NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley [email protected]...

Copyright Steve Yurek March 17, 2014 1

NHTM Spring ConferenceNashua, NH

March 17, 2014

The Harmonic Mean: Overlooked & Undervalued,

yet Way Cool!Steve Yurek Lesley University [email protected], MA


Napoli’s Yard Sale

How much do you want for that set of 2004 Red Sox Baseball Cards?$5,000I’ll give you $1,000.$4,000$2,000$3,000OK, it’s a deal



Which mean is at work here?The Arithmetic Mean(5,000 + 1,000)/23,000Let’s see what the arithmetic mean looks like geometrically.



Which mean is at work here?The Arithmetic Mean(5,000 + 1,000)/23,000Let’s see what the arithmetic mean looks like geometrically.Arithmetic Mean in Sketchpad


Thanksgiving ChairsThe ratio of the height of the adult chairs in my Mom’s house to that of the kids’ chairs, is the same as that of the kids’ chairs compared to that of the doll’s chairs. If the adult chairs are 49” tall and the dolls’ chairs are 16” tall, then what is the height of the kids’ chairs.


Thanksgiving Chairs249

h = 49 16 16

7 4 28

h

h

h h


Thanksgiving Chairs2h = 49

49

16 16

7 4 28

h

h

h h



h =

7

49 16 16

4 28h

h

h

h


Thanksgiving Chairs

249 h = 49 16

16 7 4

28

h

h

h

h


Thanksgiving Chairs

249 h = 49 16

16 7 4

28

h

hh

h



h = 49 16 16

7 4

28

h

hh

h

Geometric Mean in Sketchpad


Let’s look at this often miscalculated problem

For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.

For the next 10 mi. it registered 100 mpg. What was the Mini’s gas mileage

for the entire trip?Your students would probably answer ……

55 mpg






55 mpg




For the next 10 mi. it registered 100 mpg. What was the Mini’s average gas mileage


55 mpg






55 mpg


But we know better.



1 gallon of fuel used

For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used

Total distance traveled = 20 milesTotal fuel used = 1.1 gallons

20/1.1 = 18.18 mpg






20/1.1 = 18.18 mpg






20 / 1.1 = 18.18 mpg


But let’s look at this one a little differentlyLet R = Rate for the 1st D milesLet r = Rate for the 2nd D milesThen T for the 1st D miles = D/R and t for the 2nd D miles = D/rTotal Rate = Total Distance / Total Time

2D 2DR’ = -------------- = ------------------

D D Dr + DR--- + --- --------------- R r Rr


But let’s at this one a little differentlyLet R = Rate for the 1st D milesLet r = Rate for the 2nd D milesThen T for the 1st D miles = D/R and t for the 2nd D miles = D/rTotal Rate = Total Distance / Total Time

2D 2DR’ = -------------- = ------------------

D D Dr + DR--- + --- --------------- R r Rr


But let’s at this one a little differentlyLet R = Rate for the 1st D milesLet r = Rate for the 2nd D milesThen for the 1st D miles T = and for the 2nd D miles t =

Total Rate = = =



Total Rate = = =


Total Rate = R’ =

R’ = R’ =

The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get


Total Rate = R’ =

R’ = R’ =



So if D is irrelevant, then let’s choose D = 1That makes

1 1, (the sum of the reciprocals)

1(2)2 2R' = meaning that R

T + t

'

=

1 1 1 1 12

)(

R r

R r R r



(the sum of the reciprocals)

1(2)2 2R' = meaning that R'

1

1 1 1 1 12

1T + t = ,

)(

R

R r

r

R r



1(2)2 2R' = meaning that R'

1

1 1T + t = , (the sum of the reciproc

1 1 1

al

12

s)

)(R r

R r

R r



1 1T + t = , (the sum of the reciprocals)

R

1(2)2 2 meaning that R'

1 1 1 1 12

' = )(

R

r

r

R r R




2R

1(2)

2meaning that R'' = 1

2

1 1 1 1)(

R

R r

r

R r




2R

1(2)

2meaning that R'' =

2

1

1 11 1 )(

R r

R r R r




(2)2R' = meaning tha

12t R' =

1 1 1 112

)(

R r

R r R r



1

2

1 R' =

1 1( )R r



1

2

1 R' =

1 1( )R r

the reciprocal of the (arithmetic) mean

of the reciprocals - and this is precisely the definition

of the Harmonic Mean

2Wh

Which means that the "average"

ich can be simplified as R' =

Rate

is

rR

r R



1

2

1 R' =

1 1( )R r

of the (arithmetic) mean


of the

Which means

Harmonic M

that the "average" Rate

is the r

ean

2Which can be simplified as R'

eciprocal

= rR

r R



1

2

1 R' =

1 1( )R r

of the reciprocals - and this is precis

Which means that the "average" Rate

is t

ely the definition


2Wh

he reciprocal of the

ich can be simplified

(arithmetic

as R' =

) mean

rR

r R



1

2

1 R' =

1 1( )R r


is the reciprocal of the (arithmetic) mean



2Which can be simplified as

R' = rR

r R



1

2

1 R' =

1 1( )R r




of the Harm

2Which can be simplified as R' =

onic Mean

rR

r R



1

2

1 R' =

1 1( )R r





Which can be simplified as 2

R' = rR

r R



1

2

1 R' =

1 1( )R r





2Which can be simplified as R' =

rR

r R


1. What is the sum of the reciprocals of 10 & 100?

2. What is the mean of these reciprocals?

3. What is the reciprocal of the answer from part 2?

It’s a bit clumsy to do all this.Hooray for the formula




55 mpg









55 mpg

11

100









55 mpg

11

10011

200









55 mpg

11

10011

200

200

11









55 mpg

11

10011

200

218

11









55 mpg


Let’s look at a similar, yet different one




55 mpg



For the first 10 miles of a trip, Pete’s average speed was 20 mph

What was the Mini’s gas mileage for the entire trip?

Your students would probably answer ……55 mpg




For the next 10 miles it was 100 mph. What was the Mini’s gas mileage


55 mpg




For the next 10 miles it was 100 mph. What was the Mini’s average speed


55 mpg






55 mpg






60 mph


But we know better.


For the first 10 miles of a trip, Pete’s average speed was 20 mph.




20/1.1 = 18.18 mpg


For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes



20/1.1 = 18.18 mpg



For the next 10 miles it was 100 mph.1/10 gallon of fuel used


20/1.1 = 18.18 mpg



For the next 10 miles it was 100 mph.It took 6 minutes


20/1.1 = 18.18 mpg




Total time = 36 minutes (.6 hours)Total fuel used = 1.1 gallons

20/1.1 = 18.18 mpg




Total time = 36 minutes (.6 hours)Total Distance = 20 miles

20 / 1.1 = 18.18 mpg





Overall Speed =20 mi/.6 hr or





Overall Speed =20 mi/.6 hr or Overall Speed = 33 1/3 mph


Use the definition to verify your answer• Sum of reciprocals (1/20 + 1/100) = 3/50• Average of previous answer 3/100• Reciprocal from previous answer 33 1/3

Now use the formula with a = 20 and b = 100


Use the definition to verify your answer• Sum of reciprocals (1/20 + 1/100) = 3/50• Average of previous answer 3/100• Reciprocal from previous answer




2(20)(102 0H(M)

) 133

0

1=

20 0 3

ab

a b




2 2(20)(100)H(M) =

133

320 100

ab

a b




2 2(20)(100) 1H(M) = 33

20 100 3

ab

a b




2 2(20)(100) 1H(M) = 33

20 100 3

ab

a b

Harmonic Mean in Sketechpad


Whenever we are taking an average of an average (as long as the base remains constant), the harmonic mean will save

lots of time.Proving that will take more time than we have here, but we can demonstrate this concept.


Whenever we are taking an average of an average (as long as the base remains constant), the harmonic mean will save

lots of time.Even if there are more than 2

components


Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisor’s printer. What was the average cost of a single cartridge?


We can certainly determine how many of each type she bought and then divide that number into $600 ($15 15/19)

Does this correspond with the harmonic mean of 10, 20 & 25?



Does this correspond with the harmonic mean of 10, 20 & 25? Using the Definition?



Does this correspond with the harmonic mean of 10, 20 & 25?


Can we generate a formula for the harmonic mean of 3 numbers a, b & c?



arithmetic mean

reciprocal

3

1

3

1 1 bc ac ab

abcbc ac ab

abcabc

bc ac

c

a

b

b

a



arithmetic mean

reciprocal

3

1

3

1 1 bc ac ab

bc ac aba b c a

abcabc

bc ac

b

a

c

b



arithmetic mean

reciprocal

3

1

3

1 1 bc ac

bc ac ab

abcabc

b

ab

c ac

a c

b

b ab

a

c



arithmetic mean

reciprocal

1 1 1

33abc

bc ac

bc

ab

ac ab

a b c abcbc ac ab

abc



arithmetic mean

reciprocal

1 1 1

33

bc ac ab

a b c abcbc ac ab

abcabc

bc ac ab


Can we generate a formula for the harmonic mean of more than 3 numbers?



1 2 3

1 1 1 1... ???

na a a a



1 2 3

1 1 1 1... ???

Check out the final slidesna a a a


How do the 3 means

compare with one another?


The trichotomy law states that when comparing any 2 numbers, a & b, then there are 3 and only 3 possible outcomes:

a > b

a < b

a = b



a > b

a < b

a = b


So will the 3 means ever all be equal?They will if the original numbers are equal.



2

2

1( )

2

( , )

2 2( , )

2

( , ) n n n

GM n n n n n n

n n nHM n

AM n n

n nn n n



2

2

( , )

1( , ) ( )

2, )

2

2(

2

n

GM n n n n n n

n n nHM n n n

n n n

AM n n n n



2

2

( , )

2 2( , )

2

1( , ) ( )

2

GM n n n n n n

n n nHM n n n

n n n

AM n n n n n



2

2

1( , ) ( )

2 2( ,

2

, )

2

(

)

AM n n n n n

GM n n n n n n

n n nHM n n n

n n n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

n n

n n nH

AM n n n n n

GM n n

M n n nn n n

n n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

AM n n n n n

GM n n n n n n

n n nHM n n n

n n n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

AM n n n n n

GM n n n n n

n n nHM n n n

n n

n

n



2

2

1( , ) ( )

2

( , )

2 2,

2( )

AM n n n n n

GM n n n n n

n n nn

n n

n

Mn

H n n



2

2

1( , ) ( )

2

( , )

22,

2( )

AM n n n n n

GM n n n n n n

n nHM n n

n n

nn

n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

AM n n n n n

GM n n n n n n

n n nHM n n

n nn

n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

AM n n n n n

GM n n n n n n

n n nHM n n n

n n n



2

2

1( , ) ( )

2

( , )

2 2( , )

2

AM n n n n n

GM n n n n n n

n n nHM n n n

n n n

Back to Sketchpad


But what if a ≠b? (let’s keep them both >0)Let’s assume that AM(a,b) < GM(a,b)



2

2

2 2

2 2

2 2

2

2

2

4

2 4

2 0

( ) 0

Impossible, AM(a,b)>GM(a )

2

,b

a bab

a ab bab

a ab b ab

a

a ba

b

a b

b

ab



2 2

2

2 2

2 2

2

2

2

4

2 4

2 0

( ) 0

Impossible, AM(a,b)>GM(a,b)

2

2

a ab bab

a ab b ab

a

a bab

a bab

ab b

a b



2

2

2

2

2 2

2

2

2

2 4

2 0

( ) 0

Impossible, AM(a,

2

2

b)>GM(a,b

2

4

)

a ab b ab

a

a bab

a bab

a ab b

b b

ab

a

a b



2 2

2

22

2 2

2 2

2 0

( ) 0

Impossible

2

2

, AM(a,b)>GM(a

4

2

,

2

4

b)

a

a bab

a bab

a ab bab

a ab b a

ab b

a b

b



2

2

2 2

2

2

2

2 2

( ) 0

Impossible, AM(

2

2

2

4

2 4

a,b)>GM

2

(a,b)

0

a bab

a bab

a ab bab

a ab b ab

a a

b

b

a

b



2

2

2 2

2 2

2 2

2

Impossible, AM(a,b)

2

2

2

4

2 4

>GM(a,

2

0

b)

0

( )

a bab

a bab

a ab bab

a ab b ab

a ab b

a b



2

2

2 2

2 2

2 2

2

2

2

2

4

2 4

2 0

( ) 0

Impossible, AM(a,b)>GM(a,b)

a bab

a bab

a ab bab

a ab b ab

a ab b

a b


But what if a ≠b? (let’s keep them both >0)Let’s also assume that GM(a,b) < HM(a,b)



2 2

2 2

3 2 2 3 2 2

3 2 2 3

2 2

2

4

2

2 4

2 0

( 2 ) 0

( ) 0

Impossible so GM(a,b) > HM(a

2

,b)

a bab

a ab b

a b a b ab a b

a b a b

a

ab

ab a ab b

a

ba

b

b

b

a b

a



3 2 2 3 2 2

3 2 2 3

2 2

2

2

2

2

2

2 4

2 0

( 2 ) 0

( ) 0

Impossible so GM(a,b) > HM(a,b)

2 4

2

a b a b ab a b

a b a b ab

ab a ab b

ab a bab ab

a b a a b

ab a

b

b



3 2 2 3

2 2

2 2

3 2 2 3 2 2

2 2

2

2 0

( 2

2 4

2

) 0

( ) 0

Impossible so GM(a,b) > HM(a,b

4

)

2

a b a b ab

ab a a

ab a bab ab

a b a ab b

a b

b b

ab a b

a b ab a b



2 2

2 2

3 2 2 3 2 2

3 2 2 3

2 2

2

2 4

( 2 ) 0

( ) 0


2

2 4

2 0

ab a a

ab a bab ab

a b a ab b

a b a b ab a b

a b a b ab

b b

ab a b



2 2

2 2

3 2 2 3 2 2

3 2 2 3

2 2

2

2 4

2

2 4

2 0

( 2

( ) 0

Impossible so GM(a,b)

)

> H ,b

0

M(a )

ab a bab ab

a b a ab b

a b a b ab a b

a b a b ab

ab a ab b

ab a b



2 2

2 2

3 2 2 3 2 2

3 2 2 3

2 2

2

2 4

2

2 4

2 0

( 2

Impossible so GM(a,b) > HM(a,

) 0

)

) 0

(

b

ab a bab ab

a b a ab b

a b a b ab a b

a b a b ab

ab a ab b

ab a b



2 2

2 2

3 2 2 3 2 2

3 2 2 3

2 2

2

2 4

2

2 4

2 0

( 2 ) 0

( ) 0


ab a bab ab

a b a ab b

a b a b ab a b

a b a b ab

ab a ab b

ab a b


So , for any a,b where a and b both >0AM(a,b) > GM(a,b)

&GM(a,b) > HM(a,b)

thenAM(a,b) > GM(a,b) > HM(a,b)



&GM(a,b) > HM(a,b)




&GM(a,b) > HM(a,b)


Sketchpad – One Last Time


How about this one?


Really ----- Last time for Sketchpad

How about this one?


80’

50’30’


80’

50’30’

E


80’

50’30’

E

?


80’

50’30’

E

18.75


80’

50’30’

E

18.75

As it turns out, this is very cool --- watch this


The Harmonic Mean turns up in many fascinating places

Can you reconcile its place in the telephone problem?What is the harmonic mean of 30 & 50?It’s 37.5 What the…..???How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean?


THERE IS, BUT WHY THE “ADAPTATION”?


THERE IS, BUT WHY THE “ADAPTATION”?

THAT’S ONE THING I’LL LEAVE YOU WITH TODAY


H2

3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

M(a,b) = ab

a babc

bc ac acabcd

bcd acd abd abc

abcde

bcde acde abde abce abcd


3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e

2

)=

5

HM(a,b) = a b

abc

bc ac acabcd

bcd acd abd abc

abcde

bcde acde abde abce ab

b

d

a

c


3HM(a,b,c) =

4HM(a,b,c,d) =

H

2HM(a,b)

M(

=

a,b,c,d

,e)=

5

abc

bc ac acabcd

bcd acd abd abc

abcde


b

c

a

d

ab


2HM(a,b) =

HM(a,b,c) = 3

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

abc

bc ac acabcd

bcd acd abd abc

abcde


b

c

a

d

ab


2HM(a,b) =

3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

bc ac acabcd

bcd acd abd abc

abcde

bcde acde abde abce abc

ab

a babc

d


4HM(a,b,c,d) =

HM(a,b,c,d,e)=

2HM(a,b) =

3HM(a,b,c) =

5

abcd

b

a

cd acd abd abc

abcde

bc

b

a babc

bc ac a

de acde abde abce ab

b

cd


4

HM(a,b,c,d

2HM(a,b)

,e)

=

3HM(a,b,c) =

HM(a,b,c,d) =

=

5

abcd

b

a

cd acd abd abc

abcde

bc

b

a babc

bc ac a

de acde abde abce ab

b

cd


2HM(a,b) =

3HM(a,b,c) =

HM(a,b,c,d,e)=

5

4HM(a,b,c,d) =

b

ab

a babc

bc ac abab

cd acd abd abc

abcde


cd


2H

HM(a,b,c,d,e

M(a,b) =

3HM(a,b,c) =

4HM(a,b,c,d) =

)=

5

ab

a babc

bc ac ababcd

b

abcde

bc

cd acd abd a

de acde abde abc a

bc

e bcd


2HM(a,b) =

3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

ab

a babc

bc ac ababcd

b

abcde

bc

cd acd abd a

de acde abde abc a

bc

e bcd


2HM(a,b) =

3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

ab

a babc

bc ac ababcd

b

bcd

cd acd abd a

e acde abde abce

bc

abcde

abcd


2HM(a,b) =

3HM(a,b,c) =

4HM(a,b,c,d) =

HM(a,b,c,d,e)=

5

ab

a babc

bc ac ababcd

bcd acd abd abc

abcde



1 2 3 n

1 2 3

HM(a ,a ,a ,...,a )=

...

i

n

a

n a a a a


1 2 3 n

1 2 3

HM(a ,a ,a ,...,a )=

...

i

n

a

n a a a a

j=1

n

π

Σi=1

n aj


Or if we use the definition

itself


1 2 3 n

1

HM(a ,a ,a ,...,a )=

1

1 1n

i in a


So Let’s Solve some

Problems


Last season Cody got 60 hits for a Fenway batting average of .400. For his AWAY games he also got 60 hits, but his batting average was only .300. What was Cody’s batting average for the entire season?


Jack’s company will award an end-of-the-year bonus to any employee whose total yearly sales represent at least 10% of the company’s sales. During the last fiscal year, Jack’s sales were a consistent $50,000 for each quarter. However his quarter 1 sales represented 10% of the company’s sales. For quarter 2, his sales represented 6% of the company’s sales. For quarters 3 & 4 they represented 8% and 30% respectively. Did Jack earn his bonus? Defend your answer.


In a recent county election poll, voters in each of the 5 districts were asked whom they support. For each of the 5 counties, 473 voters expressed their support for Mr. James W. Beam: These results represented 19% of voters in District A, and 25%, 18%, 57% and 31% in the other 4 districts.Mr. Beam claims to have the support of 30% of the county. Is he correct? If no, then by how much is he off?


Ollie Charles Dickens (known to his friends as OCD) will have a great day if he can average exactly 60 mph on his way to work. The 1st 6.3 miles are along back roads, while the final 6.3 miles is traveled on “straight as an arrow” freeway. School buses, wet leaves and a touch of ice slowed the back road portion to only 30 mph. How fast must Mr. OCD travel on the freeway, so that he can have a great day?


Each day Violet sells 72 each of 9 fruits: Apples – Oranges – Pears – Plums – Kiwi – Nectarine – Pomegranate – Peaches & Tangellos and respectively they represent 3/4 , 2/3, 1/2, 3/5, 3/7, 8/9, 6/13, 9/11 and 1/2 of the amount of each fruit that she bought. At the end of the day she donates any unsold fruit to a food pantry. What percent of her daily fruit purchase goes to the food pantry?


The EPA has mandated that, by 2015, the total % of all models of all vehicles produced by any manufacturer must get at least 35 mpg. Four of the 5 models of the Great Wall Auto Corp have tested to get 30 mpg, 38 mpg, 42 mpg and 25 mpg. If they all use the same test track, what must the fuel mileage of the 5th model be in order for GWAC to be allowed to manufacture automobiles in the US?


Periodically the water in Sparkletown is tested for impurities. Recently 7 samples were tested: Four 25 gallon samples from spots near each of the four corners and three 50 gallon samples from varying spots in the middle of the reservoir. The corner samples registered 84%, 87%, 89% and 85% pure, while the center samples had readings of 96%, 99% and 93% pure. On the state report, Sparkletown Public Health reported that “the purity rate of the 250 gallons sampled was ________ percent.


Periodically the water in Sparkletown is tested for impurities. Recently 7 samples were tested: Each of 4 samples from spots near each of the four corners revealed 25 gallons of pure water and each of 3 samples from varying spots in the middle of the reservoir yielded 50 gallons of pure water. The corner samples registered 84%, 87%, 89% and 85% pure, while the center samples had readings of 96%, 99% and 93% pure. On the state report, Sparkletown Public Health reported that “the purity rate of the 250 gallons sampled was ________ percent.


Two telephone poles are “a” feet and “b” feet tall, and they are positioned so that they are “F” feet apart. When guide wires from the tops of each extend to the bases of the others, they intersect at a specific point “E”. How high is “E” above the ground?


If Thelma can paint a house, by herself, in 8 hours, and Louise can paint the same house in only 5 hours, then how long will it take for them to paint the house if they work together?


Harry can mow a lawn all by himself in H minutes, while David can mow the same lawn in D minutes. If they both work together, then how long will it take for them to mow the same lawn?


Thank You All For Being Here

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NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley [email protected]...

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Transcript of NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley [email protected]...