Comparing and Ordering Rational Numbers

Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Write the prime factorization of 364.

22 x 7 x 13

Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Write the prime factorization of each number.

1. 20 2. 125 3. 45

4. 186 5. 621 6. 1,575

(For help, go to Lesson 4-3.)

Check Skills You’ll Need

PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Solutions

1. 20 2. 125 3. 452 • 10 5 • 25 3 • 152 • 2 • 5 5 • 5 • 5 3 • 3 • 5 20 = 22 • 5 125 = 53 45 = 32 • 5

4. 186 5. 621 6. 1,5752 • 93 3 • 207 3 • 5252 • 3 • 31 3 • 3 • 69 3 • 3 • 175186 = 2 • 3 • 31 3 • 3 • 3 • 23 3 • 3 • 5 • 35

621 = 33 • 23 3 • 3 • 5 • 5 • 71,575 = 32 • 52 • 7

Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers

Today, the school’s baseball and soccer teams had

games. The baseball team plays every 7 days. The soccer team

plays every 3 days. When will the teams have games on the

same day again?

7, 14, 21, 28, 35, 42, . . . List the multiples of 7.

3, 6, 9, 12, 15, 18, 21, . . . List the multiples of 3.

The LCM is 21. In 21 days both teams will have games again.

Quick Check

Find the LCM of 16 and 36.

= 144 Multiply.

16 = 24

36 = 22 • 32Write the prime factorizations.

The LCM of 16 and 36 is 144.

LCM = 24 • 32 Use the greatest power of each factor.

Quick Check

Find the LCM of 5a4 and 15a.

5a4 = 5 • a4

15a = 3 • 5 • a Write the prime factorizations.

= 15a4 Multiply.

The LCM of 5a4 and 15a is 15a4.

LCM = 3 • 5 • a4 Use the greatest power of each factor.

Quick Check

Graph and compare the fractions in each pair.

is on the left, so < .38

b. – 13

, – 16

is on the right, so > .– 16

– 16

– 13

– 16

Quick Check

The softball team won of its games and the hockey

team won of its games. Which team won the greater fraction

of its games?

Step 1 Find the LCM of 7 and 9.7 = 7 and 9 = 32

LCM = 7 • 32 = 63Step 2 Write equivalent fractions with a denominator of 63.

6 • 97 • 97 • 79 • 7

=54634963

Step 3 Compare the fractions.5463

> >, so

The softball team won the greater fraction of its games.

Quick Check

Order , , and from least to greatest.37

3 • 127 • 12

1 • 214 • 21

2 • 283 • 28

The LCM of 7, 4, and 3 is 84.Use 84 as the common denominator.

< < , so < < .

Quick Check

Find the LCM of each pair of numbers.

1. 8, 6 2. 12, 16

3. Compare and order , – , and – from least to greatest. 316

– –< <316

Fractions and DecimalsFractions and Decimals

Without actually dividing, which expression is the largest?

6.1 ÷2.8 12.6 ÷ 1.1 9.305 ÷ 2.99

12.6 ÷ 1.1

Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2

(For help, go to Skills Handbook page 728.)

Write the decimals in order from least to greatest.

1. 2.41, 0.241, 24.1, 12.4

2. 1.030, 13.03, 1.300, 1.003

3. 0.1, 0.01, –0.1, –0.01

Solutions

1. 0.241, 2.41, 12.4, 24.1

2. 1.003, 1.030, 1.300, 13.03

3. –0.1, –0.01, 0.01, 0.1

The fuel tank of Scott’s new lawn mower holds gal of

gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank?

= 1 ÷ 2 = 0.5

Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank.12

Quick Check

Write each fraction as a decimal. State the block of

digits that repeats.

5 ÷ 6 = 0.83333 … Divide.

Place a bar over the digit that repeats.= 0.8356

= 0.83; the digit that repeats is 3.

7 ÷ 11 = 0.636363 … Divide.

= 0.63 Place a bar over the block of digits that repeats.

= 0.63; the block of digits that repeats is 63.7

11 Quick Check

Write the numbers in order, from least to greatest.

–0.8, , , 0.125312

–1.25 < –0.8 < 0.125 < 0.25 Compare the decimals.

3 ÷ 12 = 0.25Change the fractions to decimals.

–5 ÷ 4 = –1.25

From least to greatest, the numbers are , –0.8, 0.125, and .54

– 312

Quick Check

Write 1.72 as a mixed number in simplest form.

Keep the whole number 1.Write seventy-two hundredths as a fraction.

1.72 = 1 72100

Divide the numerator and denominator of the fraction by the GCF, 4.

72 ÷ 4100 ÷ 4= 1

Simplify.1825

1.72 = 1

Quick Check

Write 0.18 as a fraction in simplest form.

n Let the variable n equal the decimal.= 0.18

As a fraction in simplest form, 0.18 = .2

100n = 18.18 Because 2 digits repeat, multiply each side by 102, or 100.

= Divide each side by 99.

n 18 ÷ 999 ÷ 9

= Divide the numerator and denominator by theGCF, 9.

211= Simplify.

100n = 18.18n = 0.18–

99n = 18

The Subtraction Property of Equality lets you subtract the same value from each side of the equation. So, subtract to eliminate 0.18.

Quick Check

1. Order , – , 0.625, –0.35 from least to greatest.

2. Write as a decimal.

Write each decimal as a fraction or mixed number in simplest form.

3. 6.04 4. 0.21

–0.35, – , 0625,

Adding and Subtracting FractionsAdding and Subtracting Fractions

Five consecutive multiples of 6 have a sum of 270. What are they?

42, 48, 54, 60, 66

Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3

Find the LCM of each group of numbers or expressions.

1. 4, 8 2. 9, 18 3. 2n, 10

4. 3, 6, 9 5. 8, 5, 4 6. 10, n

Solutions

1. 4 = 22 2. 9 = 32 3. 2n = 2 • n

8 = 23 18 = 2 • 32 5 = 5

LCM = 23 LCM = 2 • 32 LCM = 2 • 5 • n

= 8 = 18 = 10n

4. 3 = 3 5. 8 = 23 6. 10 = 2 • 5

6 = 2 • 3 5 = 5 n = n

9 = 32 4 = 22 LCM = 2 • 5 • n

LCM = 2 • 32 LCM = 23 • 5 = 10n

= 18 = 40

Find each sum or difference. Simplify if possible.

4 + 29 Add the numerators.

= Simplify.

– =5b

12 – 5b Subtract the numerators.

7b= Simplify. Quick Check

Simplify each difference.

– 516

4 – 1824

= Use the Order of Operations to simplify.

–1424

= Simplify.

= –724

Simplify.

– =516

2 • 16 – 5 • yy • 16

Rewrite using a common denominator.

= 32 – 5y16y

Simplify.

– = 1 • 4 – 3 • 66 • 4

Quick Check

Suppose one day you rode a bicycle for 3 hours, and

jogged for 1 hours. How many hours did you exercise?14

You exercised for 4 hours.34

3 + = +1 14

Write mixed numbers as improper fractions.

= 7 • 4 + 5 • 22 • 4

= 28 + 108

Use the Order of Operations to simplify.

= 4 68

Write as a mixed number.

4 Simplify.

Quick Check

Find each sum or difference.

1. 2. 5 – 2

3. – – 4.

– 8c + 2114c

Multiplying and Dividing FractionsMultiplying and Dividing Fractions

Write each number as an improper fraction.

a. 4 b. 315

Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4

Write each mixed number as an improper fraction.

1. 2 2. 3 3. 1

4. 4 5. 7 6. 5

(For help, go to Skills Handbook page 738.)

Solutions

1. 2. 3.

2 = 2 +13

=6 + 1

=30 + 3

3 = 3 +3

= +3010

103310

1 = 1 +49

=9 + 4

Solutions (continued)

4. 5. 6.

4 = 4 +45

=20 + 4

7 = 7 +78

=56 + 7

5 = 5 +17

=35 + 1

Find • .23

• = 2 • 5 Multiply the numerators.

= 1021 Simplify.

Multiply the denominators.3 • 7

Quick Check

a. Find • .

• = 23

•1 1

12Divide the common factors.

Multiply.

Find • .

• = •5w

1Divide the common factors.

= Multiply.

Quick Check

Keesha’s desktop is a rectangle 3 ft long and 1 ft

wide. What is the area of her desktop?

A Area of a rectangle = length • width.= 3 • 1 12

• Write 3 and 1 as improper fractions, and .12

Multiply.= 214

Write as a mixed number.14= 5

The area of Keesha’s desk is 5 ft2.14

Quick Check

a. Find ÷ .35

710÷ = •

107 Multiply by the reciprocal of the divisor.

= • 351

Divide the common factors.

Multiply.

(continued)

b. Find ÷ .9

Multiply.

94q÷ = •

4q9 Multiply by the reciprocal of the divisor.

= • 1

Divide the common factors.278q

= 1 12

Write as a mixed number.

Quick Check

Find 4 ÷ (–3 ).12

4 ÷ (–3 ) = ÷ (– ) Change to improper fractions.278

= • (– ) Multiply by – , the reciprocal of – . 827

= • – Divide the common factors. 827

= – , or –1 Simplify.43

Quick Check

Simplify each expression.

3. 4.12g25

712•

1116÷ 15

– 732 , or 1

13, or 3

65 , or 1

Using Customary Units of MeasurementUsing Customary Units of Measurement

In 2002, a disc jockey broadcast her show for 27 h each week.

How many hours did she broadcast that year?

1,430 h

Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5

Find each product.

1. 4 • 2. 3 •

3. 1 • 4. 4 •

Solutions

4 • = •41

3 • = •161

1 • = •54

81 4 • = •

Choose an appropriate unit of measure. Explain your

choice.

a. weight of a hummingbird

b. length of a soccer field

Measure its weight in ounces because a hummingbird is very light.

Measure its length in yards because it is too long to measure in feet or inches and too short to measure in miles.

Quick Check

Use dimensional analysis to convert 68 fluid ounces to cups.

68 fl oz = •Use a conversion factor that changes fluid ounces to cups.

68 fl oz 1

1 c8 fl oz

= Divide the common factors and units.68 fl oz • 1 c 8 fl oz2

17 2 = c Simplify.

= 8 c Write as a mixed number.12

There are 8 c in 68 fl oz.12 Quick Check

Fred’s fruit stand sells homemade lemonade in 6 -pint

bottles for $1.99. Jill’s fruit stand stand sells homemade

lemonade in 3 -qt containers for the same price. At which stand

do you get more lemonade for your money?

Since 7 pints > 6 pints, you get more lemonade for your money

at Jill’s stand.

3 qt = qt • Use a conversion factor that changes quarts to pints

2 pt1 qt

= • Divide the common factors and units.1 2 pt

1 qt7 qt 2 1

= 7 pt Multiply.

Quick Check

Choose an appropriate unit of measure. Explain your choice.

1. weight of a dog

2. length of a football field

3. Use dimensional analysis to convert 1 mi to ft.

Pounds; dogs are too heavy to measure in ounces and too light to measure in tons.

Yards; it is too long to measure in feet or inches and too short to measure in miles.

7,920 ft

Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward

Use mental math to decide whether or not each pair of fractions is

yes, yes, no

equal: and , and , and3642

Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6

Order from least to greatest.

15, , ,

33, , – 3

7 , – 35

313, – , – 3

Solutions

1. 2.12

15, , ,

33, , – 3

7 , – 35

313, – , – 3

0.50, 0.3, 0.25, 0.20, 0.10

13, , ,

2.3, 1.6, 4.3, 1

0.43, 0.6, 0.23, 1

–0.43, –0.6, –0.23, 1

– 33 , – 3

13, – , –37

Your flight leaves the airport at 10:00 A.M. You must

arrive 2 hours early to check your luggage. The drive to the

airport takes about 90 minutes. A stop for breakfast takes about

30 minutes. It will take about 15 minutes to park and get to the

terminal. At what time should you leave home?

Move the hands of the clock to find the time you should leave home.

Write the starting time for each event.

(continued)

You should leave home at 5:45 A.M.

Quick Check

Solve by working backward.

1. Jason has a total of 212 sheep. Last spring, 52 lambs were born, but 9 died. How many sheep did he have before the lambs were born?

2. The bus picks up Tom every morning at 7:20 A.M. Tom walks his dog every morning before school for 36 minutes. It takes him 25 minutes to get ready for school and 7 minutes to walk to the bus stop. At what time does Tom need to wake up in order to get to the bus stop on time?

169 sheep

6:12 A.M.

Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting Fractions

Find the sum or difference. If possible, write each answer in lowest

terms.

a. + = ? b. 7 – 5 = ? 45

Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7

Find each sum or difference.

1. 1 – 2 2. 3 + 2

3. 5 – 3 4. –4 – 3

Solutions

1. 1 – 2 2. 3 + 434

= – = –198

78 – =

7 • 8 – 4 • 234 • 8

56 – 9232=

–3632=

348 + 44096=

78896=

5512+ =

29 • 12 + 8 • 558 • 12

= 8 = 82096

3. 5 – 3 4. –4 – 334

–225 – 17454=

–39954=

299+ =

–25 • 9 – 6 • 296 • 9

= –7 = –72154

184 – 10032=

258– =

23 • 8 – 4 • 254 • 8

= 2 = 22032

One school recycles about of its waste paper. The

student council set a goal of recycling of the school’s waste

paper by the end of the year. By how much does the school need

to increase its paper recycling to reach the goal?

Words plus is

Let n = the increase.

Equation + n =

fraction school recycles

theincrease

student goal

(continued)

+ n =13

– + n = – Subtract from each side.13

n = Use 3 • 4 as the common denominator.3 • 3 – 1 • 4 3 • 4

n = Use the Order of Operations.9 – 4 12

n = Simplify. 512

To meet the student council goal, the school needs to recycle more of its waste paper.

(continued)

Check Is the answer reasonable? The present fraction of paper

waste that is recycled plus the increase must equal the

goal. Since + = + = = , the answer

is reasonable.

Quick Check

Solve x – = .23

x – + = + Add to each side.23

x – = 23

x = Use 9 • 3 as the common denominator. 1 • 3 + 9 • 2 9 • 3

x = Use the Order of Operations.3 + 18 27

x = Divide the common factors.2127

x = Simplify.79

Quick Check

Solve q – 6 = –1 .12

q – 6 = 1 12

q = Use the Order of Operations.–16 + 65 10

q = Simplify.4910

–8 • 2 + 5 • 13 5 • 2

Use 5 • 2 as the common denominator.

q = – + Write mixed numbers as improper fractions.

q = 4 Write as a mixed number. 910

Quick Check

q – 6 + 6 = 1 + 6 Add 6 to each side.12

Solve the following equations.

1. + s = 2. k + = 3. u – =

4. – + g = 5. x – 2 = –4

1415–1

Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions

310Because of a mishap, of the oranges in a crate were ruined. If

the oranges originally weighed 50 lb, how many pounds of oranges

were ruined?

Find each product.

58 •

–1 •51–

14 •

Solutions

1. 2.•47

910 •

4 • 57 • 8

= 2056

910 •

53– =

9 • –510 • 3

=–4530

= –112

–96=

3. 4.–1 •51–

14 •

•54 =

–5 • –54 • 1–

9 • 22 • 3

Solve 7y = .13

• (7y) = • Multiply each side by , the reciprocal of 7.13

7y = 13

y = Simplify. 121

Quick Check

w = • Divide the common factors.1315

w = 2 Write as a mixed number.16

Solve w = .25

• w = • Multiply each side by , the reciprocal of .25

w = 25

w = Simplify.13 6

Quick Check

Solve – c = .49

– c = 49

c = Divide common factors.27 • 420 • 9

= – Simplify.35

– – c = – Multiply each side by – , the

reciprocal of – .

Quick Check

How many 2 -t trucks can you place on a rail car that

has a carrying capacity of 15 t?

Words times is

Let n = the number of trucks.

Equation 2 • n = 15

weight ofeach truck

the number of trucks

carrying capacity

(continued)

12 2 • n = 15

n = Divide common factors.2 • 155 • 1

n = 15 Write 2 as .52

• n = • 15 Multiply each side by , the reciprocal of .52

= 6 Simplify.

You can place 6 trucks on the rail car. Quick Check

– g =

Solve each equation.

1. 2. 3. 15 = –3 m= b

13–4

Powers of Products and QuotientsPowers of Products and Quotients

112 x 32 x 22 is the prime factorization for ?.

Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9

1. (22)3 2. (32)2 3. (15)4

4. (x3)6 5. (b2)5 6. (a7)4

Solutions

1. (22)3 2. (32)2 3. (15)4

= 22 • 3 = 32 • 2 = 15 • 4

= 26 = 34 = 120

= 64 = 81 = 1

4. (x3)6 5. (b2)5 6. (a7)4

= x3 • 6 = b2 • 5 = a7 • 4

= x18 = b10 = a28

Simplify (3z5)4.

(3z5)4 = 34 • (z5)4 Raise each factor to the fourth power.

= 34 • z5 • 4 Use the Rule for Raising a Power to a Power.

= 34 • z20 Multiply exponents.

= 81z20 Simplify.

Quick Check

a. Simplify (–3a)4.

b. Simplify –(3a)4.

(–3a)4 = (–3)4(a)4

= 81a4

–(3a)4 = (–1)(3a)4

= (–1)(3)4(a)4

= –81a4Quick Check

Find the area of a square with side length .x4

A = s2 s = length of a side

The area of the square is square units. x2

Quick Check

1. (12x2yz)2 2. (–4ab3)3 3. –(–2g2h)8

4. 5.u3

144x4y2z2 –64a3b9 –256g16h8

3q8r s7

243q r s35

Comparing and Ordering Rational Numbers

Documents

Transcript of Comparing and Ordering Rational Numbers

LESSON How do you compare and order rational numbers? Comparing and Ordering Rational Numbers 2.3.

Skills Practice Workbook - McGraw-Hill Educationglencoe.mheducation.com/sites/dl/free/0078740509/590597/m3_nat... · Skills Practice Workbook ... 2-2 Comparing and Ordering Rational

2.1 Comparing and Ordering Rational Numbers · Explore Rational Numbers 1. How are the rational numbers in the table on page 46 related. Explain your reasoning. 2. a) Choose a rational

Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers.

2.1 Comparing and Ordering Rational Numbers · You can write the numbers in decimal form.-1.2 4__ 5 ... 48 MHR • Chapter 2. ... 2.1 Comparing and Ordering Rational Numbers • MHR

Comparing and Ordering Fractions.ppt

Holt CA Course 1 3-6 Comparing and Ordering Rational Numbers Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview.

Lesson 5-8 Pages 227-231 Comparing and Ordering Rational Numbers Lesson Check 5-7.

A. Rational Numbers Math 9. Outcomes N9.2 N9.2 Demonstrate understanding of rational numbers including: comparing and ordering relating to other.

UNIT 1 LESSON 1 Comparing and Ordering Rational Numbers …2... · UNIT 1 LESSON 1 Comparing and Ordering Rational Numbers (2.1 in textbook) ... 1" ’ −2 # & 3 > - order of operations

Fractions comparing ordering

Comparing Numbers & Ordering Numbers

COMPARING, ORDERING AND ROUNDING NUMBERS€¦ · COMPARING & ORDERING When comparing or ordering numbers, you must line up the digits by their place or decimal point. The number with

Comparing Requirements Management Tools – IBM Rational ...

Comparing and Ordering Numbers - ELEMENTARY MATH€¦ · Number and Operations in Base Ten 3 Comparing and Ordering Numbers Objective Comparing and ordering numbers requires a deep

Course 3 2-2 Comparing and Ordering Rational Numbers 2-2 Comparing and Ordering Rational Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of.

Estimating , Comparing and Ordering Decimals

Comparing & Ordering Numbers2

Integers: Comparing and Ordering Today’s Objective I will learn how to compare and order rational numbers.

Comparing & Ordering Fractions