Post on 28-Mar-2015
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 syurek@lesley.eduCambridge, MA
Copyright Steve Yurek March 17, 2014 2
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
Copyright Steve Yurek March 17, 2014 3
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
Copyright Steve Yurek March 17, 2014 4
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
Copyright Steve Yurek March 17, 2014 5
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
Copyright Steve Yurek March 17, 2014 6
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
Copyright Steve Yurek March 17, 2014 7
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
Copyright Steve Yurek March 17, 2014 8
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
Copyright Steve Yurek March 17, 2014 9
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
Copyright Steve Yurek March 17, 2014 10
Napoli’s Yard Sale
Which mean is at work here?The Arithmetic Mean(5,000 + 1,000)/23,000Let’s see what the arithmetic mean looks like geometrically.
Copyright Steve Yurek March 17, 2014 11
Napoli’s Yard Sale
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
Copyright Steve Yurek March 17, 2014 12
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.
Copyright Steve Yurek March 17, 2014 13
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.
Copyright Steve Yurek March 17, 2014 14
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.
Copyright Steve Yurek March 17, 2014 15
Thanksgiving Chairs249
h = 49 16 16
7 4 28
h
h
h h
Copyright Steve Yurek March 17, 2014 16
Thanksgiving Chairs2h = 49
49
16 16
7 4 28
h
h
h h
Copyright Steve Yurek March 17, 2014 17
Thanksgiving Chairs249
h =
7
49 16 16
4 28h
h
h
h
Copyright Steve Yurek March 17, 2014 18
Thanksgiving Chairs
249 h = 49 16
16 7 4
28
h
h
h
h
Copyright Steve Yurek March 17, 2014 19
Thanksgiving Chairs
249 h = 49 16
16 7 4
28
h
hh
h
Copyright Steve Yurek March 17, 2014 20
Thanksgiving Chairs249
h = 49 16 16
7 4
28
h
hh
h
Geometric Mean in Sketchpad
Copyright Steve Yurek March 17, 2014 21
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
Copyright Steve Yurek March 17, 2014 22
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
Copyright Steve Yurek March 17, 2014 23
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
Copyright Steve Yurek March 17, 2014 24
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 average gas mileage
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 25
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 average gas mileage
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 26
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 average gas mileage
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 27
But we know better.
Copyright Steve Yurek March 17, 2014 28
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 29
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 30
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 31
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 32
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 33
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 34
For the first 10 miles of a trip, Pete’s gas mileage on his Harley Mini was 10 mpg.
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
Copyright Steve Yurek March 17, 2014 35
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
Copyright Steve Yurek March 17, 2014 36
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
Copyright Steve Yurek March 17, 2014 37
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
Copyright Steve Yurek March 17, 2014 38
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 = = =
Copyright Steve Yurek March 17, 2014 39
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 = = =
Copyright Steve Yurek March 17, 2014 40
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 = = =
Copyright Steve Yurek March 17, 2014 41
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 = = =
Copyright Steve Yurek March 17, 2014 42
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 = = =
Copyright Steve Yurek March 17, 2014 43
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 = = =
Copyright Steve Yurek March 17, 2014 44
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 = = =
Copyright Steve Yurek March 17, 2014 45
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 = = =
Copyright Steve Yurek March 17, 2014 46
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 = = =
Copyright Steve Yurek March 17, 2014 47
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 = = =
Copyright Steve Yurek March 17, 2014 48
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 49
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 50
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 51
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 52
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 53
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 54
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 55
Total Rate = R’ =
R’ = R’ =
The distance has absolutely no bearing on the answer. Let r = 10 & R = 100 --- See what you get
Copyright Steve Yurek March 17, 2014 56
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
Copyright Steve Yurek March 17, 2014 57
So if D is irrelevant, then let’s choose D = 1That makes
(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
Copyright Steve Yurek March 17, 2014 58
So if D is irrelevant, then let’s choose D = 1That makes
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
Copyright Steve Yurek March 17, 2014 59
So if D is irrelevant, then let’s choose D = 1That makes
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
Copyright Steve Yurek March 17, 2014 60
So if D is irrelevant, then let’s choose D = 1That makes
1 1T + t = , (the sum of the reciprocals)
2R
1(2)
2meaning that R'' = 1
2
1 1 1 1)(
R
R r
r
R r
Copyright Steve Yurek March 17, 2014 61
So if D is irrelevant, then let’s choose D = 1That makes
1 1T + t = , (the sum of the reciprocals)
2R
1(2)
2meaning that R'' =
2
1
1 11 1 )(
R r
R r R r
Copyright Steve Yurek March 17, 2014 62
So if D is irrelevant, then let’s choose D = 1That makes
1 1T + t = , (the sum of the reciprocals)
(2)2R' = meaning tha
12t R' =
1 1 1 112
)(
R r
R r R r
Copyright Steve Yurek March 17, 2014 63
So if D is irrelevant, then let’s choose D = 1That makes
1 1T + t = , (the sum of the reciprocals)
(2)2R' = meaning tha
12t R' =
1 1 1 112
)(
R r
R r R r
Copyright Steve Yurek March 17, 2014 64
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
Copyright Steve Yurek March 17, 2014 65
So if D is irrelevant, then let’s choose D = 1That makes
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
Copyright Steve Yurek March 17, 2014 66
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
of the (arithmetic) mean
of the reciprocals - and this is precisely the definition
of the
Which means
Harmonic M
that the "average" Rate
is the r
ean
2Which can be simplified as R'
eciprocal
= rR
r R
Copyright Steve Yurek March 17, 2014 67
So if D is irrelevant, then let’s choose D = 1That makes
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
of the Harmonic Mean
2Wh
he reciprocal of the
ich can be simplified
(arithmetic
as R' =
) mean
rR
r R
Copyright Steve Yurek March 17, 2014 68
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
Which means that the "average" Rate
is the reciprocal of the (arithmetic) mean
of the reciprocals - and this is precisely the definition
of the Harmonic Mean
2Which can be simplified as
R' = rR
r R
Copyright Steve Yurek March 17, 2014 69
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
Which means that the "average" Rate
is the reciprocal of the (arithmetic) mean
of the reciprocals - and this is precisely the definition
of the Harm
2Which can be simplified as R' =
onic Mean
rR
r R
Copyright Steve Yurek March 17, 2014 70
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
Which means that the "average" Rate
is the reciprocal of the (arithmetic) mean
of the reciprocals - and this is precisely the definition
of the Harmonic Mean
Which can be simplified as 2
R' = rR
r R
Copyright Steve Yurek March 17, 2014 71
So if D is irrelevant, then let’s choose D = 1That makes
1
2
1 R' =
1 1( )R r
Which means that the "average" Rate
is the reciprocal of the (arithmetic) mean
of the reciprocals - and this is precisely the definition
of the Harmonic Mean
2Which can be simplified as R' =
rR
r R
Copyright Steve Yurek March 17, 2014 72
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
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
Copyright Steve Yurek March 17, 2014 73
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
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
11
100
Copyright Steve Yurek March 17, 2014 74
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
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
11
100
Copyright Steve Yurek March 17, 2014 75
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
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
11
10011
200
Copyright Steve Yurek March 17, 2014 76
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
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
11
10011
200
Copyright Steve Yurek March 17, 2014 77
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
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
11
10011
200
200
11
Copyright Steve Yurek March 17, 2014 78
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
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
11
10011
200
218
11
Copyright Steve Yurek March 17, 2014 79
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
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
Copyright Steve Yurek March 17, 2014 80
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
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
Copyright Steve Yurek March 17, 2014 81
Let’s look at a similar, yet different one
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
Copyright Steve Yurek March 17, 2014 82
Let’s look at a similar, yet different one
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
Copyright Steve Yurek March 17, 2014 83
Let’s look at a similar, yet different one
For the first 10 miles of a trip, Pete’s average speed was 20 mph
For the next 10 miles it was 100 mph. What was the Mini’s gas mileage
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 84
Let’s look at a similar, yet different one
For the first 10 miles of a trip, Pete’s average speed was 20 mph
For the next 10 miles it was 100 mph. What was the Mini’s gas mileage
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 85
Let’s look at a similar, yet different one
For the first 10 miles of a trip, Pete’s average speed was 20 mph
For the next 10 miles it was 100 mph. What was the Mini’s average speed
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 86
Let’s look at a similar, yet different one
For the first 10 miles of a trip, Pete’s average speed was 20 mph
For the next 10 miles it was 100 mph. What was the Mini’s average speed
for the entire trip?Your students would probably answer ……
55 mpg
Copyright Steve Yurek March 17, 2014 87
Let’s look at a similar, yet different one
For the first 10 miles of a trip, Pete’s average speed was 20 mph
For the next 10 miles it was 100 mph. What was the Mini’s average speed
for the entire trip?Your students would probably answer ……
60 mph
Copyright Steve Yurek March 17, 2014 88
But we know better.
Copyright Steve Yurek March 17, 2014 89
For the first 10 miles of a trip, Pete’s average speed was 20 mph.
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
Copyright Steve Yurek March 17, 2014 90
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
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
Copyright Steve Yurek March 17, 2014 91
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.1/10 gallon of fuel used
Total distance traveled = 20 milesTotal fuel used = 1.1 gallons
20/1.1 = 18.18 mpg
Copyright Steve Yurek March 17, 2014 92
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.It took 6 minutes
Total distance traveled = 20 milesTotal fuel used = 1.1 gallons
20/1.1 = 18.18 mpg
Copyright Steve Yurek March 17, 2014 93
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.It took 6 minutes
Total time = 36 minutes (.6 hours)Total fuel used = 1.1 gallons
20/1.1 = 18.18 mpg
Copyright Steve Yurek March 17, 2014 94
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.It took 6 minutes
Total time = 36 minutes (.6 hours)Total Distance = 20 miles
20 / 1.1 = 18.18 mpg
Copyright Steve Yurek March 17, 2014 95
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.It took 6 minutes
Total time = 36 minutes (.6 hours)Total Distance = 20 miles
Overall Speed =20 mi/.6 hr or
Copyright Steve Yurek March 17, 2014 96
For the first 10 miles of a trip, Pete’s average speed was 20 mph.It took 30 minutes
For the next 10 miles it was 100 mph.It took 6 minutes
Total time = 36 minutes (.6 hours)Total Distance = 20 miles
Overall Speed =20 mi/.6 hr or Overall Speed = 33 1/3 mph
Copyright Steve Yurek March 17, 2014 97
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
Copyright Steve Yurek March 17, 2014 98
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
Copyright Steve Yurek March 17, 2014 99
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
Copyright Steve Yurek March 17, 2014 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 33 1/3
Now use the formula with a = 20 and b = 100
Copyright Steve Yurek March 17, 2014 101
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
Copyright Steve Yurek March 17, 2014 102
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
Copyright Steve Yurek March 17, 2014 103
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
Copyright Steve Yurek March 17, 2014 104
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
Copyright Steve Yurek March 17, 2014 105
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
Copyright Steve Yurek March 17, 2014 106
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
2(20)(102 0H(M)
) 133
0
1=
20 0 3
ab
a b
Copyright Steve Yurek March 17, 2014 107
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
2 2(20)(100)H(M) =
133
320 100
ab
a b
Copyright Steve Yurek March 17, 2014 108
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
2 2(20)(100) 1H(M) = 33
20 100 3
ab
a b
Copyright Steve Yurek March 17, 2014 109
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
2 2(20)(100) 1H(M) = 33
20 100 3
ab
a b
Harmonic Mean in Sketechpad
Copyright Steve Yurek March 17, 2014 110
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.
Copyright Steve Yurek March 17, 2014 111
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
Copyright Steve Yurek March 17, 2014 112
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?
Copyright Steve Yurek March 17, 2014 113
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?
Copyright Steve Yurek March 17, 2014 114
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?
Copyright Steve Yurek March 17, 2014 115
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?
Copyright Steve Yurek March 17, 2014 116
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?
Copyright Steve Yurek March 17, 2014 117
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?
Copyright Steve Yurek March 17, 2014 118
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? Using the Definition?
Copyright Steve Yurek March 17, 2014 119
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? Using the Definition?
Copyright Steve Yurek March 17, 2014 120
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?
Copyright Steve Yurek March 17, 2014 121
Can we generate a formula for the harmonic mean of 3 numbers a, b & c?
Copyright Steve Yurek March 17, 2014 122
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
Copyright Steve Yurek March 17, 2014 123
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
bc ac aba b c a
abcabc
bc ac
b
a
c
b
Copyright Steve Yurek March 17, 2014 124
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
bc ac ab
abcabc
b
ab
c ac
a c
b
b ab
a
c
Copyright Steve Yurek March 17, 2014 125
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
bc ac ab
abcabc
b
ab
c ac
a c
b
b ab
a
c
Copyright Steve Yurek March 17, 2014 126
Can we generate a formula for the harmonic mean of 3 numbers a, b & c?
arithmetic mean
reciprocal
1 1 1
33abc
bc ac
bc
ab
ac ab
a b c abcbc ac ab
abc
Copyright Steve Yurek March 17, 2014 127
Can we generate a formula for the harmonic mean of 3 numbers a, b & c?
arithmetic mean
reciprocal
1 1 1
33abc
bc ac
bc
ab
ac ab
a b c abcbc ac ab
abc
Copyright Steve Yurek March 17, 2014 128
Can we generate a formula for the harmonic mean of 3 numbers a, b & c?
arithmetic mean
reciprocal
1 1 1
33
bc ac ab
a b c abcbc ac ab
abcabc
bc ac ab
Copyright Steve Yurek March 17, 2014 129
Can we generate a formula for the harmonic mean of more than 3 numbers?
Copyright Steve Yurek March 17, 2014 130
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
Copyright Steve Yurek March 17, 2014 131
Can we generate a formula for the harmonic mean of more than 3 numbers?
1 2 3
1 1 1 1... ???
Check out the final slidesna a a a
Copyright Steve Yurek March 17, 2014 132
How do the 3 means
compare with one another?
Copyright Steve Yurek March 17, 2014 133
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
Copyright Steve Yurek March 17, 2014 134
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
Copyright Steve Yurek March 17, 2014 135
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
Copyright Steve Yurek March 17, 2014 136
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
Copyright Steve Yurek March 17, 2014 137
So will the 3 means ever all be equal?They will if the original numbers are equal.
Copyright Steve Yurek March 17, 2014 138
So will the 3 means ever all be equal?They will if the original numbers are equal.
Copyright Steve Yurek March 17, 2014 139
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
Copyright Steve Yurek March 17, 2014 140
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
GM n n n n n n
n n nHM n n n
n n n
AM n n n n
Copyright Steve Yurek March 17, 2014 141
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 142
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 143
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 n nH
AM n n n n n
GM n n
M n n nn n n
n n
Copyright Steve Yurek March 17, 2014 144
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 145
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 146
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 147
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 148
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 149
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 150
So will the 3 means ever all be equal?They will if the original numbers are equal.
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
Copyright Steve Yurek March 17, 2014 151
But what if a ≠b? (let’s keep them both >0)Let’s assume that AM(a,b) < GM(a,b)
Copyright Steve Yurek March 17, 2014 152
But what if a ≠b? (let’s keep them both >0)Let’s assume that AM(a,b) < GM(a,b)
Copyright Steve Yurek March 17, 2014 153
But what if a ≠b? (let’s keep them both >0)Let’s assume that AM(a,b) < GM(a,b)
Copyright Steve Yurek March 17, 2014 154
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
Copyright Steve Yurek March 17, 2014 155
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
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
Copyright Steve Yurek March 17, 2014 156
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 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
Copyright Steve Yurek March 17, 2014 157
But what if a ≠b? (let’s keep them both >0)Let’s assume that AM(a,b) < GM(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
Copyright Steve Yurek March 17, 2014 158
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
( ) 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
Copyright Steve Yurek March 17, 2014 159
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
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
Copyright Steve Yurek March 17, 2014 160
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
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
Copyright Steve Yurek March 17, 2014 161
But what if a ≠b? (let’s keep them both >0)Let’s also assume that GM(a,b) < HM(a,b)
Copyright Steve Yurek March 17, 2014 162
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
Copyright Steve Yurek March 17, 2014 163
But what if a ≠b? (let’s keep them both >0)Let’s also assume that GM(a,b) < HM(a,b)
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
Copyright Steve Yurek March 17, 2014 164
But what if a ≠b? (let’s keep them both >0)Let’s also assume that GM(a,b) < HM(a,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
Copyright Steve Yurek March 17, 2014 165
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
2 4
( 2 ) 0
( ) 0
Impossible so GM(a,b) > HM(a,b)
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
Copyright Steve Yurek March 17, 2014 166
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
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
Copyright Steve Yurek March 17, 2014 167
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
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
Copyright Steve Yurek March 17, 2014 168
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
2 4
2
2 4
2 0
( 2 ) 0
( ) 0
Impossible so GM(a,b) > HM(a,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
Copyright Steve Yurek March 17, 2014 169
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)
Copyright Steve Yurek March 17, 2014 170
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)
Copyright Steve Yurek March 17, 2014 171
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)
Copyright Steve Yurek March 17, 2014 172
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)
Copyright Steve Yurek March 17, 2014 173
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)
Sketchpad – One Last Time
Copyright Steve Yurek March 17, 2014 174
How about this one?
Copyright Steve Yurek March 17, 2014 175
Really ----- Last time for Sketchpad
How about this one?
Copyright Steve Yurek March 17, 2014 176
Copyright Steve Yurek March 17, 2014 177
80’
50’30’
Copyright Steve Yurek March 17, 2014 178
80’
50’30’
E
Copyright Steve Yurek March 17, 2014 179
80’
50’30’
E
?
Copyright Steve Yurek March 17, 2014 180
80’
50’30’
E
18.75
Copyright Steve Yurek March 17, 2014 181
80’
50’30’
E
18.75
As it turns out, this is very cool --- watch this
Copyright Steve Yurek March 17, 2014 182
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?
Copyright Steve Yurek March 17, 2014 183
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?
Copyright Steve Yurek March 17, 2014 184
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?
Copyright Steve Yurek March 17, 2014 185
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?
Copyright Steve Yurek March 17, 2014 186
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?
Copyright Steve Yurek March 17, 2014 187
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?
Copyright Steve Yurek March 17, 2014 188
THERE IS, BUT WHY THE “ADAPTATION”?
Copyright Steve Yurek March 17, 2014 189
THERE IS, BUT WHY THE “ADAPTATION”?
THAT’S ONE THING I’LL LEAVE YOU WITH TODAY
Copyright Steve Yurek March 17, 2014 190
Copyright Steve Yurek March 17, 2014 191
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
Copyright Steve Yurek March 17, 2014 192
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
Copyright Steve Yurek March 17, 2014 193
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
bcde acde abde abce ab
b
c
a
d
ab
Copyright Steve Yurek March 17, 2014 194
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
bcde acde abde abce ab
b
c
a
d
ab
Copyright Steve Yurek March 17, 2014 195
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
Copyright Steve Yurek March 17, 2014 196
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
Copyright Steve Yurek March 17, 2014 197
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
Copyright Steve Yurek March 17, 2014 198
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
bcde acde abde abce abcd
cd
Copyright Steve Yurek March 17, 2014 199
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
Copyright Steve Yurek March 17, 2014 200
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
Copyright Steve Yurek March 17, 2014 201
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
Copyright Steve Yurek March 17, 2014 202
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
bcde acde abde abce abcd
Copyright Steve Yurek March 17, 2014 203
1 2 3 n
1 2 3
HM(a ,a ,a ,...,a )=
...
i
n
a
n a a a a
Copyright Steve Yurek March 17, 2014 204
1 2 3 n
1 2 3
HM(a ,a ,a ,...,a )=
...
i
n
a
n a a a a
Copyright Steve Yurek March 17, 2014 205
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
Copyright Steve Yurek March 17, 2014 206
Or if we use the definition
itself
Copyright Steve Yurek March 17, 2014 207
1 2 3 n
1
HM(a ,a ,a ,...,a )=
1
1 1n
i in a
Copyright Steve Yurek March 17, 2014 208
So Let’s Solve some
Problems
Copyright Steve Yurek March 17, 2014 209
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?
Copyright Steve Yurek March 17, 2014 210
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.
Copyright Steve Yurek March 17, 2014 211
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?
Copyright Steve Yurek March 17, 2014 212
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?
Copyright Steve Yurek March 17, 2014 213
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?
Copyright Steve Yurek March 17, 2014 214
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?
Copyright Steve Yurek March 17, 2014 215
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.
Copyright Steve Yurek March 17, 2014 216
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.
Copyright Steve Yurek March 17, 2014 217
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?
Copyright Steve Yurek March 17, 2014 218
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?
Copyright Steve Yurek March 17, 2014 219
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?
Copyright Steve Yurek March 17, 2014 220
Thank You All For Being Here
This presentation will be uploaded to the NHTM Conference Website
Or you can write to me at syurek@lesley.edu