitedStates)intoavoirdupoisounces(b)avoirdupoisouncesintofluidounces

(a)Avoirdupoisounces=104fluidounces(UnitedStates)

(b)Fluidounces(UnitedStates)=avoirdupoisounces104

799Howmanyouncesofaluminumchlorideshouldbedissolvedtomakeagallonof25wv aqueoussolution

OneUnitedStatesgallon = 128fluidounces

25 = 25

25times128 = 32

there432times104 = 3328avoirdupoisouncesofaluminumchlorideAns

800Howmuchof445potassiumsulfiteand67ofmorpholineofspecificgravity10016shouldbeusedtomakeagallonofsolution


445 = 0445

0445times128 = 5696fluidounces

5696times104 = 592avoirdupoisouncesofpotassiumsulfite

67 = 067

067times128 = 8576=858

Sincemorpholineisafluiditismoreconvenienttomeasurethantoweighsotofindtheequivalentvolumedivide858bythespgr10016andby104toconverttofluidounces

801Howmuchpurelysol(100)isneededtomake1000ccof3lysolsolution

Writethisintheformofaproportion

or

802Amixtureof54pintsofacidandwatercontains24pintsofpureacidand30pintsofwaterHowmuchwatermustbeaddedtomakeamixturethatis25pureacid

Theconstantingredientistheacid=24pints24pints=25offinalsolutionor

CTanksandReceptacles(FillingEmptying)

803Whenwearegiventhetimeittakestofillatankhowcanweexpressthepartofthetankfilledinaunitoftime

Expressedbyafraction1dividedbythetime

EXAMPLEIfittakes10minutestofillatankhowmuchofthetankisfilledin1minute

ofthetankisfilledin1minuteAns

804Whenwearegiventhefractionofthetankfilledinaunitoftimehowcanwefindthetimeittakestofillthewholetank

Divide1bythefractionofthetank

EXAMPLEIfin1minuteapipecanfill ofatankthen

805Howdowefindthetimeittakestofillatankwhenwehaveseveralpipesactingatthesametimeandwearegiventhetimeeachtakestofillitwhenactingalone

(a)Findthepartofthetankfilledin1minutebyeachpipeinfractionform

(b)Addthefractions

(c)Invertthesumtogetthetimeneededwhenallacttogether

EXAMPLEA2-inchpipefillsatankin8minutesa3-inchpipefillsitin5minutesHowlongwillittaketofillthetankwithbothpipesactingtogether

The2-inpipefills ofthetankin1min


there4 = minforbothpipestofillthetankactingtogetherAns

806Whatistheprocedureforsolvingatankproblemwhenfillingandemptyingtakeplaceatthesametime

(a)Foreachpipeactingalonefindthefractionalpartofthetankbeingfilledoremptiedinaunitoftime

(b)Addthefractionsforfilling

(c)Addthefractionsforemptying

(d)ComparethesumsbyfindingthelowestcommondenominatorofbothfractionsTheonewiththegreaternumeratorwillbethelargerquantityandthefasterprocess

EXAMPLEWillatankeventuallyremainfilledorbeemptiedifithasapipe(1)whichcanfillitin10hoursapipe(2)whichcanfillitin6hoursapipe(3)whichcanemptyitin7hoursandapipe(4)whichcanemptyitin5hoursandallpipesareinsimultaneousoperation

Pipe1canfill ofthetankin1hr=rateoffilling


Pipe3canempty ofthetankin1hr=rateofemptying


Sumoffillingrates

Sumofemptyingrates

15=3times535=7times5there4LCD=3times5times7=105

Thus and

ofthetankisfilledin1hour ofthetankisemptiedin1hour

Thetankwilleventuallybeemptiedwhenallthepipesareopen

ofthetankwillbeemptiedin1hour hourstoemptythetankAns

Notethatherewheretheemptyingfractionisgreaterthanthefillingfractionthetankmustbefilledatthebeginningoftheoperation

807Howcanwefindthenumberofgallonsacontainercanhold

Multiplyitscontents(expressedincubicfeet)by

1cuft=12intimes12intimes12in=1728cuin

1standardUnitedStatesgalloncontains231cuin

there4 = gallonsinacuft= gallons(approx)Ans

EXAMPLEHowmanygallonsinacontainer6primetimes10primetimes4prime

DScalesforModelsandMaps

808Whendowehaveatruescalemodelofanystructure

Whentheratioofthelengthofanypartofamodeltothelengthofthesamepartintheactualstructureisthesameforallpartsthenwehaveatruescalemodelofthestructure

EXAMPLEWhatisthescaleofamodelofatoweronasuspensionbridgeiftheactualheightis200ftandtheheightonthemodelis10inches

10in=200ftor1in=20ft

Thismeansthat1inanywhereonthemodelrepresents20ftor

12times20=240inonthestructurethere41240or isthescaleofthemodelAns

809Iftheuniformrecommendationforairplanemodelsis172whatisthewingspanofamodelifthewingspanoftheactualplaneis80ft

Scaleis172or

Thismeans1inonthemodelrepresents72inonthestructurethen

810Ifthescaleofthemodelofanairplaneis172how

farawayfromthemodelwouldyouhavetostandsothatitwouldappearthesameasifyouwere900yardsfromtheactualplane

Scaleis172or1yd72ydThen

811Whatisamapandhowisitsscaleexpressed

Amapisascalediagramshowinggeographicfeaturesontheearthlocatedwithreferencetooneanother

Thescaleissometimesgivendiagrammaticallyas

andissometimesexpressedasaratio

InsectionalchartsoftheUnitedStatesthescaleis1500000

InregionalchartsoftheUnitedStatesthescaleis11000000

Ex(a)Whatistheratioofamapthatisdrawntothescaleof1inchtothemile

Thescaleorratioisthus163360

Ex(b)Howmanymilesdoes1inchrepresentonasectionalchart

Scaleis1500000or1inrepresents500000inontheearth

812Howmanymileswill inchesrepresentonamapdrawntoascaleof15000000

Scaleis15000000or1inrepresents5000000in

813Ifthescaleofamapis121120whatwouldbethedistancebetweentwotownswhichare24inapartonthemap

(a)Bytheratiomethod

(b)Bythemethodofgettingthevalueof1inchonthemapfirstandthenmultiplyingbythenumberofinchesonthemap

1inonmaprepresents21120inontheearth24inonmaprepresents24times21120inontheearth

814Ifthescaleofamapis inchestothemilewhatwouldbetheactualareainacresofalakeonthismapthathasbeenfoundbyaplanimetertohaveanareaof56squareinches

Scaleis inchesImileTherefore intimes in(=2025sqin)=1sqmiNow1sqmi=640acres

(a)Byratiomethod

(b)Bygettingthevalueof1squareinchonthemapfirstandthenmultiplyingbythegivensquareinches

RuleIfyouwanttogetthevalueofoneunitofanyelementintheproblemyoushoulddividebythatelement

Wewantthenumberofacresin1squareinchsowedividebysquareinchesThus

EAnglemeasurement

815Whatisanangle

TheopeningbetweentwolinesintersectingatapointiscalledanangleThegableofaroofandtheintersectionoftwostreetsarepracticalexamplesofangles

Alsoangle=amountofturningrequiredtorotateBAtopositionBC

816Whatarethepartsofanangle

Aninitiallineaterminallineandavertexconstituteanangle

BA=initiallineBC=terminallineB=vertex

817Whatismeantby(a)anangleof1degree(b)anarcof1degree

(a)Dividethecircumferenceofacircleinto360equalpartsanddrawlinesfromthecenterofthecircletothepointsofdivision360smallangleswillbeformedeachofwhichiscalledanangleof1degreeor1deg= ofcircumference

(b)Eachofthe360equalpartsofthecircumferenceiscalledanarcof1degree

Aquarterofacircle=arightangle=90deg=ninety1-degreeanglessidebysideHalfacircle=180degrees

Thesymbolforadegreeis[deg]Thus90deg=90degrees

818Whatismeantbyanangleof1minute

Divideanangleof1deginto60equalanglesEachoftheseiscalledanangleof1minuteThesymbolforaminuteis[prime]Thus Eachcorrespondingarcdivisioniscalledanarcof1minute

819Whatismeantbyanangleof1second

Divideanangleof1minuteinto60equalanglesEachoftheseiscalledanangleof1secondThesymbolforasecondis[ldquo]Thus Eachcorrespondingarcdivisioniscalledanarcof1second

820Howcanananglebemeasured

AnanglecanbemeasuredwithaninstrumentcalledaprotractorPlacetheprotractorontheanglewith00ononesideandpoint0onthevertexReadthescalewheretheothersidecrossesitThisgivesdegreesofangularmeasurement

PROBLEMS

1Workingatequalratesofspeed8mentake12daystofinishajobHowlongwillittakeonemantodothejob

2Onemanworks10daystofinishajobHowlongwillittakefivementodothesamejoballworkingatequalratesofspeed

3Howlongwillittake8mentodoajobthatisdoneby12menin40daysworkingatequalratesofspeed

4If3mendoajobin12daysandtwoofthemenarethreetimesasfastasthethirdhowlongwillittakeoneofthefastermentodothejob

5Ifittakesaman12daystodoajobhowmuchoftheworkwillhedoin3days

6If ofajobisdonein15dayshowlongwillittaketocompletethejob

7IfittakesA4daystobuildaboatB6daysforthesamejobandC10dayshowlongwillittakethemtodothejobworkingtogether

8Acandoajobin5daysandBin7daysABandCworkingtogethercandoitin3daysHowlongwillittakeCtodoitbyhimself

9Howmuchalcoholwouldyouaddtoa25alcoholmixtureof160gallonsofalcoholandammoniatomakea40alcoholmixture

10Howmanypoundsofricethatsellsfor25nsubperlbshouldbemixedwithricethatsellsfor35centperlbtogetatotalmixtureof120lbtosellfor28centperlb

11Whatisthepercentagestrengthofalcoholinamixtureof8galof14alcohol10galof22alcoholand16galof40alcohol

12Howmanyquartsofwatermustbeaddedto8quartsofa40solutionofhydrochloricacidtoreduceittoa16solution


14Inwhatproportionshould35and65mixturesofalcoholbemixedtogiveanalcoholmixtureof54strengthPercentagesarebyvolumes

15ShowhowProblem14canbesolvedbyadiagrammaticmethod

16Howmanygallonsofwatershouldbemixedwith14galofglycerineofspecificgravity122togetadesired105specificgravityUsediagrammaticmethodtogettheratioofvolumesofwatertoglycerine

17Howmanyouncesavoirdupoisofaluminumchlorideshouldbedissolvedtomakeagallonof30wvaqueoussolution

18Howmuch335potassiumsulfiteand82morpholineofspecificgravity1002shouldbeusedtomakeagallonofsolution

19Howmuchpurelysol(100)isrequiredtomake2500ccof5lysolsolution


21Ifittakes12minutestofillatankhowmuchofthetankisfilledin1minute

22Ifin1minuteapipecanfill ofatankhowlongwillittaketofilltheentiretank

23A2-inpipecanfillatankin12mina3-inpipecanfillitin4minHowlongwillittaketofillthetankwithbothpipesactingtogether

24WillatankeventuallybefilledoremptiedifithasapipeAwhichcanfillitin8hoursapipeBwhichcanfillitin6hoursapipeCwhichcanemptyitin5hoursapipeDwhichcanemptyitin6hoursandallpipesareinoperationsimultaneously

25Howmanygallonsarethereinacontainer8fttimes12fttimes6ft

26Whatisthescaleofthemodelofaradiotoweriftheactualheightis450ftandtheheightofthemodelis15in

27Ifthescaleis172whatisthewingspanofamodelwhenthewingspanoftheplaneis105ft

28Ifthescaleofthemodelofaplaneis172howfarfromthemodelshouldyoubesothatitwillappearthesamesizeastherealplaneatadistanceof1500yd

29Whatistheratioofamapthatisdrawntothescaleof1into4miles

30Howmanymilesdoes inrepresentonasectionalUnitedStateschart

31Howmanymileswill inrepresentonamapdrawntoascaleof15000000

32Ifthescaleonamapis131680whatwouldbethedistancebetweentwotownswhichare30inapart

33Ifthescaleofamapis5intothemilewhatwouldbetheactualareainacresofalakeonthismapthathasbeenplanimeteredtobe38sqin

34Howmanyminutesarethereinanangleof34degrees

35Howmanydegreesaretherein2revolutionsoftheterminalline

36Howmanysecondsarethereinanangleof34minutes

37Howmanysecondsaretherein

CHAPTERXX

INTRODUCTIONTOALGEBRA

821WhatisalgebraTheArabicwordal-jabrissaidtomeanthereunionofbrokenpartsAlgebra

thusunifiesarithmeticcompletesitandshortensmathematicalsolutionsItisthesciencetreatingthecorrectuseofmathematicsByitsuseunknownquantitiesmaybecomeknown

822WhyisalgebrasaidtobeashorthandextensionofarithmeticInarithmeticweareconcernedwiththenumbersofthingsas15molecules

20applesand80dollarsIneachcasewehaveanumberrepresentingthequantityofthisandtheparticularthingitselfwithitsnamewrittenout

InalgebrawestillhavethenumberrepresentingthequantitybutweselectasymboltorepresentthethingasxmoleculesyapplesandzdollarsThen15xrepresents15molecules20yrepresents20applesand80zrepresents80dollars

Thesymbolsprovideuswithashorthandmethodofexpressingfacts

Whenaletterisusedtorepresentanumberitisknownasaliteralnumber

EXAMPLEWhatismeantbyxpoundsorydollars

Thexorymayrepresentanyamountdependinguponthecircumstancesintheproblemthatisbeingconsidered

823HowarethelettersymbolsinalgebraselectedAsymbolmaybeusedtorepresentanythingwepleaseThesamelettermay

beusedtorepresentacertainthinginoneproblemandadifferentthinginadifferentproblembutinanyoneproblemonesymbolisalwayskeptforonethingandadifferentsymbolforadifferentthing

Aletterfromthebeginningofthealphabetsuchasabcdetcischosen

foraquantitythatisconstantinanyoneproblem

Aletterfromtheendofthealphabetasvwxyorzischosenforaquantitythatisavariableinanyoneproblem

Howeverthesymbolsarefrequentlyarbitraryasayearsbdollarsppoundsandxfeet

Somesymbolsarefrequentlyconventionalandareself-suggestiveofwhattheyrepresentsuchasR=rateP=principalt=timeA=arear=radiusw=weightV=volumev=velocitya=accelerationetc

SmallnumbersknownassubscriptsareoftenusedtodistinguishonesymbolfromanotherrepresentingthesamekindofquantityForexamplev1andv2areusedtorepresenttwodifferentvelocitiesinthesameproblemt1andt2mayrepresenttwodifferenttemperaturesandA1andA2mayindicatetwodifferentareas

824WhatismeantwhentwolettersoranumberandaletterareplacedalongsideeachotherItmeansthattheyaretobemultipliedtogether

EXAMPLEab=atimesbxy=xtimesy3m=3timesmand20p=20timesp

Ifp= then20p=20times =5

825WhatismeantbyacoefficientThenumberorarithmeticalpartinfrontofthesymboliscalledacoefficient

EXAMPLEIn20p20isthecoefficientofp

826WhatismeantbyatermThenumberandsymboltakentogetherarecalledatermOnetermiscalleda

monomial

EXAMPLE20p=aterm

Notethatwhenwedealwithonearticleweusuallyomitthecoefficient1

EXAMPLEIfwewanttorepresentonedollarwewritesimplyxinsteadof1x

827WhatisabinomialAnexpressionthatcontainstwoterms(fromLatinbi-meaningtwo)

EXAMPLE(a+b)(3xndash2y)and(6ndash4x)arebinomials

828Whatismeantby(a)afactorofaproduct(b)literalfactorsornumbers(c) specificnumbers(a)Eachofseveralnumbersorlettersthataremultipliedisafactorofthe

product

EXAMPLEInabaandbarefactorsoftheproductabIn3ab3aandbarefactorsoftheproduct3abIn5times6=305and6arefactorsof30

(b)Lettersusedtoexpressnumbersarecalledliteralfactorsorliteralnumbers

EXAMPLEIn3abaandbareliteralfactors

(c)Signednumbersareoftencalleddirectedorspecificnumbers

EXAMPLEmdash3mdash7andmdash9arespecificnumbers

829Whatismeantby(a)analgebraicquantity(b) analgebraicexpression(a)Analgebraicquantityisonethathasallliteralfactorsoracombinationof

literalandspecificnumbers

EXAMPLEab2c3isanalgebraicquantitywithallliteralfactorsmdash3a2b2isanalgebraicquantitywithacombinationofliteralandspecificnumbers

(b)Analgebraicexpressioncontainstwoormorefactorsorquantitiesoracombinationofbothconnectedbysignsofoperation

EXAMPLE2ab+x2+5d9ymdash5andx2mdash2yx+y2arealgebraicexpressions

Thusanalgebraicexpressionismadeupofterms

830WhatismeantbythecoefficientsofaproductInanyproducteachfactoristhecoefficientofeveryotherfactororgroupof

factors

Ex(a)Intheproduct3x3isthecoefficientofxandxisthecoefficientof3

Ex(b)Inay2aisthecoefficientofy2andy2isthecoefficientofa

Ex(c)In(andash1)b(amdash1)isthecoefficientofbandbisthecoefficientof(amdash1)

Ex(d)In12xy12isthecoefficientofxy12xisthecoefficientofyand12yisthecoefficientofx

831WhatisapolynomialAquantityoftwoormoretermsconnectedbyplusorminussignsisa

polynomial

EXAMPLE3x+5y4ab2ndash3bc2+bcd2arepolynomials

832WhatsymbolsareusedinalgebratoindicateadditionandsubtractionThesamesymbolsusedinarithmetic

Letxdenoteathing

(a)Then4x+7x=11x=addition

(b)And7xmdash4x=3x=subtraction

833Whatsymbolsareusedtoindicatemultiplicationanddivision(a)5xtimes3=15x(multiplicationwithamultiplicationsignbetweenthe

factors)or5xmiddot3=15x(usingadotforthemultiplicationsign)

Twoormoreletterswrittentogetherwithnosignbetweenthemmeansthattheyaretobemultipliedtogetheras

atimesb=amiddotb=ab=amultipliedbyb

xtimesytimesz=xmiddotymiddotz=xyz=timesmultipliedbyymultipliedbyz

(b)

834Whatarethefourelementsofeveryalgebraicterm(a)Asign(b)acoefficient(c)asymboland(d)anindex

EXAMPLEInndash4x3thesignisndashthecoefficientis4thesymbolisxandtheindexis3

Thetermisreadldquominus4xcubedrdquo

835Onwhatoccasionsaresomeoftheelements

omitted(a)Whenthecoefficientis1itisomitted

Thusndashx2isactuallyndash1x2=ldquominusonexsquaredrdquo

(b)Whentheindexis1itisomitted

Thusndash5xisactuallyndash5x1=ldquominusfivextothefirstpowerrdquo

(c)Aplussignisomittedwhenthetermstandsaloneoratthebeginningofanexpression

Thus5x2isactually+5x2=ldquoplusfivexsquaredrdquo

(d)Accordingto(a)(b)and(c)

xisactually+1x1=ldquoplusonextothefirstpowerrdquo

Thesigncoefficientandindexareomitted

ndashxisactuallymdash1x1=ldquominusonextothefirstpowerrdquo

Hereweomitthecoefficientandindexbutnotthesign

836Howis+x1 ndash5x2 +1x4 ndash3y3 writteninpractice

xndash5x2+x4ndash3y3Ans

837Whatlawsofadditionsubtractionmultiplicationanddivisionofnumbersarealsoapplicabletoalgebraicprocesses(a)Cumulativelawforaddition

Inarithmetic5+9=9+5=14

Inalgebraa+b=b+a

Thesumisthesameregardlessoftheorderinwhichthetermsareadded

(b)Associativelawforaddition

Inarithmetic(5+9)+12=5+(9+12)=26

Inalgebra(a+b)+c=a+(b+c)=a+b+c

Thesumisthesameregardlessofthegroupsthatareformed

(c)Cumulativelawformultiplication

Inarithmetic5times9=9times5=45

Inalgebraab=ba

Theproductisthesameregardlessoftheorderofthefactors

(d)Associativelawformultiplication

Inarithmetic(5times9)times12=5times(9times12)=540

Inalgebra(ab)c=a(bc)=abc

Theproductisthesameregardlessofthegroupingofthefactors

(e)Whenyoumultiplyafactorbythesumofseveraltermsitisthesameastakingthesumoftheproductsofthefactormultipliedbyeachterm

Inarithmetic5(9+12)=5times9+5times12

Inalgebraa(b+c)=ab+ac

(f)Whenyoumultiplyafactorbythedifferencebetweentwotermsitisthesameastakingthedifferencebetweentheproductsofthefactormultipliedbyeachterm

Inarithmetic5(9mdash12)=5times9mdash5times12

Inalgebraa(bmdashc)=abmdashac

Cases(e)and(f)areknownasthedistributivelawsformultiplicationwithrespecttoadditionandsubtraction

838HowmayweregardtwoormorelettersornumbersenclosedinparenthesesWemayregardthemallasonequantity

Ex(a)In3(a+b)wefirstadda+bandthenmultiplyby3

Ex(b)In5(amdash3)wefirstsubtract3fromaandthenmultiplyby5

Ex(c)In8(m+n+p)wefirstaddmnandpandthenmultiplyby8

Ex(d)In wefirstaddPandRandthendivideby4

839InalgebraicfractionswhymaythefractionbeconsideredtoactasasetofparenthesesBecausetheentirenumeratoristobedividedbythedenominator

Ex(a)In youfirstadd3toaandthendivideby4

Ex(b)In 2a+3bisconsideredonequantitywhichistobedividedby5

Ex(c)In firstsubtract5from20xandthendivideby5

Itisnot

Howeverifyoubreakupthenumeratoryoumustdivideeachpartbythedenominatoror

Ifx=2then

or

4xmdash1=4times2ndash1=7

840Inwhatwaysmay xbewritten

(a) times(b) (c)75x

841Howareverbalexpressionstranslatedtoalgebraic

symbolsandtermsBysubstitutingcoefficientssymbolsandsignsforwords

(a)Threetimesanumber=3a

(b)One-sixththebaseB= timesB

(c)Threetimesanumberincreasedby5=3a+5

(d)Anumberlessone-thirditself=amdash

(e)Costplus8=c+8

(f)Thesumofanythreenumbers=a+b+c

(g)Heighthless15=hmdash15

(h)Twicethesumofanytwonumbers=2(a+b)

(i)One-thirdthedifferenceofanytwonumbers= (amdashb)

(j)Fivetimesanumberlesstwiceanothernumber=5amdash2b

(k)Theproductofanythreenumbers=amiddotbmiddotc

(l)Anyevennumber=2a

(m)Anyoddnumber=2a+1

(n)Fourtimestheproductofanytwonumbersdividedbyathirdnumber=

842Howarealgebraicsymbolsconvertedtoverbalexpressions(a)andash5=fivelessthana

(b)a+5=fivemorethana

(c)5mn=fivetimestheproductofmandn

(d)5x+4y=fivetimesxincreasedbyfourtimesy

(e)3pmdash7=threetimespdiminishedbyseven

(f) =one-fifthoftheproductofaandb

(g) =three-eighthsofkorone-eighthofthreetimesk

(h)2a+3bndash5c=fivetimesanumbersubtractedfromthesumoftwice

anothernumberandthreetimesathirdnumber

(i)6(a+3)=sixtimesthesumofaand3

(j) (m+n)=onesixthofhmultipliedbythesumofmandn

(k) =onethirdthesumofaandb

(l) Bh=onethirdtheproductofBandh

(m)radic2gh=thesquarerootoftheproductof2gandh

843WhatisthegeneralprocedureforexpressingthoughtsalgebraicallyDonotsetupacompleteproblemldquoinonesteprdquoTakecareofeachphraseor

sentencethatexpressesaconditionindividuallyThencombinetheseparatepartsintooneormoreexpressions

Ex(a)Whatisthetotalcostofgolfballstoadealerifhebuys10dozenat$6adozenand30dozenat$8adozen

10doztimes$6= $60 = costoffirstlot

30doztimes$8= $240 = costofsecondlot

there4 $300 = totalcostAns

Sinceallfactorsarespecificnumberswegetaspecificanswer

Ex(b)Whatisthetotalvalueofsaleswhenamerchantsellsashirtsat$1250pershirtandbshirtsat$1050pershirt

$1250a=valueoffirstlot$1050b=valueofsecondlot

there4$1250a+$1050b=totalvalueofshirts

Theanswerisnotaspecificnumberbecausesomeofthetermsareliteral

Theanswercannotbesimplifiedbutifweleta=48andb=72

$1250times48+$1050times72=$600+$756=$1356Ans

844Howdoweindicatealettermultipliedbyitselfanumberoftimes

atimesa=aa=a2atimesatimesa=a3

atimesatimesatimesatimesa=a5etc

Smallfigurescalledexponentsareplacedtotherightabovetheletterandindicatehowmanytimesthefactorismultipliedbyitself

Thereforea5doesnotmean5timesabutamultipliedbyitselffivetimesover

5timesa=5abuta5=atimesatimesatimesatimesathere45times2=10but25=2times2times2times2times2=32

Theproductofafactortimesitselfiscalledthepowerofthefactor

845Whyisa2 calledldquoa squaredrdquoWhenallfoursidesofarectangleareofequallengthitiscalledasquare

Theareaisthenabulla=a2squnits

there4a2iscalledldquoasquaredrdquo

846Whyisa3 calledldquoa cubedrdquoArectangularsolidwithequalsidesoflengthbreadthandheightiscalleda

cube

Thevolumeofsuchacubeisabullaabull=a3

there4a3iscalledldquoacubedrdquo

Bythesameprocesswecanobtainexpressionswithhigherexponentssuchasa4=atothefourthpower

Wehaveseenthatraisingquantitiesortermstogivenpowersiscalledinvolution

847HowdoweraiseanalgebraictermtoanypowerAnalgebraictermconsistsofanumberandasymbol

(a)Raisethenumbertothepowerindicated

(b)Raisethesymboltothesamepower

(c)Multiplytheresults

Ex(a)3xsquaredmeans32timesx2

Ex(b)3xcubedmeans33timesx3

Ex(c)3xraisedtothefourthpower=34timesx4=81x4

848WhatistheruleformultiplyingthesamekindoflettersorexpressionstogetherAddtheexponentsin

(a+b)3times(a+b)4=(a+b)3+14=(a+b)7

Now

x3=xbullxbullxandx4=xbullxbullxbullx

Therefore

x3Xx4=xbullxbullxbullxbullxbullxbullx=x7orx3timesx4=x3+4=x7

849HowdowemultiplylettersthathavecoefficientsaffixedFirstmultiplythecoefficientsthenmultiplytheletters

Ex(a)3xtimes4x=3times4timesxxx=12x2Notethatx=x1

Ex(b)6x3b2times3xb5=6times3timesx3+1timesb2+5=18x4b7

Ex(c)6a2b+5times3abndash3=6times3timesa2b+5+b-3=18a3b+2

850Whatisthemeaningofsquareroot

TheareaofasquareisderivedfromthelengthofanyoneofitssidesWemaythusconsiderthesideastherootfromwhichthesquarehasevolvedWethuscallthelengthofthesideofasquarethesquarerootoftheareaofthatsquare

851WhatistheruleforgettingthesquarerootofanypowerofaletterTakeonehalftheexponentunderthesquarerootsigntogettheexponentof

thesquaretoot

EXAMPLES

852WhatismeantbytherootofagivennumberortermEachoftheequalnumbersortermsusedtoproduceapowerofaquantityor

termissaidtobearootofthepowerquantityorterm

Ex(a)Ifx3=xbullxbullxthenxisarootofx3or =x=cuberootofx3

Ex(b)If27x3=3bull3bull3timesxbullxbullx=33timesx3then3xisarootof27x3or =3x=cuberootof27x3

Ex(c) =3x=squarerootof9x2

853WhatistherulefordivisionofthesamekindofsymbolsSubtracttheexponentofthedenominatorfromthatofthenumerator

Ex(a)Dividex5byx3

Ex(b)

Ex(c) (notx2)

854Howcanweshowthataquantitytothezeropower=1

Butweknowthat =1(anythingdividedbyitself=1)

there4x0=1oranyquantitytothezeropower=1

EXAMPLE(a2middoty3radicx)0=1

855Whatistheresultof(a) (b) (c) (d)

(e) (a)

(b)

(c)

(d)

(e)

Thenumericalcoefficientsaredividedbythemselves

856Whatdoes mean

But (dividingnumeratoranddenominatorbyy4)

there4ymdash2and meanthesamething

Similarly

ymdash3= 10mdash1= =1

ymdash4= 10mdash2= = =01

ymdash1= etc10mdash3= = =001etc

857WhenmayweregardtwotermsasliketermsWhentheycontainlikesymbolswithlikeindicesandarethusofequalvalue

Ex(a)xandx2arenotliketermsTheindicesaredifferentxissimplyxwhilex2=xmiddotxIfx=3thenonetermis3andtheotheris9andarethusnotalikeinvalue

Ex(b)b2andb3xy2andx2ya2b3anda3b2arealsonotalike

Ex(c)7xand12x9yand17y3a2and5a2arealike

858DoestheorderinwhichthesymbolsoccurmatteratallNo

EXAMPLExyzhasthesamevalueasxzyorasyxzIfx=3y=4andz=5then

xyz=3times4times5=60or3times5times4=60or4times3times5=60

859WhatisasimpletestastowhethertwotermsareorarenotalikeinvalueWriteouteachtermwithoutindicesandcompare

Ex(a)Isa2b3likea3b2

there4Theyarenotalike

Ex(b)Isa2b3c2likea2c2b3

there4Theyarealike

860Whatdo[+]and[mdash]signsmeaninalgebraThesign[+]meansamovementinacertaindirection

Thesign[mdash]meansamovementintheoppositedirection

Ex(a)Ifyoumove300fttowardtherightfromAtoBinthefollowingdiagramandthenmoveback100fttoCyouarenowonly200ftfromA

Ifmovementtotherightis[+]andmovementtotheleftis[mdash]then

+300ftndash100ft=200ftrelativetoA

Ex(b)Ifyoumove300fttotherighttoBandthenmoveback400fttoCthen

+300ftndash400ft=mdash100ftrelativetoA

Ex(c)Ifyourose5000ftintheairthencamedown1000ftandyouassumethatupis[+]anddownis[mdash]thenalgebraically

+5000ftndash1000ft=+4000ft

Nowyouareonly4000ftaboveground

Ex(d)Ifyouwentdowninamine1500ftthencameup800ftandyouassumethatupis[+]anddownis[mdash]thenalgebraically

mdash1500ft+800ft=mdash700ft

Youareonly700ftdown

861Howare[+]and[mdash]quantitiesappliedtodebtandincome

Let[+]=incomeLet[mdash]=debt(orwhatyouhavespent)

Ex(a)Whatwouldbeyourfinancialpositionifyouspend$25then$10andthengetyoursalaryof$150

Algebraically

ndash$25mdash$10+$150=$115=whatyouhaveleftover

Ex(b)Ifyouhave25xdollarsandyouowe35xdollarswhatisthealgebraicsum

+25xmdash35x=mdash10xdollarsAns

Thismeansthatyoupayasmuchofthedebtasyoucanandyoustillowe10xdollars

862WhatistheruleforsubtractionofoneplusquantityfromanotherplusquantityChangethesignoftheplusquantitytobesubtractedandaddasusual

Ex(a)Subtract+8from+15

15mdash(+8)=15mdash8=7(Changesignof+8tomdash8)

Ex(b)Subtract25from60


Ex(c)12abmdash(+8ab)=12abmdash8ab=4ab

863HowcanweshowthattwominusesmeanaplusEx(a)Ifwesubtractmdash8from15weget

15mdash(mdash8)=15+8=23(Minusamdash8=+8)

As[+]meansamovementinacertaindirectionand[mdash]meansamovementintheoppositedirectionthenmdash(mdash8)meansastepintheoppositedirectionto(mdash8)whichmustmeanastepinthe[+]direction

there4mdash(mdash8)means+8and15+8=23

Ex(b)12abmdash(mdash8ab)=12ab+8ab=20ab

864WhatistheruleforsignsLikesignsgiveplus

Unlikesignsgiveminus

EXAMPLES

+(+8)=+8+(+a)=+a=amdash(mdash8)=+8mdash(mdasha)=+a=a+(mdash8)=mdash8+(mdasha)=mdashamdash(+8)=mdash8mdash(+a)=mdasha

865Whatistherulefornumbers(orletters)thataremultipliedtogetheroraredividedTheruleforsignsmustbeapplied

Ex(a)

+3(+8)=+24+a(+b)=+ab=abmdash3(mdash8)=+24mdasha(mdashb)=+ab=ab+3(mdash8)=-24+a(mdashb)=mdashabmdash3(+8)=-24mdasha(+b)=mdashab

+3(mdash8)means3stepseachof8unitsinthesamedirectionastheminusdirection

mdash3(mdash8)means3stepsofunitsintheoppositedirectiontotheminusdirectionthatisintheplusdirection

Ex(b)Sincedivisionisthereverseofmultiplicationtheruleofsignsalsoapplies

866Howdowedistinguishbetween+3(mdash8)and+3mdash8

+3(mdash8)=3stepseachof8unitstotheleft=mdash24+3mdash8=3stepstotherightandthen8stepstotheleft=mdash5

867Whatistheresultof8(a mdashb)mdash12(3a mdash4b)Removeparenthesesbymultiplicationandruleofsigns

8amdash8bmdash36a+48b

CombinearsquosandbrsquosNotethatnosigninfrontofaletterornumbermeans[+]

there4mdash28a+40bAns

868Whatistheresultof7[3a mdash4(5b mdash6a)mdash2b]Firstremovetheinnerparentheses

7[3amdash20b+24amdash2b]there47[27amdash22b]=189amdash154bAns

869Whatistheresultof3[4x mdash(2x +y)+5(3x +y)mdash6y]Removeinnerparenthesesfirst

3[4xmdash2x+y+15x+5ymdash6y]

Removeinnerbrackets

3[4xmdash2xmdashymdash15xmdash5ymdash6y]=3[mdash13xmdash12y]there439xmdash36yAns

870HowcanyoucheckyourselftoknowwhetheryoursolutioniscorrectSubstitutesmallvaluesforthedifferentlettersintheproblemandinthe

answer

EXAMPLEThusinQuestion8678(amdashb)mdash12(3amdash4b)assumea=1andb=2

8(1mdash2)mdash12(3mdash8)=mdash8+60=52

Nowintheanswerndash28a+40b

ndash28+80=52Check

871WhatistheprocedureforevaluatingalgebraictermsSubstitutetheappropriatenumbersfortheletters

Ex(a)Ifa=3b=4c=mdash6andx=5thenthevalueofa2mdash2ax+x2is

32mdash2times3times5+52=9mdash30+25=4

Ex(b)

Ex(c)

872WhatisthefirstimportantfacttorememberinaddingorsubtractingalgebraictermsOnlythosetermswhicharealikemaybeaddedorsubtracted

EXAMPLEWemaycombine

7xy+4xymdash3xyinto(7+4mdash3)xy=8xy

Wemaynotcombine

12x2mdash9y3+6z4beyond3(4x2mdash3y3+2z4)

873Whatistheprocedureforgettingthealgebraicsumofanumberofterms(a)Arrangethesignedtermswithlikesymbolsinseparatecolumns

(b)Ineachcolumngetthesumoftheminustermsandthesumoftheplustermsseparately

(c)Subtractthesmallersumfromthegreaterandaffixthesignofthegreaterabsolutenumber

Ex(a)Findthealgebraicsumof15xmdash16y8zmdash17x15ymdash12z16ymdash20x14z11xmdash6zandmdash5y

+15x mdash16y +8z

mdash17x +15y mdash12z

mdash20x +16y +14z

+11x mdash5y mdash6z

mdash11x +10y +4z =algebraicsums

Subtractsmallerfromgreaterandaffixsignofgreater

there4ndash11x+10y+4zAns

Ex(b)Add6a3bc4a2dndash3andash4bc7a2d7aandndash6bcd

874WhyisitthattoanytermyoumayaddonlyotherliketermsifyouwanttogivetheresultasasingletermIfxisapplesthenthesumof8apples15applesand6applesis8x+15x+

6x=29xButwemaynotrepresentthesumof8x15yand6zasasingletotal(term)anymorethanwecanrepresentthesumof8apples15pearsand6peachesasasingletotal

875WhatistheprocedureforsubtractionofalgebraicquantitiesChangethesignsofthesubtrahendandproceedasinaddition

EXAMPLEFrom10andash4b+5csubtract5a+7b+3d

10andash4b+5cminus(5a+7b+3d)

Thisbecomes

876WhatistheprocedureforremovingparenthesesorbracketsenclosinganumberofalgebraictermsOnremovingparenthesesprecededbya[ndash]signchangethesignsofall

termswithintheparentheses

Ex(a)6+(10ndash6)ndash(5+3)becomes

6+10ndash6ndash5ndash3=16ndash14=2

Ex(b)5a+(7ndash[3andash8])Firstremovetheinnerbracketsthenremovetheparentheses

5a+7ndash3a+8=2a+15

Ex(c)5andash(7ndash[3andash8])Removetheinnerbracketsfirst

5andash(7ndash3a+8)Nowremovetheparentheses5andash7+3andash8=8andash15

877Howmayweillustratethemultiplicationofapolynomialalgebraically

Ex(a)Tomultiply7by14wehave

Nowsubstituteletters

Multiplyeachtermofthepolynomialbythemultiplier

Ex(b)Tomultiplyalgebraically26times12


878Whatistheproductof(a)ndash3a2b4by5a3c3

(b)4a2+6abndash8c2by7a3

(c)2a2b3ndash3b2c3+5c2d3ndash4a2bc2d2byndash5a2b3c2

(d)6a2+3bby3a+4b2

(a)

(b)4a2+6abndash8c2times7a3=28a5+42a4bndash56a3c2Ans

(d)

879Howcanweshowthatthesquareofthesumoftwotermsisequaltothesquareofthefirsttermplustwicetheproductofthetwotermsplusthesquareofthesecondterm

Ex(a)(a+5)2=a2+2timesatimes5+52=a2+10a+25Ans

Ex(b)

NoteAlineoveratermwithanindexoverittotherightmeansthattheentiretermisraisedtothepoweroftheindex

880Howcanweshowthatthesquareofthedifferenceoftwotermsisequaltothesquareofthefirsttermminustwicetheproductofthetwotermsplusthesquareofthesecondterm

Ex(a)(andash5)2=a2ndash2timesatimes5+52=a2ndash10a+25Ans

Ex(b)

881Howcanweshowthattheproductofthesumanddifferenceoftwotermsisequaltothedifferenceoftheirsquares

Ex(a)

Ex(b)

Ex(c)

Ex(d)

Ex(e)

882Whatistheprocedureforgettingthedirectanswertothemultiplicationofanybinomialbyanotherbinomial

(a)Multiplythelefttermsforthefirstproduct4xsdot3x=12x2(b)Multiplytheoutertermsandaddtheproducttotheproductoftheinner

termsforthesecondproduct

4xtimes(ndash5y)+2ytimes3x=ndash20xy+6xy=ndash14xy

(c)Multiplytherighttermsforthethirdproduct

2ysdot(ndash5y)=ndash10y2Ans=12x2ndash14xyndash10y2

883Whatistheresultofsimplifying2x(x +5y)+3y(x+4y)

2x2+10xy+3xy+12y2or2x2+13xy+12y2

884Whatistheresultofsimplifying

(a+2)(a+4)+(a+3)(a+4)+(a+2)(a+5)a2+6a+8+a2+7a+12+a2+7a+10

or

3a2+20a+30Ans

UsemethodofQuestion882

Checkbyassumingthata=1andsubstitutinginoriginalexpressionandinanswer

Inoriginalexpression


(2andash2b)(2a+4b)ndash(2a+3x)(2andash5x)ndash2b(2andash4b)4a2+4abndash8b2ndash(4aandash4axndash15x2)ndash4ab+8b2

Changesignsonremovingparentheses

4a2+4abndash8b2ndash4a2+4ax+15x2ndash4ab+8b2there415x2+4axAns

886WhatistheprocedurefordividingapolynomialbyasingletermDivideeachterminthepolynomialbythesingleterm

Ex(a)Divide24x3ndash12x2+6xby3x

Ex(b)Divide96xndash56yndash88zbyndash8

Useruleofsigns

Ex(c)Divide18a4b5ndash13ab+7ab4by3a2b2

Ex(d)Divide2a+3b+4cbyy

Theresultineachcaseisthenumeratordividedbythedenominator

Ex(e)Dividex2ndasha2b2c3bya2b2c3

Divisionofasymbolwithanindexbyalikesymbolandindexisequalto1

887WhatistheprocedurefordivisionofapolynomialbyapolynomialProceedasinlongdivisioninarithmetic

EXAMPLEDividea2+4andash45byandash5

aofdivisorgoesintoa2ofdividendatimes

Nowmultiplyaby(andash5)gettinga2ndash5a

Subtractthisfroma2+4agetting9a

Bringdownndash45getting9andash45=remainder

aofdivisorgoesinto9a9times

Multiply9byandash5getting9andash45

Subtractthisfrom9andash45gettingzero

888Whatisthequotientofa2 +2a2b +4ab +2ab2 +3b2 dividedbya +2ab +3b

889Whatistheresultofdivisionofa3 ndasha2b -7ab2 -20bg byandash4b

890WhatistheldquocommontermrdquomethodofgettingthefactorsofanexpressionTakeanytermwhichisafactorcommontoeachtermoftheexpression

Dividetheexpressionbythiscommonfactortogettheotherfactor

Ex(a)Findthefactorsof12xndash16

4isafactorof12xand-16Dividebyfactor4

Checkbymultiplyingfactorstogethertogettheoriginalexpression

Ex(b)Factor36x3y2minus12x2y

12x2yisafactorof36x3y2andndash12x2yDivideby12x2y

Ex(c)Factorab+acndashad

aiscommontoallthreeterms

there4a(b+cndashd)=algebraicexpressionintermsoffactors

Ex(d)Factor4a2b4ndash8ab2+10a5b3

2ab2iscommontoallterms

there42ab2(2ab2ndash4+5a4b)=factors

Ex(e)Factor5(a+b)bndash6(a+b)a

(a+b)iscommontobothterms

there4(a+b)(5bndash6a)=factors

Ex(f)Factor4b2ndash12b5

4b2iscommontobothterms

there44b2(1ndash3b3)=factors

891WhatistheldquocommonparenthesesrdquomethodofgettingthefactorsofanexpressionTakeoutanyparenthesiswhichisafactorcommontothetermsofthe

expressionandusethisasafactorDividebythistogettheotherfactororfactors

Ex(a)Factor2a(3x+y)+3a(3x+y)

(3x+y)iscommontobothpartsDivideby(3x+y)

there4(3x+y)(2a+3a)=factors

Ex(b)2a(3x+y)ndash3a(3x+y)minus(3x+y)(2aminus3a)=factors

Ex(c)b(2y+x)+k(x+2y)=(x+2y)(b+k)=factors

892WhatistheprocedureforfactoringbythecombinationofthecommontermandthecommonparenthesesmethodsFirstfindacommontermfactor

Thenfindacommonparenthesesfactor

Ex(a)Factor2x2+3xy+2xz+3yz

Takeoutacommontermx(2x+3y)+z(2x+3y)

Takeoutthecommonparentheses(2x+3y)(x+z)=factors

Ex(b)Factor2x2ndash3xyndash2xz+3yz

x(2xndash3y)ndashz(2xndash3y)

there4(2xndash3y)(xndashz)=factors

Ex(c)Factor2x2+4xyndashxndash2y

2x(x+2y)ndash1(x+2y)

there4(x+2y)(2xndash1)=factors

893Whatistheldquoproductoftwobinomialsrdquomethodofgettingthefactorsofathree-termexpression(a)Writetotheleftwithineachparenthesistwofactorsofthefirsttermof

theexpression

(b)Writetotherightwithineachparenthesistwofactorsofthelasttermoftheexpression

(c)MultiplytogethertheextremetermsoftheparenthesesmultiplytogetherthemiddletermsoftheparenthesesandaddthetwoproductsChecktoseethatthisequalsthemiddletermoftheexpressionandtryanothersetifthesedonotgivethemiddleterm

Ex(a)Factorx2+17x+60

Factorx2into(x+)(x+)

Factor60into(+12)+5)

Multiplyextremesxand5=5x

Multiplymiddleterms12andx=12x

Sum= =middletermcorrect

Ex(b)Factorx2+6x+8

xandxarethefactorsofthefirstterm

Nowtry8andIasthefactorsofthelastterm

Extremesxtimes8=8x

Middleterms1timesx=1x

doesnotequalmiddleterm

Nowtry4and2asthefactorsofthelastterm

Extremesxtimes2=2x



Ex(c)Factorx2ndash8xndash20

894WhatarethefactorswhentheexpressionisrecognizedasaperfectsquareWhenthemiddletermistwicetheproductofthesquarerootsoftheothertwo

termsthenwehaveaperfectsquare

Ex(a)Factor9a2ndash30ab+25b2

Squarerootof9a2is3a

Squarerootof25b2is5b

Twicetheirproductis2times3times5ab=30ab=middleterm

there4(3andash5b)(3andash5b)=(3andash5b)2=factors

Ex(b)Factorx2+6x+9

(x+3)(x+3)=(x+3)2=factors

895WhatarethefactorswhentheexpressionisintheformofthedifferenceoftwosquaresOnefactoristhesquarerootofthefirsttermminusthesquarerootofthe

secondtermandtheotherfactoristhesquarerootofthefirsttermplusthesquarerootofthesecondterm

Ex(a)Factorx2ndash16

(xndash4)(x+4)=factors

Ex(b)Factorx8ndash625

(x4ndash25)(x4+25)

But(x4ndash25)isalsothedifferenceoftwosquares

there4(x2ndash5)(x2+5)=(x4ndash25)=factors

Ex(c)Factor256a8b8+c8

(16a4b4ndashc4)(16a4b4+c4)(4a2b2ndashc2)(4a2b2+c2)(16a4b4+c4)

there4(2abndashc)(2ab+c)(4a2b2+c2)(16a4b4+c4)=factors

Ex(d)Factor(x+y)2ndash1

(x=yndash1)(x+y+1)=factors

Ex(e)Factor(x+y)2ndash(2a+3b)2

(x+yndash2andash3b)(x+y+2a+3b)=factors

Sincethe[ndash]signisinfrontof(2a+3b)theentireexpressionisminusforoneofthefactors

Ex(f)Factor(x+y)2ndash(cndashp)

(x+yndashc+p)(x+y+cndashp)=factors

Ex(g)Factor1ndash4x2+8xyndash4y2or1ndash(4x2ndash8xy+4y2)

Try(2xndash2)(2xndash2)ndash4xndash4x=ndash8x=middleterm

there41ndash(2xndash2)2and(1ndash2x+2)(1+2xndash2)=factors

Ex(h)Factor(9x2ndash12xy+4y2ndash4c2+4cdndashd2)

(9x2ndash12xy+4y2)ndash(4c2ndash4cd+d2)

or

(3xndash2yndash2c+d)(3xndash2y+2cndashd)=factors

896Whatisthevalueof whena =3andb =2Factorthenumeratorinto(4andash5b)(4a+5b)

897WhatisanequationAnequationisabalancingofexpressionsorquantitiesoneachsideofan

equalssignBecausethetwosidesmustbalanceitresemblesasetofbalancescaleswiththeequalssignasthepivotpoint

EXAMPLES15ndashx=10A=πr2H= υ= A= hay2+by+c=0areallequations

898Howcanweshowthebalance-scaleresemblanceofanequationEx(a)If10lbisontherightpanofthescalesand(x+3)lbontheleftand

theybalancethenx+3=10istheexpressionoftheequation

Nowifyoutakeaway3lbfromtheleftpanyoumustalsotakeaway3lbfromtherightpantokeepthebalanceor

(x+3)ndash3=10ndash3

andx=7lbwhichisthesolutionofxMoresimply

x+3=10there4x=10ndash3=7

Ex(b)

(1)15ndash5=10=balance=equation

(2)15=10ndash(-5)=10+5=balance=equation

[ndash5fromequation(1)ismovedtorightside]

Ex(c)(1)15=12+3=balance=equation(2)15ndash12=3=balance=equation

[12fromequation(1)ismovedtoleftside]



Ex(d)(1)8+5=13=balance=equation(2)8=13ndash5=balance=equation

[5fromequation(1)ismovedtorightside]

(3)5=13ndash8=balance=equation

[8fromequation(2)ismovedtorightsideand5fromequation(2)ismovedtoleftside]

899WhatisthechiefuseofanequationItisameansoffindinganunknownnumberinaproblem

Ex(a)Solve3x=21

3xmustbalance21Then of3xmustbalance of21or

x=7Ans

Ex(b)Solve x=36

of3xmustbalance of36or

900WhatismeantbytherootofanequationThesolutionorthevalueoftheunknownthatmakestheequationbalanceis

therootThismaybeexpressedasanintegeradecimaloracommonfraction

EXAMPLEIfthecircumferenceofacirculartankis260ftwhatisitsdiameter

C=πd=260ftwhereπ=31416d=diameter ofπdmustbalance of260or

901WhatismeantbyanidentityWhentheleftpartoftheequationisidenticalwiththeparttotherightofthe

equalssignthenwehaveanidentityTheequilibriumistrueforallvaluesofthesymbol(orofthevariables)

Ex(a)a(bndashc)=abndashac(trueforallvaluesofabandc)

Ex(b)2x+3y=3y+2x(trueforallvaluesofxandy)

Ex(c)5a+7a=12a(trueforanyvalueofa)

902Whatismeantbyaconditionalequation

OnethatimposesaconditionuponthenumbervaluesofthelettersintheequationTheequalityistrueforonlyonevalueofthevariableorforalimitednumberofvalues

Ex(a)12orangescost60centTheequationis12x=60ifweletx=costofoneorange

HereonlyonevalueofxmakestheequationbalanceTheconditionisthatx=5Thustheequationisaconditionalequation

Since12x=60 of12xmustbalance of60

Ex(b)5y=60(trueonlyfory=12)

Ex(c)

903WhatisalinearorsimpleequationWhenthehighestpowerofthevariableis1theequationiscalledlinear

simpleorfirstdegree

Onlyonevaluewillmaketheequalitytrueinasimpleorfirstdegreeequation

EXAMPLE3x+4=22isalinearequation

3x=22ndash4=18 of3x= of18orx=6(theonlyvalueofxthatwillmaketheequationtrue)

Thereisthusonerootorsolution

904WhatisaquadraticequationWhenthehighestpowerofthevariableis[2]thentheequationisquadratic

EXAMPLE4x2=64

of4x2= of64forbalanceorx2=16andx=+4orminus4(tworoots)

Therearealways2rootsorsolutionstoaquadraticequation

905Whatmaybedonetobothsidesofanequationwithoutaffectingitsbalance(a)Wemayaddthesamequantitytobothsides

(b)Wemaysubtractthesamequantityfrombothsides

(c)Wemaymultiplybothsidesbythesamequantity

(d)Wemaydividebothsidesbythesamequantity

(e)Wemayraisebothsidestothesamepowerorwemaytakethesamerootofbothsides

906WhatistheruleofsignsformovingtermsfromonesideoftheequalssigntotheotherOnmovingatermfromonesideofanequationtotheothersideyoumust

changeitssignIfitisplusitbecomesminusandifitisminusitbecomesplus

Ex(a)xndash5=0

x=0+5=5Moveminus5torightandchangeitto+5

Ex(b)x+5=12

x=12ndash5Move+5torightsideandchangeittominus5x=7

Ex(c)xndash7=8

x=8+7Movendash7torightsideandchangeitto+7x=15

907WhatistheresultwhenbothsidesofanequationaremultipliedordividedbythesamequantityAnotherequivalentequationresults

Ex(a)Solve

Ex(b)Solve =64

Ex(c)If ofanumberis18whatisthenumber

Letx=thenumber

Then

(1)

Moredirectlydividebothsidesof(1)by

Ex(d)Solve06x=18

Ex(e)Solve3 =30

Ex(f)Solve08x=1000

908HowcanwesolvesimpleequationsbyadditionorsubtractionAddorsubtractanappropriatenumberorquantityiftheequationcannotbe

solvedbymultiplicationordivision

ThisprocessissimilartothatofQuestion906formovingtermsfromonesideoftheequationtotheother

Ex(a)Solvex+4=10

x+4ndash4=10ndash4Subtract4fromeachsidex=6Ans

Ex(b)Solve16=7+y

16minus7=7+yminus7Subtract7fromeachsidey=9Ans

Ex(c)Solve20=yndash3

20+3=yminus3+3Add3toeachsidethere4y=23Ans

Ex(d)Solve8=14ndashx

8+x=14ndashx+xAddxtoeachside8+xminus8=14ndash8Subtract8fromeachside

there4x=6Ans

Ofcoursethiscanbedonemoredirectlybytheruleofsignsformovingtermstotheoppositesideoftheequalssign

From8=14ndashxweget

x=14ndash8Moveminusxtoleftandmove8toright

909Whatarethestepsinthesolutionofanequation(a)Clearequationoffractions

(b)Removeanyparentheses

(c)Collectalltermscontainingtheunknownfactorontheleft(preferably)oftheequalssignandallothertermsontherightoftheequalssign

(d)Changethesignfrom[+]to[-]orfrom[-]to[+]whenmovingatermtotheoppositesideoftheequalssign

(e)Factortheexpressioncontainingtheunknowntomakeallothervaluesintheexpressionthecoefficientoftheunknown

(f)Dividetheentireequationbythecoefficientoftheunknown

Ex(a)Solveforxin7xndash5=9+3x

7xndash3x=9+5Allxrsquosonleftnumbersonright4x=14

Tochecksubstitute forxoneachside

Leftside

Rightside Check

Ex(b)Solve

ConvertanywholenumberintoafractionFindthelowestcommondenominatorofallthedenominatorsandarrangeeachsideoftheequationonthenewdenominator

YouneednotwritedownthiscommondenominatorinworkingequationsbecauseifafractionofonequantityequalsthesamefractionofanotherquantitythenthequantitiesthemselvesmustbeequalIf ofa= ofbthena=b

ConvertwholenumbersintofractionsLCM=4times5=20

Then

Substitutex=ndash20

Leftside

Rightside

Ex(c)Solve5(xminus6)=9(x+3)

Giveanswerinformof+x=minus14

910WhatisthesolutionforyofPMultiplybothsidesby(tminusy)

P(tminusy)=a(cminusy)

Then

PtndashPy=acndashay Removeparentheses

minusPy+ay=acminusPt TransposeminusaytoleftandPttorightside

y(aminusP)=acminusPt

Factorleftsidesothatystandsasasinglefactortimesthebinomialfactor(aminusP)

there4y Dividebothsidesby(aminusP)thecoefficientofy

911Whatisthesolutionford inAminuspd=b minusd

912WhatisthesolutionforWin W=T

Multiplybothsidesby

913Whatisthesolutionforx in = Firstcross-multiplyThisisthesameasmultiplyingbothsidesby16tand

thendividingbothsidesbyb+P

914Whatisthesolutionofx +7minus3xminus5=12minus4x

915Whatisthesolutionof(x +5)2 minus(x +4)2 =x +12

916Whatisthesolutionof7(x +5)minus9(x minus2)=8x +3

917Whatisthevalueofx in8(x minus3)(x +3)=x(8x minus8)

918Whatisthevalueofx in minus(x minus2)2 =22Dividex3minus4xminus15byxminus3togetx2+3x+5Then

919Whatistheprocedureforsolvingequationsinvolvingdecimals(a)Considerthetermcontainingthelargestnumberofdecimalplaces

(b)Makethatawholenumberbymovingthedecimalpointtotheright

(c)Movethedecimalpointineachothertermintheentireequationthesamenumberofplacestotherighttobalancetheequation

Note

6xcontains1decimalplace65xcontains2decimalplaces

EXAMPLESolve6x+05=5ndash3x

Move2placestotherightineachtermThen

Tocheck

Leftside6times55+05=335 CheckRightside5ndash3times55=335

920WhatisaformulaItisanalgebraicexpressiongivingtherelationofmathematicalfactsabout

variousquantities

Ex(a)A=πr2=formulaexpressingtheareaofanycircle

A=areaofanycircleπ=31416=constantandr=radiusofcircle

ForeveryradiusrwehaveanareaAtocorrespond

Ex(b)υ2=2gh=formulaforafallingbody

υ=velocityinftpersech=heightinfeetfromwhichbodyfallsandg=322=constantofgravity

Ex(c)d=υt=distancecoveredbyanobjectmovingataconstantspeedυforatimet

Whatisthedistancewhenthespeedis60milesperhourandthetimeis2hours

Whatisthedistancewhenυ=44ftsecandt=10sec

d=44x10=440ftAns

Ex(d)Ifthedistancescoveredbyafreelyfallingbodyintsecondsisgivenbyformulas=16t2whatisswhent=5andwhent=20

921IftherelationbetweentheFahrenheittemperaturereadingsofathermometerandtheCentigradereadingsisexpressedasF= C+32whatistheFahrenheitreadingwhen(a)C=50deg(b)C=30deg(c)C=10deg(a)F= x50+32=122degF(b)F= x30+32=86degF(c)F= x10+32=50degF

922WhatismeantbysolvingforanothervariableinaformulaEx(a)IntheformulaA=ltimeswwherel(=length)andw(=width)are

variablesandA=resultingareawecaneasilyfindAwhenweknowlandwbuttofindldirectlywehavetosolveforthevariablel

DividebothsidesofA=ltimeswbywtogetl=

Ex(b)Whatistheexpressionforυandtheexpressionfortind=vt

Divideeachsidebyυtoget

Divideeachsidebyttoget

Ex(c)Intheformula whereI=currentinamperes

E=voltageinvoltsandR=resistanceinohmswhatis(1)E(2)R1 MultiplybothsidesbyRtogetIR=E

2 Dividebothsidesof(1)byItoget

Ex(d)From findC

Fndash32= CMove32toleftside times(Fndash32)= times CMultiplybothsidesby (Fndash32)=CAns

Ex(e)Froms=gt22findgandt

923Whatisthegeneralprocedureforputtingwordsintoequationformtoexpresssimpleequationswithoneunknown(a)Expresseachphraseorsentencethatstatesaconditionandthencombine

thesetoformoneormoreexpressions

(b)Representtheunknownbyaletterfromtheendofthealphabet

(c)Expresseachstatementpertainingtotheunknownandanyotherunknownintermsofthisletter

(d)Expressionsrepresentingstatementsofequalvaluearethenplacedequaltoeachother

EXAMPLEIfyoumultiplyanumberby5andaftertaking9fromtheresult16remainswhatisthenumber

Letx=thenumber

924Ifthesumofthreeconsecutiveevennumbersis90whatarethenumbers

Then

there4thenumbersare2830and32Ans

925Ifatankis fullofwaterandafterrunningoff300gallonsitis fullwhatisthecapacityofthetank

Letx=capacityofthetank

xgallonsofwaterndash300gallons= gallons xndash =300Transposeandchangesigns

926Ifyouare45yearsoldandyoursonis12yearsold(a)whenwillyoursonbehalfyourage(b)howlongagowereyou5timesasoldasyourson(a)Letx=numberofyearsuntilyoursonwillbehalfyourage

Atthattimeyoursonwillbex+12yearsoldandyouwillbex+45yearsoldThen

x+45 = 2(x+12) = 2x+24

xminus2x = 24minus45

minusx = minus21

there4x = 21

In21yearsrsquotimeyoursonwillbehalfyouragethenyouwillbe45+21=66andyourson12+21=33yearsoldAns

(b)Lety=thenumberofyearsagowhenyouwere5timesyoursonrsquosageThen

45minusy = 5(12minusy)=60minus5y

minusy+5y = 60minus45

4y = 15

there4y = yearsago

Thenyouwere45ndash3 =41 yearsoldandyourson12ndash3 8 yearsold

Ans

5x8 =41 yearsCheck

927Iftwomachineoperatorspunchout1400plasticpartsperhourandoneproduces asmanypartsastheotherwhatistheproductionofeachLetx=partsproducedbyfasterworker=base

Then =partsproducedbyslowerworker

Andx+ x=partsproducedbybothworkersperhour

928Ifyouandyourwifetogetherhold$7800inUnitedStatesgovernmentbondsandyourshareis$1100morethanyourwifersquoshowmuchdoyoueachhaveLetx=yourwifersquosshare

Thenx+1100=yourshare

Andx+x+1100=combinedholdings

2x+1100=78002x=7800ndash1100=6700x= $3350=yourwifersquossharex+1100=3350+1100=$4450=yourshare

929Ifyoubought3suitsfor$226andthefirstcosttwiceasmuchasthesecondwhilethethirdcost$10morethanthesecondwhatisthecostofeachsuitLetx=costofsecondsuit=base

Then2x=costoffirstsuit

Andx+10=costofthirdsuit

930Ifyouhave$245innickelsanddimesandyouhave30coinsinallhowmanyofeachdoyouhaveLetx=numberofnickels

Then30ndashx=numberofdimes

5x=numberofcentsrepresentedbythenickels

10(30ndashx)=numberofcentsrepresentedbythedimes

931Atwhattimebetween4and5orsquoclockarethehandsofawatchoppositeeachotherLetx=distanceornumberofminutespacestraveledbytheminute-handfrom

4orsquoclocktotherequiredtime

Nowthehour-handis20minutespacesaheadoftheminute-handatexactly4orsquoclockandwhenthehandsareoppositeeachotheritwillbe30minutespacesawayfromtheminute-hand

Thusthehour-handwillhavetraveled30+20=50minutespaceslessthantheminute-handThereforexndash50=numberofminutespacesordistancetraveledbythehour-handfrom4orsquoclockuptothetimewhenthehandsareoppositeeachother

Buttheminute-handtravels12timesasmuch(or12timesthedistance)asthehour-hand

932Ifyouwanttosaleprice3001bofcoffeeat78centa1bandyouhaveonekindthatnormallysellsfor90centa1bandanotherthatsellsfor70centa1bhowmanylbofeachmustyoumixsothatyouwillnotlosemoneyLetx=numberoflbofthe90centkind=base

Then90x=salesvalueofthiskind

And300ndashx=numberoflbofthe70centminusaminus1bkind

And70(300ndashx)=salesvalueofthiskind

933Ifyousell3taxicabsandbuy2newonesfor$7800andyouthenhave$2400lefthowmuchdidyougetforeachtaxicabyousoldLetx=amountreceivedpertaxicabsold

Then3x=amountreceivedfor3taxicabs

And3xndash$7800=amountleftafterbuying2newtaxicabs

934Duringtheyearyouyourwifeandyourdaughtersavedatotalof$1200Yousaved$100lessthantwiceyourdaughterrsquossavingsandyourdaughtersaved$10morethantwiceyourwifersquosHowmuchdideachsaveLetx=yourwifersquossavings=base

Then2x+10=yourdaughterrsquossavings

And2(2x+10)ndash100=yoursavings

x+2x+10+2(2x+10)ndash100=$1200x+2x+10+4x+20ndash100=$12007x=1200ndash10ndash20+100=$1270

=wifersquossavings

2x+10=2times18143+10=$37286=daughterrsquossavings2(2x+10)ndash100

=2times37286ndash100=$64572=yoursavings

18143+37286+64572=$120001Check

Theonepennymoreisduetofractionalmanipulationsofthefigures

935Whatisthenumberwhichwhenmultipliedby4equalstheoriginalnumberplus36Letx=thenumberThen

936IfatrainleavesWashingtonDCforChicagoandtravelsattherateof50milesperhourandhourlateranautoleavesforChicagofromWashingtontravelingattherateof55milesperhourhowlongwillittaketheautotoovertakethetrainLetx=traveltimeinhoursofautountilitovertakestrain

Thenx+ =traveltimeoftrain

Now5x=distanceautotravels(mphxhours=distance)and =distancetraintravels

BothhavetraveledthesamedistanceatmeetingpointThen

Autotravels5hoursbeforeovertakingtrain

937Youstartouttowalktoyourfriendrsquoshouseattherateof4mphYourfriendstartsatthesametimeforyourhouseat3mphYoulive14milesfromeachotherHowfardoeseachofyouwalkbeforemeeting

Letx=timeofwalkingforeachbeforemeeting

Then4x=numberofmilesyouwalk

And3x=numberofmilesyourfriendwalks

4x+3x=14milestotaldistance7x=14X= 2hours4times2=8milesYouwalk8miles

3times2=6milesYourfriendwalks6miles

PROBLEMS

1Whatarethefactorsoftheproduct6cdp

2Whatisthenumericalcoefficientof36k

3Whataretheliteralfactorsof20xyz2

4Isndash8abaspecificnumber

5Whatarethecoefficientsoftheproduct15y(andashb)

6Whatisthedifferencebetween8+7and7+8c+dandd+c

7Isthereadifferencebetweenbkkandkb

8Is6(5ndash4)=6times5ndash6times4

9Inwhatwaysmay ybewritten

10Translatethefollowingverbalexpressionsintoalgebraicsymbols(a)Sixtimesanumber(b)One-thirdthebaseB(c)Seventimesanumberincreasedby8(d)Anumberlessone-eighthofitself(e)Costplus10(f)Thedifferenceoftwonumbers(g)Weightwless20(h)Threetimesthesumofanytwonumbers(i)One-sixththedifferenceofanytwonumbers(j)Eighttimesanumberlessthreetimesanothernumber(k)Theproductofanyfournumbers

(l)Thenextevennumbersaboveandbelow2x(m)Thenexthighernumberafterx(n)Fiveconsecutivenumbersofwhichxisthemiddlenumber(o)Fivetimestheproductofanytwonumbersdividedbyathirdnumber(p)Thesquarerootoftheproductoftwonumbers

11Convertthefollowingsymbolstoverbalexpressions(a)bndash6(b)a+7(c)9pg(d)7x+3y(e)4wndash8(f)(g)3a+bndash6c(h)(i)9(c+5)(j) (A+4B+C)(k)(l) bh(m)(n)A=P(l+r)n(o)(p)

12Whatisthealgebraicexpressionforthetotalvalueofsaleswhenamerchantsellsapairsoftrousersat$15apairandbpairsoftrousersat$1895apair

13Whatisthedifferencebetween4aanda4

14Whatis(a)5ysquared(b)5ycubed(c)5ytothefourthpower

15Whatistheresultof(a)(a+b)2+(a+b)6(b)x4timesx7(c)5xtimes6x2

(d)7x4c3times8xc4(e)9a3a+2times5a4andash1

16Evaluate(a)(b)(c)(d)(e)(f)(g)

17Evaluate(a)(b)(c)

18Divide(a)y6byx2

(b)y9byy3

(c)8y5by2y2(d)y5byy5

19Whatistheresultof

(a) (b)

(c)

(d)

(e)

20Are andxndash5thesameWhy

21Area3c2andc2a3alikeIsc3b2a5likec3b5a2

22Ifyouwentdowninamine2400ftandcameup1100ftwhatwouldbeyourpositionalgebraically

23(a)Whatisyourfinancialpositionalgebraicallyifyouspend$50then

$25andthengetacheckfor$200

(b)Ifyouhave50xdollarsandyouowe75xdollarswhatisthealgebraicsum

24Subtract(a)9from16(b)ndash9from16(c)ndash6abfrom13ab

25Whatistheresultof(a)+5x+9(b)-5x-9(c)+5xndash9(d))ndash5x+9(e)2ax3b(f)ndash2axndash3b(g)+2axndash3b(h)ndash2atimes+3b

26Whatistheresultof

(a) (b)(c)

(d) (e)

(f) (g)+ (h)

27Whatistheresultof9(andashb)ndash15(2andash5b)

28Whatistheresultof8[5andash6(4bndash7a)ndash3b]

29Whatistheresultof4[5xndash(3x+2y)+7(5x+2y)ndash3y]

30Ifa=2b=3c=ndash4andx=8whatisthevalueof(a)2a3ndash3a2x2+x3(b)

(c)

31Findthealgebraicsumof10xndash12y9zndash15x14y11z19yndash23x15z

9xndash8zandndash3y

32Add7a4bc5a2dndash5andash3bc9a2d8andash116bcd

33From12andash6b+8csubtract4a+6b+2d

34Whatisthevalueof(a)7+(12ndash5)ndash(8+4)(b)7a+(9ndash[5andash10])(c)8andash(11ndash[4andash9])

35Multiply(a)5a+8bby3a(b)5a+9bby3andash2b(c)ndash4a3b2by6a4c2

(d)5a2+7abndash9c2by8a4

(e)3a2b4ndash5b3c2+6c3d2ndash5a3b2cd3byndash7a3b2c4

(f)8a2+4b2by5a+362

36Whatistheresultof(a)(a+4)2(b)(4x+6y)2(c)(andash4)2(d)(9xndash4y)2(e)(andash7)(a+7)(f)(6xndash2)(6x+2)(g)(8xndash3y)(8x+3y)

37Multiply(5x+3y)by(4xndash6y)directlyasshownintext

38Simplify(a)3x(x+8y)+4y(x+7y)((b)(a+3)(a+5)+(a+2)(a+4)+(a+2)(a+7)(c)(3andash2b)(3a+5b)ndash(3a+4x)(3andash6x)

39Divide(a)48x4ndash36x3+12x2ndash6xby3x2(b)70xndash42yndash56zbyndash7(c)24a5b4ndash15a2b3+16ab2by4a2b3

(d)3a+4b+5cbyx(e)y3ndasha3b3c5bya3b2c2

(f)a2ndash2andash35byandash7(g)6a2+8a2b+17ab+12ab2+12b2by3a+4ab+4b(h)6a3ndash11a2bndash2ab2ndash20b3by2andash5b

40Factor(a)16xndash20(b)24x4y3ndash6x3y2(c)2a+3acndash4ad(d)8a3b5ndash4a2b3+12a6b2(e)6(a+2b)andash7(a+2b)b(f)6b3ndash18b7(g)3a(4x+2y)+5a(4x+2y)(h)3a(4x+2y)ndash5a(4x+2y)(i)c(2x+3y)+p(3y+2x)(j)6x2+10xy+12xz+20yz(k)6x2ndash30xyndashxz+5yz(l)8x2+4xyndash2xndash4y

41Factorbytheproductoftwobinomialsmethod(a)10x2+14xndash24(b)x2+12x+35(c)x2ndash9xndash36(d)xandash11x+28(e)18x2ndash18xndash20

42Factorbyperfectsquaremethod(a)4a2ndash12ab+9b2

(b)xa+16x+64

43Factorbythedifferenceoftwosquaresmethod(a)x4ndash25(b)y2ndash49(c)225a4b2ndashc6

(d)(2x+3y)2ndash1

(e)(andashb)2ndash(3andash2b)2

(f)(x+y)4ndash(kndashl)8

(g)1ndash9y2+24yndash16(h)25x2ndash30xy+9y2ndash16c2ndash16cdndash4d2

44Whatisthevalueof18a2ndash20b2whena=4andb=3

45Solve(a)5x=35(b)(c)C=πdwhend=12π=31416(d)6x+7=25(e)16x2+7=96(f)xndash7=0(g)x+8=15(h)xndash8=15(i)(j) =24

46If ofanumberis49whatisthenumberUsetheequationmethod

47Solve(a)08x=24(b) =62(c)07x=22400(d)x+3=12(e)26=8+y(f)18=yndash5(g)7=12ndashx(h)9xndash6=11+4x(i)(j)6(xndash7)=8(x+4)

(k) (forw)()Bndashsm=cndashm5(form)(m) (forP)

(n) (fory)(o)x+9ndash4xndash7=12ndash5x(p)(x+9)2ndash(x+7)2=x+6(q)6(x+4)ndash8(xndash1)=9x+2(r)6(xndash4)(x+4)=x(6xndash6)

(s)

(t)8x+09=9minus4x

48Ifd=vtwhatisthedistancedwhenvis30mphandt=4hr

49If whatisFwhenC=minus4deg

50If whatisa

51Ifyoumultiplyanumberby7and55remainsafteryouhavetakenaway15fromtheresultwhatisthenumber



54Ifyouare30yearsoldandyoursonis8yearsold(a)whenwillyoursonbehalfyourage(b)howlongagowereyou7timesasoldasyourson

55Iftwomachineoperatorspunch2600plasticpartsperhourandoneproduces asmanypartsastheotherwhatistheproductionofeach

56Ifyouandyoursontogetherhave$12000inbondsandyourshareis$2500morethanyoursonrsquoshowmuchdoyoueachhave

57Ifyoubought3suitsofclothesfor$277andthefirstcost timesasmuchasthesecondwhilethethirdcost$25morethanthesecondwhatisthecostofeachsuit

58Ifyouhave$445indimesandquartersandyouhave25coinsinallhowmanyofeachdoyouhave

59Atwhattimebetween2and3orsquoclockarethehandsofawatchoppositeeachother

60Ifyouwanttosaleprice400lbofgroatsat30centalbandyouhaveonegradethatsellsfor35centalbandanotherthatsellsfor25centalbhowmanylbofeachmustyoumixsothatyouwillnotlosemoney

61Ifyousell3safesandbuytwonewonesfor$26000andthenhave$7000lefthowmuchdidyougetforeachsafeyousold

62IfABandCsaved$6001totalandAsaved$500lessthantwiceCrsquossavingswhileCsaved$200morethantwiceBrsquoshowmuchdideachsave

63Whatisthenumberwhichwhenmultipliedby5willbeequaltotheoriginalnumberincreasedby44

64IfatrainleavesWashingtonDCforChicagotravelingattherateof52milesperhourand ofanhourlateranautoleavesforChicagofromWashingtonDCtravelingattherateof58milesperhourhowlongwillittaketheautotoovertakethetrain

65Youstartouttowalktoyourfriendrsquoshouseattherateof mphYourfriendstartsatthesametimeforyourhouseat mphYoulive16milesapartHowfardoeseachofyouwalkbeforemeeting

APPENDIXA

ANSWERSTOPROBLEMS

Introduction(pp10-12)1379

31937467296

57bundlesofhundreds6bundlesoftens5bundlesofunits2bundlesofhundreds3bundlesoftens4bundlesofunitsetc

77bundlesofthousands4bundlesofhundreds8bundlesoftens6bundlesofunits8bundlesofthousands0bundlesofhundreds9bundlesoftens0bundlesofunitsetc

96bundlesoftenthousands0bundlesofthousands3bundlesofhundreds0bundlesoftens8bundlesofunits4bundlesoftenthousands6bundlesofthousands9bundlesofhundreds5bundlesoftens1bundleofunitsetc

113bundlesofhundredthousands6bundlesoftenthousands9bundlesofthousands2bundlesofhundreds4bundlesoftens3bundlesofunitsetc

131bundleofmillions7bundlesofhundredthousands5bundlesoftenthousands3bundlesofthousands0bundlesofhundreds0bundlesoftens2bundlesofunits75bundlesofmillions(mayalsobecalled7bundlesoftenmillionsand5bundlesofmillions)2bundlesofhundredthousands0bundlesoftenthousands6bundlesofthousands0bundlesofhundreds0bundlesoftens8bundlesofunitsetc

1527bundlesofbillions(mayalsobecalled2bundlesoftenbillionsand7bundlesofbillions)3bundlesofhundredmillions9bundlesoftenmillions2bundlesofmillions4bundlesofhundredthousands9bundlesoftenthousands6bundlesofthousands0bundlesofhundredstensandunitsetc

17(a)073586

(b)8000008(c)050321(d)70090000012(e)023504910000630003(f)4792086500005(g)04090306001000700008(h)0364575(i)0908006034

19(a)Sixteenandfivethousandths(b)Fiftyandsixhundredseventhousandths(c)Twoten-thousandths(d)Eighty-sevenandninethousandthreehundredseventy-fiveten-thousandths

(e)Thirty-fiveandtwohundredonethousandths(f)Eighty-sixandfivethousandthreehundredninety-twoten-thousandths(g)Twoandthreethousandfourhundredforty-oneten-thousandths(h)Twohundredandthreethousandfourhundredeighty-seventen-thousandths

(i)Twentyandtwothousandseventy-fourten-thousandths(j)Twohundredsixandtenthousandfifty-sevenhundred-thousandths(k)Thirtyandfivehundredsixty-fourthousandths(l)Ninety-sevenandfourthousandthreehundredfifty-sixten-thousandths

21Threehundredfifty-sixdollarsthirty-fivecentssixmills

23(a)$066(b)$080(c)$047(d)$010(e)$120(f)$712

25(a)475(b)5621(c)22(d)10540(e)10765

(f)2555(g)100(h)4444

27(a)XII(b)XVIII(c)XIX(d)XLIII(e)XXXIII(f)XXVIII(g)LVI(h)LXXXII(i)LXXVI(j)XCVII(k)CXVII(l)CCCLXXXV(m)CCXL(n)DXII(o)CDLXX(p)DCCXLII(q)CDXXII(r)CMXLII(s)MCDXXVI(t)MDCCCLXXIV(u)VDCCCLXXII(v)XXIVDCCLXIV(w)CCLVIIDCCCXLVI(x)MCDLDCCXXIX(y)MMMMDCCCXLV(z)MMMMMMMMMMDLXIICMXLII

ChapterI(pp21-23)136912

36121824

535793711153915213111927

79131721916233091827369111315

91997

11$203265

1360

1595

1721

1990

21(a)255(b)244(c)209(d)263(e)270(f)250

23(a)169(b)155(c)140(d)141(e)1879(f)1457(g)1667(h)2039

25907gallons

271039miles

29$525

31

(a)280778(b)295263(c)292690(d)242893

33(a)195564(b)293220(c)208675(d)142415

ChapterII(pp34-36)12242192225612425

3633432392134320372

5152294215059811720

74stepstotheleftorminus4

9minus4deglat

11$24

12(a)124959(b)151833(c)74296(d)161574(e)$305907(f)$873883(g)$38849254(h)$60579179

14(a)25697(b)49779(c)92922(d)$22015250(e)$100035090(f)$91357818

17(a)4228(b)4214(c)4319(d)5659(e)3357(f)2165

19(a)2443393(b)888

(c)1669(d)178556

21(a)1421(b)41135

23$40746

251548576

27(a)53514947534945415347413553453729(b)89868380898479748982756889807162(c)74696459746760537471686574655647

ChapterIII(pp54-56)1540540054000

3176201762001762000

518001800018000018000000

71917000

9(a)28428(b)7136(c)63851(d)54008(e)43362(f)55859(g)43776(h)2700578(i)443772(j)7589594(k)3050260(l)3794186(m)3157596(n)2615057(o)2893230(p)28201925(q)3047385(r)75874332(s)18083583(t)75490868(u)3571632(v)9602484(w)428505(x)4346136(y)3455412(z)7346628

11$14425$6347013$1886

1517424001b17$420

19(a)238(b)272(c)306

(d)304

21(a)7395(b)2352(c)3074(d)1184(e)4355(f)9306(g)5328(h)728(i)306

23(a)945(b)8295(c)6435(d)630(e)4005

25(a)2709(b)2625(c)1316(d)3149(e)3364(f)2016(g)2236

27(a)4275(b)4875(c)5525(d)1925(e)3325(f)4125(g)1225(h)6375

(i)9425(j)$6075(k)$12375(l)$20425

29(a)(b)(c)(d)(e)(f)(g)(h)(i)

31(a)(b)(c)(d)

33$1400

35(a)$21250(b)$12325(c)$2875(d)$1200(e)$1200(f)$2100(g)$1800(h)$41600(i)$900

37$9000

39(a)768(b)1632(c)30008

(d)1368

41(a)516456(b)528849(c)38952(d)890901(e)7628688

43(a)5496(b)4809(c)3456(d)3024(e)7856(f)6874

45(a)8232(b)9024(c)7998(d)7505(e)7216(f)960376

47(a)6384(b)63672(c)3196(d)49088(e)7128(f)2964(g)7392(h)64528

ChapterIV(pp72-74)17

34

520

71acrepermansand acreperboy


117hours

13

(a)214(b)402(c)428

17(a)3(b)Yes2

19(a)2958(b)60(c)80868(d)365(e)1680(f)6912(g)72(h)42(i)139(j)36(k)112

21(a)Subtract2(b)Subtract1

23842

25$101522

ChapterV(pp80-81)

12369235610152346121839272345681012152024304060

3234612

512357111317192329313741434753596167717379838997

72257222252231327222223337222355235217(Note1isaprimefactorofallintegers)

9(a)2531(b)33311(c)2567(d)31319(e)277(f)5711(g)25553(h)333335(i)3107(j)33557(k)3-seventimes5(l)22357(m)771(n)23337(o)23711(p)2333335(q)2-eighttimes3311(r)223351137(s)2237111159(t)2233317149(u)5-sixtimes(v)55761(w)555101(x)222261(y)255723

(z)2221337

11(a)918(b)3570(c)1836(d)2142(e)4080(f)612(g)816(h)72144(i)918(j)2448

13(a)21(b)15(c)28(d)24(e)161536lb1718days

ChapterVI(pp102-106)1

3One-thirdone-sixthone-sixteenthone-twelfthone-twentieththedenominator

5(a)allproper(b) proper improper4 mixednumber(c) proper improper(d) proper improper8 16 mixednumbers(e) proper improper8 17 mixednumbers(f) proper improper6 mixednumber

7(a)8(b)5(c)6(d)8(e)5(f)9(g)7(h)10(i)26(j)6(k)24(l)1(m)1(n)72(o)13

9(a)14(b)28(c)7(d)No

11(a)

(b)(c)(d)(e)(f)

13(a)(b) (c) (d) (e) (f) (g) (h) (i)

15(a) or1(b) or1(c) or3

17(a)(b)(c)(d)(e)(f)

19(a)2(b)15(c)2(d)67(e)1 or(f)1 or(g)1(h)1

(i)28(j)42 (k)139 (l)129

21(a)(b)(c)(d)(e) or8(f)(g)(h)27(i)60(j)(k)9(l)14 23(a)52 1 1 (b)1474 3 2 2

25(a)(b)(c)(d)16(e)2(f)216(g)(h)(i)(j)

27(a)100(b)$688

29

31

33

35$44

371 ozperslice

39286miles

41 $1350$1350$900

43$246$6150$9225$3075$2050

45

4717 rods

ChapterVII(pp121-124)1(a)060421(b)70099053000030000011(c)015504920000630004(d)6782086500006(e)004 0036 8000 8004

3Twelveandfivehundredeighty-fourmillionsixty-twothousandeighteenbillionths

51000100000

7Ten

9(a)8=80=800(b)046=0460=04600(c)738=7380=73800=0738

110040004

13246246

15246024600246000

17246576246576

19(a)032(b)0625(c)014(d)0392(e)01875(f)065(g)04(h)0175(i)03125(j)0115(k)046875(l)0232

21(a)(b)(c)(d)(e)(f)(g)(h)

23(a)1274735(b)18125608(c)22135538(d)7202238

25(a)4234408(b)4494375(c)38316(d)35425(e)553308

27(a)52655625(b)2582398(c)39130222(d)2012315(e)0638027

29(a)15895794(b)38884176(c)17517890(d)112489886(e)54923664(f)21073016

31(a)5977(b)5976

33012

352918

37(a)0078125(b)015625(c)0375(d)03125(e)028125(f)0171875(g)028(h)0184

39$042$007

41$568750

43$34000$7480$10880$12240

45$282

470968lb

4911cents832mills

5132lb

53A0750B0714

ChapterVIII(pp136-139)127

3

502502020002500020002

7(a)900(b)60(c)25(d)(e)(f)28(g)85(h)(i)(j)16(k)04(l)(m)84(n)(o)65(p)60(q)80(r)(s)7(t)(u)

916

11(a)25(b)64(c)100(d)325(e)30(f)420

13$3000$11040$9960

151904votes

1720

19$132389

21(a)406(b)131(c)1278(d)40(e)(f)2323(g)0135(h)2188(i)1662(j)364(k)7150(l)4442(m)5138

23 sqft

2740032

29506675911

3119000054

33$29143

35$13636

3721

39

41$1838

43$150

45$35235

474200students26smaller

49$20588

51 25

53(a)72(b)60(c)006696

55(a)304(b)720(c)2300

ChapterIX(pp155-158)1(a)6(b)6(c)6

3$928$128

5(a)March4(b)March3

7(a)249(b)84(c)118(d)248(e)142

9$789

11$240

13$4919

15(a)$4717(b)$38111(c)$291(d)$1186(e)$28603(f)$370(g)$3431(h)$363(i)$4912

17$4310

19$1438

21$247

23(a)1566(b)6015(c)3063(d)60306(e)3010(f)601566(g)606063(h)603015(i)60601510(j)156(k)60606(l)603063

25$469

27$150

29$9653

31

3385days

35(a)120days(b)140days(c)47days(d)229days

37(a)$120(b)$068(c)$829(d)$240(e)$028(f)$425

39$444500

41

Accumulationfactorfor8yearsat2=1171659Accumulationfactorfor4yearsat4=1169859

43$260000

45$104040

47$1643615

49$6289

51$2693706$693706

ChapterX(pp181-185)1(a)13(b)31(c)17(d)43(e)56(f)65(g)12(h)34(i)56

359

5116

7118811

906

11(a)35(b)85(c)13(d)98

13(a)13(b)12(c)13(d)110(e)1379(f)19(g)(h)1625(i)140(j)1571(k)1115(l)1222(m)14

(n)165(o)160(p)1136(q)1114

15

1741

1914and21

21507080

235134

2515

27 inches

29$43875

31

3322ftx ft

3531

37(a)4(b)7(c)

398

41(a)6(b)2(c)18(d)24(e)18(f)3(g)12(h)32

43$3750

4572feet

471057lb

49(a)10(b)15(c)

512171b

5342men

55821$3528$1536

57 days

5923

61082ohm

63400feet

65x=6

6790psi

6966men

ChapterXI(pp198-199)1$9267

347mph

58562

7435minutes

9$1784

11$209067

1359

15(a)13(b)19

17$340

19$300to$399

21No

23Thereareasmanygradesabove81astherearebelow

ChapterXII(pp214-216)1(a)39inches(b)12feet(c)33yards(d) feet(e)1600rods(f)396inches(g) yards(h) yards(i)5576rods(j) feet(k)6602miles(l)31680feet

38rods2feet

5 cubicinches

783688lbofwater

93025bbl

11$1816

1349280lb

15366

17184

1942doz

2130years

23(a)288sheets(b)1440sheets(c)1920sheets(d)14400sheets

25(a)735dm

(b)74126meters

27(a)0048261sqmeters(b)7480sqdm

2939122dg

316944grains

33102058cg

350664grains

37240Prime360Prime7200Prime

39392pt

41 bu

430883bu

4500181gal

471yr9mo18days4hr44min52sec

493A76sqrd13sqyd6sqft108sqin

517504610meters

537976meters

5511664kg

ChapterXIII(pp238-240)1(a)25(b)64(c)400(d)1(e)121(f)1(g)1000(h)81(i)625(j)4913y(k)571787(l)1953125(m)05625(n)(o)(p)(q)x4

(r)16x2

(s)8b3(t)1953125

34000sqft

548sqyd

7(a)256(b)19683(c)16(d)3(e)axminusy

(f)ax+y(g)4096(h)15625(i)1(j)1

(k)1(l)24(m)(n)(o)

92176782336

11(a)784(b)4489(c)5776(d)7921

13950625

15(a)256(b)2025(c)65025

17(a)99980001(b)9801(c)999998000001

19(a)12(b)4b4

(c)a3b32

(d)x2y4(e)(f)(g)8(h)2646=(i)

21(a)

(b)12(c)

23

(a)

(b)

(c)

25(a)(b)(c)

27(a)4a2y54(b)

(c)

291287feet

316314

33(a)(b)(c)

(d)

(e)(f)01334(g)(h)0949(i)(j)9709(k)00255

35

ChapterXIV(pp258ndash261)1(a)3log5(b)6(c)ndash5log3(d)radic2log9(e)4log3(f)ndash2log2

3(a)44=256(b)xb=a(c)bdeg=1(d)10minus6=0000001(e)104=10000(f)64=1296

501234minus1minus2minus3minus4

7(a)1000(b)64(c)minus5(d)512(e)(f)10(g)(h)(i)7

9(a)0(b)3(c)minus1(d)6(e)1(f)minus5

(g)2(h)8(i)minus3(j)1(k)0(l)0(m)minus14(n)ndash8(o)6(p)minus1

11(a)07740(b)29910(c)88075minus10(d)79441minus10(e)15790(f)01396(g)84857minus10(h)58321minus10(i)57539minus10(j)18048

132log7+log4

15log1944

17(a)log432+log748-log566(b)

19

(a)(b)

(c)(d)

21

(a)3170(b)2633(c)1490(d)1057(e)2681(f)minus2861(g)minus1661(h)1661(i)0792(j)0921(k)minus2861(l)2861

23(a)0340(b)3679(c)00036(d)4016(e)000027(f)164

ChapterXV(pp264ndash265)1(a)5mileswestonthescale(b)Atsea-levelorElev0(c)Atzeroor0degonthescale(d)Zerochangenogainandnoloss

3(a)12(b)6(c)(d)16(e)350

5(a)1464(b)minus378(c)(d)12(e)84(f)minus84

ChapterXVI(pp276ndash277)1(a)24(b)4(c)16(d)5(e)(f)98(g)64(h)16

3a=15S=645

533 4 5 5 6 7 8

7

9250500

11l=39366S=29524

132

15

172

1915

212 12ndash18ndash ndash3

23$70388

ChapterXVII(pp294-296)132174cent(averagecostperquart)

15$5425

ChapterXVIII(pp331-335)1$42

33846

5$3

7$108$1692

9$27354

113825

13$141221

15$54

17$2693

19426

21$4421$269

23$4815

25$3708

27$400

29$229665$10335

31$213359

33$30

35$2650

3733

39373

41$2330036

43$3000$1000

45$799056$199056

47242

49204

511082

53436

55457

57$95940

59$225

61$28712

6323075$230 per$100$2307 per$1000

65$3073196

ChapterXIX(pp353-355)196days

360days

5

7 or194days

940gal

11286

133 qt

141119

173994avoz

19125cc

21

233min

254320gal

27

291253440

31276miles

339728acres

35720deg

374rdquo

ChapterXX(pp398-402)16cdp

3xyzz

515

7No

3y83(y8) etc

11(a)Anumberblesssix(b)Anumberaplusseven(c)Theproductofninethequantitypandthequantityq(d)Seventimesthequantityxplusthreetimesthequantityy(e)Fourtimesthequantitywtheproductlesseight(f)One-sevenththeproductoftwonumbersaandb(g)Threetimesanumberaplusasecondnumberblesssixtimesathirdnumberc

(h)Five-eighthsofacertainlength(i)Ninetimesthesumofthenumbercandthenumber5(j)One-sixthofltimesthesumofthethreetermsAfourtimesBandC(Prismoidalformulaforvolume)

(k)One-fourththesumofcandd(l)One-halftheproductofbandh(Areaofatriangle)(m)One-halftheproductofmandthesquareofv(Formulaforkineticenergy)(n)Thecompoundamount(A)isequaltotheprincipal(P)multipliedbyabinomialoneplustherate(r)saidbinomialhavingbeenmultipliedbyitselfntimes(Compoundinterestformula)

(o)dtimesthesquarerootofthebinomialasquaredplusbsquared(p)One-halfofhtimesthesumofaandb(Areaofatrapezoid)

134a=thenumber4timesaa4=atimesatimesatimesa

15(a)(a+b)8(b)xrdquo(c)30x3

(d)56x5c7

(e)45a7a+1

17(a)x3

(b)3x3(c)

19(a)a4b2c2

(b)x2y4z(c)x3yz3

(d)y-2b2

(e)9xy2z-2

21Yesno

23(a)+$125(b)ndash$25x25(a)45(b)45(c)ndash45(d)ndash45(e)6ab(f)6ab(g)ndash6ab(h)ndash6ab

27ndash21a+66b

29ndash132xndash76y

31-19x+18y+27z

338andash12b+8cndash2d

35(a)15a2+24ab(b)15a2+17abndash18b2

(c)ndash24a7b2c2

(d)40a6+56a5bndash72a4c2

(e)ndash21a5b6c4+35a3b5c6ndash42a3b2c7d2+35a6b4c5d3

(f)40a3+24a2b2+20ab2+12b4

3720x2ndash18xyndash18y2

39(a)16x2ndash12x+4ndash2xndash1(b)ndash10x+6y+8z(c)6a3bndash 4andash1bndash1

(d)

(e)(f)a+5(g)2a+3b(h)3a2+2ab+4b2

41(a)2(5x+12)(xndash1)(b)(x+5)(x+7)(c)(x+3)(xndash12)(d)(xndash7)(xndash4)(e)2(3xndash5)(3x+2)

43(a)(x2ndash5)(x2+5)(b)(yndash7)(y+7)(c)(15a2bndashc3)(15a2b+c3)(d)(2x+3yndash1)(2x+3y+1)(e)(-2a+b)(4andash3b)(f)[(x+y)ndash(kndashl)2][(x+y)+(kndashl)2][(x+y)2+(kndashl)4](g)-3(yndash1)(3yndash5)(h)(5xndash3yndash4cndash2d)(5xndash3y+4c+2d)

45(a)x=7(b)y=32(c)c=37699(d)x=3(e)x=plusmnradic6

(f)x=77(g)x=7(h)x=23(i)x=70(j)x=72

47(a)x=300(b)x=12(c)x=320000(d)x=9(e)y=18(f)y=23(g)x=5(h)(i)(j)x=ndash37(k)(l)(m)(n)y=(o)x=5(p)(q)x=(r)x=16(s)(t)x=7425

49248degF

5110

53400gal

5510001600

571405681

59 orsquoclock

61$9000

6311

659miles7miles

APPENDIXBTABLES

TABLEI

NUMBEROFEACHDAYOFTHEYEAR

TABLE2

AMOUNTATCOMPOUNDINTEREST(I+i)n

TABLE3

FOUR-PLACECOMMONLOGARITHMS

TABLE4

PRESENTVALUEI(I+i)n

INDEX

acseearithmeticalcomplementabscissaaxisof

absolutevalueabstractnumberaccumulationfactoraccuratemethod(ofsimpleinterest)addition

algebraicrulesofassociativelawofbymultiplicationofanaveragecheckingcorrectnessofcumulativelawofdecimalizedofdecimalsofdenominatenumbersoffractionsofpercentsofpositiveandnegativenumbersruleforsymbolof

inalgebraalgebra

symbolsforoperationsinalgebraicexpressionalgebraicquantityalgebraicsymbolraisedtoapoweraliquotparts

fractionalequivalentofindivisioninmultiplication

alternationproportionbyamount

compoundininterestinpercentagetax

anglemeasurementantecedent(inratios)antilogarithm(antilog)apothecariesrsquoweightsapproximationofdecimalsArabicnumeralsystemarcarithmeticandpassim

fundamentaloperationsofarithmeticalcomplement(ac)useinsubtraction

arithmeticmeanseealsoaverage

arithmeticprogressionascendingprogression(series)assessedvaluationassessmentassociativelawforadditionformultiplication

Austrianmethodofsubtractionaverageadvantagesofdeviationfromdisadvantagesofhowtosimplifytwogeneralclassesofweighted

avoirdupoisweightsaxisofabscissasofcoordinatesx

ybankdiscount

bankersrsquomethod(ofsimpleinterest)barchartdivided

100percent

longbargraph

horizontal

verticalbase

definedasfactorraisedtopowerinpercentage

inprofitandloss

intaxation

oflogarithmbasicnumbers

binomial

blockgraph

Boylersquoslaw

Briggssystemoflogs

Britishmoney

broken-linegraph

bundlesofunits

businessusesofpercentagein

buyingcommission

calculation

cancellation

carryingcharge

Cartesiancoordinates

cashdiscount

castingoutelevensinsubtraction

castingoutninesincheckingaddition

insubtraction

tocheckmultiplicationchainfractions

characteristic(oflog)negative

chargecarrying

financingchart

dividedbar

100percentbar

longbar

seealsographcheckingcorrectness

inaddition

inalgebra

indivision

inmultiplication

insubtractionchecknumber(figure)

cipher

circlegraph

circularmeasure

circulatingdecimal

circumference

coefficient

cologarithm(colog)

commissionbuying

salescommondivisor

greatestcommonfactor

greatestcommonfractions

powersofcommonlog

commonmultipleleast

commonparenthesesmethodoffactoringcommonsystemoflogs

commontermmethodoffactoringcomplementarithmetical

complementmultiplication

complexdecimal

complexfraction

compositenumber

compositionproportionby

compoundamount

compound-amount-of-1tables

compoundfraction

compoundinterestaccumulationfactor

compoundproportion

compoundratio

computation

concretenumber

conditionalequation

consequent(inratios)

constant

constant-ratiomethodforinstallmentinterestratecontinuedfraction

conversionofcommonfractionsanddecimalsofdecimalsintopowersoftenofinterest

frequencyofofpercentsintofractionsanddecimals

conversionperiod(ofinterest)coordinatesaxesof

Cartesiancost

gross

net

primecountingmeasures

crossmultiplicationinadditionandsubtractionoffractions

cube

cuberootextractionof

cubicmeasureinmetricsystem

cumulativelawforaddition

formultiplicationcurve(d)graph

decimaladditionof

andUSmoney

approximationof

circulating

complex

conversionoftocommonfractions

topercentdivisionof

equationswith

multiplicationof

powersof

recurring

repeating

simple

subtractionofdecimaldivision

decimalfractionseedecimaldecimalizationinsubtractiondecimalizedaddition

decimalplace

decimalpoint

degree

denominatenumbersadditionof

reductionofascending

descendingsubtractionof

denominatorlowestcommon

depreciation

descendingprogression(series)deviationfromaverage

diagramline

staircasedifference

inpercentage

insubtractiondifferencemethodofcomparinglikequantitiesdigit

directednumber

directionconceptof

negative

positivedirectproportion

directtax

directvariation

discountbank

cash

trade

truedistributionlawsformultiplicationdividedbarchart

dividend

divisibilitybyvariousnumbersdivisionalgebraicrulesfor

bylogs

checkingcorrectnessof

decimal

factoring-of-the-divisormethodofhowtosimplify

long

ofdecimals

offractions

ofpercents

ofpolynomials

ofpositiveandnegativenumbersofpowers

ofpowersoften

ofsamekindofsymbols

ofUSmoney

proportionby

pureproofof

short

symbolofinalgebra

divisionsign

divisorcommon

greatestcommon

trialdrymeasure

inmetricsystem

effectiverateofinterest

elevenasachecknumber

emptyingproblems

ldquoequaladditionsrdquomethodofsubtractionequals(to)

equalssign

equationconditional

linear

quadratic

rootof

simple

solutionof

withdecimalsevennumber

evolutionsymbolof

exactmethod(ofsimpleinterest)excess-of-ninesmethodofcheckingdivisionexponentfractional

lawsof

logarithmdefinedas

negative

raisedtoapower

signof

zero

seealsopowersexpressionalgebraic

extrapolation

ldquoextremesrdquo(ofproportion)

factorcommon

greatestcommon

literal

primefactoring

commonparenthesesmethod

commontermmethod

inalgebra

productoftwobinomialsmethodfactoring-of-the-divisormethodindivisionfillingproblems

finance

financingcharge

fluidounces

formula

fourthroot

fractionadditionof

chain

commonseefractioncomplex

compound

continued

conversionoftodecimals

topercents

decimalseedecimaldivisionof

improper

multiplicationof

powersof

proper

reductiontolowestterms

rootof

simple

subtractionof

unit

vulgarfractionalequivalentofaliquotpartsfractionalexponent

fractionalplaces

Frenchmoney

frequencydistributiongraphfrequencypolygon

futurevalue(worth)

GCD(gcd)seegreatestcommondivisorgeometricmean

geometricprogression

Germanmoney

gram

graphadvantagesanddisadvantagesofbar

horizontal

verticalblock

broken-line

circle

curve(d)

frequencydistribution

ofquadraticformula

pie

rectangle

smooth-linegreatestcommondivisor(GCD)

ruleforfindinggreatestcommonfactor

grosscost

grossprofit

grosspurchases

grosssalesharmonicmean

harmonicprogression

Hookersquoslaw

horizontalbargraph

100percentbarchart

ldquohundredsrdquoposition

identity

imaginarynumber

imperfectpower

improperfraction

incometax

index

indexnumber

indirecttax

initialline(ofangle)

installmentpurchaseproblems

constant-ratiomethodofinteger

integralnumber

interestcompound

accumulationfactorinconversionof

rateofeffective

nominalsimple

formulafor

methodsoffiguringsix-day6percentmethodofsixty-day6percentmethodof

interestcost

interestearned

interpolation

inverseproportion

inverseratio

inversevariation

inversionproportionby

invertedmultiplication

invertedsubtraction

involutionsymbolof

irrationalnumberjointvariation

keynumber(figure)

LCDseelowestcommondenominatorlanguageofvariation

lawsofadditionseeadditionofmultiplicationseemultiplication

leastcommonmultiple(LCM)left-handmultiplication

left-handsubtraction

leverprincipleof

licence

liketerms

lineinitial(ofangle)

terminal(ofangle)linearequation

linearmeasureinmetricsystem

linediagram

liquidmeasureinmetricsystem

liter

literalfactor

literalnumber

loans

logarithm(log)

accuracyofcomputationby

Briggssystemof

characteristicofnegative

common

commonsystemof

divisionby

extractionofrootsby

mantissaof

multiplicationby

Napieriansystemof

natural

naturalsystemof

proportionalpartof

raisingtopowersby

tableoflongbarchart

longdivisionrulefor

loss

lowestcommondenominator(LCD)

makingchangemethodofseeAustrianmethodmantissa

mapsstatistical

marginofprofit

meanarithmeticseealsoaveragegeometric

harmonic

proportional

squareofldquomeansrdquo(ofproportion)

meanvalue

measurecircular

counting

cubic

dry

linear

liquid

metricsystemof

paper

square

timemedian

advantagesof

disadvantagesofmerchantsrsquoruleinpartialpaymentproblemsmeter

metricsystemofweightsandmeasuresmillintaxmatters

minuend

minussigninalgebra

minute(partofdegree)

miscellaneousseries

mixednumber

mixtures

modeadvantagesof

disadvantagesofmodelsscale

moneyBritish

French

GermanUnitedStates

anddecimals

divisionof

howwrittenmonomial

multiplecommon

leastcommonmultiplicand

multiplicationalgebraicrulesfor

associativelawfor

bylogs


complement

cross

cumulativelawof

distributivelawsfor

howtosimplify

inverted

left-hand

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften

ruleforlikeandunlikesignsinsymbolofinalgebra

multiplicationsign

multiplicationtable

multiplier

Napieriansystemoflogs

naturallogs

naturalsystemoflogs

negativedirection

negativeexponent

negativenumbersadditionof

divisionof

multiplicationof

subtractionofnetcost

netprofit

netpurchases

netsales

nineaschecknumberinadditionseealsocastingoutnines

nominalrateofinterest

nought

number

abstract

Arabic

basic

composite

concrete

denominateadditionof

reductionof

subtractionofdirected

even

imaginary

index

integral

irrational

literal

mixed

negativeadditionof

divisionof

multiplicationof

subtractionofodd

positiveadditionof

divisionof

multiplicationof

subtractionofprime

real

Roman

signed

specific

wholenumberscale

numerator

oddnumber

ldquoontimerdquoseeinstallmentpurchaseproblemsoperationsofarithmeticfundamental

direct

inverse

symbolsoforders

ordinarymethod(ofsimpleinterest)ordinateaxisof

origin

papermeasure

parabola

parentheses

partialpaymentsmerchantsrsquorulefor

USruleforpercent(percentage)additionof

businessusesof

conversiontodecimalsandfractionsdivisionof

inprofitandloss

lessthan1percent

multiplicationof

relationtoratio

subtractionofperfectpower

period

pictograph

piegraph(chart)

placesdecimal

fractionalplottinggraphofquadraticformula

straightlinerelationshipplussigninalgebra

pointdecimal

polltax

polygonfrequency

polynomialmultiplicationof

positivedirection

positivenumbersadditionof

divisionof

multiplicationof

subtractionofpowersdivisionof

imperfect

multiplicationof

ofcommonfractions

ofdecimals

oftenconvertingdecimalsinto

divisionof

multiplicationofperfect

raisingtobylogspowerszeroseealsoexponentpresentvalue(worth)

priceselling

primecost

primefactor

primenumber

principalininterest

product

ldquoproductoftwobinomialsrdquomethodoffactoringprofitgross

marginof

netprogressionarithmetic

ascending

descending

geometric

harmonic

seealsoseriesproofpureofdivision

properfraction

propertytax

proportionbyalternation

bycomposition

bydivision

byinversion

compound

direct

inverseproportionalmean

proportionalpartoflog

protractor

purchasesgross

net

return

quadrants

quadraticequation

quadraticformulagraphof

quantityalgebraic

constant

variablequotient

radical

reducedtosimplestform

similarradicalsign

radicand

rateinpercentage

inprofitandloss

ofinteresteffective

nominaltax

workingofspeedratio

compound

howtosimplify

inaseries

inverse

relationtopercent

rulesforcalculationof

symbolofratiomethodofcomparinglikequantitiesrealnumber

receptacles

reciprocal

rectanglegraph

recurringdecimal

reductionofdenominatenumbers

ascending

descendingoffractions

ofradicalstosimplestformremainder

indivision

insubtractionrepeatingdecimal

returnpurchases

Romannumeralsystem

rootcube

extractionof

extractionofbylogs

fourth

ofequation

offraction

squareextractionof

salesgross

netsalescommission

scaleformodelsandmaps

numbersecond(partofdegree)

sellingprice

seriesascending

descending

miscellaneous

sumtoinfinity

seealsoprogressionshortdivision

signofexponents

ruleforsignednumber

similarradicals

simpledecimal

simpleequation

simplefraction

simpleinterestbankersrsquomethodfor

exactmethodfor

formulafor

ordinarymethodforsimplifying

algebraicexpressions

averages

division

multiplication

ratios

squaringofnumbers

subtractionsmooth-linegraph

solutionofequations

solutions(mixtures)

solvingforvariableinformulaldquosomuchperhundredrdquo

specificnumber

speedworkingratesof

squareofanumber

ofthemeansquaremeasure

inmetricsystemsquareroot

extractionofsquaringofnumbershowtosimplifystaircasediagram

statisticalmap

statistics

straightlinerelationship

subtractionalgebraicrulesfor

Austrianmethodof


howtosimplify

inverted

left-hand

methodofldquoequaladditionsrdquoinofdecimals

ofdenominatenumbers

offractions

ofpercents

ofplusquantities

ofpositiveandnegativenumbersrulefor

symbolofinalgebra

subtractiontable

subtrahend

sumofseriestoinfinity

surtax

symboloffundamentaloperations

tablescompound-amount-of-1

multiplication

oflogs

subtractiontanks

taxdirect

income

indirect

poll

property

totaltaxamount

taxmatters

taxrate

tenpowersofseepowersoftenldquotensrdquoposition

term(algebraic)like

terminalline(ofangle)

ldquothereforerdquosymbol

timeininterest

measurementofldquotimesrdquosign

totaltax

tradediscount

trialdivisor

troyweights

truediscount

unit

UnitedStatesmoneyseemoneyUnitedStatesUnitedStatesruleinpartialpaymentproblemsUnited

StatesweightsseeweightsUnitedStatesunitfraction

ldquounitsrdquoposition

valuationassessed

valueabsolute

future

mean

presentvariable

solvingforinformulavariation

direct

inverse

joint

languageofvertex(ofangle)

verticalbargraph

vulgarfraction

weightedaverage

weightsmetricsystem

UnitedStatesapothecariesrsquo

avoirdupois

troywholenumber

workingratesofspeed

worthfuture

present

xaxis

yaxis

zeroeffectondecimals

effectonnumberszeropower(exponent)

ACATALOGOFSELECTEDDOVERBOOKS

INALLFIELDSOFINTEREST

ACATALOGOFSELECTEDDOVERBOOKSINALLFIELDSOFINTEREST

CONCERNINGTHESPIRITUALINARTWassilyKandinskyPioneeringworkbyfatherofabstractartThoughtsoncolortheorynatureofartAnalysisofearliermasters12illustrations80ppoftext5times8frac12

0-486-23411-8

CELTICARTTheMethodsofConstructionGeorgeBainSimplegeometrictechniquesformakingCelticinterlacementsspiralsKells-typeinitialsanimalshumansetcOver500illustrations160pp9times12(AvailableinUSonly)

0-486-22923-8

ANATLASOFANATOMYFORARTISTSFritzSchiderMostthoroughreferenceworkonartanatomyintheworldHundredsofillustrationsincludingselectionsfromworksbyVesaliusLeonardoGoyaIngresMichelangeloothers593illustrations192pp7⅛times10frac14

0-486-20241-0

CELTICHANDSTROKE-BY-STROKE(IrishHalf-UncialfromldquoTheBookofKellsrdquo)AnArthurBakerCalligraphyManualArthurBakerCompleteguideto

creatingeachletterofthealphabetindistinctiveCelticmannerCovershandpositionstrokespensinkspapermoreIllustrated48pp8frac14times11

0-486-24336-2

EASYORIGAMIJohnMontrollCharmingcollectionof32projects(hatcuppelicanpianoswanmanymore)speciallydesignedforthenoviceorigamihobbyistClearlyillustratedeasy-to-followinstructionsinsurethatevenbeginningpaper-crafterswillachievesuccessfulresults48pp8frac14times11

0-486-27298-2

BLOOMINGDALErsquoSILLUSTRATED1886CATALOGFashionsDryGoodsandHousewaresBloomingdaleBrothersFamedmerchantsrsquoextremelyrarecatalogdepictingabout1700productsclothinghousewaresfirearmsdrygoodsjewelrymoreInvaluablefordatingidentifyingvintageitemsAlsocopyright-freegraphicsforartistsdesignersCo-publishedwithHenryFordMuseumampGreenfieldVillage160pp8frac14times11

0-486-25780-0

THEARTOFWORLDLYWISDOMBaltasarGracianldquoThinkwiththefewandspeakwiththemanyrdquoldquoFriendsareasecondexistencerdquoandldquoBeabletoforgetrdquoareamongthis1637volumersquos300pithymaximsAperfectsourceofmentalandspiritualrefreshmentitcanbeopenedatrandomandappreciatedeitherinbrieforatlength128pp5⅜times8frac12

0-486-44034-6

JOHNSONrsquoSDICTIONARYAModernSelectionSamuelJohnson(ELMcAdamandGeorgeMilneeds)Thismodernversionreducestheoriginal1755editionrsquos2300pagesofdefinitionsandliteraryexamplestoamoremanageablelengthretainingtheverbalpleasureandhistoricalcuriosityoftheoriginal480pp5 times8frac14

0-486-44089-3

ADVENTURESOFHUCKLEBERRYFINNMarkTwainIllustratedbyEWKembleAworkofeternalrichnessandcomplexityasourceofongoingcriticaldebateandaliterarylandmarkTwainrsquos1885masterpieceaboutabarefootboyrsquosjourneyofself-discoveryhasenthralledreadersaroundtheworldThishandsomeclothboundreproductionofthefirsteditionfeaturesall174oftheoriginalblack-and-whiteillustrations368pp5times8frac12

0-486-44322-1

STICKLEYCRAFTSMANFURNITURECATALOGSGustavStickleyandLampJGStickleyBeautifulfunctionalfurnitureintwoauthenticcatalogsfrom1910594illustrationsincluding277photosshowsettlesrockersarmchairsrecliningchairsbookcasesdeskstables183pp6frac12times9frac14

0-486-23838-5

AMERICANLOCOMOTIVESINHISTORICPHOTOGRAPHS1858to1949RonZiel(ed)Ararecollectionof126meticulouslydetailedofficialphotographscalledldquobuilderportraitsrdquoofAmericanlocomotivesthatmajesticallychronicletheriseofsteamlocomotivepowerinAmericaIntroductionDetailedcaptionsxi+129pp9times12

0-486-27393-8

AMERICArsquoSLIGHTHOUSESAnIllustratedHistoryFrancisRossHollandJrDelightfullywrittenprofuselyillustratedfact-filledsurveyofover200Americanlighthousessince1716Historyanecdotestechnologicaladvancesmore240pp8times10frac34

0-486-25576-X

TOWARDSANEWARCHITECTURELeCorbusierPioneeringmanifestobyfounderofldquoInternationalSchoolrdquoTechnicalandaesthetictheoriesviewsofindustryeconomicsrelationofformtofunctionldquomass-productionsplitrdquoandmuchmoreProfuselyillustrated320pp6⅛times9frac14(AvailableinUSonly)0-486-25023-7

HOWTHEOTHERHALFLIVESJacobRiisFamousjournalisticrecordexposingpovertyanddegradationofNewYorkslumsaround1900bymajorsocialreformer100strikingandinfluentialphotographs233pp10times7⅞

0-486-22012-5

FRUITKEYANDTWIGKEYTOTREESANDSHRUBSWilliamMHarlowOneofthehandiestandmostwidelyusedidentificationaidsFruitkeycovers120deciduousandevergreenspeciestwigkey160deciduousspeciesEasilyusedOver300photographs126pp5times8frac12

0-486-20511-8

COMMONBIRDSONGSDrDonaldJBorrorSongsof60mostcommonUSbirdsrobinssparrowscardinalsbluejaysfinchesmore-arrangedinorderofincreasingcomplexityUpto9variationsofsongsofeachspecies

Cassetteandmanual0-486-99911-4

ORCHIDSASHOUSEPLANTSRebeccaTysonNorthenGrowcattleyasandmanyotherkindsoforchids-inawindowinacaseorunderartificiallight63illustrations148pp5⅜times8frac12

0-486-23261-1

MONSTERMAZESDavePhillipsMasterfulmazesatfourlevelsofdifficultyAvoiddeadlyperilsandevilcreaturestofindmagicaltreasuresSolutionsforall32excitingillustratedpuzzles48pp8frac14times11

0-486-26005-4

MOZARTrsquoSDONGIOVANNI(DOVEROPERALIBRETTOSERIES)WolfgangAmadeusMozartIntroducedandtranslatedbyEllenHBleilerStandardItalianlibrettowithcompleteEnglishtranslationConvenientandthoroughlyportablemdashanidealcompanionforreadingalongwitharecordingortheperformanceitselfIntroductionListofcharactersPlotsummary121pp5frac14times8frac12

0-486-24944-1

FRANKLLOYDWRIGHTrsquoSDANAHOUSEDonaldHoffmannPictorialessayofresidentialmasterpiecewithover160interiorandexteriorphotosplanselevationssketchesandstudies128pp9frac14times10frac34

0-486-29120-0

THECLARINETANDCLARINETPLAYINGDavidPinoLivelycomprehensiveworkfeaturessuggestionsabouttechniquemusicianshipandmusicalinterpretationaswellasguidelinesforteachingmakingyourownreedsandpreparingforpublicperformanceIncludesanintriguinglookatclarinethistoryldquoAgodsendrdquoTheClarinetJournaloftheInternationalClarinetSocietyAppendixes7illus320pp5⅜times8frac12

0-486-40270-3

HOLLYWOODGLAMORPORTRAITSJohnKobal(ed)145photosfrom1926-49HarlowGableBogartBacall94starsinallFullbackgroundonphotographerstechnicalaspects160pp8times11frac14

0-486-23352-9

THERAVENANDOTHERFAVORITEPOEMSEdgarAllanPoeOver40oftheauthorrsquosmostmemorablepoemsldquoTheBellsrdquoldquoUlalumerdquoldquoIsrafelrdquoldquoToHelenrdquoldquoTheConquerorWormrdquoldquoEldoradordquoldquoAnnabelLeerdquomanymoreAlphabeticlistsoftitlesandfirstlines64pp5 times8frac14

0-486-26685-0

PERSONALMEMOIRSOFUSGRANTUlyssesSimpsonGrantIntelligentdeeplymovingfirsthandaccountofCivilWarcampaignsconsideredbymanythefinestmilitarymemoirseverwrittenIncludeslettershistoricphotographsmapsandmore528pp6⅛times9frac14

0-486-28587-1

ANCIENTEGYPTIANMATERIALSANDINDUSTRIESALucasandJHarrisFascinatingcomprehensivethoroughlydocumentedtextdescribesthisancientcivilizationrsquosvastresourcesandtheprocessesthatincorporatedthemindailylifeincludingtheuseofanimalproductsbuildingmaterialscosmeticsperfumesandincensefibersglazedwareglassanditsmanufacturematerialsusedinthemummificationprocessandmuchmore544pp6⅛times9frac14(AvailableinUSonly)0-486-40446-3

RUSSIANSTORIESRUSSKIERASSKAZYADual-LanguageBookeditedbyGlebStruveTwelvetalesbysuchmastersasChekhovTolstoyDostoevskyPushkinothersExcellentword-for-wordEnglishtranslationsonfacingpagesplusteachingandstudyaidsRussianEnglishvocabularybiographicalcriticalintroductionsmore416pp5⅜times8frac12

0-486-26244-8

PHILADELPHIATHENANDNOW60SitesPhotographedinthePastandPresentKennethFinkelandSusanOyamaRarephotographsofCityHallLoganSquareIndependenceHallBetsyRossHouseotherlandmarksjuxtaposedwithcontemporaryviewsCaptureschangingfaceofhistoriccityIntroductionCaptions128pp8frac14times11

0-486-25790-8

NORTHAMERICANINDIANLIFECustomsandTraditionsof23Tribes

ElsieClewsParsons(ed)27fictionalizedessaysbynotedanthropologistsexaminereligioncustomsgovernmentadditionalfacetsoflifeamongtheWinnebagoCrowZuniEskimoothertribes480pp6⅛times9frac14

0-486-27377-6

TECHNICALMANUALANDDICTIONARYOFCLASSICALBALLETGailGrantDefinesexplainscommentsonstepsmovementsposesandconcepts15-pagepictorialsectionBasicbookforstudentviewer127pp5times8frac12

0-486-21843-0

THEMALEANDFEMALEFIGUREINMOTION60ClassicPhotographicSequencesEadweardMuybridge60true-actionphotographsofmenandwomenwalkingrunningclimbingbendingturningetcreproducedfromrare19th-centurymasterpiecevi+121pp9times12

0-486-24745-7

ANIMALS1419Copyright-FreeIllustrationsofMammalsBirdsFishInsectsetcJimHarter(ed)Clearwoodengravingspresentinextremelylifelikeposesover1000speciesofanimalsOneofthemostextensivepictorialsourcebooksofitskindCaptionsIndex284pp9times12

0-486-23766-4

1001QUESTIONSANSWEREDABOUTTHESEASHORENJBerrillandJacquelynBerrillQueriesansweredaboutdolphinsseasnailsspongesstarfishfishesshorebirdsmanyothersCoversappearancebreedinggrowthfeedingmuchmore305pp5frac14times8frac14

0-486-23366-9

ATTRACTINGBIRDSTOYOURYARDWilliamJWeberEasy-to-followguideoffersadviceonhowtoattractthegreatestdiversityofbirdsbirdhousesfeederswaterandwaterersmuchmore96pp5 times8frac14

0-486-28927-3

MEDICINALANDOTHERUSESOFNORTHAMERICANPLANTSAHistoricalSurveywithSpecialReferencetotheEasternIndianTribesCharlotteErichsen-BrownChronologicalhistoricalcitationsdocument500yearsofusageofplantstreesshrubsnativetoeasternCanadanortheasternUSAlsocompleteidentifyinginformation343illustrations544pp6frac12times9frac14

0-486-25951-X

STORYBOOKMAZESDavePhillips23storiesandmazesontwo-pagespreadsWizardofOzTreasureIslandRobinHoodetcSolutions64pp8frac14times11

0-486-23628-5

AMERICANNEGROSONGS230FolkSongsandSpiritualsReligiousandSecularJohnWWorkThisauthoritativestudytracestheAfricaninfluencesofsongssungandplayedbyblackAmericansatworkinchurchandasentertainmentTheauthordiscussesthelyricsignificanceofsuchsongsasldquoSwingLowSweetChariotrdquoldquoJohnHenryrdquoandothersandoffersthewordsandmusicfor230songsBibliographyIndexofSongTitles272pp6frac12times9frac14

0-486-40271-1

MOVIE-STARPORTRAITSOFTHEFORTIESJohnKobal(ed)163glamorstudiophotosof106starsofthe1940sRitaHayworthAvaGardnerMarlonBrandoClarkGablemanymore176pp8times11frac14

0-486-23546-7

YEKLandTHEIMPORTEDBRIDEGROOMANDOTHERSTORIESOFYIDDISHNEWYORKAbrahamCahanFilmHesterStreetbasedonYekl(1896)NovelotherstoriesamongfirstaboutJewishimmigrantsonNYrsquosEastSide240pp5⅜times8frac12

0-486-22427-9

SELECTEDPOEMSWaltWhitmanGeneroussamplingfromLeavesofGrassTwenty-fourpoemsincludeldquoIHearAmericaSingingrdquoldquoSongoftheOpenRoadrdquoldquoISingtheBodyElectricrdquoldquoWhenLilacsLastintheDooryardBloomrsquodrdquoldquoOCaptainMyCaptainrdquo-allreprintedfromanauthoritativeeditionListsoftitlesandfirstlines128pp5 times8frac140-486-26878-0

SONGSOFEXPERIENCEFacsimileReproductionwith26PlatesinFullColorWilliamBlake26full-colorplatesfromarare1826editionIncludesldquoTheTygerrdquoldquoLondonrdquoldquoHolyThursdayrdquoandotherpoemsPrintedtextofpoems48pp5frac14times7

0-486-24636-1

THEBESTTALESOFHOFFMANNETAHoffmann10ofHoffmannrsquosmostimportantstoriesldquoNutcrackerandtheKingofMicerdquoldquoTheGoldenFlowerpotrdquoetc458pp5⅜times8frac12

0-486-21793-0

THEBOOKOFTEAKakuzoOkakuraMinorclassicoftheOriententertainingcharmingexplanationinterpretationoftraditionalJapanesecultureintermsofteaceremony94pp5⅜times8frac12

0-486-20070-1

FRENCHSTORIESCONTESFRANCcedilAISADual-LanguageBookWallaceFowlieTenstoriesbyFrenchmastersVoltairetoCamusldquoMicromegasrdquobyVoltaireldquoTheAtheistrsquosMassrdquobyBalzacldquoMinuetrdquobydeMaupassantldquoTheGuestrdquobyCamussixmoreExcellentEnglishtranslationsonfacingpagesAlsoFrench-Englishvocabularylistexercisesmore352pp5times8frac12

0-486-26443-2

CHICAGOATTHETURNOFTHECENTURYINPHOTOGRAPHS122HistoricViewsfromtheCollectionsoftheChicagoHistoricalSocietyLarryAViskochilRarelarge-formatprintsofferdetailedviewsofCityHallStateStreettheLoopHullHouseUnionStationmanyotherlandmarkscirca1904-1913IntroductionCaptionsMaps144pp9times12frac14

0-486-24656-6

OLDBROOKLYNINEARLYPHOTOGRAPHS1865-1929WilliamLeeYoungerLunaParkGravesendracetrackconstructionofGrandArmyPlazamovingofHotelBrightonetc157previouslyunpublishedphotographs165pp8⅞times11frac34

0-486-23587-4

THEMYTHSOFTHENORTHAMERICANINDIANSLewisSpenceRichanthologyofthemythsandlegendsoftheAlgonquinsIroquoisPawneesandSiouxprefacedbyanextensivehistoricalandethnologicalcommentary36illustrations480pp5times8frac12

0-486-25967-6

ANENCYCLOPEDIAOFBATTLESAccountsofOver1560Battlesfrom1479BCtothePresentDavidEggenbergerEssentialdetailsofeverymajor

battleinrecordedhistoryfromthefirstbattleofMegiddoin1479BCtoGrenadain1984ListofBattleMapsNewAppendixcoveringtheyears1967-1984Index99illustrations544pp6frac12times9frac14

0-486-24913-1

SAILINGALONEAROUNDTHEWORLDCaptainJoshuaSlocumFirstmantosailaroundtheworldaloneinsmallboatOneofgreatfeatsofseamanshiptoldindelightfulmanner67illustrations294pp5times8frac12

0-486-20326-3

ANARCHISMANDOTHERESSAYSEmmaGoldmanPowerfulpenetratingpropheticessaysondirectactionroleofminoritiesprisonreformpuritanhypocrisyviolenceetc271pp5⅜times8frac12

0-486-22484-8

MYTHSOFTHEHINDUSANDBUDDHISTSAnandaKCoomaraswamyandSisterNiveditaGreatstoriesoftheepicsdeedsofKrishnaShivatakenfrompuranasVedasfolktalesetc32illustrations400pp5⅜times8frac12

0-486-21759-0

MYBONDAGEANDMYFREEDOMFrederickDouglassBornaslaveDouglassbecameoutspokenforceinantislaverymovementThebestofDouglassrsquoautobiographiesGraphicdescriptionofslavelife464pp5times8frac12

0-486-22457-0

FOLLOWINGTHEEQUATORAJourneyAroundtheWorldMarkTwainFascinatinghumorousaccountof1897voyagetoHawaiiAustraliaIndiaNewZealandetcIronicbemusedreportsonpeoplescustomsclimatefloraandfaunapoliticsmuchmore197illustrations720pp5⅜times8frac12

0-486-26113-1

THEPEOPLECALLEDSHAKERSEdwardDAndrewsDefinitivestudyofShakersoriginsbeliefspracticesdancessocialorganizationfurnitureandcraftsetc33illustrations351pp5⅜times8frac12

0-486-21081-2

THEMYTHSOFGREECEANDROMEHAGuerberAclassicofmythologygenerouslyillustratedlongprizedforitssimplegraphicaccurateretellingoftheprincipalmythsofGreeceandRomeandforitscommentaryontheiroriginsandsignificanceWith64illustrationsbyMichelangeloRaphael

TitianRubensCanovaBerniniandothers480pp5⅜times8frac12

0-486-27584-1

PSYCHOLOGYOFMUSICCarlESeashoreClassicworkdiscussesmusicasamediumfrompsychologicalviewpointCleartreatmentofphysicalacousticsauditoryapparatussoundperceptiondevelopmentofmusicalskillsnatureofmusicalfeelinghostofothertopics88figures408pp5⅜times8frac12

0-486-21851-1

LIFEINANCIENTEGYPTAdolfErmanFullestmostthoroughdetailedolderaccountwithmuchnotinmorerecentbooksdomesticlifereligionmagicmedicinecommercemuchmoreManyillustrationsreproducetombpaintingscarvingshieroglyphsetc597pp5⅜times8frac12

0-486-22632-8

SUNDIALSTheirTheoryandConstructionAlbertWaughFarandawaythebestmostthoroughcoverageofideasmathematicsconcernedtypesconstructionadjustinganywhereSimplenontechnicaltreatmentallowsevenchildrentobuildseveralofthesedialsOver100illustrations230pp5⅜times8frac12

0-486-22947-5

THEORETICALHYDRODYNAMICSLMMilne-ThomsonClassicexpositionofthemathematicaltheoryoffluidmotionapplicabletobothhydrodynamicsandaerodynamicsOver600exercises768pp6⅛times9frac14

0-486-68970-0

OLD-TIMEVIGNETTESINFULLCOLORCarolBelangerGrafton(ed)Over390charmingoftensentimentalillustrationsselectedfromarchivesofVictoriangraphicsmdashprettywomenposingchildrenplayingfoodflowerskittensandpuppiessmilingcherubsbirdsandbutterfliesmuchmoreAllcopyright-free48pp9frac14times12frac14

0-486-27269-9

PERSPECTIVEFORARTISTSRexVicatColeDepthperspectiveofskyandseashadowsmuchmorenotusuallycovered391diagrams81reproductionsofdrawingsandpaintings279pp5⅜times8frac12

0-486-22487-2

DRAWINGTHELIVINGFIGUREJosephSheppardInnovativeapproachto

artisticanatomyfocusesonspecificsofsurfaceanatomyratherthanmusclesandbonesOver170drawingsoflivemodelsinfrontbackandsideviewsandinwidelyvaryingposesAccompanyingdiagrams177illustrationsIntroductionIndex144pp8x11frac14

0-486-26723-7

GOTHICANDOLDENGLISHALPHABETS100CompleteFontsDanXSoloAddpowerelegancetoposterssignsothergraphicswith100stunningcopyright-freealphabetsBlackstoneDolbeyGermania97moremdashincludingmanylower-casenumeralspunctuationmarks104pp8⅛times11

0-486-24695-7

THEBOOKOFWOODCARVINGCharlesMarshallSayersFinestbookforbeginnersdiscussesfundamentalsandoffers34designsldquoAbsolutelyfirstratewellthoughtoutandwellexecutedrdquo-EJTangerman118pp7frac34times10⅝

0-486-23654-4

ILLUSTRATEDCATALOGOFCIVILWARMILITARYGOODSUnionArmyWeaponsInsigniaUniformAccessoriesandOtherEquipmentSchuylerHartleyandGrahamRareprofuselyillustrated1846catalogincludesUnionArmyuniformanddressregulationsarmsandammunitioncoatsinsigniaflagsswordsriflesetc226illustrations160pp9times12

0-486-24939-5

WOMENrsquoSFASHIONSOFTHEEARLY1900sAnUnabridgedRepublicationofldquoNewYorkFashions1909rdquoNationalCloakampSuitCoRarecatalogofmail-orderfashionsdocumentswomenrsquosandchildrenrsquosclothingstylesshortlyaftertheturnofthecenturyCaptionsofferfulldescriptionspricesInvaluableresourceforfashioncostumehistoriansApproximately725illustrations128pp8⅜times11frac14

0-486-27276-1

HOWTODOBEADWORKMaryWhiteFundamentalbookoncraftfromsimpleprojectstofive-beadchainsandwovenworks106illustrations142pp5⅜times8

0-486-20697-1

THE1912AND1915GUSTAVSTICKLEYFURNITURECATALOGSGustavStickleyWithover200detailedillustrationsanddescriptionsthesetwo

catalogsareessentialreadingandreferencematerialsandidentificationguidesforStickleyfurnitureCaptionscitematerialsdimensionsandprices112pp6frac12times9frac14

0-486-26676-1

EARLYAMERICANLOCOMOTIVESJohnHWhiteJrFinestlocomotiveengravingsfromearly19thcenturyhistorical(1804-74)main-line(after1870)specialforeignetc147plates142pp11times8frac14

0-486-22772-3

LITTLEBOOKOFEARLYAMERICANCRAFTSANDTRADESPeterStockham(ed)1807childrenrsquosbookexplainscraftsandtradesbakerhattercooperpotterandmanyothers23copperplateillustrations140pp4⅝times6

0-486-23336-7

VICTORIANFASHIONSANDCOSTUMESFROMHARPERrsquoSBAZAR1867-1898StellaBlum(ed)Daycostumeseveningwearsportsclothesshoeshatsotheraccessoriesinover1000detailedengravings320pp9times12frac14

0-486-22990-4

THELONGISLANDRAILROADINEARLYPHOTOGRAPHSRonZielOver220rarephotosinformativetextdocumentorigin(1844)anddevelopmentofrailserviceonLongIslandVintageviewsofearlytrainslocomotivesstationspassengerscrewsmuchmoreCaptions8⅞times11frac34

0-486-26301-0

VOYAGEOFTHELIBERDADEJoshuaSlocumGreat19th-centurymarinerrsquosthrillingfirsthandaccountofthewreckofhisshipoffSouthAmericathe35-footboathebuiltfromthewreckageanditsremarkablevoyagehome128pp5times8frac12

0-486-40022-0

TENBOOKSONARCHITECTUREVitruviusThemostimportantbookeverwrittenonarchitectureEarlyRomanaestheticstechnologyclassicalorderssiteselectionallotheraspectsMorgantranslation331pp5times8frac12

0-486-20645-9

THEHUMANFIGUREINMOTIONEadweardMuybridgeMorethan4500stopped-actionphotosinactionseriesshowingundrapedmenwomenchildren

jumpinglyingdownthrowingsittingwrestlingcarryingetc390pp7⅞times10

0-486-20204-6Clothbd

TREESOFTHEEASTERNANDCENTRALUNITEDSTATESANDCANADAWilliamMHarlowBestone-volumeguideto140treesFulldescriptionswoodlorerangeetcOver600illustrationsHandysize288pp4frac12times6

0-486-20395-6

GROWINGANDUSINGHERBSANDSPICESMiloMiloradovichVersatilehandbookprovidesalltheinformationneededforcultivationanduseofalltheherbsandspicesavailableinNorthAmerica4illustrationsIndexGlossary236pp5times8frac12

0-486-25058-X

BIGBOOKOFMAZESANDLABYRINTHSWalterShepherd50mazesandlabyrinthsinall-classicalsolidrippleandmore-inonegreatvolumePerfectinexpensivepuzzlerforcleveryoungstersFullsolutions112pp8⅛times11

0-486-22951-3

PIANOTUNINGJCreeFischerClearestbestbookforbeginneramateurSimplerepairsraisingdroppednotestuningbyeasymethodofflattenedfifthsNopreviousskillsneeded4illustrations201pp5times8frac12

0-486-23267-0

HINTSTOSINGERSLillianNordicaSelectingtherightteacherdevelopingconfidenceovercomingstagefrightandmanyotherimportantskillsreceivethoughtfuldiscussioninthisindispensibleguidewrittenbyaworld-famousdivaoffourdecadesrsquoexperience96pp5times8frac12

0-486-40094-8

THECOMPLETENONSENSEOFEDWARDLEAREdwardLearAllnonsenselimerickszanyalphabetsOwlandPussycatsongsnonsensebotanyetcillustratedbyLearTotalof320pp5times8frac12(AvailableinUSonly)

0-486-20167-8

VICTORIANPARLOURPOETRYAnAnnotatedAnthologyMichaelRTurner117gemsbyLongfellowTennysonBrowningmanylesser-knownpoetsldquoTheVillageBlacksmithrdquoldquoCurfewMustNotRingTonightrdquoldquoOnlya

BabySmallrdquodozensmoreoftendifficulttofindelsewhereIndexofpoetstitlesfirstlinesxxiii+325pp5⅝times8frac14

0-486-27044-0

DUBLINERSJamesJoyceFifteenstoriesoffervividtightlyfocusedobservationsofthelivesofDublinrsquospoorerclassesAtleastoneldquoTheDeadrdquoisconsideredamasterpieceReprintedcompleteandunabridgedfromstandardedition160pp times8frac14

0-486-26870-5

GREATWEIRDTALES14StoriesbyLovecraftBlackwoodMachenandOthersSTJoshi(ed)14spellbindingtalesincludingldquoTheSinEaterrdquobyFionaMcLeodldquoTheEyeAbovetheMantelrdquobyFrankBelknapLongaswellasrenownedworksbyRHBarlowLordDunsanyArthurMachenWCMorrowandeightothermastersofthegenre256pp5times8frac12(AvailableinUSonly)0-486-40436-6

THEBOOKOFTHESACREDMAGICOFABRAMELINTHEMAGEtranslatedbySMacGregorMathersMedievalmanuscriptofceremonialmagicBasicdocumentinAleisterCrowleyGoldenDawngroups268pp5⅜times8frac12

0-486-23211-5

THEBATTLESTHATCHANGEDHISTORYFletcherPrattEminenthistorianprofiles16crucialconflictsancienttomodernthatchangedthecourseofcivilization352pp5times8frac12

0-486-41129-X

NEWRUSSIAN-ENGLISHANDENGLISH-RUSSIANDICTIONARYMAOrsquoBrienThisisaremarkablyhandyRussiandictionarycontainingasurprisingamountofinformationincludingover70000entries366pp4frac12times6⅛

0-486-20208-9

NEWYORKINTHEFORTIESAndreasFeininger162brilliantphotographsbythewell-knownphotographerformerlywithLifemagazineCommutersshoppersTimesSquareatnightmuchelsefromcityatitspeakCaptionsbyJohnvonHartz181pp9frac14times10frac34

0-486-23585-8

INDIANSIGNLANGUAGEWilliamTomkinsOver525signsdevelopedbySiouxandothertribesWritteninstructionsanddiagramsAlso290pictographs

111pp6⅛times9frac14

0-486-22029-X

ANATOMYACompleteGuideforArtistsJosephSheppardAmasteroffiguredrawingshowsartistshowtorenderhumananatomyconvincinglyOver460illustrations224pp8times11frac14

0-486-27279-6

MEDIEVALCALLIGRAPHYItsHistoryandTechniqueMarcDroginSpiritedhistorycomprehensiveinstructionmanualcovers13styles(ca4thcenturythrough15th)Excellentphotographsdirectionsforduplicatingmedievaltechniqueswithmoderntools224pp8times11frac14

0-486-26142-5

DRIEDFLOWERSHowtoPrepareThemSarahWhitlockandMarthaRankinCompleteinstructionsonhowtousesilicagelmealandboraxperliteaggregatesandandboraxglycerineandwatertocreateattractivepermanentflowerarrangements12illustrations32pp5⅜times8frac12

0-486-21802-3

EASYTO-MAKEBIRDFEEDERSFORWOODWORKERSScottDCampbellDetailedsimple-to-useguidefordesigningconstructingcaringforandusingfeedersTextillustrationsfor12classicandcontemporarydesigns96pp5times8frac12

0-486-25847-5

THECOMPLETEBOOKOFBIRDHOUSECONSTRUCTIONFORWOODWORKERSScottDCampbellDetailedinstructionsillustrationstablesAlsodataonbirdhabitatandinstinctpatternsBibliography3tables63illustrationsin15figures48pp5frac14times8frac12

0-486-24407-5

SCOTTISHWONDERTALESFROMMYTHANDLEGENDDonaldAMackenzie16livelytalestellofgiantsrumblingdownmountainsidesofamagicwandthatturnsstonepillarsintowarriorsofgodsandgoddessesevilhagspowerfulforcesandmore240pp5times8frac12

0-486-29677-6

THEHISTORYOFUNDERCLOTHESCWillettCunningtonandPhyllisCunningtonFascinatingwell-documentedsurveycoveringsixcenturiesof

Englishundergarmentsenhancedwithover100illustrations12th-centurylaced-upbodicefootedlongdrawers(1795)19th-centurybustles19th-centurycorsetsformenVictorianldquobustimproversrdquomuchmore272pp5⅝times8frac14

0-486-27124-2

ARTSANDCRAFTSFURNITURETheCompleteBrooksCatalogof1912BrooksManufacturingCoPhotosanddetaileddescriptionsofmorethan150nowverycollectiblefurnituredesignsfromtheArtsandCraftsmovementdepictdavenportssetteesbuffetsdeskstableschairsbedsteadsdressersandmoreallbuiltofsolidquarter-sawedoakInvaluableforstudentsandenthusiastsofantiquesAmericanaandthedecorativearts80pp6frac12times9frac14

0-486-27471-3

WILBURANDORVILLEABiographyoftheWrightBrothersFredHowardDefinitivecrisplywrittenstudytellsthefullstoryofthebrothersrsquolivesandworkAvividlywrittenbiographyunparalleledinscopeandcolorthatalsocapturesthespiritofanextraordinaryera560pp6⅛times9frac14

0-486-40297-5

THEARTSOFTHESAILORKnottingSplicingandRopeworkHerveyGarrettSmithIndispensableshipboardreferencecoverstoolsbasicknotsandusefulhitcheshandsewingandcanvasworkmoreOver100illustrationsDelightfulreadingforsealovers256pp5times8frac12

0-486-26440-8

FRANKLLOYDWRIGHTrsquoSFALLINGWATERTheHouseandItsHistorySecondRevisedEditionDonaldHoffmannAtotalrevision-bothintextandillustrations-ofthestandarddocumentonFallingwatertheboldestmostpersonalarchitecturalstatementofWrightrsquosmatureyearsupdatedwithvaluablenewmaterialfromtherecentlyopenedFrankLloydWrightArchivesldquoFascinatingrdquomdashTheNewYorkTimes116illustrations128pp9frac14times10frac34

0-486-27430-6

PHOTOGRAPHICSKETCHBOOKOFTHECIVILWARAlexanderGardner100photostakenonfieldduringtheCivilWarFamousshotsofManassasHarperrsquosFerryLincolnRichmondslavepensetc244pp10times8frac14

0-486-22731-6

FIVEACRESANDINDEPENDENCEMauriceGKainsGreatback-to-the-

landclassicexplainsbasicsofself-sufficientfarmingTheonebooktoget95illustrations397pp5times8frac12

0-486-20974-1

AMODERNHERBALMargaretGrieveMuchthefullestmostexactmostusefulcompilationofherbalmaterialGiganticalphabeticalencyclopediafromaconitetozedoarygivesbotanicalinformationmedicalpropertiesfolkloreeconomicusesmuchelseIndispensabletoseriousreader161illustrations888pp6frac12times9frac142-volset(AvailableinUSonly)VolI0-486-22798-7VolII0-486-22799-5

HIDDENTREASUREMAZEBOOKDavePhillipsSolve34challengingmazesaccompaniedbyheroictalesofadventureEvildragonspeople-eatingplantsbloodthirstygiantsmanymoredangerousadversarieslurkateverytwistandturn34mazesstoriessolutions48pp8frac14times11

0-486-24566-7

LETTERSOFWAMOZARTWolfgangAMozartRemarkablelettersshowbawdywithumorimaginationmusicalinsightscontemporarymusicalworldincludessomelettersfromLeopoldMozart276pp5times8frac12

0-486-22859-2

BASICPRINCIPLESOFCLASSICALBALLETAgrippinaVaganovaGreatRussiantheoreticianteacherexplainsmethodsforteachingclassicalballet118illustrations175pp5times8frac12

0-486-22036-2

THEJUMPINGFROGMarkTwainRevengeeditionTheoriginalstoryofTheCelebratedJumpingFrogofCalaverasCountyahaplessFrenchtranslationandTwainrsquoshilariousldquoretranslationrdquofromtheFrench12illustrations66pp5times8frac12

0-486-22686-7

BESTREMEMBEREDPOEMSMartinGardner(ed)The126poemsinthissuperbcollectionof19th-and20th-centuryBritishandAmericanverserangefromShelleyrsquosldquoToaSkylarkrdquototheimpassionedldquoRenascencerdquoofEdnaStVincentMillayandtoEdwardLearrsquoswhimsicalldquoTheOwlandthePussycatrdquo224pp5⅜times8frac12

0-486-27165-X

COMPLETESONNETSWilliamShakespeareOver150exquisitepoemsdealwithlovefriendshipthetyrannyoftimebeautyrsquosevanescencedeathandotherthemesinlanguageofremarkablepowerprecisionandbeautyGlossaryofarchaicterms80pp times8frac14

0-486-26686-9

HISTORICHOMESOFTHEAMERICANPRESIDENTSSecondRevisedEditionIrvinHaasAtravelerrsquosguidetoAmericanPresidentialhomesmostopentothepublicdepictinganddescribinghomesoccupiedbyeveryAmericanPresidentfromGeorgeWashingtontoGeorgeBushWithvisitinghoursadmissionchargestravelroutes175photographsIndex160pp8frac14times11

0-486-26751-2

THEWITANDHUMOROFOSCARWILDEAlvinRedman(ed)Morethan1000ripostesparadoxeswisecracksWorkisthecurseofthedrinkingclassesIcanresisteverythingexcepttemptationetc258pp5⅜times8frac12

0-486-20602-5

SHAKESPEARELEXICONANDQUOTATIONDICTIONARYAlexanderSchmidtFulldefinitionslocationsshadesofmeaningineverywordinplaysandpoemsMorethan50000exactquotations1485pp6frac12times9frac142-volset

Vol10-486-22726-XVol20-486-22727-8

SELECTEDPOEMSEmilyDickinsonOver100best-knownbest-lovedpoemsbyoneofAmericarsquosforemostpoetsreprintedfromauthoritativeearlyeditionsNocomparableeditionatthispriceIndexoffirstlines64pp times8frac14

0-486-26466-1

THEINSIDIOUSDRFU-MANCHUSaxRohmerThefirstofthepopularmysteryseriesintroducesapairofEnglishdetectivestotheirarchnemesisthediabolicalDrFu-ManchuFlavorfulatmospherefast-pacedactionandcolorfulcharactersenliventhisclassicofthegenre208pp times8frac14

0-486-29898-1

THEMALLEUSMALEFICARUMOFKRAMERANDSPRENGERtranslatedbyMontagueSummersFulltextofmostimportantwitchhunterrsquosldquobiblerdquousedbybothCatholicsandProtestants278pp6⅝times10

0-486-22802-9

SPANISHSTORIESCUENTOSESPANtildeOLESADual-LanguageBook

AngelFlores(ed)Uniqueformatoffers13greatstoriesinSpanishbyCervantesBorgesothersFaithfulEnglishtranslationsonfacingpages352pp5⅜times8frac12

0-486-25399-6

GARDENCITYLONGISLANDINEARLYPHOTOGRAPHS1869-1919MildredHSmithHandsometreasuryof118vintagepicturesaccompaniedbycarefullyresearchedcaptionsdocumenttheGardenCityHotelfire(1899)theVanderbiltCupRace(1908)thefirstairmailflightdepartingfromtheNassauBoulevardAerodrome(1911)andmuchmore96pp8⅞times11

0-486-40669-5

OLDQUEENSNYINEARLYPHOTOGRAPHSVincentFSeyfriedandWilliamAsadorianOver160rarephotographsofMaspethJamaicaJacksonHeightsandotherareasVintageviewsofDeWittClintonmansion1939WorldrsquosFairandmoreCaptions192pp8⅞times11

0-486-26358-4

CAPTUREDBYTHEINDIANS15FirsthandAccounts1750-1870FrederickDrimmerAstoundingtruehistoricalaccountsofgrislytorturebloodyconflictsrelentlesspursuitsmiraculousescapesandmorebypeoplewholivedtotellthetale384pp5⅜times8frac12

0-486-24901-8

THEWORLDrsquoSGREATSPEECHES(FourthEnlargedEdition)LewisCopelandLawrenceWLammandStephenJMcKennaNearly300speechesprovidepublicspeakerswithawealthofupdatedquotesandinspirationmdashfromPericlesrsquofuneralorationandWilliamJenningsBryanrsquosldquoCrossofGoldSpeechrdquotoMalcolmXrsquospowerfulwordsontheBlackRevolutionandEarlofSpenserrsquostributetohissisterDianaPrincessofWales944pp5times8

0-486-40903-1

THEBOOKOFTHESWORDSirRichardFBurtonGreatVictorianscholaradventurerrsquoseloquenteruditehistoryoftheldquoqueenofweaponsrdquo-fromprehistorytoearlyRomanEmpireEvolutionanddevelopmentofearlyswordsvariations(sabrebroadswordcutlassscimitaretc)muchmore336pp6⅛times9frac14

0-486-25434-8

AUTOBIOGRAPHYTheStoryofMyExperimentswithTruthMohandasKGandhiBoyhoodlegalstudiespurificationthegrowthoftheSatyagraha(nonviolentprotest)movementCriticalinspiringworkofthemanresponsibleforthefreedomofIndia480pp5⅜times8frac12(AvailableinUSonly)0-486-24593-4

CELTICMYTHSANDLEGENDSTWRollestonMasterfulretellingofIrishandWelshstoriesandtalesCuchulainKingArthurDeirdretheGrailmanymoreFirstpaperbackedition58full-pageillustrations512pp5times8frac12

0-486-26507-2

THEPRINCIPLESOFPSYCHOLOGYWilliamJamesFamouslongcoursecompleteunabridgedStreamofthoughttimeperceptionmemoryexperimentalmethodsgreatworkdecadesaheadofitstime94figures1391pp5⅜times8frac122-volset

VolI0-486-20381-6VolII0-486-20382-4

THEWORLDASWILLANDREPRESENTATIONArthurSchopenhauerDefinitiveEnglishtranslationofSchopenhauerrsquoslifeworkcorrectingmorethan1000errorsomissionsinearliertranslationsTranslatedbyEFJPayneTotalof1269pp5times8frac122-volsetVol10-486-21761-2Vol20-486-21762-0

MAGICANDMYSTERYINTIBETMadameAlexandraDavid-NeelExperiencesamonglamasmagicianssagessorcerersBonpawizardsAtruepsychicdiscovery32illustrations321pp5times8frac12(AvailableinUSonly)

0-486-22682-4

THEEGYPTIANBOOKOFTHEDEADEAWallisBudgeCompletereproductionofAnirsquospapyrusfinesteverfoundFullhieroglyphictextinterlineartransliterationword-for-wordtranslationsmoothtranslation533pp6frac12times9frac14

0-486-21866-X

HISTORICCOSTUMEINPICTURESBraunampSchneiderOver1450costumedfiguresinclearlydetailedengravings-fromdawnofcivilizationtoendof19thcenturyCaptionsManyfolkcostumes256pp8⅜times11frac34

0-486-23150-X

MATHEMATICSFORTHENONMATHEMATICIANMorrisKlineDetailedcollege-leveltreatmentofmathematicsinculturalandhistorical

contextwithnumerousexercisesRecommendedReadingListsTablesNumerousfigures641pp5times8frac12

0-486-24823-2

PROBABILISTICMETHODSINTHETHEORYOFSTRUCTURESIsaacElishakoffWell-writtenintroductioncoverstheelementsofthetheoryofprobabilityfromtwoormorerandomvariablesthereliabilityofsuchmultivariablestructuresthetheoryofrandomfunctionMonteCarlomethodsoftreatingproblemsincapableofexactsolutionandmoreExamples502pp5times8frac12

0-486-40691-1

THERIMEOFTHEANCIENTMARINERGustaveDoreacuteSTColeridgeDoreacutersquosfinestwork34platescapturemoodssubtletiesofpoemFlawlessfull-sizereproductionsprintedonfacingpageswithauthoritativetextofpoemldquoBeautifulSimplybeautifulrdquomdashPublisherrsquosWeekly77pp9frac14times12

0-486-22305-1

SCULPTUREPrinciplesandPracticeLouisSlobodkinStep-by-stepapproachtoclayplastermetalsstoneclassicalandmodern253drawingsphotos255pp8⅛times11

0-486-22960-2

THEINFLUENCEOFSEAPOWERUPONHISTORY1660-1783ATMahanInfluentialclassicofnavalhistoryandtacticsstillusedastextinwarcollegesFirstpaperbackedition4maps24battleplans640pp5times8frac12

0-486-25509-3

THESTORYOFTHETITANICASTOLDBYITSSURVIVORSJackWinocour(ed)WhatitwasreallylikePanicdespairshockinginefficiencyandalittleheroismMorethrillingthananyfictionalaccount26illustrations320pp5times8frac12

0-486-20610-6

ONETWOTHREEINFINITYFactsandSpeculationsofScienceGeorgeGamowGreatphysicistrsquosfascinatingreadableoverviewofcontemporarysciencenumbertheoryrelativityfourthdimensionentropygenesatomicstructuremuchmore128illustrationsIndex352pp5⅜times8frac12

0-486-25664-2

DALIacuteONMODERNARTTheCuckoldsofAntiquatedModernArtSalvadorDaliacuteInfluentialpainterskewersmodernartanditspractitionersOutrageousevaluationsofPicassoCeacutezanneTurnermore15renderingsofpaintingsdiscussed44calligraphicdecorationsbyDali96pp5times8frac12(AvailableinUSonly)0-486-29220-7

ANTIQUEPLAYINGCARDSAPictorialHistoryHenryReneacuteDrsquoAllemagneOver900elaboratedecorativeimagesfromrareplayingcards(14th-20thcenturies)Bacchusdeathdancingdogshuntingscenesroyalcoatsofarmsplayerscheatingmuchmore96pp9frac14times12frac14

0-486-29265-7

MAKINGFURNITUREMASTERPIECES30ProjectswithMeasuredDrawingsFranklinHGottshallStep-by-stepinstructionsillustrationsforconstructinghandsomeusefulpiecesamongthemaSheratondeskChippendalechairSpanishdeskQueenAnnetableandaWilliamandMarydressingmirror224pp8⅛times11frac14

0-486-29338-6

NORTHAMERICANINDIANDESIGNSFORARTISTSANDCRAFTSPEOPLEEvaWilsonOver360authenticcopyright-freedesignsadaptedfromNavajoblanketsHopipotterySiouxbuffalohidesmoreGeometriessymbolicfiguresplantandanimalmotifsetc128pp8⅜times11(NotforsaleintheUnitedKingdom)0-486-25341-4

THEFOSSILBOOKARecordofPrehistoricLifePatriciaVRichetalProfuselyillustrateddefinitiveguidecoverseverythingfromsingle-celledorganismsanddinosaurstobirdsandmammalsandtheinterplaybetweenclimateandmanOver1500illustrations760pp7frac12times10⅛

0-486-29371-8

VICTORIANARCHITECTURALDETAILSDesignsforOver700StairsMantelsDoorsWindowsCornicesPorchesandOtherDecorativeElementsAJBicknellampCompanyEverythingfromdormerwindowsandpiazzastobalconiesandgableornamentsAlsoincludeselevationsandfloorplansforhandsomeprivateresidencesandcommercialstructures80pp9⅜times12frac14

0-486-44015-X

WESTERNISLAMICARCHITECTUREAConciseIntroductionJohnD

HoagProfuselyillustratedcriticalappraisalcomparesandcontrastsIslamicmosquesandpalacesmdashfromSpainandEgypttootherareasintheMiddleEast139illustrations128pp6times9

0-486-43760-4

CHINESEARCHITECTUREAPictorialHistoryLiangSsu-chrsquoengMorethan240rarephotographsanddrawingsdepicttemplespagodastombsbridgesandimperialpalacescomprisingmuchofChinarsquosarchitecturalheritage152halftones94diagrams232pp10frac34times9

0-486-43999-2

THERENAISSANCEStudiesinArtandPoetryWalterPaterOneofthemosttalked-aboutbooksofthe19thcenturyTheRenaissancecombinesscholarshipandphilosophyinaninnovativeworkofculturalcriticismthatexaminestheachievementsofBotticelliLeonardoMichelangeloandotherartistsldquoTheholywritofbeautyrdquo-OscarWilde160pp5times8frac12

0-486-44025-7

ATREATISEONPAINTINGLeonardodaVinciThegreatRenaissanceartistrsquospracticaladviceondrawingandpaintingtechniquescoversanatomyperspectivecompositionlightandshadowandcolorAclassicofartinstructionitfeatures48drawingsbyNicholasPoussinandLeonBattistaAlberti192pp5⅜times8frac12

0-486-44155-5

THEMINDOFLEONARDODAVINCIEdwardMcCurdyMorethanjustabiographythisclassicstudybyadistinguishedhistoriandrawsuponLeonardorsquosextensivewritingstooffernumerousdemonstrationsoftheRenaissancemasterrsquosachievementsnotonlyinsculptureandpaintingbutalsoinmusicengineeringandevenexperimentalaviation384pp5times8frac12

0-486-44142-3

WASHINGTONIRVINGrsquoSRIPVANWINKLEIllustratedbyArthurRackhamLovelyprintsthatestablishedartistasaleadingillustratorofthetimeandforeveretchedintothepopularimaginationaclassicofCatskilllore51full-colorplates80pp8times11

0-486-44242-X

HENSCHEONPAINTINGJohnWRobichauxBasicpaintingphilosophy

andmethodologyofagreatteacherasexpoundedinhisfamousclassesandworkshopsonCapeCod7illustrationsincoloroncovers80pp5times8frac12

0-486-43728-0

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0-486-44143-1

ASTROLOGYANDASTRONOMYAPictorialArchiveofSignsandSymbolsErnstandJohannaLehnerTreasuretroveofstoriesloreandmythaccompaniedbymorethan300rareillustrationsofplanetstheMilkyWaysignsofthezodiaccometsmeteorsandotherastronomicalphenomena192pp8⅜times11

0-486-43981-X

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0-486-44043-5

MAKINGBIRDHOUSESEasyandAdvancedProjectsGladstoneCaliffEasy-to-followinstructionsincludediagramsforeverythingfromaone-roomhouseforbluebirdstoaforty-two-roomstructureforpurplemartins56plates4figures80pp8times6

0-486-44183-0

LITTLEBOOKOFLOGCABINSHowtoBuildandFurnishThemWilliamSWicksHandyhow-tomanualwithinstructionsandillustrationsforbuildingcabinsintheAdirondackstylefireplacesstairwaysfurniturebeamedceilingsandmore102linedrawings96pp8times6⅜

0-486-44259-4

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0-486-44220-9

THEMETROPOLISOFTOMORROWHughFerrissGenerouspropheticvisionofthemetropolisofthefutureasperceivedin1929Powerfulillustrationsoftoweringstructureswideavenuesandrooftopparks-allfeaturesinmanyoftodayrsquosmoderncities59illustrations144pp8frac14times11

0-486-43727-2

THEPATHTOROMEHilaireBellocThis1902memoiraboundsinlivelyvignettesfromavanishedtimerecountingapilgrimageonfootacrosstheAlpsandApenninesinordertoldquoseeallEuropewhichtheChristianFaithhassavedrdquo77oftheauthorrsquosoriginallinedrawingscomplementhissparklingprose272pp5times8frac12

0-486-44001-X

THEHISTORYOFRASSELASPrinceofAbissiniaSamuelJohnsonDistinguishedEnglishwriterattackseighteenth-centuryoptimismandmanrsquosunrealisticestimatesofwhatlifehastooffer112pp5times8frac12

0-486-44094-X

AVOYAGETOARCTURUSDavidLindsayAbrilliantflightofpurefancywherewildcreaturescrowdthefantasticlandscapeanddementedtorturersdominatevictimswiththeirbizarrementalpowers272pp5times8frac12

0-486-44198-9

PaperboundunlessotherwiseindicatedAvailableatyourbookdealeronlineatwwwdoverpublicationscomorbywritingtoDeptGIDoverPublicationsInc31East2ndStreetMineolaNY11501Forcurrentpriceinformationorforfreecatalogs(pleaseindicatefieldofinterest)writetoDoverPublicationsorlogontowwwdoverpublicationscomandseeeveryDoverbookinprintDoverpublishesmorethan500bookseachyearonscienceelementaryandadvancedmathematicsbiologymusicartliteraryhistorysocialsciencesandotherareas

1 Answerstoodd-numberedproblemsbeginonp403

2 CalculationshereareshowntofiveplacesBecauseoflimitationsofspaceithasnotbeenpossibletoincludeatableoffive-placelogarithmsAtableoffour-placelogarithmshowevermaybefoundonpp424-425(Table3AppendixB)

Dover Books on Mathematics

BOOKS BY A ALBERT KLAF

Title Page

Copyright Page

FOREWORD

Table of Contents

INTRODUCTION

CHAPTER I - ADDITION

34 Why is addition merely a short way of counting

35 What is our standard group or bundle

36 What is thus meant by addition

37 What is meant by sum

38 Of the total number of 45 additions of two digits at a time for all the nine digits which give single numbers as a sum and which give double numbers

39 What is the rule for addition

40 What is the proper way of adding

41 What is the simplest but slowest way of adding

42 What is a variation of the above

43 How can grouping of numbers help you in addition

44 How is addition accomplished by multiplication of the average of a group

45 What is the procedure for adding two columns at a time

46 How are three columns added at one time

47 What is a convenient way of adding two small quantities by making a decimal of one of them

48 How may decimalized addition be carried out to a fuller development

49 How may sight reading be used in addition

50 What simple method is used to check the correctness of addition of a column of numbers

51 What is meant by a check figure in addition

52 What are the interesting facts on the use of the check number 9

53 What is the procedure in checking addition by the use of the check figure 9 often called ldquocasting out ninesrdquo

54 Why is ldquocasting out ninesrdquo not a perfect test of accuracy in addition


56 Why is the checking of addition work by the use of the check figure 11 (often called ldquocasting out elevensrdquo) superior to that of ldquocasting out ninesrdquo

57 What is the procedure in checking addition by the use of the check figure 11

CHAPTER II - SUBTRACTION

58 What is subtraction

59 Why may subtraction be said to be a form of addition

60 What three questions will lead to the process of subtraction

61 What are the terms of a subtraction

62 Why is it said that we can always add but we cannot always subtract

63 When is it possible to subtract with the number expressing the subtrahend greater than the number expressing the minuend

64 What is the subtraction table that should be studied until the answers can be given quickly and correctly

65 What is the rule for subtraction

66 What is known as the method of ldquoequal additionsrdquo in subtraction

67 What is the mode of thinking of subtraction that is called the Austrian method or the method of making change

68 How may subtraction be simplified

69 How may the above be extended

70 How can the subtraction of two-figure numbers be done by simple inspection using decimalization

71 How can inverted or left-hand subtraction be done

72 What is meant by the arithmetical complement of a number

73 What is the simplest way of calculating the ac of a number

74 When and how is the ac used in subtraction

75 How do we proceed to give change to a customer by the use of the so-called ldquoAustrian methodrdquo of subtraction

76 What is the best check in subtraction

77 Is ldquocasting out ninesrdquo a practical check in subtraction

78 May casting out of elevens be used as a check

CHAPTER III - MULTIPLICATION

79 What is multiplication

80 What are the terms of a multiplication

81 What is (a) a concrete number (b) an abstract number (c) the type of number of the multiplier in multiplication

82 What are the most useful products that should be committed to memory

83 When several numbers are multiplied does it matter in what order the multiplication is performed

84 What is the rule in multiplication when (a) the two signs of the numbers are both plus [+] (b) both signs are minus [ ndash ] (c) the two signs are unlike

85 What is the effect upon a number when you move it one two three places to the left in the period

86 What is the rule for multiplying when either multiplier or multiplicand ends in zeros

87 How is ordinary simple multiplication performed

88 What is the procedure when the numbers to be multiplied contain more than one digit

89 How can the fact that either number may be used as the multiplier serve to provide a check on our multiplication

90 How can we extend the multiplication table beyond 12 times 12 by making use of the smaller products by 2 or by 4

91 How can multiplication by two-digit numbers be simplified

92 How can the multiplication of two 2-digit numbers having the same figure in the tens place be simplified

93 How can multiplication be simplified by multiplying one factor and dividing the other factor by the same quantity

94 What can be done when multiplication may simplify one of the factors but when the other factor is not divisible by the same number

95 When the tens digits are alike and the units digits add up to 10 how is multiplication simplified

96 When the units digits are alike and the tens digits add up to 10 how is multiplication simplified

97 When neither of above combinations is applicable how may so-called cross multiplication be applied to advantage

98 When the units digits are 5 and the sum of the tens digits is even how is multiplication simplified

99 When the units digits are 5 and the sum of the tens digits is odd how is multiplication simplified

100 What is meant by left-hand multiplication or what is sometimes called inverted multiplication

101 What is meant by an aliquot (ălrsquoi-kwŏt) part of a number

102 What is meant by a fractional equivalent of an aliquot part

103 When are some numbers useful while not aliquot parts themselves

104 What are some of the aliquot parts of 100 and their fractional equivalents

105 How may aliquot parts of 100 be written as decimals

106 Why are aliquot parts useful in calculations involving dollars

107 How may aliquot parts of 100 be used in multiplication

108 What is the practical use of aliquot parts in multiplication

109 May the number of articles and the price be interchanged as a means of simplifying a problem in aliquot parts

110 What is the cost of 1780 lb of feed at $1500 a ton

111 How can we simplify the multiplication by 24


113 How can we multiply a number by 9 using subtraction

114 How can we multiply by 11 using addition

115 How can we multiply by 111 by using addition

116 How can we simplify the multiplication by 8 and by 7

117 How do we multiply by 99 98 97 or by 999 998 997

118 What is meant by the complement of a number

119 How is complement multiplication performed

120 How can we multiply by a number between 12 and 20 using only one line in the product

121 What is meant by cross multiplication

122 What is the result of 76 times 64 using cross multiplication


124 How can we check a multiplication by ldquocasting out ninesrdquo

CHAPTER IV - DIVISION

125 What is meant by division

126 In what other ways may division be thought of

127 What are the terms of a division

128 When the dividend is concrete and the divisor is abstract what is the quotient

129 What is the result when both the dividend and divisor are concrete

130 What is meant by a remainder in division

131 Why may we think of division as the process of finding one factor when the product and the other factor are given

132 How can we make use of the fact that division is the opposite of multiplication

133 If we wanted to divide 3492 men into 4 groups how would we proceed

134 What is meant by ldquoshort divisionrdquo and what is the process in simple form

135 How do we divide 3762 by 7 using short division

136 How do we proceed with long division

137 What do we do when the last subtraction is not zero

138 What is the principle of the trial divisor in long division

139 What is the rule for long division

140 What is a pure proof of any division

141 What is the procedure for division with United States money

142 What is the quotient of the division of $4536 by $027

143 How can factoring of the divisor be used to reduce a problem of long division to a series of short divisions

144 What is the procedure for the above when there is a remainder

145 What is the quotient of 65349 by 126 using the factoring-of the-divisor method

146 What is the procedure for dividing by 10 100 1000 etc

147 How do we apply the excess-of-nines method to prove the correctness of a division

148 What is meant by an even number

149 How can we know when a number is divisible by 3

150 If we have an even number and it is divisible by 3 by what other number is it also divisible

151 When is a number divisible by 4


153 What number or any multiples of it can be divided by 7 11 or 13





158 What is the criterion for a number divisible by 11

159 How can we tell in advance what the remainder will be when the divisor is 9

160 What is a short-cut way of dividing by 5

161 What is a simple way of dividing by 25

162 What is a simple way to divide by 125

163 What is the short-cut way of dividing by any aliquot part of 100


165 How can we make a number divisible by 3


167 How do we obtain an average of a number of items

168 What is the rule for finding the value of one of anything

CHAPTER V - FACTORSmdashMULTIPLESmdashCANCELLATION

169 What is a prime number

170 What is a composite number

171 What is a factor of a number

172 What is meant by factoring

173 What is a prime factor

174 What do we call a number that has the factor 2

175 What is meant by a common divisor or factor

176 What facts regarding the divisibility of numbers are of assistance in factoring

177 How do we find the prime factors of a number

178 What is meant by the greatest common divisor or factor abbreviated GCD or gcd

179 What is the rule for finding the GCD of two or more numbers

180 What is a more convenient method of finding GCD

181 What is meant by a multiple of a number

182 What is meant by a common multiple of two or more numbers

183 What is meant by the least common multiple (LCM) of two or more numbers

184 What is a method of finding the least common multiple (LCM) of 18 28 and 36

185 What is another method of getting the LCM of 18 28 and 36

186 What is meant by cancellation

CHAPTER VI - COMMON FRACTIONS

187 What does a fraction mean

188 What are the terms of a fraction

189 What is assumed in expressing fractional division

190 What is meant when we say that a thing is divided equally into two parts and how is the fraction expressed

191 What is meant by

192 What is meant by a unit fraction

193 What is a vulgar fraction and how is it classified

194 What are the parts of a vulgar fraction and how is it written

195 What other meaning has the bar in a fraction

196 What are the three ways in which a fraction may be interpreted

197 When we add up all the fractional parts of a unit what do we get as a result

198 What is a simple fraction

199 What is a compound fraction

200 What is a complex fraction

201 What is a proper fraction

202 What is an improper fraction

203 What is a mixed number

204 How may we shorten the process of finding the value of an improper fraction

205 How do we change a mixed number into an improper fraction

206 What happens to the value of a fraction when we multiply or divide both the numerator and the denominator by the same number

207 When is a fraction said to be reduced to its lowest terms

208 How do we reduce a fraction to its lowest terms

209 How can we change a fraction to higher terms

210 What must be done to fractions in giving the answer to a problem

211 How can we increase the value of a fraction

212 How can we decrease the value of a fraction

213 How do we change a compound fraction to a simple fraction

214 How do we change a complex fraction to a simple fraction

215 What is another method of simplifying a complex fraction

216 What is the condition for adding or subtracting of fractions

217 What is the procedure when the denominators are not the same

218 What is the procedure for subtraction of fractions

219 How do we subtract mixed numbers when they are large

220 Can a whole number always be expressed in a fractional form Yes

221 In adding or subtracting two fractions how can we use cross multiplication to get the same result as with the LCD method

222 What is the procedure in multiplying one proper fraction by another

223 How do we multiply a proper fraction by a whole number

224 What is the procedure for multiplying one mixed number by another

225 What is the four-step method of multiplying onemixed number by another

226 How do we multiply a mixed number by a proper fraction

227 What word is frequently used instead of the multiplication sign or the word ldquomultiplyrdquo

228 What is meant by the reciprocal of a number

229 When the product of two numbers equals 1 what is each of the two numbers called

230 How can we show that to multiply by the reciprocal of a number is the same as to divide by that number

231 How many times are (a) and contained in 1

232 In each case what can we do when we want to divide a whole number by a fraction or a fraction by a whole number or a fraction by a fraction

233 Specifically how do we divide a proper fraction by a whole number

234 How do we divide a whole number by a fraction

235 How do we divide one mixed number by another

236 How do we divide a mixed number by a whole number

237 What is another method to use for the above case when the dividend is a large number

238 What are some other methods of dividing whole mixed numbers

239 What is the difference between a fraction applicable to an abstract number and one applicable to a concrete number

240 How do we find what part the second of two numbers is of the first

241 If you are given a number that is a certain fraction of a whole how would you find the whole

242 How do we tell which one of two fractions is the greater

243 What is a chain (or a continued) fraction

244 What chain fractions are of interest to us

245 How is a proper fraction converted into a chain fraction

246 How can the above be simplified

247 How is a chain fraction converted to a proper fraction

248 Of what practical use are chain fractions

249 What fraction in smaller terms nearly expresses

250 How can we get a closer approximation

251 What feature of a chain fraction makes it valuable to us

CHAPTER VII - DECIMAL FRACTIONS

252 What is decimal division

253 What is a decimal fraction

254 What do we call the decimal point

255 How may decimal fractions be expressed

256 What are the names of the decimal places and how are decimals written

257 How is a decimal read

258 What is the relation of the number of figures in a decimal to the number of zeros in its denominator when expressed as a common fraction

259 Is the value of a decimal fraction changed by adding or omitting zeros on the right No

260 What is the effect on decimal fractions of moving the decimal point to the left

261 What is the effect of moving the decimal point to the right

262 What must be done when there is not a sufficient number of figures in the numerator to indicate the denominator of a decimal fraction

263 How are decimals classified

264 Do we need a decimal point after every whole number

265 How do we divide any number by a decimal number

266 How do we multiply any number by a decimal number

267 What is a mixed number in decimal form and how do we multiply and divide it by a decimal

268 How can we change a common fraction to a decimal

269 How can we extend a complex decimal

270 How can we convert a decimal expression to a common fraction

271 What is the procedure for adding whole numbers and simple decimals

272 What is the procedure for adding whole numbers and complex decimals

273 What is the procedure for subtracting simple decimals

274 What is the procedure for subtracting a simple decimal and a complex decimal

275 What is the procedure for multiplying simple decimals

276 What is the procedure for multiplying complex decimals

277 What is the procedure for dividing one simple decimal by another

278 What is the procedure for dividing one complex decimal by another

279 How is a decimal number shortened for all practical purposes

280 What other method of decimal approximation has been internationally approved

281 What is the least number of significant figures that must be kept when the decimal is purely fractional and contains a number of zeros to the right of the decimal point

282 What is the result of 03024 times 0196 correct to 2 significant figures

283 Why is it the rule to work a problem to one more decimal place than we need

284 What can we do to simplify things when we want to get an answer correct to two decimal places in multiplying 4879 by 3765

285 What is another way of approximating the desired result involving decimals

286 What is a recurring decimal

287 How are recurring circulating or repeating decimals denoted

288 How can we convert pure recurring decimals to fractions

289 How can we convert mixed recurring decimals to fractions

290 Why in particular should you know the decimal equivalents of and

291 How can we sometimes produce a decimal equivalent by multiplying both numerator and denominator by a suitable number

292 How do we find the whole number when a decimal part of it is given

293 How is United States money related to decimal fractions

294 If a British pound (pound) is worth $280 and there are 20 shillings to the pound and 12 pence to the shilling how much is (a) 1 shilling worth (b) 1 penny worth

295 A manufacturer submitted a bid to the United States government for military insignia in the sum of $6839970 at 31 cents mills per dozen How many dozen would be delivered

CHAPTER VIII - PERCENTAGE

296 What is meant by (a) per cent (b) percentage

297 What is the symbol used to represent the denominator 100

298 In what ways may a given per cent or a given number of hundredths of a number be expressed

300 How do we reduce a number written with a per cent sign to a decimal

301 How do we convert to a decimal when the per cent is expressed as a number and a fraction

302 How can we convert a whole number a decimal fraction a fraction or a mixed number to a per cent

303 What are the per cent equivalents of very common fractions

304 What per cent of the large square is the shaded part

305 What is the most common method of finding a given per cent of a number

306 What is another method of finding a given per cent of a number

307 What is the third method of finding a given per cent of a number

308 What terms are commonly used in percentage

309 What is the rule for finding the percentage when the base and rate are given

310 What is the rule for finding the rate when the percentage and base are given

311 What is the rule for finding the base when the rate and the percentage are given

312 What is meant by (a) amount (b) difference in percentage problems

313 How can we find the base when the rate and amount are given

314 How do we find the base when the rate and difference are given

315 On what do we always base the per cent of increase in some quantity

316 On what do we always base the per cent of decrease in some quantity

317 How are per cents less than 1 per cent or fractional parts of 1 per cent written and used in business and financial matters

318 How is the expression of ldquoso much per hundredrdquo commonly used in business

319 How is the mill used in tax matters

320 How are per cents added subtracted multiplied or divided

321 If a number is increased by a certain per cent to get an amount what per cent must be subtracted from this amount to get the original number again

322 If Boston has a population of 2000000 and Philadelphia is 50 larger how much smaller is Boston than Philadelphia

323 If a man spends 30 of his income for rent and 10 of the remainder for clothes what is his salary if the landlord gets $1150 more than the clothier

324 A man sells his car to his friend and takes a loss of 20 His friend sells the car later to a third party for $1500 losing 25 How much did the original owner pay for the car

CHAPTER IX - INTEREST

325 What is meant by interest

326 What are the three factors to consider in calculating interest

327 How do we express a rate of interest

328 What is meant by simple interest

329 What is meant by compound interest

330 What is the formula for figuring simple interest

331 What is meant by the ldquoamountrdquo and what is its symbol

332 In figuring simple interest for less than a year what is the rule for establishing (a) the terminal days (b) the due date

333 How are the methods for figuring simple interest commonly referred to

334 How do we find the time by the ordinary method

335 How do we find the time by the exact method

336 How do we figure time by the bankersrsquo method

337 Find the interest on $3000 at 6 from November 18 1958 to April 6 1959 (a) by the ordinary method (b) by the exact method (c) by the bankersrsquo method

338 What is the constant relationship of exact interest to ordinary or bankersrsquo interest based on exact number of days

339 What is the 60-day 6 per cent method of calculating interest

340 A businessman borrowed $850 for 75 days at 6 How much interest did he pay

341 How are the aliquot parts of 60 used when the time is greater or less than 60 days in finding interest by the 60-day 6 method

342 What is the interest on $95370 for 124 days at 6


344 How can we sometimes simplify the 60-day 6 process

345 How do we find the interest at a rate other than 6

346 How can we make use of the interest formula in finding one of the four factorsmdashinterest principal rate and timemdashwhen the other three are given

347 What is the 6-day 6 method of finding interest and what is its principal value

348 What is the significance of compound interest

349 What is meant by (a) compound amount (b) compound interest (c) conversion period (d) frequency of conversion

350 What will $450 amount to in three years at 4 if interest is compounded annually

351 What is a shorter method of figuring the compound amount

352 What is the formula for the amount at compound interest

353 In order to have $6000 at the end of 3 years how much must you invest now at 5 compounded annually

354 What is used in actual business and financial practice to save a great deal of time labor and computation in figuring compound interest

355 What would $12000 amount to if invested for 7 years at 4 compounded annually

356 What amount of money invested at 5 for nine years would amount to $589505

357 If you deposited $1800 in a bank which pays 4 per annum how long will it take for this deposit to grow to $227758 if interest is compounded annually

358 What is meant by the nominal rate of interest

359 What is meant by the effective annual rate of interest

360 When are nominal and effective rates equivalent

361 What is the formula showing the relationship between an effective rate i and an equivalent nominal rate rp compounded p times a year

362 What is the formula for the compound amount of 1 at a rate rp compounded p times per annum for t years

363 What is the rule for use of compound-amount-of-1 tables where interest is compounded at a nominal rate more than once a year

364 A man invests $8000 for 12 years at 5 compounded quarterly What amount will he get after 12 years

CHAPTER X - RATIOmdashPROPORTIONmdashVARIATION

365 What are the two ways of comparing like quantities

366 What is meant by a ratio

367 What two terms are given in all ratio calculations

368 What symbol is used to indicate ratio

369 How may ratios be expressed

370 Can there be a ratio of unlike things

371 Is a ratio dependent upon the units of measure

372 Does multiplying or dividing both terms of a ratio by the same number change its value No

373 What is the relation between ratio and per cent

374 How is a ratio simplified

375 What can be done in order to compare readily two or more ratios

376 What would you do when required to work out a complicated ratio containing fractions per cents or decimals

377 How do we divide some given number in a given ratio

378 How can we divide 65 in the ratio

379 How do we solve a ratio problem in which the ratio is not given

380 If the wing span of a plane is 76 ft 6 in what will the wing span of a model have to be when the ratio of the length of any part of the model to the length of the corresponding part of the actual plane is 172

381 If a bankrupt firm can pay 60cent on the dollar and if its assets amount to $28000 what are its liabilities

382 What selling price should be placed on a TV set if the cost is $250 and the dealer operates on a margin of 30 of cost

383 If you allow 12 of your income for clothing and 21 for rent (a) what is the ratio of the cost of rent to the cost of clothing (b) how much do you spend for rent per month when your income is $8400 per year

384 If a town estimates that it has to raise $300000 in taxes and the assessed valuation of its real property is $9000000 what is the tax rate

385 A certain concrete mixture is to be made up of 1 part cement 3 parts sand and 5 parts stone What is (a) ratio of sand to stone (b) the ratio of cement to sand (c) per cent of sand in the concrete mixture

386 If the bedroom of a house is shown on the print to be in times in and if the scale of the blueprint is in = 1 ft what are the actual dimensions of the room

387 What is meant by an ldquoinverse ratiordquo

388 What would be your share in an automobile that cost you and your brother $880 if of your share is equal to of your brotherrsquos

389 What are some general rules for ratio calculation

390 How do we compound ratios

391 How do we solve in a manner similar to that of a ratio problem a problem in which the same number of articles are bought each at a different price

392 How do we solve in a manner similar to that of a ratio problem a problem in which a different number of articles are bought at different prices

393 What is meant by a proportion

394 How are proportions written

395 What are the terms of a proportion

396 What is the test as to whether the terms are in proportion

397 From the above how do we find either mean that is not given

398 From the above how do we find a missing extreme

399 You buy 8 tons of coal for $208 What will 12 tons cost

400 A 9-foot-high tree casts a shadow of feet What is the height of a radio tower that casts a shadow of 203 feet

401 When are quantities said to be in direct proportion

402 What is meant by a mean proportional

403 How does stating a problem as a simple proportion simplify the finding of an unknown term in a problem

404 An alloy consists of 4 parts of tin and 6 parts of copper How many pounds of copper would be needed with 120 pounds of tin to maintain the given ratio

405 What is meant by an inverse proportion

406 Driving to your office at 45 mph you make it in 55 minutes At what speed would you have to travel to get there in 50 minutes

407 How is an inverse proportion set up

408 If 130 yards of a copper wire offer 18 ohm resistance what will be the resistance of 260 yards of copper wire of times the cross-sectional area

409 What is a compound proportion

410 What is the rule for solving a compound proportion

411 If 20 men working 6 hours per day can dig a trench 80 feet long in 30 days how many men working 10 hours a day can dig a trench 120 feet long in 12 days

412 Why is it possible to set up the second member of the proportion as a single ratio

413 If 2 men cut 8 cords of wood in 4 days how long will it take 12 men to cut 36 cords

414 If the eggs laid by 30 hens in 15 weeks are worth $108 what will be the value of the eggs laid by 60 hens in 10 weeks

415 What are some of the properties of proportion that can be obtained by elementary algebraic changes in the form of the equation which expresses the proportion

416 What proportions of 3 milk and 5 milk must be mixed to get milk

417 How is proportion applied to the principle of the lever

418 What is the relation between ratio and proportion and the language of variation

419 What may be said about each of the statements of ratio and proportion

420 What is implied in a direct variation and how is a direct variation expressed

421 What is implied in an inverse variation and how is an inverse variation expressed

422 What is meant by a joint variation and how is it expressed

423 What is the electrical resistance of 1000 feet of copper wire inch in diameter using k = 103

CHAPTER XI - AVERAGES

424 What is meant by an average in statistics

425 What are the uses of averages in statistics

426 Why may an average be a more reliable figure to represent a group than a sample figure selected from the group

427 Can averages be compared when they are derived from data representing widely different conditions and groups

428 What is meant by a deviation from the average

429 What is the significance of a small total amount of deviations

430 What are the two classes of averages in general

431 How do we find the arithmetic average or mean value of a number of similar quantities

432 When is an average an excellent way of showing the middle or most typical figure

433 If a train takes the following times between stopsmdash48 minutes 55 minutes 1 hour 8 minutes and 42 minutes mdashwhat is the average time between stops

434 A car travels 10 miles up a steep grade at 30 mph and then 90 miles on a level road at 50 mph What is its average speed

435 Two planes leave at the same time from Seattle Washington for El Paso Texasmdasha distance of 1381 miles One plane A flies at 400 mph and returns at 400 mph The other plane B flies at 600 mph from Seattle and returns at 200 mph because of defective engines If each plane remains 12 hours in El Paso which comes back first

436 If you paid an income tax of 22 on $3400 one year and 28 on $4600 the following year how much did you pay altogether

437 How would you find the total given the average with ordinary numbers (not ratios)

438 An appliance dealer sells 15 TV sets that cost $180 per set at an average profit of 30 and 20 other TV sets that cost him $260 per set at an average profit of 35 What is the total profit assuming the percentages are based on the cost price

439 What is meant by a weighted average

440 How can we find the value of one quantity that is not given when the weights and the final average are known

441 There are 8 manufacturing plants having 453 699 341 621 383 562 741 and 214 employees respectively If the employees in plants 1 2 and 3 worked 38 hours per week in plants 4 5 and 6 40 hours per week and in plants 7 and 8 42 hours per week how could we (a) get a true comparison of their productivity expressed in man-hours (b) determine the average number of hours each man worked in the given week

442 How can we simplify the process of getting an average of several numbers that differ from one another by a comparatively small amount

443 For scattered data what two other ways are there of finding the ldquomiddlerdquo that stand for more than an average

444 What is meant by the median

445 How is the median located

446 If 25 salesmen in an organization report their average weekly incomes as $260 $200 $95 $200 $220 $160 $160 $800 $240 $240 $235 $350 $150 $260 $200 $275 $450 $275 $175 $200 $500 $225 $250 $650 and $200 what is the average weekly income of the group and is this average representative of the group

447 What is the median of the above and does this median give a reasonable idea of the group income

448 What is meant by the mode

449 What is the mode of the weekly incomes of Question 446

450 How can we widen the concept that the mode is the most typical figure and get a better measure of the group

451 What are the best measures of typical earnings of the group of salesmen

452 What are the advantages of the arithmetic mean or average

453 What are the disadvantages of the arithmetic mean or average

454 What are the advantages of the median

455 What are the disadvantages of the median

456 What are the advantages of the mode

457 What are the disadvantages of the mode

CHAPTER XII - DENOMINATE NUMBERS

458 What is a denominate number

459 What is meant by reduction of denominate numbers

460 What is meant by (a) reduction descending (b) reduction ascending

461 What are the standard linear measures

462 What is the result of the reduction of the following

463 What is the procedure for reduction to lower denominations when the length is expressed in several denominations

464 What is the procedure for reduction to higher denominations

465 What are the units used in measuring the areas of surfaces (square measure)


467 What are the measurements for solids (cubic measure)

468 What are the units applicable to liquid measure

469 What are the units applicable to dry measure

470 How many kinds of weight are in use in the United States

471 What constitutes the avoirdupois table of weights

472 What constitutes the troy table of weights

473 What constitutes the apothecariesrsquo table of weights

474 What are some comparisons of weights

475 What are the units for measurement of time

476 What are the measures of counting

477 What are the units for paper measure

478 What are some measures of value

479 What is the metric system of weights and measures

480 What is the linear measure table in the metric system

481 What is the area measure table in the metric system

482 What is the volume or cubic measure table in the metric system

483 What is the table for measures of liquid and dry capacity in the metric system

484 What is the table for measures of weight in the metric system

485 What are the units for circular measure

486 In reducing 4 bu 3 pk 5 qt 2 pt to pints what is the procedure

487 What is the result of reducing gal to lower denominations

488 What is the result of reducing 10 qt 2 pt to the fraction of a bushel

489 What is the result of reducing ft to the fraction of a rod

490 What is the result of reducing 2 pk 6 qt pt to a decimal of a bushel

491 What is the result of reducing 27 lb apothecariesrsquo to lower denominations

492 What is the result of reducing 62 gill to a decimal of a gallon

493 What is the procedure for addition of denominate numbers

494 What is the procedure for subtraction of denominate numbers

495 What is the result of multiplying 26 sq rd 10 sq yd 5 sq ft 34 sq in by 8

496 What is the result of dividing 18 A 142 sq rd 24 sq yd by 7

497 How many pounds of avoirdupois are 25 pounds troy weight

498 How can we reduce 6 km 4 hm 3 m 5 dm 9 mm to meters

499 How can we reduce 5327698 dm to km

500 What is the result of adding 48 m 284 cm and 5 Dm 2 dm with the answer expressed in meters

501 How many centimeters remain when from a pipe 283 m long 167 cm is cut off

502 What is the total weight in kg of 3450 cartons when each carton weighs 3600 g

CHAPTER XIII - POWERmdashROOTSmdashRADICALS

503 How can we show that the square of a number is the product of a number with itself

504 How can we show that the cube of a number is the product of the number taken 3 times as a factor

505 What is meant by raising a number to a power

506 What is meant by (a) an exponent (b) a base

507 How do we raise an algebraic symbol to a power

508 What is the operation of raising quantities or terms to given powers called

509 How can we show that the square of the sum of any two numbers is the square of the first plus the square of the second plus twice the product of the two numbers

510 How can the above be shown graphically

511 How do we find the number of square units in the surface of any plane figure or flat surface

512 How do we calculate a higher power of a common fraction

513 What are the rules affecting the powers of decimal fractions

514 Why is a decimal fraction raised to a power of a smaller value than the original fraction

515 What is the procedure when two powers of the same base or number are to be multiplied

516 What is the procedure when two powers of the same base or number are to be divided

517 What limits the above processes

518 What is the procedure when the power of a number is itself to be raised to a power

519 How can we show that any number or base to the zero power equals 1

520 How can we show that the sign of an exponent may be changed by changing the position of the number from one side of the denominator line to the other

521 Why is a decimal fraction raised to a negative power of greater value than the original decimal fraction

522 Why are the negative powers of whole numbers smaller than the original numbers

523 How can we simplify the raising of a number to a power that can be factored

524 What is the basis for a short method of squaring a number from 1 to 100

525 What then is the procedure for a short method of squaring a number from 1 to 100

526 How does the procedure of Question 509 compare with the above as a short method of squaring a number from 1 to 100

527 How can we apply the procedure of Question 509 to mixed numbers as etc

528 How may aliquot parts be applied to the above method

529 How is the squaring of a number that is divisible by factor 2 3 or 5 made simpler

530 What is the procedure for getting the square of the mean between two numbers

531 What is an easy way of squaring a number ending in

532 What is the procedure when the number ends in 5 instead of

533 What is the procedure for squaring a number consisting of 9rsquos

534 What does the exponent of any power of 10 indicate

535 Does the above apply to negative exponents of base 10

536 How can we express decimals as powers of 10

537 What is done with the exponents in multiplying powers of 10

538 What is done with the powers of 10 in division

539 What is meant by a root of a number or power

540 What is meant by evolution

541 What is the symbol of evolution

542 What is meant by (a) a perfect power (b) an imperfect power

543 What is the simplest method of extracting a root

544 What is the rule for extracting the required root of a quantity

545 What is the rule for fractional exponents

546 When are radicals similar

547 When may a factor of the radicand be removed from under the radical sign

548 How may a factor in the coefficient of a radical be introduced under the radical sign

549 How may a fraction with a radical in the denominator be reduced to a fraction with a rational denominator

550 How may a radical with a fractional radicand be reduced to a fraction whose denominator has no radical

551 How may a radical be changed to one of a higher order with an index that is a multiple of the original index

552 When may a radical be reduced to a radical of a lower order

553 When is a radical expression said to be in simplest form

554 What is the result of reduced to its simplest form

555 What is the result of reducing (a) (b) to the simplest form

556 How many figures does it take to express the square root of a number of (a) 1 or 2 figures (b) 3 or 4 figures (c) 5 or 6 figures

557 (a) What is the relation of the number of decimal places in the square of a decimal to that of the decimal itself and (b) what is the relation of the number of decimal places in the square root of a decimal to that of the decimal itself

558 What is the square root of 676

559 What is the rule for the extraction of a square root





564 How do we get the root of a fraction

565 What is the rule for the extraction of the cube root

566 What is the cube root of 245314376

567 In summary what are the principles applying to exponents

CHAPTER XIV - LOGARITHMS

568 What is meant by (a) logarithm (abbreviated ldquologrdquo) (b) exponent (c) base

569 What are the two forms of expressing the relationship between the base the power and the exponent

570 What two systems of logarithms are in general use

571 To what exponent (logarithm) must the base 10 be raised to produce a number between 1 and 10

572 To what exponent (log) must the base 10 be raised to produce a number between 10 and 100


574 How does this condition apply to higher powers of 10 for any number you may want to produce

575 Why is the log of a number between 1 and 1 expressed as mdash 1 plus the same positive decimal fraction as for Question 571 with the same sequence of digits in the number

576 How does this apply to finding the log of still smaller decimal fractions

577 Why may numbers between 1 and 10 be considered as basic numbers for a system of logs having 10 as a base

578 What is meant by the characteristic of a logarithm

579 What is meant by the mantissa of a logarithm

580 What is the rule for finding the characteristic of the logarithm of a number

581 What is the rule for finding the characteristic of a purely decimal number

582 Why is a negative characteristic kept distinct from the mantissa of a logarithm

583 How are negative characteristics generally expressed

584 May a negative characteristic be expressed in other ways

585 What is a table of common logarithms

586 How do we look up a log in a table

587 What is meant by a proportional part of a log

588 What is meant by an antilogarithm

589 How do we obtain an antilog or number from a table of logs

590 Upon what laws do computations with logs depend

591 What is the procedure for multiplying two or more quantities by logs

592 What is the procedure for getting the quotient of two numbers by logs

593 What is the procedure for raising a number to a power by logs

594 What is the procedure for getting the root of a number by logs

595 How can we express the log of 75 in terms of the log of 5 and the log of 3

596 How can we express as an algebraic sum of logs

597 How can we reduce log 7 + 3 log 5 to the log of a single number

598 What is the log of 1 to any base

599 What is the log of the base itself in any system

600 What is the log of 0 in any system whose base is greater than 1

601 How can we find the log of a number to a new base when the logs of numbers to a particular base are given

602 How are natural and common logs related as seen from the above

603 What is meant by the cologarithm of a number

604 What is the rule for obtaining the colog of a number to base 10

605 When are cologs used to advantage

606 What is the result of 005864 times 2726 times 8465

607 What is the result of (262)4

608 What is the result of

609 What is the value of (1834) ndash 3

610 What is the value of (2718)-14





615 What is the result of (3846)-16

616 What is the result of (42)71 x (76)-62 x (432 ndash 69)



619 What is the result of (58)y = 567

620 How accurate are results of numerical computations by logs

PROBLEMS

CHAPTER XV - POSITIVE AND NEGATIVE NUMBERS

621 What is meant by ldquosignedrdquo numbers

622 What is meant by ldquopositiverdquo and ldquonegativerdquo numbers

623 What is meant by the absolute value of a number

624 How can the relations between the plus numbers the minus numbers and zero be shown by the number scale

625 What are the two meanings of plus and minus signs

626 What is the procedure for addition of positive and negative numbers

627 What is the procedure for subtraction of positive and negative numbers

628 What is the procedure for multiplication of positive and negative numbers

629 What is the procedure for division of positive and negative numbers

PROBLEMS

CHAPTER XVI - PROGRESSIONSmdashSERIES

630 What is a series

631 What is an arithmetic progression

632 What is a geometric progression

633 What is a harmonic progression

634 What is known as a miscellaneous series

635 What is the procedure for solving an ascending arithmetic progression

636 What is the procedure for solving a descending arithmetic progression

637 How can we obtain a general formula for solving an arithmetic progression

638 How can we find an expression for the sum of the terms of an arithmetic progression

639 What is the sum of the first twenty-seven terms of 14 11 8 5 2 mdash1 mdash4

640 When any three of the five elements of an arithmetic progression are given how are the other two found

641 How can we insert any number of arithmetic means between two given terms

642 How can we show that the arithmetic mean between two quantities is equal to one half their sum

643 How can we find an expression for the last term l of a geometric progression when given the first term a the ratio r and the number of terms n

644 How can we find an expression for the sum S of a geometric progression when given the first term a the last term l and the ratio r

645 How can we find two of the five elements of a geometric progression when any three are given

646 (a) What do we call the limit to which the sum of the terms of a decreasing geometric progression approaches when the number of terms is indefinitely increased (b) How can we find an expression for this limit

647 How can we find the value of a repeating decimal by the use of the sum of a series to infinity

648 What is the procedure for inserting any number of geometric means between two given terms

649 How can we show that the geometric mean between two quantities is equal to the square root of their product

650 What is the procedure for solving a harmonic progression

651 How can we insert six harmonic means between 2 and

652 How can we find an expression for the harmonic mean between two terms

653 How is the sum of an arithmetic series applied in certain installment purchase problems

CHAPTER XVII - GRAPHSmdashCHARTS

654 What are graphs

655 What are the advantages of graphs

656 What are the disadvantages of graphs

657 What questions should we ask about graphs

658 What types of graphs are commonly used

659 What are horizontal bar graphs and when are they used

660 How is a bar graph constructed

661 What are vertical bar graphs and when are they used

662 What types of charts or graphs are used to show the relation of the parts to the whole of an item and which type is preferred

663 When is a circle graph or pie chart used and how is it drawn

664 How is the same information shown in the form of a long bar chart

665 What is a block graph

666 What is a broken-line graph or line diagram and when is it used

667 What is a curved graph (smooth-line graph) and when is it used

668 What are pictographs and when are they used

669 What are frequency distribution graphs (frequency polygons sometimes called ldquostaircaserdquo diagrams)

670 What is meant by an index number and how is it obtained

671 What are the advantages of index numbers

672 What is meant by interpolation

673 What is meant by extrapolation

674 When are interpolation and extrapolation advisable

676 What is easier to compare two areas or the lengths of two lines

677 When and how are statistical maps used

678 What is meant by Cartesian coordinates

679 What is meant by the axis of abscissas

680 What is meant by the axis of ordinates

681 In what order are the four quadrants formed by the axes of coordinates designated

682 What directions are considered positive and what directions negative

683 How are points located in Cartesian coordinates

684 How do we plot a straight line relationship

685 How do we plot the graph of a quadratic formula

CHAPTER XVIII - BUSINESSmdashFINANCE

686 What are the two types of cost

687 Into what two groups is profit divided

688 What constitutes cost of doing business

689 What is meant by (a) gross sales (b) net sales (c) gross purchases (d) return purchases (e) net purchases (f) depreciation

690 What are (a) trade discounts (b) cash discounts

691 What is (a) a sales commission (b) a buying commission

692 When is there (a) a profit (b) a loss

693 In figuring profit or loss what is (a) the base (b) the rate (c) the percentage

694 How do we find the selling price when the net cost and the rate of profit are given

695 How do we find the selling price when there is a loss and you are given the net cost and the rate of loss

696 How do we find the per cent of profit given the cost and selling price

697 How do we find the per cent of loss given the cost and the selling price

698 How do we figure a discount or a commission

699 How do we find the cash discount when the amount of the bill and the rate of discount are given

700 What is meant by bank discount

701 How is simple bank discount figured

702 How do we figure the net price of an item when there is a series of discounts as 40 5 and 2 (meaning 40 5 and 2)

703 How may the above process be shortened by obtaining a single equivalent of the remainder after deducting all the discounts

704 What is the procedure for getting a single discount which is equal to two discounts by mental calculation

705 Using this method how can we get a single discount which is equal to a series of discounts

706 If after 8 and 4 discounts are deducted the net cost of an invoice of goods is $168436 what is the list price

707 If the amount of discount is $39842 and the discounts are 40 and 2 what is the net cost of the goods

708 If the terms on a $2680 invoice of goods are 410 n60 how much do you gain if you borrow money from a bank at 6 for 60 days and pay cash for the merchandise

709 If the gross cost of an article is $672 and the article is sold at a profit of 30 on the selling price how much is the net profit if 21 is charged to the cost of doing business

710 What is the procedure for getting the selling price given the net cost percentage of profit and cost of selling

711 How can we find the relation of net profit to selling price in percentage

712 How can we find the relation of net profit to gross cost or to net cost expressed as a percentage

713 If shirts are bought for $560 less 14 and 8 and are sold for $740 less 10 and the buying expenses are 4 of the net cost and selling expenses are 5 of net sales what of the gross cost is the net profit

714 If we know the amount of profit the per cent of profit on the gross cost and the per cent of buying cost how do we get the net cost and the cost of buying

715 If we know the net cost per cent of buying expenses and the amount of profit how do we find the per cent of profit and the selling price

716 If you buy an article invoiced at $3460 less 3 discount and sell it at 30 profit what is the selling price

717 If a dealer buys a TV set for $360 pays $12 freight and cartage and sells it at a profit of what is the selling price

718 If a merchant pays $1860 for an article and sells it at a profit of 25 of the selling price what is the selling price

719 If the gross cost of an article is $865 and it is sold at a profit of 25 on the selling price what is the net profit if the cost of doing business is 12

720 If a merchant sells apples at $550 a bushel at commission and his commission amounts to $14850 while other charges are 35cent a bushel how many bushels does he sell and how much are the net proceeds

721 The cost of a TV set to an appliance dealer is $360 less 40 and 2 What should he mark the set if he wants to make a profit of 25 on the net cost and allow the customer a 15 discount on the marked price

722 What is meant by the ldquofuture worthrdquo or value of a sum of money

723 What is meant by the ldquopresent worthrdquo or value of a sum of money

724 What is meant by the true discount

725 What are the present worth and the true discount of a debt for $1800 due in 8 months if money is worth 6 interest

726 If A owes B $1000 which is not due until 3 years from now and A offers to pay B today what sum should A pay now at compound interest assuming the money to be worth 4

727 What is meant by the present value of 1 and how is it used

728 In what two ways may consumer finance be considered

729 What is meant by installment buying or buying goods ldquoon timerdquo

730 If you buy a washing machine for $280 are given a $50 trade-in allowance for your old machine and agree to pay the balance in 10 monthly installments plus a final installment of $35 how much would you save by buying for cash

731 If you borrow $2400 from a bank and pay it back in monthly payments of $3805 over 6 years how much do you pay the bank for the loan

732 Why is buying goods on credit the same as borrowing money

733 Why does credit or installment buying cost more

734 Why do some merchants prefer the credit plan to cash despite all this

735 What are some of the ranges of interest charged in consumer finance

736 What is the 6 method offered by some credit companies and how do we find the monthly payment

737 If you as a merchant decide to charge an additional 14 on the goods you sell ldquoon timerdquo what would be the price on a 10-equal-payment plan and the amount of each payment on a clock radio that sells for $8860 cash

738 What is the key in figuring the annual rate of interest charge you pay when you buy on the installment plan or when you borrow money from a finance company to be repaid in monthly installments

739 How much interest and financing charge do you pay when you buy a TV set for $280 if you are allowed $50 for your old set as trade-in allowance and you agree to pay the balance in 10 monthly installments of $23 plus a final installment of $35

740 What precaution must you take in getting the sum of the number of months you keep or borrow the installment payment

741 How can we solve for the rate of interest by getting the total amount of the installment money you keep or borrow for one month in the example of Question 740

742 If you borrow $300 from a finance company to pay a surgical bill and you are charged 3 per month interest on the unpaid balance of the loan while you are required to repay the loan in 12 monthly installments of $25 each how much do you pay back for the $300 loan and what is the annual interest rate using the installment plan method

743 If you borrow $300 from a credit union where the interest charge is 1 a month on the unpaid balance and you pay back the loan in 12 monthly payments of $25 plus interest charge how much do you pay back and what is the annual interest rate How does this compare with a secured bank loan of $300 for 1 year at 6

744 If you get a loan of $2500 at 5 interest per year and you agree to pay it back in 20 years at $1650 per month how much is the total amount of repayment and how much does it cost you

745 How does the above cost compare with a bank loan of $2500 for 20 years at 5

746 If you get a loan of $7000 at 5 a year on the unpaid balance from a mortgage company to finance your home and you agree to pay it back in 8 years at $8862 per month what is the total repayment on the loan and how much does it cost you

747 What is a commonly used method of determining the annual rate of interest when you buy or borrow on the installment plan

748 What is the formula for the equal installment constant-ratio method of finding annual interest rate in installment plans

749 How is the constant-ratio formula obtained

750 If a TV set is priced at $150 cash and the advertised payment plan is $25 down and $3 a week for 45 weeks what is the interest rate

751 A clock radio is offered for $45 cash or on time payments for 10 more with a down payment of $950 and the balance in 13 weekly payments What is the annual rate of interest

752 A hi-fi set can be bought for $380 cash with a discount of $19 or in 12 equal monthly installments by paying $130 and adding a $30 carrying charge What is the annual rate of interest

753 If you borrow $150 from a loan company for 10 months and repay it in 10 equal installments of $1734 what rate of interest do you pay

754 How can we get the annual rate paid in Question 753 by finding the amount of money the borrower had the use of for 1 month

755 How can we get the annual rate paid in Question 753 by finding the total time the borrower had the amount of the installment available for use

756 If you borrow $300 from a bank for 15 months and pay back $2157 per month what annual rate are you paying as figured by the three methods shown

757 If you buy on time a set of dishes that costs $86 cash and $12 is added for carrying charges on a payment plan of $14 down and $14 a month for 6 months what is the rate of interest you pay

758 What is the interest on the time plan if a clothes dryer sells for $189 cash or $20 down and $21 per month for 10 months

759 What is the constant-ratio formula for finding the interest rate when all payments are equal except the last one

760 What is the interest rate per year if a clock costs $25 cash or $5 down and $5 per month for 4 months with a $375 payment the fifth month

761 What is meant by partial payments

762 What two rules are used to solve partial payment problems and upon what does the method used depend

763 How do banks accepting partial payments of notes submitted for discount collect compound interest and yet avoid the Supreme Court ruling

764 For how long do notes and accounts on which no payments have been made remain in full force

765 Must mortgages made for a definite time be paid on maturity

766 What is the procedure for solving partial payment problems by the merchantsrsquo rule

767 What is the procedure for solving partial payment problems by the United States rule

768 By the United States rule how much is required to settle on August 1 1961 a demand note for $10000 dated February 1 1960 with interest at 6 and with the following payments endorsed upon it April 10 1960 $2000 August 4 1960 $100 February 1 1961 $4000 June 1 1961 $1000

769 What are the two general kinds of taxes

770 What is (a) a poll tax (b) a property tax (c) an income tax (d) a surtax

771 What is (a) a licence (b) an assessment

772 In what form are assessments usually stated

773 What are the three items that are usually involved in taxation

774 What is the tax on a property assessed for $7500 if the rate is $2885 per $100 and the collectorsrsquo fee is 2

775 How do we find the tax rate when given the base (assessed valuation) and the tax amount

776 How do we find the assessed valuation when given the tax rate and the tax

777 How do we calculate (a) surtax (b) total tax

CHAPTER XIX - VARIOUS TOPICS

A Working rates of speed

B MixturesmdashSolutions

C Tanks and Receptacles (Filling Emptying)

D Scales for Models and Maps

E Angle measurement

CHAPTER XX - INTRODUCTION TO ALGEBRA

821 What is algebra

822 Why is algebra said to be a shorthand extension of arithmetic

823 How are the letter symbols in algebra selected

824 What is meant when two letters or a number and a letter are placed alongside each other

825 What is meant by a coefficient

826 What is meant by a term

827 What is a binomial

828 What is meant by (a) a factor of a product (b) literal factors or numbers (c) specific numbers

829 What is meant by (a) an algebraic quantity (b) an algebraic expression

830 What is meant by the coefficients of a product

831 What is a polynomial

832 What symbols are used in algebra to indicate addition and subtraction

833 What symbols are used to indicate multiplication and division

834 What are the four elements of every algebraic term

835 On what occasions are some of the elements omitted

836 How is + x1 ndash 5x2 + 1x4 ndash 3y3 written in practice

837 What laws of addition subtraction multiplication and division of numbers are also applicable to algebraic processes

838 How may we regard two or more letters or numbers enclosed in parentheses

839 In algebraic fractions why may the fraction be considered to act as a set of parentheses

840 In what ways may x be written

841 How are verbal expressions translated to algebraic symbols and terms

842 How are algebraic symbols converted to verbal expressions

843 What is the general procedure for expressing thoughts algebraically

844 How do we indicate a letter multiplied by itself a number of times

845 Why is a2 called ldquoa squaredrdquo

846 Why is a3 called ldquoa cubedrdquo

847 How do we raise an algebraic term to any power

848 What is the rule for multiplying the same kind of letters or expressions together

849 How do we multiply letters that have coefficients affixed

850 What is the meaning of square root

851 What is the rule for getting the square root of any power of a letter

852 What is meant by the root of a given number or term

853 What is the rule for division of the same kind of symbols

854 How can we show that a quantity to the zero power = 1

855 What is the result of (a) (b) (c) (d) (e)

856 What does mean

857 When may we regard two terms as like terms

858 Does the order in which the symbols occur matter at all

859 What is a simple test as to whether two terms are or are not alike in value

860 What do [+] and [mdash] signs mean in algebra

861 How are [+] and [mdash] quantities applied to debt and income

862 What is the rule for subtraction of one plus quantity from another plus quantity

863 How can we show that two minuses mean a plus

864 What is the rule for signs

865 What is the rule for numbers (or letters) that are multiplied together or are divided

866 How do we distinguish between +3(mdash8) and +3 mdash 8

867 What is the result of 8(a mdash b) mdash 12(3a mdash 4b)

868 What is the result of 7[3a mdash 4(5b mdash 6a) mdash 2b]

869 What is the result of 3[4x mdash (2x + y) + 5(3x + y) mdash 6y]

870 How can you check yourself to know whether your solution is correct

871 What is the procedure for evaluating algebraic terms

872 What is the first important fact to remember in adding or subtracting algebraic terms

873 What is the procedure for getting the algebraic sum of a number of terms

874 Why is it that to any term you may add only other like terms if you want to give the result as a single term

875 What is the procedure for subtraction of algebraic quantities

876 What is the procedure for removing parentheses or brackets enclosing a number of algebraic terms

877 How may we illustrate the multiplication of a polynomial algebraically

878 What is the product of

879 How can we show that the square of the sum of two terms is equal to the square of the first term plus twice the product of the two terms plus the square of the second term

880 How can we show that the square of the difference of two terms is equal to the square of the first term minus twice the product of the two terms plus the square of the second term

881 How can we show that the product of the sum and difference of two terms is equal to the difference of their squares

882 What is the procedure for getting the direct answer to the multiplication of any binomial by another binomial

883 What is the result of simplifying 2x(x + 5y) + 3y(x + 4y)

884 What is the result of simplifying


886 What is the procedure for dividing a polynomial by a single term

887 What is the procedure for division of a polynomial by a polynomial

888 What is the quotient of a2 + 2a2b + 4ab + 2ab2 + 3b2 divided by a + 2ab + 3b

889 What is the result of division of a3 ndash a2b - 7ab2 - 20bg by a ndash 4b

890 What is the ldquocommon termrdquo method of getting the factors of an expression

891 What is the ldquocommon parenthesesrdquo method of getting the factors of an expression

892 What is the procedure for factoring by the combination of the common term and the common parentheses methods

893 What is the ldquoproduct of two binomialsrdquo method of getting the factors of a three-term expression

894 What are the factors when the expression is recognized as a perfect square

895 What are the factors when the expression is in the form of the difference of two squares

896 What is the value of when a = 3 and b = 2

897 What is an equation

898 How can we show the balance-scale resemblance of an equation

899 What is the chief use of an equation

900 What is meant by the root of an equation

901 What is meant by an identity

902 What is meant by a conditional equation

903 What is a linear or simple equation

904 What is a quadratic equation

905 What may be done to both sides of an equation without affecting its balance

906 What is the rule of signs for moving terms from one side of the equals sign to the other

907 What is the result when both sides of an equation are multiplied or divided by the same quantity

908 How can we solve simple equations by addition or subtraction

909 What are the steps in the solution of an equation

910 What is the solution for y of P

911 What is the solution for d in A minus pd = b minus d

912 What is the solution for W in W = T

913 What is the solution for x in =

914 What is the solution of x + 7 minus 3x minus 5 = 12 minus 4x

915 What is the solution of (x + 5)2 minus (x + 4)2 = x + 12

916 What is the solution of 7(x + 5) minus 9(x minus 2) = 8x + 3

917 What is the value of x in 8(x minus 3)(x + 3) = x(8x minus 8)

918 What is the value of x in minus (x minus 2)2 = 22

919 What is the procedure for solving equations involving decimals

920 What is a formula

921 If the relation between the Fahrenheit temperature readings of a thermometer and the Centigrade readings is expressed as F = C + 32 what is the Fahrenheit reading when (a) C = 50deg (b) C = 30deg (c) C = 10deg

922 What is meant by solving for another variable in a formula

923 What is the general procedure for putting words into equation form to express simple equations with one unknown

924 If the sum of three consecutive even numbers is 90 what are the numbers

925 If a tank is full of water and after running off 300 gallons it is full what is the capacity of the tank

926 If you are 45 years old and your son is 12 years old (a) when will your son be half your age (b) how long ago were you 5 times as old as your son

927 If two machine operators punch out 1400 plastic parts per hour and one produces as many parts as the other what is the production of each

928 If you and your wife together hold $7800 in United States government bonds and your share is $1100 more than your wifersquos how much do you each have

929 If you bought 3 suits for $226 and the first cost twice as much as the second while the third cost $10 more than the second what is the cost of each suit

930 If you have $245 in nickels and dimes and you have 30 coins in all how many of each do you have

931 At what time between 4 and 5 orsquoclock are the hands of a watch opposite each other

932 If you want to sale price 300 1b of coffee at 78cent a 1b and you have one kind that normally sells for 90cent a 1b and another that sells for 70cent a 1b how many lb of each must you mix so that you will not lose money

933 If you sell 3 taxicabs and buy 2 new ones for $7800 and you then have $2400 left how much did you get for each taxicab you sold

934 During the year you your wife and your daughter saved a total of $1200 You saved $100 less than twice your daughterrsquos savings and your daughter saved $10 more than twice your wifersquos How much did each save

935 What is the number which when multiplied by 4 equals the original number plus 36

936 If a train leaves Washington DC for Chicago and travels at the rate of 50 miles per hour and hour later an auto leaves for Chicago from Washington traveling at the rate of 55 miles per hour how long will it take the auto to overtake the train

937 You start out to walk to your friendrsquos house at the rate of 4 mph Your friend starts at the same time for your house at 3 mph You live 14 miles from each other How far does each of you walk before meeting

APPENDIX A - ANSWERS TO PROBLEMS

APPENDIX B TABLES

INDEX

A CATALOG OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST

TECHNIQUESANDVARIOUSAPPLICATIONSZAMelzak(0-486-45781-8)MATHEMATICALPROGRAMMINGStevenVajda(0-486-47213-2)

FOUNDATIONSOFGEOMETRYCRWylieJr(0-486-47214-0)

SeeeveryDoverbookinprintatwwwdoverpublicationscom

BOOKSBYAALBERTKLAF


ArithmeticRefresher

AAlbertKlaf




9780486141930


FOREWORD




FRANKLINSKLAFMD

TableofContents



INTEREST

INTRODUCTION





89arecalleddigits









































thatnumber





integralnumbers





towardstheleft




beginningattheright






EXAMPLE




EXAMPLE400536080209












theunitsposition








100000000=10times10000000




EXAMPLES











smallerfraction

EXAMPLE





EXAMPLE


EXAMPLE



readldquoandrdquo




forty-threecents









EXAMPLES



EXAMPLES


EXAMPLES


EXAMPLES

PROBLEMS1

1Howmanyunitsin379























21Howis$356356read


(e)$444(f)$888



(a)

(b)

(c)

(d)








CHAPTERI

ADDITION







groupedtogether







EXAMPLE



EXAMPLE



EXAMPLE(asabove)


EXAMPLE


groupthen














EXAMPLES(a)

(b)


theother

EXAMPLE96+78



increments

EXAMPLE



EXAMPLES

(a)Add26to53

(b)Add67to86




EXAMPLE





Ex(a)

Ex(b)






EXAMPLE








EXAMPLES

(a)

(b)












EXAMPLE

PROBLEMS









9Add269745and983


11Add$525$1760$085$175$4565

12Findthesumof

(a ) (b ) (c)

$380865 $987367 $887406

37692 38898 51856

38623 573200 129897

48008 898719 54265

88842 782492 38600

75182 608604 4209























(a)

(b)

CHAPTERII

SUBTRACTION




8minus5=3 8minus3=5





Ex(b)16minus9=7












Ex(a)



Ex(b)






Ex(a)








SubtractionTable


etc






Ex(a)

Ex(b)


Ex(c)



Ex(a)


Ex(b)



EXAMPLE




Ex(a)


Ex(b)


Ex(c)


Ex(b)



columnattheright

Ex(a)

Ex(b)



EXAMPLE





Startatleft














EXAMPLES


Ex(a)


Ex(b)


Ex(a)



Ex(b)

PROBLEMS


1

2

3

4

5

6






12Subtract

















CHAPTERIII

MULTIPLICATION



4+4+4+4+4+4+4+4=32or8times4=32






EXAMPLE






MultiplicationTable















EXAMPLE



EXAMPLE



added457times
















Ex





(d)Add(a)+(b)+(c)

EXAMPLES(1)

(2)

(3)






Ex(a)45times29



Ex(b)323times35


Ex(c)271times55


multiplybytheother

Ex(a)

Ex(b)

Ex(c)


Ex(a)

Ex(b)

Ex(c)


Ex(b)

Ex(c)


Ex(a)

Ex(b)


discardingfraction

Ex(a)


Ex(b)

Ex(c)

Ex(d)


Ex(a)

Ex(b)













there4 is of100


there4 is of100


there475is of100



decimalThebaseis100

EXAMPLE(asabove)


020405062506660833


125133316662025






EXAMPLE


EXAMPLE


EXAMPLE



EXAMPLE


















numberfromtheresult

Ex(a)


Then


numbertothis

Ex(a)

Ex(b)




Inoneline



Inoneline









EXAMPLE


EXAMPLE








(d)Add(b)to(c)

Ex(a)




88times100=88008800+32=8832Ans



7900+98=7998Ans





EXAMPLE













EXAMPLE




PROBLEMS




4Multiply631by60



7Multiply6390by300

8Multiply


9Whatistheproductof


10Multiply




























35Whatisthecostof















CHAPTERIV

DIVISION




















EXAMPLES


(b) or

(c) orDivisor(=7




EXAMPLE



remainder

EXAMPLE





In =7wehavedivision



972=900+70+2


howwouldweproceed


4rsquos=3200 4)3492





(e)3times4=12



800+70+3=873




































getting2asquotient












EXAMPLE











Quotientis







Quotient=





Ex(c)




Ex(d)








EXAMPLE



in2468orinazero


in21273834485





and2times3=6






Ex(a)

Ex(b)


Ex(a)

Ex(b)


by8

Ex(a)




Ex(c)








Ex(a)

Ex(b)


Ex(a)

Ex(b)


Ex(a)


Ex(b)

Ex(c)


Ex(a)867


Ex(b)973285









Ex(b)





placestotheleft












Ex(1)


Ex(2)











Monday $26860

Tuesday $32985

Wednesday $9745

Thursday $23990

Friday $29670


$24650 (=Average)









For10cent



then



PROBLEMS



















9Divide

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

(u) (v) (w) (x)

(y) (z)

10Findthequotientof





(m) (n) (o)


12Findthequotientof





14Divide






16










17


18



(a) (b)

(c)

(d) (e)










CHAPTERV











Ex





EXAMPLE
















GCD





Ex(a)


there42times3=6=GCD

Ex(b)





















7appearsonce
















PROBLEMS










(a)310(b)297(c)670(d)741(e)981(f)385(g)2650

(h)1215((i)321(j)1575(k)10935(l)420(m)497

(n)378(o)462(p)2430(q)25344(r)73260(s)599676

(t)273564(u)15625(v)10675(w)12625(x)976

(y)8050(z)3848


(a)68112240(b)2184126147(c)212877

(d)457281(e)4477121(f)1498112(g)248096

(h)284236(i)457281(j)31522679012

(k)144576(l)820697(m)1251751792(n)60043318

(o)125423618163(p)1086905




(j)86and4



(e)3642and48(f)3918and27(g)51525and35


(l)212426and28(m)92142and63

(n)367548and24(o)71456and84(p)2472128and240



(d)88368divide3682(e)32768divide2048







CHAPTERVI

COMMONFRACTIONS





EXAMPLE


exactlyequalsize



Thus

191Whatismeantby

(a)

(b)

(c)

(d)










Ex(a)


Ex(c)


equalparts





(b)



Wegetthewholeunit

Ex(a)

Ex(b)

Ex(c)







numbers










Ex(a)

Ex(b)






Ex(b)



Ex(a)

Ex(b)

Ex(c)

Thereasoningis



Ex(a)

Ex(b)






commonfactors

Ex(a)

Ex(b)







EXAMPLE


EXAMPLE


















Ex(a)



Ex(a)

Ex(b)



EXAMPLE


mustallbethesame


Ex(a)Add and


Add


Addfractions

Then




Now

Ex(b)Add (LCD=20)








Ex(b)Subtract from



Ex(a)






Ex(b)

Ex(c)


Ex(a)

Ex(b)


Ex(c)

Ex(d)


Ex(a)

Ex(b)Multiply by11

Ex(c)


inposition



Ex(a)

Ex(b)Multiply




EXAMPLEMultiply



Ex(a)

Ex(b)Multiply by

Ex(c)Multiply by



EXAMPLE













(a)

(b)


EXAMPLEDivide by




Ex(a)Divide by2


Ex(b)


Ex(a)Divide24by

Ex(b)Divide17by or






EXAMPLEDivide by3



EXAMPLEDivide by6



Ex(b)Divide by



partsareexpressed






Ex(c)Whatpartof is

Ex(d)Whatpartof is7
















EXAMPLE


fractions












Next

Next

Finally
















Startfromthebottom







PROBLEMS






(a) (b)

(c) (d)

(e) (f)










(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b)

(c) (d)

15


16





(a) (b) (c) (d) (e) (f)

19Add

(a) (b) (c)(d) (e) (f) (g)(h) (i) (j)(k) (l)

20Subtract














31














encloseit




CHAPTERVII

DECIMALFRACTIONS


EXAMPLES




EXAMPLES




Ex(a)0207008024017

Ex(b)


EXAMPLES








EXAMPLE5631056923









EXAMPLES




EXAMPLES



numerator





048386356



EXAMPLE6=6=60=600




EXAMPLES





EXAMPLES



EXAMPLES

(a) or

(b) or

(c) or

(d) or

(e)

(f)














Ex(b)










usualway









Ex(a)Multiply38by6

Ex(b)

Ex(c)


improperfractions

EXAMPLE






Ex(a)Divide192by06











theleft




















4879times3765=18369435



mostimportant





EXAMPLE



EXAMPLES

(a)(b)(c)


infinity



group




whichdonotrecur



EXAMPLES(a)

(b)

(c)

(d)

(e)












(b)


PROBLEMS






3Read12584062018








260260026260260













23Add(a) and







(e)64575by165(f)9686by136



31Shorten(a)59767(b)59755
























CHAPTERVIII

PERCENTAGE





EXAMPLE


EXAMPLES(a)(b)

(c)

(d)

(e)(f)(g)










EXAMPLES



EXAMPLES








Ex(b)Find14of$9751(14=14)



Ex(a)Find6of$6700

1of


6of$6700=6times$67=$402

Ex(b)Find4of$1860

1of$1860=$1860there44is4times$1860=$7440

Ex(c)Find of$7000



Ex(a)Find25of$51

Ex(b)Find of$8475







Ex(b)Find9of50


fraction












adecimalSince



or





If$285is2then

or





or






or












$15750times0025=$39375=$3938Ans



120times$55=$66Ans




$46375times43=$19941Ans






fromtheamount


06times85+85=51+85=901=Amount











PROBLEMS



31400iswhatof3600

4Reducetoadecimal




(k) (l)



















25Whatpercentis of6
































CHAPTERIX

INTEREST









EXAMPLE




EXAMPLE

005=fivepercent=




EXAMPLE



















=$500(115)=$575Ans










eachor360days



Year Month Day

1959 2 8

1957 5 15

_____ ____ _____

1 8 23









May 23days

June 30days

July 31days

August 31days

September 30days

October 31days

November 30days

December 31days

January 12days

249days











Year Month Day

1959 4 6

1958 11 18

4 18 =138days










Then

and

Then

and







Now6daysare







(b)58days(c)77days



4days 6days 2days

49days 2days 77days

58days

58days



2

















or
































$468=$450+$450times04

or


and





or






Therefore


S=$50619


$50619Ans














S=$12000times131593=$1579116Ans






n=6=t=6yearsAns














Inabove




Inabove













126898555Ans



S=$8000times181535485=145228388


$1452284Ans

PROBLEMS
























































CHAPTERX

















EXAMPLES

$54000$18000=31412=13(to)(to)(to)(to)











numbers



Ex(b)




$100to$500is














Dividebothtermsby

















there4320560720=479Ans





there4 Ans







or




Ormargin



(b) forrent















EXAMPLE3=9


EXAMPLE36=12

















Value=4


Value=2










EXAMPLE



[]=ldquoasrdquo














othertwo

Ex(a)36612


Ex(b)


Ex(c)3xx12




cost

3x=5times$864


there4 Ans















doitin5days


























Thus






andsolvedasfollows








3times8=4times6

(b)If then


EXAMPLEIf =then 3

(c)If then


EXAMPLEIf then

(d)If then


EXAMPLEIf then or

(e)If then


EXAMPLEIf then or

(f)If then


EXAMPLEIf then or




or










variation



C=πd=variationform
















Sαr2



S=4πr2







andinversetypes




From

then



where


PROBLEMS












(a)1525(b)2415(c)824(d)2724

























36(a)4=16(b)5=4(c)

37(a)24=6(b)49=7(c)






(a)(b)

(c) (d)

(e)

(f)

(g)

(h)

(i)








(a)

(b)

(c)
























CHAPTERXI

AVERAGES






















things











48+55+68+42=213minutes








there4















Theweightedmarksare






Remainder=199























Themeanaverageis



Themeanaverageis




200200200220225235240240250260260275275350450500650and800





amount



$700 and over (1)

$400 to $699 (3)

$300 to $399 (1)

$250 to $299 (5)

$200 to $249 (10)

$150 to $199 (4)

less than $150 (1)



Themedianis$235


























destroyed


whichdonoteffectit










PROBLEMS























$500(1)$50(12)$300(2)$25(22)$100(3)$15(6)$75(4)




CHAPTERXII

DENOMINATENUMBERS






EXAMPLES



descending







161feet=1rod



Note(c)













































Pound Ounce









24hours = 1day(da)

7days = 1week(wk)


52weeks = 1year(yr)




10years = 1decade

20years = 1score






20units = 1score

12units = 1dozen











TheLatinprefixesare


TheGreekprefixesare











= 0001sqmeter


= 01sqmeter


= 1sqmeter=1centare

= 1sqmeter=1centare


= 100sqmeters=1are




= 1000000sqmeters








= 000001cumeter





= 1000cumeters


= 1000000cumeters


































andfindthetotal







8qt

(or25)qt+6qt=625qt


4pk

78125pk+2pk=278125pk


6953125buAns







EXAMPLEAdd








EXAMPLE




















23sqyd+1sqyd=24sqyd



turn







6403509metersAns






5327698kmAns





PROBLEMS















































46Add











CHAPTERXIII







EXAMPLES





EXAMPLES





used







26x26=676





400+240+36=676


(a+b)2=a2+2ab+b2









EXAMPLES

22=0452=25


EXAMPLES



EXAMPLE


EXAMPLE


Now25=32and23=8




Ex(a)Divide32by8

Ex(b)Divide243by9






(42)3=42bull42bull42=42times3=46=4096




Ex








Now




EXAMPLES




Ex(a)

(b)


Ex(a)

(b)










Raise52=25Then252=625Then6253=244140625=512




(29+1)(29mdash1)=(292ndash1)

or

30times28=840=(292ndash1)292=840+1=841


number




(b)




(a+b)2=a2+2ab+b2=a(a+2b)+b2




Ex(a)

(b)

(c)






(b)Square35 =772=4949times52=1225

(c) =772=4949times22=196(d) =992=8181times22=324







Ex(a)

(b)


Ex(a) issimilarto

(75)2=70times80+25=5625(b)(125)2=120times130+25=15600+25=15625




Thenfigure8


Ex(a)

(b)






(b)4000000=4times1000000=4times106

(c)36300000=363times10000000=363times107









Ex(a)

(b)

(c)

(d)

(e)



Ex(a)

(b)

(c)












leavingaremainder









3)9

3




(c) =a2b(d)

(e)(f)





Ex(b)(c)(d)

(e)(f)

(g)




Ex(a)(b)


Ex(a)(b)



Ex(a)

(b)



EXAMPLE


bythesamenumber

EXAMPLE


EXAMPLE









(b)






decimalitself

EXAMPLES



EXAMPLES






(a+b)2=a2+2ab+b2=a2+(2a+b)b









radic676=26



















there4




remainder





















Ex(a)

(b)




















there4 =626Ans






=1 =1

PROBLEMS

1Find(a)52

(b)82

(c)202

(d)14

(e)H2

(f)19

(g)103

(h)34

(i)252

(j)173

(k)833












(c)(d)3times4-1

(e)(05)-2

(ƒ)(a-m)-n







(d)65(e)225




(e)

(f) (g)(h) (i)


(f)




(a)

(b)

(c)


(c)



(a)(b)(c)









35Whatisthevalueof

CHAPTERXIV

LOGARITHMS









Ex(b)43=64







logarithm=exponent

















Ex(b)








4642isabasicnumber











Ex(a)


1to9 1 1mdash1=0

1to9 1 1mdash1=0

10to99 2 2mdash1=1






log000865=ndash4+9370







Ex(a)log000865=49370(b)log00427=36304





3+8+6304mdash8=56304mdash8

or

+30+6304ndash30=276304ndash30




















Then83315+0003883353













105times times = =104=10000






=1093=103=1000


EXAMPLE160=1022041236=102372928=1014472






EXAMPLE135834=10213301896=1095230

Thus

Theantilogis151605



EXAMPLE374=1015729


=1015728x3






EXAMPLE




75=52times3














EXAMPLE



EXAMPLE

log10100=4343logε100=4343times46052=2



cologa=log















Then

and





there4ndash09683Ans










Nowaddallthefactors








Then




(c) =(d)9radic2=2235(e)34=81(f)2-2=







(h)x=log1001000(i)logx49=2




(n)00054times10ndash4(o)3755000(p)4343




(j)00638times104






(c)

(d)


(a)log

(b)log










(c)(d)(2112)minus3(e)(2718)ndash12(f)

(g)(06493)minus(h)(5937)minus13


(j)

(k)(l)(42)x=649

(m)(n)

CHAPTERXV









numbersas32+5+711+



EXAMPLES












(orsame)sign






(Question626)



5+(mdash7)=mdash2
















CHAPTERXVI






Ex(a)24681012etc






Ex(a)28321285122048etc


Ex(b)20485121283282





Ex(a)358101315182023










EXAMPLE13579




EXAMPLE25211713














Then













Ex(a)


Then

Now

Ex(b)


Now

(1)


Substitutingin(1)










series



Thustheseriesis

4 minus1 minus6


progression

xmdasha=bmdashx

or

2x=a+b

and

x= =halftheirsum




there4l=arnminus1


41 to7terms




(1)


(2)


rSmdashS=arnmdasha



Then

But

l=arnndash1orrl=arn



Therel= r= anda=4



FindrandS



Thiscanbewrittenas




Herea=3and


Now


Then








progression

Hence










Here andn=8Then





and



346





(15times01=0015)





Then


PROBLEMS


(d)11119977(e)8127931 (f)2818325072(g)12481632(h)403430282218























CHAPTERXVII

GRAPHSmdashCHARTS




verbaldescription
























$4000-$4999 12400

$5000-$5999 10200

$6000-$6999 8100

$7000-$7999 3040

$8000-$8999 2200

$9000-$9999 1160

$10000andover 208






Year Sales

1954 $38260000

1955 $47840000

1956 $43190000

1957 $45000000

1958 $39080000

1959 $47040000

1960 $51000000









Freshmen = 520

Sophomores = 410

Juniors = 380

Seniors = 290




Then
































91to100lb 12

101to110lb 124

111to120lb 268

121to130lb 107

131to140lb 26

141to150lb 8

141to150lb 8

151to160lb 4





















mathematicallaw




















first









EXAMPLES




PROBLEMS




$4000-$4999 8400

$5000-$5999 3200

$6000-$6999 2100

$7000-$7999 1800

$8000-$8999 760

$9000-$9999 139

$10000andover 68


1950mdash$54000000 1956mdash$46000000

1951mdash$52000000 1957mdash$45000000

1952mdash$51000000 1958mdash$39000000

1953mdash$47000000 1959mdash$47000000

1954mdash$37000000 1960mdash$52000000

1955mdash$48000000





$26500000 228






Total= $1161650000 10000


Freshmen 650

Sophomores 530

Juniors 480

Seniors 390




Year Cost

1950 $8675

1951 $9300

1952 $9475

1953 $9950

1954 $10625

1955 $11350

1956 $12225





January92hr

February106hr

March112hr

April122hr

May136hr

June146hr

July154hr

August142hr

September136hr

October114hr

November102hr

December94hr










CHAPTERXVIII










































discount


=tradediscountAns





























Theirsumis30+4=34

oftheirproductis

Thedifferenceis




(a)Combine40with10




Now

$39842=412=discount

Then







and











(a)Netcost













(f)ofprofit= =100





(b)Sellingprice





(f)of


profit

(a)Netcost14+8ndash




(d)Grosscost=$443+$18=$461

(e)Netprofit


=$666ndash$100=$566




cost
















Grosscost=$360+$12=$372



Cost=100minus25=75=$1860










=85=sellingprice








Forsimpleinterest

Forcompoundinterest






Then



Toprovethiswehave


and




Alsoitfollowsthat


or

$88900times112486=$1000






Soinsteadoffinding


there4$1000times88900=$88900









$25 = downpayment

$135 = 45weeksat$3


$150 = cashprice





= finalpayment

$35 = finalpayment


$230 = cash










































P=principal=$10here
















Sumofmonths







Youkeeporborrowthe

















$84for1month









Thenbyproportion







Totalmonths

Totalinterest

P=principal=$25




Totalmonths

Totalinterest
















Repaymentschedule



















upto thyearm























Theborrowerhad

















remainingbalance



$5down+5times$5+$2 = $32

Cashcost = $28




Here

















full







payment




thenoteisdesired


Faceofnote $100000






+$969










Weseethat


Faceofnote $1000000





















business









muchper$1000









$68750divide$2383015=02885=28852885=$2885per$1002885=$2885per$1000





$2885per$100=2885=02885



Add(a)and(b)


$16000times05 =$800

$16000times03 =$480





$8000minus$3000 = $5000

$5000times02 = $100 = surtax


PROBLEMS






































































CHAPTERXIX

VARIOUSTOPICS



























Expressasafraction






































$1800minus$1600=$200














ittoa25solution

Asabove























Then

and









Thenbyproportion















25 = 25

25times128 = 32




445 = 0445



67 = 067

067times128 = 8576=858




or













(b)Addthefractions
















Sumoffillingrates

Sumofemptyingrates


Thus and



















Scaleis172or


















(a)Bytheratiomethod





(a)Byratiomethod




EAnglemeasurement

815Whatisanangle

















PROBLEMS






































CHAPTERXX

























monomial

EXAMPLE20p=aterm






product













factors






polynomial



Letxdenoteathing








(b)




















Inalgebraa+b=b+a



Inarithmetic(5+9)+12=5+(9+12)=26





Inalgebraab=ba






















Itisnot


Ifx=2then

or



(a) times(b) (c)75x







(e)Costplus8=c+8







(l)Anyevennumber=2a





























$1250times48+$1050times72=$600+$756=$1356Ans












cube













(a+b)3times(a+b)4=(a+b)3+14=(a+b)7

Now


Therefore









thesquaretoot

EXAMPLES







Ex(a)Dividex5byx3

Ex(b)

Ex(c) (notx2)






(e) (a)

(b)

(c)

(d)

(e)


856Whatdoes mean



Similarly












Ex(a)Isa2b3likea3b2



there4Theyarealike













Youareonly700ftdown




Algebraically


















EXAMPLES



Ex(a)















Removeinnerbrackets



answer




ndash28+80=52Check




Ex(b)

Ex(c)


EXAMPLEWemaycombine


Wemaynotcombine






+15x mdash16y +8z


mdash20x +16y +14z











Thisbecomes






5a+7ndash3a+8=2a+15












(d)6a2+3bby3a+4b2

(a)


(d)



Ex(b)




Ex(b)


Ex(a)

Ex(b)

Ex(c)

Ex(d)

Ex(e)








2x2+10xy+3xy+12y2or2x2+13xy+12y2


(a+2)(a+4)+(a+3)(a+4)+(a+2)(a+5)a2+6a+8+a2+7a+12+a2+7a+10

or

3a2+20a+30Ans











Useruleofsigns























































theexpression









Ex(b)Factorx2+6x+8



Extremesxtimes8=8x




Extremesxtimes2=2x







Squarerootof9a2is3a




Ex(b)Factorx2+6x+9







(x4ndash25)(x4+25)


















or












Ex(b)













Ex(a)Solve3x=21


x=7Ans

Ex(b)Solve x=36

















Ex(c)


simpleorfirstdegree






EXAMPLE4x2=64










Ex(a)xndash5=0


Ex(b)x+5=12


Ex(c)xndash7=8



Ex(a)Solve

Ex(b)Solve =64


Letx=thenumber

Then

(1)


Ex(d)Solve06x=18

Ex(e)Solve3 =30

Ex(f)Solve08x=1000




Ex(a)Solvex+4=10


Ex(b)Solve16=7+y






there4x=6Ans


From8=14ndashxweget











Leftside

Rightside Check

Ex(b)Solve




Then

Substitutex=ndash20

Leftside

Rightside





Then








Multiplybothsidesby











Note




Tocheck



variousquantities









d=44x10=440ftAns












Ex(d)From findC









Letx=thenumber


Then







x+45 = 2(x+12) = 2x+24


minusx = minus21

there4x = 21





4y = 15

there4y = yearsago


Ans

5x8 =41 yearsCheck































=wifersquossavings



18143+37286+64572=$120001Check














PROBLEMS



















17Evaluate(a)(b)(c)

18Divide(a)y6byx2

(b)y9byy3


19Whatistheresultof

(a) (b)

(c)

(d)

(e)









26Whatistheresultof

(a) (b)(c)

(d) (e)

(f) (g)+ (h)





(c)


9xndash8zandndash3y







(f)8a2+4b2by5a+362










(b)xa+16x+64


(d)(2x+3y)2ndash1










(s)

(t)8x+09=9minus4x



50If whatisa
















APPENDIXA

ANSWERSTOPROBLEMS


31937467296







17(a)073586

(b)8000008(c)050321(d)70090000012(e)023504910000630003(f)4792086500005(g)04090306001000700008(h)0364575(i)0908006034





23(a)$066(b)$080(c)$047(d)$010(e)$120(f)$712

25(a)475(b)5621(c)22(d)10540(e)10765

(f)2555(g)100(h)4444



36121824

535793711153915213111927

79131721916233091827369111315

91997

11$203265

1360

1595

1721

1990

21(a)255(b)244(c)209(d)263(e)270(f)250

23(a)169(b)155(c)140(d)141(e)1879(f)1457(g)1667(h)2039

25907gallons

271039miles

29$525

31

(a)280778(b)295263(c)292690(d)242893

33(a)195564(b)293220(c)208675(d)142415

ChapterII(pp34-36)12242192225612425

3633432392134320372

5152294215059811720


9minus4deglat

11$24

12(a)124959(b)151833(c)74296(d)161574(e)$305907(f)$873883(g)$38849254(h)$60579179

14(a)25697(b)49779(c)92922(d)$22015250(e)$100035090(f)$91357818

17(a)4228(b)4214(c)4319(d)5659(e)3357(f)2165

19(a)2443393(b)888

(c)1669(d)178556

21(a)1421(b)41135

23$40746

251548576

27(a)53514947534945415347413553453729(b)89868380898479748982756889807162(c)74696459746760537471686574655647

ChapterIII(pp54-56)1540540054000

3176201762001762000

518001800018000018000000

71917000


11$14425$6347013$1886

1517424001b17$420

19(a)238(b)272(c)306

(d)304

21(a)7395(b)2352(c)3074(d)1184(e)4355(f)9306(g)5328(h)728(i)306

23(a)945(b)8295(c)6435(d)630(e)4005

25(a)2709(b)2625(c)1316(d)3149(e)3364(f)2016(g)2236

27(a)4275(b)4875(c)5525(d)1925(e)3325(f)4125(g)1225(h)6375

(i)9425(j)$6075(k)$12375(l)$20425

29(a)(b)(c)(d)(e)(f)(g)(h)(i)

31(a)(b)(c)(d)

33$1400

35(a)$21250(b)$12325(c)$2875(d)$1200(e)$1200(f)$2100(g)$1800(h)$41600(i)$900

37$9000

39(a)768(b)1632(c)30008

(d)1368

41(a)516456(b)528849(c)38952(d)890901(e)7628688

43(a)5496(b)4809(c)3456(d)3024(e)7856(f)6874

45(a)8232(b)9024(c)7998(d)7505(e)7216(f)960376

47(a)6384(b)63672(c)3196(d)49088(e)7128(f)2964(g)7392(h)64528


34

520



117hours

13

(a)214(b)402(c)428

17(a)3(b)Yes2

19(a)2958(b)60(c)80868(d)365(e)1680(f)6912(g)72(h)42(i)139(j)36(k)112


23842

25$101522

ChapterV(pp80-81)

12369235610152346121839272345681012152024304060

3234612

512357111317192329313741434753596167717379838997



(z)2221337

11(a)918(b)3570(c)1836(d)2142(e)4080(f)612(g)816(h)72144(i)918(j)2448

13(a)21(b)15(c)28(d)24(e)161536lb1718days





9(a)14(b)28(c)7(d)No

11(a)

(b)(c)(d)(e)(f)

13(a)(b) (c) (d) (e) (f) (g) (h) (i)

15(a) or1(b) or1(c) or3

17(a)(b)(c)(d)(e)(f)

19(a)2(b)15(c)2(d)67(e)1 or(f)1 or(g)1(h)1

(i)28(j)42 (k)139 (l)129


25(a)(b)(c)(d)16(e)2(f)216(g)(h)(i)(j)

27(a)100(b)$688

29

31

33

35$44

371 ozperslice

39286miles

41 $1350$1350$900

43$246$6150$9225$3075$2050

45

4717 rods



51000100000

7Ten

9(a)8=80=800(b)046=0460=04600(c)738=7380=73800=0738

110040004

13246246

15246024600246000

17246576246576

19(a)032(b)0625(c)014(d)0392(e)01875(f)065(g)04(h)0175(i)03125(j)0115(k)046875(l)0232

21(a)(b)(c)(d)(e)(f)(g)(h)

23(a)1274735(b)18125608(c)22135538(d)7202238

25(a)4234408(b)4494375(c)38316(d)35425(e)553308

27(a)52655625(b)2582398(c)39130222(d)2012315(e)0638027

29(a)15895794(b)38884176(c)17517890(d)112489886(e)54923664(f)21073016

31(a)5977(b)5976

33012

352918

37(a)0078125(b)015625(c)0375(d)03125(e)028125(f)0171875(g)028(h)0184

39$042$007

41$568750

43$34000$7480$10880$12240

45$282

470968lb

4911cents832mills

5132lb

53A0750B0714


3

502502020002500020002


916

11(a)25(b)64(c)100(d)325(e)30(f)420

13$3000$11040$9960

151904votes

1720

19$132389

21(a)406(b)131(c)1278(d)40(e)(f)2323(g)0135(h)2188(i)1662(j)364(k)7150(l)4442(m)5138

23 sqft

2740032

29506675911

3119000054

33$29143

35$13636

3721

39

41$1838

43$150

45$35235


49$20588

51 25

53(a)72(b)60(c)006696

55(a)304(b)720(c)2300


3$928$128

5(a)March4(b)March3

7(a)249(b)84(c)118(d)248(e)142

9$789

11$240

13$4919

15(a)$4717(b)$38111(c)$291(d)$1186(e)$28603(f)$370(g)$3431(h)$363(i)$4912

17$4310

19$1438

21$247

23(a)1566(b)6015(c)3063(d)60306(e)3010(f)601566(g)606063(h)603015(i)60601510(j)156(k)60606(l)603063

25$469

27$150

29$9653

31

3385days


37(a)$120(b)$068(c)$829(d)$240(e)$028(f)$425

39$444500

41


43$260000

45$104040

47$1643615

49$6289

51$2693706$693706


359

5116

7118811

906

11(a)35(b)85(c)13(d)98

13(a)13(b)12(c)13(d)110(e)1379(f)19(g)(h)1625(i)140(j)1571(k)1115(l)1222(m)14

(n)165(o)160(p)1136(q)1114

15

1741

1914and21

21507080

235134

2515

27 inches

29$43875

31

3322ftx ft

3531

37(a)4(b)7(c)

398

41(a)6(b)2(c)18(d)24(e)18(f)3(g)12(h)32

43$3750

4572feet

471057lb

49(a)10(b)15(c)

512171b

5342men

55821$3528$1536

57 days

5923

61082ohm

63400feet

65x=6

6790psi

6966men


347mph

58562

7435minutes

9$1784

11$209067

1359

15(a)13(b)19

17$340

19$300to$399

21No



38rods2feet

5 cubicinches

783688lbofwater

93025bbl

11$1816

1349280lb

15366

17184

1942doz

2130years


25(a)735dm

(b)74126meters


2939122dg

316944grains

33102058cg

350664grains


39392pt

41 bu

430883bu

4500181gal



517504610meters

537976meters

5511664kg


(r)16x2

(s)8b3(t)1953125

34000sqft

548sqyd

7(a)256(b)19683(c)16(d)3(e)axminusy

(f)ax+y(g)4096(h)15625(i)1(j)1

(k)1(l)24(m)(n)(o)

92176782336

11(a)784(b)4489(c)5776(d)7921

13950625

15(a)256(b)2025(c)65025

17(a)99980001(b)9801(c)999998000001

19(a)12(b)4b4

(c)a3b32

(d)x2y4(e)(f)(g)8(h)2646=(i)

21(a)

(b)12(c)

23

(a)

(b)

(c)

25(a)(b)(c)

27(a)4a2y54(b)

(c)

291287feet

316314

33(a)(b)(c)

(d)

(e)(f)01334(g)(h)0949(i)(j)9709(k)00255

35




7(a)1000(b)64(c)minus5(d)512(e)(f)10(g)(h)(i)7




132log7+log4

15log1944

17(a)log432+log748-log566(b)

19

(a)(b)

(c)(d)

21


23(a)0340(b)3679(c)00036(d)4016(e)000027(f)164


3(a)12(b)6(c)(d)16(e)350



3a=15S=645

533 4 5 5 6 7 8

7

9250500

11l=39366S=29524

132

15

172

1915


23$70388


15$5425


33846

5$3

7$108$1692

9$27354

113825

13$141221

15$54

17$2693

19426

21$4421$269

23$4815

25$3708

27$400

29$229665$10335

31$213359

33$30

35$2650

3733

39373

41$2330036

43$3000$1000

45$799056$199056

47242

49204

511082

53436

55457

57$95940

59$225

61$28712

6323075$230 per$100$2307 per$1000

65$3073196


360days

5

7 or194days

940gal

11286

133 qt

141119

173994avoz

19125cc

21

233min

254320gal

27

291253440

31276miles

339728acres

35720deg

374rdquo


3xyzz

515

7No

3y83(y8) etc







(d)56x5c7

(e)45a7a+1

17(a)x3

(b)3x3(c)

19(a)a4b2c2

(b)x2y4z(c)x3yz3

(d)y-2b2

(e)9xy2z-2

21Yesno


27ndash21a+66b

29ndash132xndash76y

31-19x+18y+27z



(c)ndash24a7b2c2



(f)40a3+24a2b2+20ab2+12b4



(d)

(e)(f)a+5(g)2a+3b(h)3a2+2ab+4b2




(f)x=77(g)x=7(h)x=23(i)x=70(j)x=72


49248degF

5110

53400gal

5510001600

571405681

59 orsquoclock

61$9000

6311

659miles7miles

APPENDIXBTABLES

TABLEI


TABLE2


TABLE3


TABLE4

PRESENTVALUEI(I+i)n

INDEX




inalgebraalgebra











ybankdiscount


100percent

longbargraph

horizontal

verticalbase


inprofitandloss

intaxation


binomial

blockgraph

Boylersquoslaw

Briggssystemoflogs

Britishmoney

broken-linegraph

bundlesofunits


buyingcommission

calculation

cancellation

carryingcharge


cashdiscount



insubtraction



chargecarrying

financingchart

dividedbar

100percentbar

longbar


inaddition

inalgebra

indivision

inmultiplication


cipher

circlegraph

circularmeasure

circulatingdecimal

circumference

coefficient

cologarithm(colog)

commissionbuying

salescommondivisor



powersofcommonlog

commonmultipleleast




complexdecimal

complexfraction

compositenumber


compoundamount


compoundfraction


compoundproportion

compoundratio

computation

concretenumber

conditionalequation


constant





Cartesiancost

gross

net



cube





decimaladditionof

andUSmoney

approximationof

circulating

complex


topercentdivisionof

equationswith

multiplicationof

powersof

recurring

repeating

simple



decimalplace

decimalpoint

degree





depreciation


diagramline

staircasedifference

inpercentage


directednumber

directionconceptof

negative


directtax

directvariation

discountbank

cash

trade


dividend


bylogs


decimal


long

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften

ofsamekindofsymbols

ofUSmoney

proportionby

pureproofof

short

symbolofinalgebra

divisionsign

divisorcommon

greatestcommon

trialdrymeasure

inmetricsystem



emptyingproblems


equalssign

equationconditional

linear

quadratic

rootof

simple

solutionof


evolutionsymbolof


lawsof

logarithmdefinedas

negative

raisedtoapower

signof

zero


extrapolation


factorcommon

greatestcommon

literal

primefactoring


commontermmethod

inalgebra


finance

financingcharge

fluidounces

formula

fourthroot

fractionadditionof

chain


compound

continued


topercents


improper

multiplicationof

powersof

proper


rootof

simple

subtractionof

unit


fractionalplaces

Frenchmoney


futurevalue(worth)



Germanmoney

gram


horizontal

verticalblock

broken-line

circle

curve(d)


ofquadraticformula

pie

rectangle



grosscost

grossprofit

grosspurchases


harmonicprogression

Hookersquoslaw

horizontalbargraph

100percentbarchart


identity

imaginarynumber

imperfectpower

improperfraction

incometax

index

indexnumber

indirecttax




integralnumber

interestcompound


rateofeffective

nominalsimple

formulafor


interestcost

interestearned

interpolation

inverseproportion

inverseratio

inversevariation



invertedsubtraction

involutionsymbolof


keynumber(figure)





leverprincipleof

licence

liketerms




linediagram


liter

literalfactor

literalnumber

loans

logarithm(log)


Briggssystemof


common

commonsystemof

divisionby

extractionofrootsby

mantissaof

multiplicationby

Napieriansystemof

natural

naturalsystemof

proportionalpartof

raisingtopowersby

tableoflongbarchart

longdivisionrulefor

loss



mapsstatistical

marginofprofit


harmonic

proportional


meanvalue

measurecircular

counting

cubic

dry

linear

liquid

metricsystemof

paper

square

timemedian

advantagesof



minuend

minussigninalgebra


miscellaneousseries

mixednumber

mixtures

modeadvantagesof


moneyBritish

French

GermanUnitedStates

anddecimals

divisionof

howwrittenmonomial

multiplecommon



associativelawfor

bylogs


complement

cross

cumulativelawof

distributivelawsfor

howtosimplify

inverted

left-hand

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften


multiplicationsign

multiplicationtable

multiplier


naturallogs

naturalsystemoflogs

negativedirection

negativeexponent


divisionof

multiplicationof


netprofit

netpurchases

netsales



nought

number

abstract

Arabic

basic

composite

concrete


reductionof


even

imaginary

index

integral

irrational

literal

mixed

negativeadditionof

divisionof

multiplicationof

subtractionofodd

positiveadditionof

divisionof

multiplicationof

subtractionofprime

real

Roman

signed

specific

wholenumberscale

numerator

oddnumber


direct

inverse

symbolsoforders


origin

papermeasure

parabola

parentheses



businessusesof


inprofitandloss

lessthan1percent

multiplicationof

relationtoratio


period

pictograph

piegraph(chart)

placesdecimal



pointdecimal

polltax

polygonfrequency


positivedirection


divisionof

multiplicationof


imperfect

multiplicationof

ofcommonfractions

ofdecimals


divisionof



priceselling

primecost

primefactor

primenumber

principalininterest

product


marginof


ascending

descending

geometric

harmonic


properfraction

propertytax


bycomposition

bydivision

byinversion

compound

direct



protractor

purchasesgross

net

return

quadrants

quadraticequation


quantityalgebraic

constant

variablequotient

radical


similarradicalsign

radicand

rateinpercentage

inprofitandloss

ofinteresteffective

nominaltax

workingofspeedratio

compound

howtosimplify

inaseries

inverse

relationtopercent



receptacles

reciprocal

rectanglegraph

recurringdecimal


ascending



indivision


returnpurchases

Romannumeralsystem

rootcube

extractionof

extractionofbylogs

fourth

ofequation

offraction

squareextractionof

salesgross

netsalescommission



sellingprice

seriesascending

descending

miscellaneous

sumtoinfinity


signofexponents

ruleforsignednumber

similarradicals

simpledecimal

simpleequation

simplefraction


exactmethodfor

formulafor



averages

division

multiplication

ratios

squaringofnumbers


solutionofequations

solutions(mixtures)


specificnumber

speedworkingratesof

squareofanumber




statisticalmap

statistics



Austrianmethodof


howtosimplify

inverted

left-hand


ofdenominatenumbers

offractions

ofpercents

ofplusquantities


symbolofinalgebra

subtractiontable

subtrahend


surtax



multiplication

oflogs

subtractiontanks

taxdirect

income

indirect

poll

property

totaltaxamount

taxmatters

taxrate


term(algebraic)like



timeininterest


totaltax

tradediscount

trialdivisor

troyweights

truediscount

unit




valuationassessed

valueabsolute

future

mean

presentvariable


direct

inverse

joint


verticalbargraph

vulgarfraction

weightedaverage

weightsmetricsystem


avoirdupois

troywholenumber

workingratesofspeed

worthfuture

present

xaxis

yaxis







0-486-23411-8


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Title Page

Copyright Page

FOREWORD

Table of Contents

INTRODUCTION

























































































































































































































































































































































































































































































































































































































PROBLEMS











PROBLEMS



























654 What are graphs

































































































































E Angle measurement


821 What is algebra



































856 What does mean



















































































APPENDIX B TABLES

INDEX


BOOKSBYAALBERTKLAF


ArithmeticRefresher

AAlbertKlaf




9780486141930


FOREWORD




FRANKLINSKLAFMD

TableofContents



INTEREST

INTRODUCTION





89arecalleddigits









































thatnumber





integralnumbers





towardstheleft




beginningattheright






EXAMPLE




EXAMPLE400536080209












theunitsposition








100000000=10times10000000




EXAMPLES











smallerfraction

EXAMPLE





EXAMPLE


EXAMPLE



readldquoandrdquo




forty-threecents









EXAMPLES



EXAMPLES


EXAMPLES


EXAMPLES

PROBLEMS1

1Howmanyunitsin379























21Howis$356356read


(e)$444(f)$888



(a)

(b)

(c)

(d)








CHAPTERI

ADDITION







groupedtogether







EXAMPLE



EXAMPLE



EXAMPLE(asabove)


EXAMPLE


groupthen














EXAMPLES(a)

(b)


theother

EXAMPLE96+78



increments

EXAMPLE



EXAMPLES

(a)Add26to53

(b)Add67to86




EXAMPLE





Ex(a)

Ex(b)






EXAMPLE








EXAMPLES

(a)

(b)












EXAMPLE

PROBLEMS









9Add269745and983


11Add$525$1760$085$175$4565

12Findthesumof

(a ) (b ) (c)

$380865 $987367 $887406

37692 38898 51856

38623 573200 129897

48008 898719 54265

88842 782492 38600

75182 608604 4209























(a)

(b)

CHAPTERII

SUBTRACTION




8minus5=3 8minus3=5





Ex(b)16minus9=7












Ex(a)



Ex(b)






Ex(a)








SubtractionTable


etc






Ex(a)

Ex(b)


Ex(c)



Ex(a)


Ex(b)



EXAMPLE




Ex(a)


Ex(b)


Ex(c)


Ex(b)



columnattheright

Ex(a)

Ex(b)



EXAMPLE





Startatleft














EXAMPLES


Ex(a)


Ex(b)


Ex(a)



Ex(b)

PROBLEMS


1

2

3

4

5

6






12Subtract

















CHAPTERIII

MULTIPLICATION



4+4+4+4+4+4+4+4=32or8times4=32






EXAMPLE






MultiplicationTable















EXAMPLE



EXAMPLE



added457times
















Ex





(d)Add(a)+(b)+(c)

EXAMPLES(1)

(2)

(3)






Ex(a)45times29



Ex(b)323times35


Ex(c)271times55


multiplybytheother

Ex(a)

Ex(b)

Ex(c)


Ex(a)

Ex(b)

Ex(c)


Ex(b)

Ex(c)


Ex(a)

Ex(b)


discardingfraction

Ex(a)


Ex(b)

Ex(c)

Ex(d)


Ex(a)

Ex(b)













there4 is of100


there4 is of100


there475is of100



decimalThebaseis100

EXAMPLE(asabove)


020405062506660833


125133316662025






EXAMPLE


EXAMPLE


EXAMPLE



EXAMPLE


















numberfromtheresult

Ex(a)


Then


numbertothis

Ex(a)

Ex(b)




Inoneline



Inoneline









EXAMPLE


EXAMPLE








(d)Add(b)to(c)

Ex(a)




88times100=88008800+32=8832Ans



7900+98=7998Ans





EXAMPLE













EXAMPLE




PROBLEMS




4Multiply631by60



7Multiply6390by300

8Multiply


9Whatistheproductof


10Multiply




























35Whatisthecostof















CHAPTERIV

DIVISION




















EXAMPLES


(b) or

(c) orDivisor(=7




EXAMPLE



remainder

EXAMPLE





In =7wehavedivision



972=900+70+2


howwouldweproceed


4rsquos=3200 4)3492





(e)3times4=12



800+70+3=873




































getting2asquotient












EXAMPLE











Quotientis







Quotient=





Ex(c)




Ex(d)








EXAMPLE



in2468orinazero


in21273834485





and2times3=6






Ex(a)

Ex(b)


Ex(a)

Ex(b)


by8

Ex(a)




Ex(c)








Ex(a)

Ex(b)


Ex(a)

Ex(b)


Ex(a)


Ex(b)

Ex(c)


Ex(a)867


Ex(b)973285









Ex(b)





placestotheleft












Ex(1)


Ex(2)











Monday $26860

Tuesday $32985

Wednesday $9745

Thursday $23990

Friday $29670


$24650 (=Average)









For10cent



then



PROBLEMS



















9Divide

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

(u) (v) (w) (x)

(y) (z)

10Findthequotientof





(m) (n) (o)


12Findthequotientof





14Divide






16










17


18



(a) (b)

(c)

(d) (e)










CHAPTERV











Ex





EXAMPLE
















GCD





Ex(a)


there42times3=6=GCD

Ex(b)





















7appearsonce
















PROBLEMS










(a)310(b)297(c)670(d)741(e)981(f)385(g)2650

(h)1215((i)321(j)1575(k)10935(l)420(m)497

(n)378(o)462(p)2430(q)25344(r)73260(s)599676

(t)273564(u)15625(v)10675(w)12625(x)976

(y)8050(z)3848


(a)68112240(b)2184126147(c)212877

(d)457281(e)4477121(f)1498112(g)248096

(h)284236(i)457281(j)31522679012

(k)144576(l)820697(m)1251751792(n)60043318

(o)125423618163(p)1086905




(j)86and4



(e)3642and48(f)3918and27(g)51525and35


(l)212426and28(m)92142and63

(n)367548and24(o)71456and84(p)2472128and240



(d)88368divide3682(e)32768divide2048







CHAPTERVI

COMMONFRACTIONS





EXAMPLE


exactlyequalsize



Thus

191Whatismeantby

(a)

(b)

(c)

(d)










Ex(a)


Ex(c)


equalparts





(b)



Wegetthewholeunit

Ex(a)

Ex(b)

Ex(c)







numbers










Ex(a)

Ex(b)






Ex(b)



Ex(a)

Ex(b)

Ex(c)

Thereasoningis



Ex(a)

Ex(b)






commonfactors

Ex(a)

Ex(b)







EXAMPLE


EXAMPLE


















Ex(a)



Ex(a)

Ex(b)



EXAMPLE


mustallbethesame


Ex(a)Add and


Add


Addfractions

Then




Now

Ex(b)Add (LCD=20)








Ex(b)Subtract from



Ex(a)






Ex(b)

Ex(c)


Ex(a)

Ex(b)


Ex(c)

Ex(d)


Ex(a)

Ex(b)Multiply by11

Ex(c)


inposition



Ex(a)

Ex(b)Multiply




EXAMPLEMultiply



Ex(a)

Ex(b)Multiply by

Ex(c)Multiply by



EXAMPLE













(a)

(b)


EXAMPLEDivide by




Ex(a)Divide by2


Ex(b)


Ex(a)Divide24by

Ex(b)Divide17by or






EXAMPLEDivide by3



EXAMPLEDivide by6



Ex(b)Divide by



partsareexpressed






Ex(c)Whatpartof is

Ex(d)Whatpartof is7
















EXAMPLE


fractions












Next

Next

Finally
















Startfromthebottom







PROBLEMS






(a) (b)

(c) (d)

(e) (f)










(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b)

(c) (d)

15


16





(a) (b) (c) (d) (e) (f)

19Add

(a) (b) (c)(d) (e) (f) (g)(h) (i) (j)(k) (l)

20Subtract














31














encloseit




CHAPTERVII

DECIMALFRACTIONS


EXAMPLES




EXAMPLES




Ex(a)0207008024017

Ex(b)


EXAMPLES








EXAMPLE5631056923









EXAMPLES




EXAMPLES



numerator





048386356



EXAMPLE6=6=60=600




EXAMPLES





EXAMPLES



EXAMPLES

(a) or

(b) or

(c) or

(d) or

(e)

(f)














Ex(b)










usualway









Ex(a)Multiply38by6

Ex(b)

Ex(c)


improperfractions

EXAMPLE






Ex(a)Divide192by06











theleft




















4879times3765=18369435



mostimportant





EXAMPLE



EXAMPLES

(a)(b)(c)


infinity



group




whichdonotrecur



EXAMPLES(a)

(b)

(c)

(d)

(e)












(b)


PROBLEMS






3Read12584062018








260260026260260













23Add(a) and







(e)64575by165(f)9686by136



31Shorten(a)59767(b)59755
























CHAPTERVIII

PERCENTAGE





EXAMPLE


EXAMPLES(a)(b)

(c)

(d)

(e)(f)(g)










EXAMPLES



EXAMPLES








Ex(b)Find14of$9751(14=14)



Ex(a)Find6of$6700

1of


6of$6700=6times$67=$402

Ex(b)Find4of$1860

1of$1860=$1860there44is4times$1860=$7440

Ex(c)Find of$7000



Ex(a)Find25of$51

Ex(b)Find of$8475







Ex(b)Find9of50


fraction












adecimalSince



or





If$285is2then

or





or






or












$15750times0025=$39375=$3938Ans



120times$55=$66Ans




$46375times43=$19941Ans






fromtheamount


06times85+85=51+85=901=Amount











PROBLEMS



31400iswhatof3600

4Reducetoadecimal




(k) (l)



















25Whatpercentis of6
































CHAPTERIX

INTEREST









EXAMPLE




EXAMPLE

005=fivepercent=




EXAMPLE



















=$500(115)=$575Ans










eachor360days



Year Month Day

1959 2 8

1957 5 15

_____ ____ _____

1 8 23









May 23days

June 30days

July 31days

August 31days

September 30days

October 31days

November 30days

December 31days

January 12days

249days











Year Month Day

1959 4 6

1958 11 18

4 18 =138days










Then

and

Then

and







Now6daysare







(b)58days(c)77days



4days 6days 2days

49days 2days 77days

58days

58days



2

















or
































$468=$450+$450times04

or


and





or






Therefore


S=$50619


$50619Ans














S=$12000times131593=$1579116Ans






n=6=t=6yearsAns














Inabove




Inabove













126898555Ans



S=$8000times181535485=145228388


$1452284Ans

PROBLEMS
























































CHAPTERX

















EXAMPLES

$54000$18000=31412=13(to)(to)(to)(to)











numbers



Ex(b)




$100to$500is














Dividebothtermsby

















there4320560720=479Ans





there4 Ans







or




Ormargin



(b) forrent















EXAMPLE3=9


EXAMPLE36=12

















Value=4


Value=2










EXAMPLE



[]=ldquoasrdquo














othertwo

Ex(a)36612


Ex(b)


Ex(c)3xx12




cost

3x=5times$864


there4 Ans















doitin5days


























Thus






andsolvedasfollows








3times8=4times6

(b)If then


EXAMPLEIf =then 3

(c)If then


EXAMPLEIf then

(d)If then


EXAMPLEIf then or

(e)If then


EXAMPLEIf then or

(f)If then


EXAMPLEIf then or




or










variation



C=πd=variationform
















Sαr2



S=4πr2







andinversetypes




From

then



where


PROBLEMS












(a)1525(b)2415(c)824(d)2724

























36(a)4=16(b)5=4(c)

37(a)24=6(b)49=7(c)






(a)(b)

(c) (d)

(e)

(f)

(g)

(h)

(i)








(a)

(b)

(c)
























CHAPTERXI

AVERAGES






















things











48+55+68+42=213minutes








there4















Theweightedmarksare






Remainder=199























Themeanaverageis



Themeanaverageis




200200200220225235240240250260260275275350450500650and800





amount



$700 and over (1)

$400 to $699 (3)

$300 to $399 (1)

$250 to $299 (5)

$200 to $249 (10)

$150 to $199 (4)

less than $150 (1)



Themedianis$235


























destroyed


whichdonoteffectit










PROBLEMS























$500(1)$50(12)$300(2)$25(22)$100(3)$15(6)$75(4)




CHAPTERXII

DENOMINATENUMBERS






EXAMPLES



descending







161feet=1rod



Note(c)













































Pound Ounce









24hours = 1day(da)

7days = 1week(wk)


52weeks = 1year(yr)




10years = 1decade

20years = 1score






20units = 1score

12units = 1dozen











TheLatinprefixesare


TheGreekprefixesare











= 0001sqmeter


= 01sqmeter


= 1sqmeter=1centare

= 1sqmeter=1centare


= 100sqmeters=1are




= 1000000sqmeters








= 000001cumeter





= 1000cumeters


= 1000000cumeters


































andfindthetotal







8qt

(or25)qt+6qt=625qt


4pk

78125pk+2pk=278125pk


6953125buAns







EXAMPLEAdd








EXAMPLE




















23sqyd+1sqyd=24sqyd



turn







6403509metersAns






5327698kmAns





PROBLEMS















































46Add











CHAPTERXIII







EXAMPLES





EXAMPLES





used







26x26=676





400+240+36=676


(a+b)2=a2+2ab+b2









EXAMPLES

22=0452=25


EXAMPLES



EXAMPLE


EXAMPLE


Now25=32and23=8




Ex(a)Divide32by8

Ex(b)Divide243by9






(42)3=42bull42bull42=42times3=46=4096




Ex








Now




EXAMPLES




Ex(a)

(b)


Ex(a)

(b)










Raise52=25Then252=625Then6253=244140625=512




(29+1)(29mdash1)=(292ndash1)

or

30times28=840=(292ndash1)292=840+1=841


number




(b)




(a+b)2=a2+2ab+b2=a(a+2b)+b2




Ex(a)

(b)

(c)






(b)Square35 =772=4949times52=1225

(c) =772=4949times22=196(d) =992=8181times22=324







Ex(a)

(b)


Ex(a) issimilarto

(75)2=70times80+25=5625(b)(125)2=120times130+25=15600+25=15625




Thenfigure8


Ex(a)

(b)






(b)4000000=4times1000000=4times106

(c)36300000=363times10000000=363times107









Ex(a)

(b)

(c)

(d)

(e)



Ex(a)

(b)

(c)












leavingaremainder









3)9

3




(c) =a2b(d)

(e)(f)





Ex(b)(c)(d)

(e)(f)

(g)




Ex(a)(b)


Ex(a)(b)



Ex(a)

(b)



EXAMPLE


bythesamenumber

EXAMPLE


EXAMPLE









(b)






decimalitself

EXAMPLES



EXAMPLES






(a+b)2=a2+2ab+b2=a2+(2a+b)b









radic676=26



















there4




remainder





















Ex(a)

(b)




















there4 =626Ans






=1 =1

PROBLEMS

1Find(a)52

(b)82

(c)202

(d)14

(e)H2

(f)19

(g)103

(h)34

(i)252

(j)173

(k)833












(c)(d)3times4-1

(e)(05)-2

(ƒ)(a-m)-n







(d)65(e)225




(e)

(f) (g)(h) (i)


(f)




(a)

(b)

(c)


(c)



(a)(b)(c)









35Whatisthevalueof

CHAPTERXIV

LOGARITHMS









Ex(b)43=64







logarithm=exponent

















Ex(b)








4642isabasicnumber











Ex(a)


1to9 1 1mdash1=0

1to9 1 1mdash1=0

10to99 2 2mdash1=1






log000865=ndash4+9370







Ex(a)log000865=49370(b)log00427=36304





3+8+6304mdash8=56304mdash8

or

+30+6304ndash30=276304ndash30




















Then83315+0003883353













105times times = =104=10000






=1093=103=1000


EXAMPLE160=1022041236=102372928=1014472






EXAMPLE135834=10213301896=1095230

Thus

Theantilogis151605



EXAMPLE374=1015729


=1015728x3






EXAMPLE




75=52times3














EXAMPLE



EXAMPLE

log10100=4343logε100=4343times46052=2



cologa=log















Then

and





there4ndash09683Ans










Nowaddallthefactors








Then




(c) =(d)9radic2=2235(e)34=81(f)2-2=







(h)x=log1001000(i)logx49=2




(n)00054times10ndash4(o)3755000(p)4343




(j)00638times104






(c)

(d)


(a)log

(b)log










(c)(d)(2112)minus3(e)(2718)ndash12(f)

(g)(06493)minus(h)(5937)minus13


(j)

(k)(l)(42)x=649

(m)(n)

CHAPTERXV









numbersas32+5+711+



EXAMPLES












(orsame)sign






(Question626)



5+(mdash7)=mdash2
















CHAPTERXVI






Ex(a)24681012etc






Ex(a)28321285122048etc


Ex(b)20485121283282





Ex(a)358101315182023










EXAMPLE13579




EXAMPLE25211713














Then













Ex(a)


Then

Now

Ex(b)


Now

(1)


Substitutingin(1)










series



Thustheseriesis

4 minus1 minus6


progression

xmdasha=bmdashx

or

2x=a+b

and

x= =halftheirsum




there4l=arnminus1


41 to7terms




(1)


(2)


rSmdashS=arnmdasha



Then

But

l=arnndash1orrl=arn



Therel= r= anda=4



FindrandS



Thiscanbewrittenas




Herea=3and


Now


Then








progression

Hence










Here andn=8Then





and



346





(15times01=0015)





Then


PROBLEMS


(d)11119977(e)8127931 (f)2818325072(g)12481632(h)403430282218























CHAPTERXVII

GRAPHSmdashCHARTS




verbaldescription
























$4000-$4999 12400

$5000-$5999 10200

$6000-$6999 8100

$7000-$7999 3040

$8000-$8999 2200

$9000-$9999 1160

$10000andover 208






Year Sales

1954 $38260000

1955 $47840000

1956 $43190000

1957 $45000000

1958 $39080000

1959 $47040000

1960 $51000000









Freshmen = 520

Sophomores = 410

Juniors = 380

Seniors = 290




Then
































91to100lb 12

101to110lb 124

111to120lb 268

121to130lb 107

131to140lb 26

141to150lb 8

141to150lb 8

151to160lb 4





















mathematicallaw




















first









EXAMPLES




PROBLEMS




$4000-$4999 8400

$5000-$5999 3200

$6000-$6999 2100

$7000-$7999 1800

$8000-$8999 760

$9000-$9999 139

$10000andover 68


1950mdash$54000000 1956mdash$46000000

1951mdash$52000000 1957mdash$45000000

1952mdash$51000000 1958mdash$39000000

1953mdash$47000000 1959mdash$47000000

1954mdash$37000000 1960mdash$52000000

1955mdash$48000000





$26500000 228






Total= $1161650000 10000


Freshmen 650

Sophomores 530

Juniors 480

Seniors 390




Year Cost

1950 $8675

1951 $9300

1952 $9475

1953 $9950

1954 $10625

1955 $11350

1956 $12225





January92hr

February106hr

March112hr

April122hr

May136hr

June146hr

July154hr

August142hr

September136hr

October114hr

November102hr

December94hr










CHAPTERXVIII










































discount


=tradediscountAns





























Theirsumis30+4=34

oftheirproductis

Thedifferenceis




(a)Combine40with10




Now

$39842=412=discount

Then







and











(a)Netcost













(f)ofprofit= =100





(b)Sellingprice





(f)of


profit

(a)Netcost14+8ndash




(d)Grosscost=$443+$18=$461

(e)Netprofit


=$666ndash$100=$566




cost
















Grosscost=$360+$12=$372



Cost=100minus25=75=$1860










=85=sellingprice








Forsimpleinterest

Forcompoundinterest






Then



Toprovethiswehave


and




Alsoitfollowsthat


or

$88900times112486=$1000






Soinsteadoffinding


there4$1000times88900=$88900









$25 = downpayment

$135 = 45weeksat$3


$150 = cashprice





= finalpayment

$35 = finalpayment


$230 = cash










































P=principal=$10here
















Sumofmonths







Youkeeporborrowthe

















$84for1month









Thenbyproportion







Totalmonths

Totalinterest

P=principal=$25




Totalmonths

Totalinterest
















Repaymentschedule



















upto thyearm























Theborrowerhad

















remainingbalance



$5down+5times$5+$2 = $32

Cashcost = $28




Here

















full







payment




thenoteisdesired


Faceofnote $100000






+$969










Weseethat


Faceofnote $1000000





















business









muchper$1000









$68750divide$2383015=02885=28852885=$2885per$1002885=$2885per$1000





$2885per$100=2885=02885



Add(a)and(b)


$16000times05 =$800

$16000times03 =$480





$8000minus$3000 = $5000

$5000times02 = $100 = surtax


PROBLEMS






































































CHAPTERXIX

VARIOUSTOPICS



























Expressasafraction






































$1800minus$1600=$200














ittoa25solution

Asabove























Then

and









Thenbyproportion















25 = 25

25times128 = 32




445 = 0445



67 = 067

067times128 = 8576=858




or













(b)Addthefractions
















Sumoffillingrates

Sumofemptyingrates


Thus and



















Scaleis172or


















(a)Bytheratiomethod





(a)Byratiomethod




EAnglemeasurement

815Whatisanangle

















PROBLEMS






































CHAPTERXX

























monomial

EXAMPLE20p=aterm






product













factors






polynomial



Letxdenoteathing








(b)




















Inalgebraa+b=b+a



Inarithmetic(5+9)+12=5+(9+12)=26





Inalgebraab=ba






















Itisnot


Ifx=2then

or



(a) times(b) (c)75x







(e)Costplus8=c+8







(l)Anyevennumber=2a





























$1250times48+$1050times72=$600+$756=$1356Ans












cube













(a+b)3times(a+b)4=(a+b)3+14=(a+b)7

Now


Therefore









thesquaretoot

EXAMPLES







Ex(a)Dividex5byx3

Ex(b)

Ex(c) (notx2)






(e) (a)

(b)

(c)

(d)

(e)


856Whatdoes mean



Similarly












Ex(a)Isa2b3likea3b2



there4Theyarealike













Youareonly700ftdown




Algebraically


















EXAMPLES



Ex(a)















Removeinnerbrackets



answer




ndash28+80=52Check




Ex(b)

Ex(c)


EXAMPLEWemaycombine


Wemaynotcombine






+15x mdash16y +8z


mdash20x +16y +14z











Thisbecomes






5a+7ndash3a+8=2a+15












(d)6a2+3bby3a+4b2

(a)


(d)



Ex(b)




Ex(b)


Ex(a)

Ex(b)

Ex(c)

Ex(d)

Ex(e)








2x2+10xy+3xy+12y2or2x2+13xy+12y2


(a+2)(a+4)+(a+3)(a+4)+(a+2)(a+5)a2+6a+8+a2+7a+12+a2+7a+10

or

3a2+20a+30Ans











Useruleofsigns























































theexpression









Ex(b)Factorx2+6x+8



Extremesxtimes8=8x




Extremesxtimes2=2x







Squarerootof9a2is3a




Ex(b)Factorx2+6x+9







(x4ndash25)(x4+25)


















or












Ex(b)













Ex(a)Solve3x=21


x=7Ans

Ex(b)Solve x=36

















Ex(c)


simpleorfirstdegree






EXAMPLE4x2=64










Ex(a)xndash5=0


Ex(b)x+5=12


Ex(c)xndash7=8



Ex(a)Solve

Ex(b)Solve =64


Letx=thenumber

Then

(1)


Ex(d)Solve06x=18

Ex(e)Solve3 =30

Ex(f)Solve08x=1000




Ex(a)Solvex+4=10


Ex(b)Solve16=7+y






there4x=6Ans


From8=14ndashxweget











Leftside

Rightside Check

Ex(b)Solve




Then

Substitutex=ndash20

Leftside

Rightside





Then








Multiplybothsidesby











Note




Tocheck



variousquantities









d=44x10=440ftAns












Ex(d)From findC









Letx=thenumber


Then







x+45 = 2(x+12) = 2x+24


minusx = minus21

there4x = 21





4y = 15

there4y = yearsago


Ans

5x8 =41 yearsCheck































=wifersquossavings



18143+37286+64572=$120001Check














PROBLEMS



















17Evaluate(a)(b)(c)

18Divide(a)y6byx2

(b)y9byy3


19Whatistheresultof

(a) (b)

(c)

(d)

(e)









26Whatistheresultof

(a) (b)(c)

(d) (e)

(f) (g)+ (h)





(c)


9xndash8zandndash3y







(f)8a2+4b2by5a+362










(b)xa+16x+64


(d)(2x+3y)2ndash1










(s)

(t)8x+09=9minus4x



50If whatisa
















APPENDIXA

ANSWERSTOPROBLEMS


31937467296







17(a)073586

(b)8000008(c)050321(d)70090000012(e)023504910000630003(f)4792086500005(g)04090306001000700008(h)0364575(i)0908006034





23(a)$066(b)$080(c)$047(d)$010(e)$120(f)$712

25(a)475(b)5621(c)22(d)10540(e)10765

(f)2555(g)100(h)4444



36121824

535793711153915213111927

79131721916233091827369111315

91997

11$203265

1360

1595

1721

1990

21(a)255(b)244(c)209(d)263(e)270(f)250

23(a)169(b)155(c)140(d)141(e)1879(f)1457(g)1667(h)2039

25907gallons

271039miles

29$525

31

(a)280778(b)295263(c)292690(d)242893

33(a)195564(b)293220(c)208675(d)142415

ChapterII(pp34-36)12242192225612425

3633432392134320372

5152294215059811720


9minus4deglat

11$24

12(a)124959(b)151833(c)74296(d)161574(e)$305907(f)$873883(g)$38849254(h)$60579179

14(a)25697(b)49779(c)92922(d)$22015250(e)$100035090(f)$91357818

17(a)4228(b)4214(c)4319(d)5659(e)3357(f)2165

19(a)2443393(b)888

(c)1669(d)178556

21(a)1421(b)41135

23$40746

251548576

27(a)53514947534945415347413553453729(b)89868380898479748982756889807162(c)74696459746760537471686574655647

ChapterIII(pp54-56)1540540054000

3176201762001762000

518001800018000018000000

71917000


11$14425$6347013$1886

1517424001b17$420

19(a)238(b)272(c)306

(d)304

21(a)7395(b)2352(c)3074(d)1184(e)4355(f)9306(g)5328(h)728(i)306

23(a)945(b)8295(c)6435(d)630(e)4005

25(a)2709(b)2625(c)1316(d)3149(e)3364(f)2016(g)2236

27(a)4275(b)4875(c)5525(d)1925(e)3325(f)4125(g)1225(h)6375

(i)9425(j)$6075(k)$12375(l)$20425

29(a)(b)(c)(d)(e)(f)(g)(h)(i)

31(a)(b)(c)(d)

33$1400

35(a)$21250(b)$12325(c)$2875(d)$1200(e)$1200(f)$2100(g)$1800(h)$41600(i)$900

37$9000

39(a)768(b)1632(c)30008

(d)1368

41(a)516456(b)528849(c)38952(d)890901(e)7628688

43(a)5496(b)4809(c)3456(d)3024(e)7856(f)6874

45(a)8232(b)9024(c)7998(d)7505(e)7216(f)960376

47(a)6384(b)63672(c)3196(d)49088(e)7128(f)2964(g)7392(h)64528


34

520



117hours

13

(a)214(b)402(c)428

17(a)3(b)Yes2

19(a)2958(b)60(c)80868(d)365(e)1680(f)6912(g)72(h)42(i)139(j)36(k)112


23842

25$101522

ChapterV(pp80-81)

12369235610152346121839272345681012152024304060

3234612

512357111317192329313741434753596167717379838997



(z)2221337

11(a)918(b)3570(c)1836(d)2142(e)4080(f)612(g)816(h)72144(i)918(j)2448

13(a)21(b)15(c)28(d)24(e)161536lb1718days





9(a)14(b)28(c)7(d)No

11(a)

(b)(c)(d)(e)(f)

13(a)(b) (c) (d) (e) (f) (g) (h) (i)

15(a) or1(b) or1(c) or3

17(a)(b)(c)(d)(e)(f)

19(a)2(b)15(c)2(d)67(e)1 or(f)1 or(g)1(h)1

(i)28(j)42 (k)139 (l)129


25(a)(b)(c)(d)16(e)2(f)216(g)(h)(i)(j)

27(a)100(b)$688

29

31

33

35$44

371 ozperslice

39286miles

41 $1350$1350$900

43$246$6150$9225$3075$2050

45

4717 rods



51000100000

7Ten

9(a)8=80=800(b)046=0460=04600(c)738=7380=73800=0738

110040004

13246246

15246024600246000

17246576246576

19(a)032(b)0625(c)014(d)0392(e)01875(f)065(g)04(h)0175(i)03125(j)0115(k)046875(l)0232

21(a)(b)(c)(d)(e)(f)(g)(h)

23(a)1274735(b)18125608(c)22135538(d)7202238

25(a)4234408(b)4494375(c)38316(d)35425(e)553308

27(a)52655625(b)2582398(c)39130222(d)2012315(e)0638027

29(a)15895794(b)38884176(c)17517890(d)112489886(e)54923664(f)21073016

31(a)5977(b)5976

33012

352918

37(a)0078125(b)015625(c)0375(d)03125(e)028125(f)0171875(g)028(h)0184

39$042$007

41$568750

43$34000$7480$10880$12240

45$282

470968lb

4911cents832mills

5132lb

53A0750B0714


3

502502020002500020002


916

11(a)25(b)64(c)100(d)325(e)30(f)420

13$3000$11040$9960

151904votes

1720

19$132389

21(a)406(b)131(c)1278(d)40(e)(f)2323(g)0135(h)2188(i)1662(j)364(k)7150(l)4442(m)5138

23 sqft

2740032

29506675911

3119000054

33$29143

35$13636

3721

39

41$1838

43$150

45$35235


49$20588

51 25

53(a)72(b)60(c)006696

55(a)304(b)720(c)2300


3$928$128

5(a)March4(b)March3

7(a)249(b)84(c)118(d)248(e)142

9$789

11$240

13$4919

15(a)$4717(b)$38111(c)$291(d)$1186(e)$28603(f)$370(g)$3431(h)$363(i)$4912

17$4310

19$1438

21$247

23(a)1566(b)6015(c)3063(d)60306(e)3010(f)601566(g)606063(h)603015(i)60601510(j)156(k)60606(l)603063

25$469

27$150

29$9653

31

3385days


37(a)$120(b)$068(c)$829(d)$240(e)$028(f)$425

39$444500

41


43$260000

45$104040

47$1643615

49$6289

51$2693706$693706


359

5116

7118811

906

11(a)35(b)85(c)13(d)98

13(a)13(b)12(c)13(d)110(e)1379(f)19(g)(h)1625(i)140(j)1571(k)1115(l)1222(m)14

(n)165(o)160(p)1136(q)1114

15

1741

1914and21

21507080

235134

2515

27 inches

29$43875

31

3322ftx ft

3531

37(a)4(b)7(c)

398

41(a)6(b)2(c)18(d)24(e)18(f)3(g)12(h)32

43$3750

4572feet

471057lb

49(a)10(b)15(c)

512171b

5342men

55821$3528$1536

57 days

5923

61082ohm

63400feet

65x=6

6790psi

6966men


347mph

58562

7435minutes

9$1784

11$209067

1359

15(a)13(b)19

17$340

19$300to$399

21No



38rods2feet

5 cubicinches

783688lbofwater

93025bbl

11$1816

1349280lb

15366

17184

1942doz

2130years


25(a)735dm

(b)74126meters


2939122dg

316944grains

33102058cg

350664grains


39392pt

41 bu

430883bu

4500181gal



517504610meters

537976meters

5511664kg


(r)16x2

(s)8b3(t)1953125

34000sqft

548sqyd

7(a)256(b)19683(c)16(d)3(e)axminusy

(f)ax+y(g)4096(h)15625(i)1(j)1

(k)1(l)24(m)(n)(o)

92176782336

11(a)784(b)4489(c)5776(d)7921

13950625

15(a)256(b)2025(c)65025

17(a)99980001(b)9801(c)999998000001

19(a)12(b)4b4

(c)a3b32

(d)x2y4(e)(f)(g)8(h)2646=(i)

21(a)

(b)12(c)

23

(a)

(b)

(c)

25(a)(b)(c)

27(a)4a2y54(b)

(c)

291287feet

316314

33(a)(b)(c)

(d)

(e)(f)01334(g)(h)0949(i)(j)9709(k)00255

35




7(a)1000(b)64(c)minus5(d)512(e)(f)10(g)(h)(i)7




132log7+log4

15log1944

17(a)log432+log748-log566(b)

19

(a)(b)

(c)(d)

21


23(a)0340(b)3679(c)00036(d)4016(e)000027(f)164


3(a)12(b)6(c)(d)16(e)350



3a=15S=645

533 4 5 5 6 7 8

7

9250500

11l=39366S=29524

132

15

172

1915


23$70388


15$5425


33846

5$3

7$108$1692

9$27354

113825

13$141221

15$54

17$2693

19426

21$4421$269

23$4815

25$3708

27$400

29$229665$10335

31$213359

33$30

35$2650

3733

39373

41$2330036

43$3000$1000

45$799056$199056

47242

49204

511082

53436

55457

57$95940

59$225

61$28712

6323075$230 per$100$2307 per$1000

65$3073196


360days

5

7 or194days

940gal

11286

133 qt

141119

173994avoz

19125cc

21

233min

254320gal

27

291253440

31276miles

339728acres

35720deg

374rdquo


3xyzz

515

7No

3y83(y8) etc







(d)56x5c7

(e)45a7a+1

17(a)x3

(b)3x3(c)

19(a)a4b2c2

(b)x2y4z(c)x3yz3

(d)y-2b2

(e)9xy2z-2

21Yesno


27ndash21a+66b

29ndash132xndash76y

31-19x+18y+27z



(c)ndash24a7b2c2



(f)40a3+24a2b2+20ab2+12b4



(d)

(e)(f)a+5(g)2a+3b(h)3a2+2ab+4b2




(f)x=77(g)x=7(h)x=23(i)x=70(j)x=72


49248degF

5110

53400gal

5510001600

571405681

59 orsquoclock

61$9000

6311

659miles7miles

APPENDIXBTABLES

TABLEI


TABLE2


TABLE3


TABLE4

PRESENTVALUEI(I+i)n

INDEX




inalgebraalgebra











ybankdiscount


100percent

longbargraph

horizontal

verticalbase


inprofitandloss

intaxation


binomial

blockgraph

Boylersquoslaw

Briggssystemoflogs

Britishmoney

broken-linegraph

bundlesofunits


buyingcommission

calculation

cancellation

carryingcharge


cashdiscount



insubtraction



chargecarrying

financingchart

dividedbar

100percentbar

longbar


inaddition

inalgebra

indivision

inmultiplication


cipher

circlegraph

circularmeasure

circulatingdecimal

circumference

coefficient

cologarithm(colog)

commissionbuying

salescommondivisor



powersofcommonlog

commonmultipleleast




complexdecimal

complexfraction

compositenumber


compoundamount


compoundfraction


compoundproportion

compoundratio

computation

concretenumber

conditionalequation


constant





Cartesiancost

gross

net



cube





decimaladditionof

andUSmoney

approximationof

circulating

complex


topercentdivisionof

equationswith

multiplicationof

powersof

recurring

repeating

simple



decimalplace

decimalpoint

degree





depreciation


diagramline

staircasedifference

inpercentage


directednumber

directionconceptof

negative


directtax

directvariation

discountbank

cash

trade


dividend


bylogs


decimal


long

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften

ofsamekindofsymbols

ofUSmoney

proportionby

pureproofof

short

symbolofinalgebra

divisionsign

divisorcommon

greatestcommon

trialdrymeasure

inmetricsystem



emptyingproblems


equalssign

equationconditional

linear

quadratic

rootof

simple

solutionof


evolutionsymbolof


lawsof

logarithmdefinedas

negative

raisedtoapower

signof

zero


extrapolation


factorcommon

greatestcommon

literal

primefactoring


commontermmethod

inalgebra


finance

financingcharge

fluidounces

formula

fourthroot

fractionadditionof

chain


compound

continued


topercents


improper

multiplicationof

powersof

proper


rootof

simple

subtractionof

unit


fractionalplaces

Frenchmoney


futurevalue(worth)



Germanmoney

gram


horizontal

verticalblock

broken-line

circle

curve(d)


ofquadraticformula

pie

rectangle



grosscost

grossprofit

grosspurchases


harmonicprogression

Hookersquoslaw

horizontalbargraph

100percentbarchart


identity

imaginarynumber

imperfectpower

improperfraction

incometax

index

indexnumber

indirecttax




integralnumber

interestcompound


rateofeffective

nominalsimple

formulafor


interestcost

interestearned

interpolation

inverseproportion

inverseratio

inversevariation



invertedsubtraction

involutionsymbolof


keynumber(figure)





leverprincipleof

licence

liketerms




linediagram


liter

literalfactor

literalnumber

loans

logarithm(log)


Briggssystemof


common

commonsystemof

divisionby

extractionofrootsby

mantissaof

multiplicationby

Napieriansystemof

natural

naturalsystemof

proportionalpartof

raisingtopowersby

tableoflongbarchart

longdivisionrulefor

loss



mapsstatistical

marginofprofit


harmonic

proportional


meanvalue

measurecircular

counting

cubic

dry

linear

liquid

metricsystemof

paper

square

timemedian

advantagesof



minuend

minussigninalgebra


miscellaneousseries

mixednumber

mixtures

modeadvantagesof


moneyBritish

French

GermanUnitedStates

anddecimals

divisionof

howwrittenmonomial

multiplecommon



associativelawfor

bylogs


complement

cross

cumulativelawof

distributivelawsfor

howtosimplify

inverted

left-hand

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften


multiplicationsign

multiplicationtable

multiplier


naturallogs

naturalsystemoflogs

negativedirection

negativeexponent


divisionof

multiplicationof


netprofit

netpurchases

netsales



nought

number

abstract

Arabic

basic

composite

concrete


reductionof


even

imaginary

index

integral

irrational

literal

mixed

negativeadditionof

divisionof

multiplicationof

subtractionofodd

positiveadditionof

divisionof

multiplicationof

subtractionofprime

real

Roman

signed

specific

wholenumberscale

numerator

oddnumber


direct

inverse

symbolsoforders


origin

papermeasure

parabola

parentheses



businessusesof


inprofitandloss

lessthan1percent

multiplicationof

relationtoratio


period

pictograph

piegraph(chart)

placesdecimal



pointdecimal

polltax

polygonfrequency


positivedirection


divisionof

multiplicationof


imperfect

multiplicationof

ofcommonfractions

ofdecimals


divisionof



priceselling

primecost

primefactor

primenumber

principalininterest

product


marginof


ascending

descending

geometric

harmonic


properfraction

propertytax


bycomposition

bydivision

byinversion

compound

direct



protractor

purchasesgross

net

return

quadrants

quadraticequation


quantityalgebraic

constant

variablequotient

radical


similarradicalsign

radicand

rateinpercentage

inprofitandloss

ofinteresteffective

nominaltax

workingofspeedratio

compound

howtosimplify

inaseries

inverse

relationtopercent



receptacles

reciprocal

rectanglegraph

recurringdecimal


ascending



indivision


returnpurchases

Romannumeralsystem

rootcube

extractionof

extractionofbylogs

fourth

ofequation

offraction

squareextractionof

salesgross

netsalescommission



sellingprice

seriesascending

descending

miscellaneous

sumtoinfinity


signofexponents

ruleforsignednumber

similarradicals

simpledecimal

simpleequation

simplefraction


exactmethodfor

formulafor



averages

division

multiplication

ratios

squaringofnumbers


solutionofequations

solutions(mixtures)


specificnumber

speedworkingratesof

squareofanumber




statisticalmap

statistics



Austrianmethodof


howtosimplify

inverted

left-hand


ofdenominatenumbers

offractions

ofpercents

ofplusquantities


symbolofinalgebra

subtractiontable

subtrahend


surtax



multiplication

oflogs

subtractiontanks

taxdirect

income

indirect

poll

property

totaltaxamount

taxmatters

taxrate


term(algebraic)like



timeininterest


totaltax

tradediscount

trialdivisor

troyweights

truediscount

unit




valuationassessed

valueabsolute

future

mean

presentvariable


direct

inverse

joint


verticalbargraph

vulgarfraction

weightedaverage

weightsmetricsystem


avoirdupois

troywholenumber

workingratesofspeed

worthfuture

present

xaxis

yaxis







0-486-23411-8


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Title Page

Copyright Page

FOREWORD

Table of Contents

INTRODUCTION

























































































































































































































































































































































































































































































































































































































PROBLEMS











PROBLEMS



























654 What are graphs

































































































































E Angle measurement


821 What is algebra



































856 What does mean



















































































APPENDIX B TABLES

INDEX


ArithmeticRefresher

AAlbertKlaf




9780486141930


FOREWORD




FRANKLINSKLAFMD

TableofContents



INTEREST

INTRODUCTION





89arecalleddigits









































thatnumber





integralnumbers





towardstheleft




beginningattheright






EXAMPLE




EXAMPLE400536080209












theunitsposition








100000000=10times10000000




EXAMPLES











smallerfraction

EXAMPLE





EXAMPLE


EXAMPLE



readldquoandrdquo




forty-threecents









EXAMPLES



EXAMPLES


EXAMPLES


EXAMPLES

PROBLEMS1

1Howmanyunitsin379























21Howis$356356read


(e)$444(f)$888



(a)

(b)

(c)

(d)








CHAPTERI

ADDITION







groupedtogether







EXAMPLE



EXAMPLE



EXAMPLE(asabove)


EXAMPLE


groupthen














EXAMPLES(a)

(b)


theother

EXAMPLE96+78



increments

EXAMPLE



EXAMPLES

(a)Add26to53

(b)Add67to86




EXAMPLE





Ex(a)

Ex(b)






EXAMPLE








EXAMPLES

(a)

(b)












EXAMPLE

PROBLEMS









9Add269745and983


11Add$525$1760$085$175$4565

12Findthesumof

(a ) (b ) (c)

$380865 $987367 $887406

37692 38898 51856

38623 573200 129897

48008 898719 54265

88842 782492 38600

75182 608604 4209























(a)

(b)

CHAPTERII

SUBTRACTION




8minus5=3 8minus3=5





Ex(b)16minus9=7












Ex(a)



Ex(b)






Ex(a)








SubtractionTable


etc






Ex(a)

Ex(b)


Ex(c)



Ex(a)


Ex(b)



EXAMPLE




Ex(a)


Ex(b)


Ex(c)


Ex(b)



columnattheright

Ex(a)

Ex(b)



EXAMPLE





Startatleft














EXAMPLES


Ex(a)


Ex(b)


Ex(a)



Ex(b)

PROBLEMS


1

2

3

4

5

6






12Subtract

















CHAPTERIII

MULTIPLICATION



4+4+4+4+4+4+4+4=32or8times4=32






EXAMPLE






MultiplicationTable















EXAMPLE



EXAMPLE



added457times
















Ex





(d)Add(a)+(b)+(c)

EXAMPLES(1)

(2)

(3)






Ex(a)45times29



Ex(b)323times35


Ex(c)271times55


multiplybytheother

Ex(a)

Ex(b)

Ex(c)


Ex(a)

Ex(b)

Ex(c)


Ex(b)

Ex(c)


Ex(a)

Ex(b)


discardingfraction

Ex(a)


Ex(b)

Ex(c)

Ex(d)


Ex(a)

Ex(b)













there4 is of100


there4 is of100


there475is of100



decimalThebaseis100

EXAMPLE(asabove)


020405062506660833


125133316662025






EXAMPLE


EXAMPLE


EXAMPLE



EXAMPLE


















numberfromtheresult

Ex(a)


Then


numbertothis

Ex(a)

Ex(b)




Inoneline



Inoneline









EXAMPLE


EXAMPLE








(d)Add(b)to(c)

Ex(a)




88times100=88008800+32=8832Ans



7900+98=7998Ans





EXAMPLE













EXAMPLE




PROBLEMS




4Multiply631by60



7Multiply6390by300

8Multiply


9Whatistheproductof


10Multiply




























35Whatisthecostof















CHAPTERIV

DIVISION




















EXAMPLES


(b) or

(c) orDivisor(=7




EXAMPLE



remainder

EXAMPLE





In =7wehavedivision



972=900+70+2


howwouldweproceed


4rsquos=3200 4)3492





(e)3times4=12



800+70+3=873




































getting2asquotient












EXAMPLE











Quotientis







Quotient=





Ex(c)




Ex(d)








EXAMPLE



in2468orinazero


in21273834485





and2times3=6






Ex(a)

Ex(b)


Ex(a)

Ex(b)


by8

Ex(a)




Ex(c)








Ex(a)

Ex(b)


Ex(a)

Ex(b)


Ex(a)


Ex(b)

Ex(c)


Ex(a)867


Ex(b)973285









Ex(b)





placestotheleft












Ex(1)


Ex(2)











Monday $26860

Tuesday $32985

Wednesday $9745

Thursday $23990

Friday $29670


$24650 (=Average)









For10cent



then



PROBLEMS



















9Divide

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

(u) (v) (w) (x)

(y) (z)

10Findthequotientof





(m) (n) (o)


12Findthequotientof





14Divide






16










17


18



(a) (b)

(c)

(d) (e)










CHAPTERV











Ex





EXAMPLE
















GCD





Ex(a)


there42times3=6=GCD

Ex(b)





















7appearsonce
















PROBLEMS










(a)310(b)297(c)670(d)741(e)981(f)385(g)2650

(h)1215((i)321(j)1575(k)10935(l)420(m)497

(n)378(o)462(p)2430(q)25344(r)73260(s)599676

(t)273564(u)15625(v)10675(w)12625(x)976

(y)8050(z)3848


(a)68112240(b)2184126147(c)212877

(d)457281(e)4477121(f)1498112(g)248096

(h)284236(i)457281(j)31522679012

(k)144576(l)820697(m)1251751792(n)60043318

(o)125423618163(p)1086905




(j)86and4



(e)3642and48(f)3918and27(g)51525and35


(l)212426and28(m)92142and63

(n)367548and24(o)71456and84(p)2472128and240



(d)88368divide3682(e)32768divide2048







CHAPTERVI

COMMONFRACTIONS





EXAMPLE


exactlyequalsize



Thus

191Whatismeantby

(a)

(b)

(c)

(d)










Ex(a)


Ex(c)


equalparts





(b)



Wegetthewholeunit

Ex(a)

Ex(b)

Ex(c)







numbers










Ex(a)

Ex(b)






Ex(b)



Ex(a)

Ex(b)

Ex(c)

Thereasoningis



Ex(a)

Ex(b)






commonfactors

Ex(a)

Ex(b)







EXAMPLE


EXAMPLE


















Ex(a)



Ex(a)

Ex(b)



EXAMPLE


mustallbethesame


Ex(a)Add and


Add


Addfractions

Then




Now

Ex(b)Add (LCD=20)








Ex(b)Subtract from



Ex(a)






Ex(b)

Ex(c)


Ex(a)

Ex(b)


Ex(c)

Ex(d)


Ex(a)

Ex(b)Multiply by11

Ex(c)


inposition



Ex(a)

Ex(b)Multiply




EXAMPLEMultiply



Ex(a)

Ex(b)Multiply by

Ex(c)Multiply by



EXAMPLE













(a)

(b)


EXAMPLEDivide by




Ex(a)Divide by2


Ex(b)


Ex(a)Divide24by

Ex(b)Divide17by or






EXAMPLEDivide by3



EXAMPLEDivide by6



Ex(b)Divide by



partsareexpressed






Ex(c)Whatpartof is

Ex(d)Whatpartof is7
















EXAMPLE


fractions












Next

Next

Finally
















Startfromthebottom







PROBLEMS






(a) (b)

(c) (d)

(e) (f)










(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


(a) (b)

(c) (d)

15


16





(a) (b) (c) (d) (e) (f)

19Add

(a) (b) (c)(d) (e) (f) (g)(h) (i) (j)(k) (l)

20Subtract














31














encloseit




CHAPTERVII

DECIMALFRACTIONS


EXAMPLES




EXAMPLES




Ex(a)0207008024017

Ex(b)


EXAMPLES








EXAMPLE5631056923









EXAMPLES




EXAMPLES



numerator





048386356



EXAMPLE6=6=60=600




EXAMPLES





EXAMPLES



EXAMPLES

(a) or

(b) or

(c) or

(d) or

(e)

(f)














Ex(b)










usualway









Ex(a)Multiply38by6

Ex(b)

Ex(c)


improperfractions

EXAMPLE






Ex(a)Divide192by06











theleft




















4879times3765=18369435



mostimportant





EXAMPLE



EXAMPLES

(a)(b)(c)


infinity



group




whichdonotrecur



EXAMPLES(a)

(b)

(c)

(d)

(e)












(b)


PROBLEMS






3Read12584062018








260260026260260













23Add(a) and







(e)64575by165(f)9686by136



31Shorten(a)59767(b)59755
























CHAPTERVIII

PERCENTAGE





EXAMPLE


EXAMPLES(a)(b)

(c)

(d)

(e)(f)(g)










EXAMPLES



EXAMPLES








Ex(b)Find14of$9751(14=14)



Ex(a)Find6of$6700

1of


6of$6700=6times$67=$402

Ex(b)Find4of$1860

1of$1860=$1860there44is4times$1860=$7440

Ex(c)Find of$7000



Ex(a)Find25of$51

Ex(b)Find of$8475







Ex(b)Find9of50


fraction












adecimalSince



or





If$285is2then

or





or






or












$15750times0025=$39375=$3938Ans



120times$55=$66Ans




$46375times43=$19941Ans






fromtheamount


06times85+85=51+85=901=Amount











PROBLEMS



31400iswhatof3600

4Reducetoadecimal




(k) (l)



















25Whatpercentis of6
































CHAPTERIX

INTEREST









EXAMPLE




EXAMPLE

005=fivepercent=




EXAMPLE



















=$500(115)=$575Ans










eachor360days



Year Month Day

1959 2 8

1957 5 15

_____ ____ _____

1 8 23









May 23days

June 30days

July 31days

August 31days

September 30days

October 31days

November 30days

December 31days

January 12days

249days











Year Month Day

1959 4 6

1958 11 18

4 18 =138days










Then

and

Then

and







Now6daysare







(b)58days(c)77days



4days 6days 2days

49days 2days 77days

58days

58days



2

















or
































$468=$450+$450times04

or


and





or






Therefore


S=$50619


$50619Ans














S=$12000times131593=$1579116Ans






n=6=t=6yearsAns














Inabove




Inabove













126898555Ans



S=$8000times181535485=145228388


$1452284Ans

PROBLEMS
























































CHAPTERX

















EXAMPLES

$54000$18000=31412=13(to)(to)(to)(to)











numbers



Ex(b)




$100to$500is














Dividebothtermsby

















there4320560720=479Ans





there4 Ans







or




Ormargin



(b) forrent















EXAMPLE3=9


EXAMPLE36=12

















Value=4


Value=2










EXAMPLE



[]=ldquoasrdquo














othertwo

Ex(a)36612


Ex(b)


Ex(c)3xx12




cost

3x=5times$864


there4 Ans















doitin5days


























Thus






andsolvedasfollows








3times8=4times6

(b)If then


EXAMPLEIf =then 3

(c)If then


EXAMPLEIf then

(d)If then


EXAMPLEIf then or

(e)If then


EXAMPLEIf then or

(f)If then


EXAMPLEIf then or




or










variation



C=πd=variationform
















Sαr2



S=4πr2







andinversetypes




From

then



where


PROBLEMS












(a)1525(b)2415(c)824(d)2724

























36(a)4=16(b)5=4(c)

37(a)24=6(b)49=7(c)






(a)(b)

(c) (d)

(e)

(f)

(g)

(h)

(i)








(a)

(b)

(c)
























CHAPTERXI

AVERAGES






















things











48+55+68+42=213minutes








there4















Theweightedmarksare






Remainder=199























Themeanaverageis



Themeanaverageis




200200200220225235240240250260260275275350450500650and800





amount



$700 and over (1)

$400 to $699 (3)

$300 to $399 (1)

$250 to $299 (5)

$200 to $249 (10)

$150 to $199 (4)

less than $150 (1)



Themedianis$235


























destroyed


whichdonoteffectit










PROBLEMS























$500(1)$50(12)$300(2)$25(22)$100(3)$15(6)$75(4)




CHAPTERXII

DENOMINATENUMBERS






EXAMPLES



descending







161feet=1rod



Note(c)













































Pound Ounce









24hours = 1day(da)

7days = 1week(wk)


52weeks = 1year(yr)




10years = 1decade

20years = 1score






20units = 1score

12units = 1dozen











TheLatinprefixesare


TheGreekprefixesare











= 0001sqmeter


= 01sqmeter


= 1sqmeter=1centare

= 1sqmeter=1centare


= 100sqmeters=1are




= 1000000sqmeters








= 000001cumeter





= 1000cumeters


= 1000000cumeters


































andfindthetotal







8qt

(or25)qt+6qt=625qt


4pk

78125pk+2pk=278125pk


6953125buAns







EXAMPLEAdd








EXAMPLE




















23sqyd+1sqyd=24sqyd



turn







6403509metersAns






5327698kmAns





PROBLEMS















































46Add











CHAPTERXIII







EXAMPLES





EXAMPLES





used







26x26=676





400+240+36=676


(a+b)2=a2+2ab+b2









EXAMPLES

22=0452=25


EXAMPLES



EXAMPLE


EXAMPLE


Now25=32and23=8




Ex(a)Divide32by8

Ex(b)Divide243by9






(42)3=42bull42bull42=42times3=46=4096




Ex








Now




EXAMPLES




Ex(a)

(b)


Ex(a)

(b)










Raise52=25Then252=625Then6253=244140625=512




(29+1)(29mdash1)=(292ndash1)

or

30times28=840=(292ndash1)292=840+1=841


number




(b)




(a+b)2=a2+2ab+b2=a(a+2b)+b2




Ex(a)

(b)

(c)






(b)Square35 =772=4949times52=1225

(c) =772=4949times22=196(d) =992=8181times22=324







Ex(a)

(b)


Ex(a) issimilarto

(75)2=70times80+25=5625(b)(125)2=120times130+25=15600+25=15625




Thenfigure8


Ex(a)

(b)






(b)4000000=4times1000000=4times106

(c)36300000=363times10000000=363times107









Ex(a)

(b)

(c)

(d)

(e)



Ex(a)

(b)

(c)












leavingaremainder









3)9

3




(c) =a2b(d)

(e)(f)





Ex(b)(c)(d)

(e)(f)

(g)




Ex(a)(b)


Ex(a)(b)



Ex(a)

(b)



EXAMPLE


bythesamenumber

EXAMPLE


EXAMPLE









(b)






decimalitself

EXAMPLES



EXAMPLES






(a+b)2=a2+2ab+b2=a2+(2a+b)b









radic676=26



















there4




remainder





















Ex(a)

(b)




















there4 =626Ans






=1 =1

PROBLEMS

1Find(a)52

(b)82

(c)202

(d)14

(e)H2

(f)19

(g)103

(h)34

(i)252

(j)173

(k)833












(c)(d)3times4-1

(e)(05)-2

(ƒ)(a-m)-n







(d)65(e)225




(e)

(f) (g)(h) (i)


(f)




(a)

(b)

(c)


(c)



(a)(b)(c)









35Whatisthevalueof

CHAPTERXIV

LOGARITHMS









Ex(b)43=64







logarithm=exponent

















Ex(b)








4642isabasicnumber











Ex(a)


1to9 1 1mdash1=0

1to9 1 1mdash1=0

10to99 2 2mdash1=1






log000865=ndash4+9370







Ex(a)log000865=49370(b)log00427=36304





3+8+6304mdash8=56304mdash8

or

+30+6304ndash30=276304ndash30




















Then83315+0003883353













105times times = =104=10000






=1093=103=1000


EXAMPLE160=1022041236=102372928=1014472






EXAMPLE135834=10213301896=1095230

Thus

Theantilogis151605



EXAMPLE374=1015729


=1015728x3






EXAMPLE




75=52times3














EXAMPLE



EXAMPLE

log10100=4343logε100=4343times46052=2



cologa=log















Then

and





there4ndash09683Ans










Nowaddallthefactors








Then




(c) =(d)9radic2=2235(e)34=81(f)2-2=







(h)x=log1001000(i)logx49=2




(n)00054times10ndash4(o)3755000(p)4343




(j)00638times104






(c)

(d)


(a)log

(b)log










(c)(d)(2112)minus3(e)(2718)ndash12(f)

(g)(06493)minus(h)(5937)minus13


(j)

(k)(l)(42)x=649

(m)(n)

CHAPTERXV









numbersas32+5+711+



EXAMPLES












(orsame)sign






(Question626)



5+(mdash7)=mdash2
















CHAPTERXVI






Ex(a)24681012etc






Ex(a)28321285122048etc


Ex(b)20485121283282





Ex(a)358101315182023










EXAMPLE13579




EXAMPLE25211713














Then













Ex(a)


Then

Now

Ex(b)


Now

(1)


Substitutingin(1)










series



Thustheseriesis

4 minus1 minus6


progression

xmdasha=bmdashx

or

2x=a+b

and

x= =halftheirsum




there4l=arnminus1


41 to7terms




(1)


(2)


rSmdashS=arnmdasha



Then

But

l=arnndash1orrl=arn



Therel= r= anda=4



FindrandS



Thiscanbewrittenas




Herea=3and


Now


Then








progression

Hence










Here andn=8Then





and



346





(15times01=0015)





Then


PROBLEMS


(d)11119977(e)8127931 (f)2818325072(g)12481632(h)403430282218























CHAPTERXVII

GRAPHSmdashCHARTS




verbaldescription
























$4000-$4999 12400

$5000-$5999 10200

$6000-$6999 8100

$7000-$7999 3040

$8000-$8999 2200

$9000-$9999 1160

$10000andover 208






Year Sales

1954 $38260000

1955 $47840000

1956 $43190000

1957 $45000000

1958 $39080000

1959 $47040000

1960 $51000000









Freshmen = 520

Sophomores = 410

Juniors = 380

Seniors = 290




Then
































91to100lb 12

101to110lb 124

111to120lb 268

121to130lb 107

131to140lb 26

141to150lb 8

141to150lb 8

151to160lb 4





















mathematicallaw




















first









EXAMPLES




PROBLEMS




$4000-$4999 8400

$5000-$5999 3200

$6000-$6999 2100

$7000-$7999 1800

$8000-$8999 760

$9000-$9999 139

$10000andover 68


1950mdash$54000000 1956mdash$46000000

1951mdash$52000000 1957mdash$45000000

1952mdash$51000000 1958mdash$39000000

1953mdash$47000000 1959mdash$47000000

1954mdash$37000000 1960mdash$52000000

1955mdash$48000000





$26500000 228






Total= $1161650000 10000


Freshmen 650

Sophomores 530

Juniors 480

Seniors 390




Year Cost

1950 $8675

1951 $9300

1952 $9475

1953 $9950

1954 $10625

1955 $11350

1956 $12225





January92hr

February106hr

March112hr

April122hr

May136hr

June146hr

July154hr

August142hr

September136hr

October114hr

November102hr

December94hr










CHAPTERXVIII










































discount


=tradediscountAns





























Theirsumis30+4=34

oftheirproductis

Thedifferenceis




(a)Combine40with10




Now

$39842=412=discount

Then







and











(a)Netcost













(f)ofprofit= =100





(b)Sellingprice





(f)of


profit

(a)Netcost14+8ndash




(d)Grosscost=$443+$18=$461

(e)Netprofit


=$666ndash$100=$566




cost
















Grosscost=$360+$12=$372



Cost=100minus25=75=$1860










=85=sellingprice








Forsimpleinterest

Forcompoundinterest






Then



Toprovethiswehave


and




Alsoitfollowsthat


or

$88900times112486=$1000






Soinsteadoffinding


there4$1000times88900=$88900









$25 = downpayment

$135 = 45weeksat$3


$150 = cashprice





= finalpayment

$35 = finalpayment


$230 = cash










































P=principal=$10here
















Sumofmonths







Youkeeporborrowthe

















$84for1month









Thenbyproportion







Totalmonths

Totalinterest

P=principal=$25




Totalmonths

Totalinterest
















Repaymentschedule



















upto thyearm























Theborrowerhad

















remainingbalance



$5down+5times$5+$2 = $32

Cashcost = $28




Here

















full







payment




thenoteisdesired


Faceofnote $100000






+$969










Weseethat


Faceofnote $1000000





















business









muchper$1000









$68750divide$2383015=02885=28852885=$2885per$1002885=$2885per$1000





$2885per$100=2885=02885



Add(a)and(b)


$16000times05 =$800

$16000times03 =$480





$8000minus$3000 = $5000

$5000times02 = $100 = surtax


PROBLEMS






































































CHAPTERXIX

VARIOUSTOPICS



























Expressasafraction






































$1800minus$1600=$200














ittoa25solution

Asabove























Then

and









Thenbyproportion















25 = 25

25times128 = 32




445 = 0445



67 = 067

067times128 = 8576=858




or













(b)Addthefractions
















Sumoffillingrates

Sumofemptyingrates


Thus and



















Scaleis172or


















(a)Bytheratiomethod





(a)Byratiomethod




EAnglemeasurement

815Whatisanangle

















PROBLEMS






































CHAPTERXX

























monomial

EXAMPLE20p=aterm






product













factors






polynomial



Letxdenoteathing








(b)




















Inalgebraa+b=b+a



Inarithmetic(5+9)+12=5+(9+12)=26





Inalgebraab=ba






















Itisnot


Ifx=2then

or



(a) times(b) (c)75x







(e)Costplus8=c+8







(l)Anyevennumber=2a





























$1250times48+$1050times72=$600+$756=$1356Ans












cube













(a+b)3times(a+b)4=(a+b)3+14=(a+b)7

Now


Therefore









thesquaretoot

EXAMPLES







Ex(a)Dividex5byx3

Ex(b)

Ex(c) (notx2)






(e) (a)

(b)

(c)

(d)

(e)


856Whatdoes mean



Similarly












Ex(a)Isa2b3likea3b2



there4Theyarealike













Youareonly700ftdown




Algebraically


















EXAMPLES



Ex(a)















Removeinnerbrackets



answer




ndash28+80=52Check




Ex(b)

Ex(c)


EXAMPLEWemaycombine


Wemaynotcombine






+15x mdash16y +8z


mdash20x +16y +14z











Thisbecomes






5a+7ndash3a+8=2a+15












(d)6a2+3bby3a+4b2

(a)


(d)



Ex(b)




Ex(b)


Ex(a)

Ex(b)

Ex(c)

Ex(d)

Ex(e)








2x2+10xy+3xy+12y2or2x2+13xy+12y2


(a+2)(a+4)+(a+3)(a+4)+(a+2)(a+5)a2+6a+8+a2+7a+12+a2+7a+10

or

3a2+20a+30Ans











Useruleofsigns























































theexpression









Ex(b)Factorx2+6x+8



Extremesxtimes8=8x




Extremesxtimes2=2x







Squarerootof9a2is3a




Ex(b)Factorx2+6x+9







(x4ndash25)(x4+25)


















or












Ex(b)













Ex(a)Solve3x=21


x=7Ans

Ex(b)Solve x=36

















Ex(c)


simpleorfirstdegree






EXAMPLE4x2=64










Ex(a)xndash5=0


Ex(b)x+5=12


Ex(c)xndash7=8



Ex(a)Solve

Ex(b)Solve =64


Letx=thenumber

Then

(1)


Ex(d)Solve06x=18

Ex(e)Solve3 =30

Ex(f)Solve08x=1000




Ex(a)Solvex+4=10


Ex(b)Solve16=7+y






there4x=6Ans


From8=14ndashxweget











Leftside

Rightside Check

Ex(b)Solve




Then

Substitutex=ndash20

Leftside

Rightside





Then








Multiplybothsidesby











Note




Tocheck



variousquantities









d=44x10=440ftAns












Ex(d)From findC









Letx=thenumber


Then







x+45 = 2(x+12) = 2x+24


minusx = minus21

there4x = 21





4y = 15

there4y = yearsago


Ans

5x8 =41 yearsCheck































=wifersquossavings



18143+37286+64572=$120001Check














PROBLEMS



















17Evaluate(a)(b)(c)

18Divide(a)y6byx2

(b)y9byy3


19Whatistheresultof

(a) (b)

(c)

(d)

(e)









26Whatistheresultof

(a) (b)(c)

(d) (e)

(f) (g)+ (h)





(c)


9xndash8zandndash3y







(f)8a2+4b2by5a+362










(b)xa+16x+64


(d)(2x+3y)2ndash1










(s)

(t)8x+09=9minus4x



50If whatisa
















APPENDIXA

ANSWERSTOPROBLEMS


31937467296







17(a)073586

(b)8000008(c)050321(d)70090000012(e)023504910000630003(f)4792086500005(g)04090306001000700008(h)0364575(i)0908006034





23(a)$066(b)$080(c)$047(d)$010(e)$120(f)$712

25(a)475(b)5621(c)22(d)10540(e)10765

(f)2555(g)100(h)4444



36121824

535793711153915213111927

79131721916233091827369111315

91997

11$203265

1360

1595

1721

1990

21(a)255(b)244(c)209(d)263(e)270(f)250

23(a)169(b)155(c)140(d)141(e)1879(f)1457(g)1667(h)2039

25907gallons

271039miles

29$525

31

(a)280778(b)295263(c)292690(d)242893

33(a)195564(b)293220(c)208675(d)142415

ChapterII(pp34-36)12242192225612425

3633432392134320372

5152294215059811720


9minus4deglat

11$24

12(a)124959(b)151833(c)74296(d)161574(e)$305907(f)$873883(g)$38849254(h)$60579179

14(a)25697(b)49779(c)92922(d)$22015250(e)$100035090(f)$91357818

17(a)4228(b)4214(c)4319(d)5659(e)3357(f)2165

19(a)2443393(b)888

(c)1669(d)178556

21(a)1421(b)41135

23$40746

251548576

27(a)53514947534945415347413553453729(b)89868380898479748982756889807162(c)74696459746760537471686574655647

ChapterIII(pp54-56)1540540054000

3176201762001762000

518001800018000018000000

71917000


11$14425$6347013$1886

1517424001b17$420

19(a)238(b)272(c)306

(d)304

21(a)7395(b)2352(c)3074(d)1184(e)4355(f)9306(g)5328(h)728(i)306

23(a)945(b)8295(c)6435(d)630(e)4005

25(a)2709(b)2625(c)1316(d)3149(e)3364(f)2016(g)2236

27(a)4275(b)4875(c)5525(d)1925(e)3325(f)4125(g)1225(h)6375

(i)9425(j)$6075(k)$12375(l)$20425

29(a)(b)(c)(d)(e)(f)(g)(h)(i)

31(a)(b)(c)(d)

33$1400

35(a)$21250(b)$12325(c)$2875(d)$1200(e)$1200(f)$2100(g)$1800(h)$41600(i)$900

37$9000

39(a)768(b)1632(c)30008

(d)1368

41(a)516456(b)528849(c)38952(d)890901(e)7628688

43(a)5496(b)4809(c)3456(d)3024(e)7856(f)6874

45(a)8232(b)9024(c)7998(d)7505(e)7216(f)960376

47(a)6384(b)63672(c)3196(d)49088(e)7128(f)2964(g)7392(h)64528


34

520



117hours

13

(a)214(b)402(c)428

17(a)3(b)Yes2

19(a)2958(b)60(c)80868(d)365(e)1680(f)6912(g)72(h)42(i)139(j)36(k)112


23842

25$101522

ChapterV(pp80-81)

12369235610152346121839272345681012152024304060

3234612

512357111317192329313741434753596167717379838997



(z)2221337

11(a)918(b)3570(c)1836(d)2142(e)4080(f)612(g)816(h)72144(i)918(j)2448

13(a)21(b)15(c)28(d)24(e)161536lb1718days





9(a)14(b)28(c)7(d)No

11(a)

(b)(c)(d)(e)(f)

13(a)(b) (c) (d) (e) (f) (g) (h) (i)

15(a) or1(b) or1(c) or3

17(a)(b)(c)(d)(e)(f)

19(a)2(b)15(c)2(d)67(e)1 or(f)1 or(g)1(h)1

(i)28(j)42 (k)139 (l)129


25(a)(b)(c)(d)16(e)2(f)216(g)(h)(i)(j)

27(a)100(b)$688

29

31

33

35$44

371 ozperslice

39286miles

41 $1350$1350$900

43$246$6150$9225$3075$2050

45

4717 rods



51000100000

7Ten

9(a)8=80=800(b)046=0460=04600(c)738=7380=73800=0738

110040004

13246246

15246024600246000

17246576246576

19(a)032(b)0625(c)014(d)0392(e)01875(f)065(g)04(h)0175(i)03125(j)0115(k)046875(l)0232

21(a)(b)(c)(d)(e)(f)(g)(h)

23(a)1274735(b)18125608(c)22135538(d)7202238

25(a)4234408(b)4494375(c)38316(d)35425(e)553308

27(a)52655625(b)2582398(c)39130222(d)2012315(e)0638027

29(a)15895794(b)38884176(c)17517890(d)112489886(e)54923664(f)21073016

31(a)5977(b)5976

33012

352918

37(a)0078125(b)015625(c)0375(d)03125(e)028125(f)0171875(g)028(h)0184

39$042$007

41$568750

43$34000$7480$10880$12240

45$282

470968lb

4911cents832mills

5132lb

53A0750B0714


3

502502020002500020002


916

11(a)25(b)64(c)100(d)325(e)30(f)420

13$3000$11040$9960

151904votes

1720

19$132389

21(a)406(b)131(c)1278(d)40(e)(f)2323(g)0135(h)2188(i)1662(j)364(k)7150(l)4442(m)5138

23 sqft

2740032

29506675911

3119000054

33$29143

35$13636

3721

39

41$1838

43$150

45$35235


49$20588

51 25

53(a)72(b)60(c)006696

55(a)304(b)720(c)2300


3$928$128

5(a)March4(b)March3

7(a)249(b)84(c)118(d)248(e)142

9$789

11$240

13$4919

15(a)$4717(b)$38111(c)$291(d)$1186(e)$28603(f)$370(g)$3431(h)$363(i)$4912

17$4310

19$1438

21$247

23(a)1566(b)6015(c)3063(d)60306(e)3010(f)601566(g)606063(h)603015(i)60601510(j)156(k)60606(l)603063

25$469

27$150

29$9653

31

3385days


37(a)$120(b)$068(c)$829(d)$240(e)$028(f)$425

39$444500

41


43$260000

45$104040

47$1643615

49$6289

51$2693706$693706


359

5116

7118811

906

11(a)35(b)85(c)13(d)98

13(a)13(b)12(c)13(d)110(e)1379(f)19(g)(h)1625(i)140(j)1571(k)1115(l)1222(m)14

(n)165(o)160(p)1136(q)1114

15

1741

1914and21

21507080

235134

2515

27 inches

29$43875

31

3322ftx ft

3531

37(a)4(b)7(c)

398

41(a)6(b)2(c)18(d)24(e)18(f)3(g)12(h)32

43$3750

4572feet

471057lb

49(a)10(b)15(c)

512171b

5342men

55821$3528$1536

57 days

5923

61082ohm

63400feet

65x=6

6790psi

6966men


347mph

58562

7435minutes

9$1784

11$209067

1359

15(a)13(b)19

17$340

19$300to$399

21No



38rods2feet

5 cubicinches

783688lbofwater

93025bbl

11$1816

1349280lb

15366

17184

1942doz

2130years


25(a)735dm

(b)74126meters


2939122dg

316944grains

33102058cg

350664grains


39392pt

41 bu

430883bu

4500181gal



517504610meters

537976meters

5511664kg


(r)16x2

(s)8b3(t)1953125

34000sqft

548sqyd

7(a)256(b)19683(c)16(d)3(e)axminusy

(f)ax+y(g)4096(h)15625(i)1(j)1

(k)1(l)24(m)(n)(o)

92176782336

11(a)784(b)4489(c)5776(d)7921

13950625

15(a)256(b)2025(c)65025

17(a)99980001(b)9801(c)999998000001

19(a)12(b)4b4

(c)a3b32

(d)x2y4(e)(f)(g)8(h)2646=(i)

21(a)

(b)12(c)

23

(a)

(b)

(c)

25(a)(b)(c)

27(a)4a2y54(b)

(c)

291287feet

316314

33(a)(b)(c)

(d)

(e)(f)01334(g)(h)0949(i)(j)9709(k)00255

35




7(a)1000(b)64(c)minus5(d)512(e)(f)10(g)(h)(i)7




132log7+log4

15log1944

17(a)log432+log748-log566(b)

19

(a)(b)

(c)(d)

21


23(a)0340(b)3679(c)00036(d)4016(e)000027(f)164


3(a)12(b)6(c)(d)16(e)350



3a=15S=645

533 4 5 5 6 7 8

7

9250500

11l=39366S=29524

132

15

172

1915


23$70388


15$5425


33846

5$3

7$108$1692

9$27354

113825

13$141221

15$54

17$2693

19426

21$4421$269

23$4815

25$3708

27$400

29$229665$10335

31$213359

33$30

35$2650

3733

39373

41$2330036

43$3000$1000

45$799056$199056

47242

49204

511082

53436

55457

57$95940

59$225

61$28712

6323075$230 per$100$2307 per$1000

65$3073196


360days

5

7 or194days

940gal

11286

133 qt

141119

173994avoz

19125cc

21

233min

254320gal

27

291253440

31276miles

339728acres

35720deg

374rdquo


3xyzz

515

7No

3y83(y8) etc







(d)56x5c7

(e)45a7a+1

17(a)x3

(b)3x3(c)

19(a)a4b2c2

(b)x2y4z(c)x3yz3

(d)y-2b2

(e)9xy2z-2

21Yesno


27ndash21a+66b

29ndash132xndash76y

31-19x+18y+27z



(c)ndash24a7b2c2



(f)40a3+24a2b2+20ab2+12b4



(d)

(e)(f)a+5(g)2a+3b(h)3a2+2ab+4b2




(f)x=77(g)x=7(h)x=23(i)x=70(j)x=72


49248degF

5110

53400gal

5510001600

571405681

59 orsquoclock

61$9000

6311

659miles7miles

APPENDIXBTABLES

TABLEI


TABLE2


TABLE3


TABLE4

PRESENTVALUEI(I+i)n

INDEX




inalgebraalgebra











ybankdiscount


100percent

longbargraph

horizontal

verticalbase


inprofitandloss

intaxation


binomial

blockgraph

Boylersquoslaw

Briggssystemoflogs

Britishmoney

broken-linegraph

bundlesofunits


buyingcommission

calculation

cancellation

carryingcharge


cashdiscount



insubtraction



chargecarrying

financingchart

dividedbar

100percentbar

longbar


inaddition

inalgebra

indivision

inmultiplication


cipher

circlegraph

circularmeasure

circulatingdecimal

circumference

coefficient

cologarithm(colog)

commissionbuying

salescommondivisor



powersofcommonlog

commonmultipleleast




complexdecimal

complexfraction

compositenumber


compoundamount


compoundfraction


compoundproportion

compoundratio

computation

concretenumber

conditionalequation


constant





Cartesiancost

gross

net



cube





decimaladditionof

andUSmoney

approximationof

circulating

complex


topercentdivisionof

equationswith

multiplicationof

powersof

recurring

repeating

simple



decimalplace

decimalpoint

degree





depreciation


diagramline

staircasedifference

inpercentage


directednumber

directionconceptof

negative


directtax

directvariation

discountbank

cash

trade


dividend


bylogs


decimal


long

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften

ofsamekindofsymbols

ofUSmoney

proportionby

pureproofof

short

symbolofinalgebra

divisionsign

divisorcommon

greatestcommon

trialdrymeasure

inmetricsystem



emptyingproblems


equalssign

equationconditional

linear

quadratic

rootof

simple

solutionof


evolutionsymbolof


lawsof

logarithmdefinedas

negative

raisedtoapower

signof

zero


extrapolation


factorcommon

greatestcommon

literal

primefactoring


commontermmethod

inalgebra


finance

financingcharge

fluidounces

formula

fourthroot

fractionadditionof

chain


compound

continued


topercents


improper

multiplicationof

powersof

proper


rootof

simple

subtractionof

unit


fractionalplaces

Frenchmoney


futurevalue(worth)



Germanmoney

gram


horizontal

verticalblock

broken-line

circle

curve(d)


ofquadraticformula

pie

rectangle



grosscost

grossprofit

grosspurchases


harmonicprogression

Hookersquoslaw

horizontalbargraph

100percentbarchart


identity

imaginarynumber

imperfectpower

improperfraction

incometax

index

indexnumber

indirecttax




integralnumber

interestcompound


rateofeffective

nominalsimple

formulafor


interestcost

interestearned

interpolation

inverseproportion

inverseratio

inversevariation



invertedsubtraction

involutionsymbolof


keynumber(figure)





leverprincipleof

licence

liketerms




linediagram


liter

literalfactor

literalnumber

loans

logarithm(log)


Briggssystemof


common

commonsystemof

divisionby

extractionofrootsby

mantissaof

multiplicationby

Napieriansystemof

natural

naturalsystemof

proportionalpartof

raisingtopowersby

tableoflongbarchart

longdivisionrulefor

loss



mapsstatistical

marginofprofit


harmonic

proportional


meanvalue

measurecircular

counting

cubic

dry

linear

liquid

metricsystemof

paper

square

timemedian

advantagesof



minuend

minussigninalgebra


miscellaneousseries

mixednumber

mixtures

modeadvantagesof


moneyBritish

French

GermanUnitedStates

anddecimals

divisionof

howwrittenmonomial

multiplecommon



associativelawfor

bylogs


complement

cross

cumulativelawof

distributivelawsfor

howtosimplify

inverted

left-hand

ofdecimals

offractions

ofpercents

ofpolynomials


ofpowersoften


multiplicationsign

multiplicationtable

multiplier


naturallogs

naturalsystemoflogs

negativedirection

negativeexponent


divisionof

multiplicationof


netprofit

netpurchases

netsales



nought

number

abstract

Arabic

basic

composite

concrete


reductionof


even

imaginary

index

integral

irrational

literal

mixed

negativeadditionof

divisionof

multiplicationof

subtractionofodd

positiveadditionof

divisionof

multiplicationof

subtractionofprime

real

Roman

signed

specific

wholenumberscale

numerator

oddnumber


direct

inverse

symbolsoforders


origin

papermeasure

parabola

parentheses



businessusesof


inprofitandloss

lessthan1percent

multiplicationof

relationtoratio


period

pictograph

piegraph(chart)

placesdecimal



pointdecimal

polltax

polygonfrequency


positivedirection


divisionof

multiplicationof


imperfect

multiplicationof

ofcommonfractions

ofdecimals


divisionof



priceselling

primecost

primefactor

primenumber

principalininterest

product


marginof


ascending

descending

geometric

harmonic


properfraction

propertytax


bycomposition

bydivision

byinversion

compound

direct



protractor

purchasesgross

net

return

quadrants

quadraticequation


quantityalgebraic

constant

variablequotient

radical


similarradicalsign

radicand

rateinpercentage

inprofitandloss

ofinteresteffective

nominaltax

workingofspeedratio

compound

howtosimplify

inaseries

inverse

relationtopercent



receptacles

reciprocal

rectanglegraph

recurringdecimal


ascending



indivision


returnpurchases

Romannumeralsystem

rootcube

extractionof

extractionofbylogs

fourth

ofequation

offraction

squareextractionof

salesgross

netsalescommission



sellingprice

seriesascending

descending

miscellaneous

sumtoinfinity


signofexponents

ruleforsignednumber

similarradicals

simpledecimal

simpleequation

simplefraction


exactmethodfor

formulafor



averages

division

multiplication

ratios

squaringofnumbers


solutionofequations

solutions(mixtures)


specificnumber

speedworkingratesof

squareofanumber




statisticalmap

statistics



Austrianmethodof


howtosimplify

inverted

left-hand


ofdenominatenumbers

offractions

ofpercents

ofplusquantities


symbolofinalgebra

subtractiontable

subtrahend


surtax



multiplication

oflogs

subtractiontanks

taxdirect

income

indirect

poll

property

totaltaxamount

taxmatters

taxrate


term(algebraic)like



timeininterest


totaltax

tradediscount

trialdivisor

troyweights

truediscount

unit




valuationassessed

valueabsolute

future

mean

presentvariable


direct

inverse

joint


verticalbargraph

vulgarfraction

weightedaverage

weightsmetricsystem


avoirdupois

troywholenumber

workingratesofspeed

worthfuture

present

xaxis

yaxis







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Vol10-486-22726-XVol20-486-22727-8


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VolI0-486-20381-6VolII0-486-20382-4



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Title Page

Copyright Page

FOREWORD

Table of Contents

INTRODUCTION

























































































































































































































































































































































































































































































































































































































PROBLEMS











PROBLEMS



























654 What are graphs

































































































































E Angle measurement


821 What is algebra



































856 What does mean



















































































APPENDIX B TABLES

INDEX


Arithmetic Refresher: Improve your working knowledge of arithmetic

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Transcript of Arithmetic Refresher: Improve your working knowledge of arithmetic