Transcript Section 3A Uses and Abuses of Percentages

Section 3A
Uses and Abuses of Percentages
Reprise
Pages 133-147
3-A
3 Ways of Using Percentages
1.
2.
3.
As fractions – “Percent of”
To describe change over time
For comparison
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2. Percents are often used to describe
how a quantity changes over time
Given: original value and new value
absolute change  newvalue  original value
absolutechange
relative change 
original value
new value  original value

original value
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3. Percents are often used to compare
two values.
Given: compared value and reference value:
absolutedifference  compared value  referencevalue
absolute difference
relative difference 
referencevalue
compared value  referencevalue

referencevalue
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The daily circulation of the Wall
Street Journal is ≈ 2.7 million. The
daily circulation of the New York
Times is ≈ 1.14 million
[Find the absolute and relative difference.
Assume that the first quantity is the
compared value and the second is the
reference value.]
3-A
The daily circulation of the Wall Street
Journal is ≈ 2.7 million. The daily
circulation of the New York Times is ≈
1.14 million
Absolute difference = 2,700,000-1,140,000 = 1,560,000
The WSJ has 1,560,000 more readers than the
NYT.
Relative difference = 1,560,000/1,140,000 = 1.37 = 137%
The WSJ has 137% more readers than the NYT.
3-A
The daily circulation of the Wall Street
Journal is ≈ 2.7 million. The daily
circulation of the New York Times is ≈
1.14 million
Absolute difference = 1,140,000-2,700,000 = -1,560,000
The NYW has 1,560,000 fewer readers than the
WSJ.
Relative difference = -1,560,000/2,700,000 = -.577 =
-57.8%
The NYT has 57.8% fewer readers than the
WSJ.
Solving Percentage Problems

You purchase a bicycle with a labeled (pre-tax)
price of $699. The local sales tax rate is 7.6%.
What is your final cost?
final cost
= 100% of labeled price + 7.6% of labeled price
= (100 + 7.6)%  labeled price
= 107.6%  $699 = 1.076×$699
= $752.12
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Solving Percentage Problems

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The final cost of your new shoes is $107.69. The
local sales tax rate is 6.2%. What was the labeled
(pre-tax) price.
final cost
= 100% labeled price + 6.2% of labeled price
= (100 + 6.2)%  labeled price
$107.69 = 106.2%  labeled price
$107.69 / 1.062 = labeled price
= $101.40
Solving Percentage Problems

Your dinner bill is $18.75. You leave $22.
What percent tip did you leave?
Total bill
$22 = dinner bill + tip
tip = $22 - $18.75 = $3.25
$3.25 is what percent of 18.75?
$3.25/18.75 = .1733
= 17.33%
3-A
3-A
Percentages of Percentages

Interest rate increases from 3% to 4%

Please DON’T say “my interest rate
increased by 1%”

Do you mean absolute interest rate? Or
relative interest rate?
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
Interest rate increased from 3% to 4%

Absolute change = 1 percentage point

Relative change
= absolute change
original value
= 4%-3%  1%
3%
 .33
 33%
3%
Example:
“The percentage of all bachelor’s degrees
awarded to women rose from 44% in
1972 to 58% in 2000.”
The percentage of degrees awarded to
women rose by 14 percentage points.
The percentage of degrees awarded to
women rose by 31.8%.
3-A
Abuses of Percentages
1.
Beware of Shifting Reference Values
2.
Less than Nothing
3.
Don’t Average Percentages
1. Shifting Reference Values:
Example:
If you accept a 10% pay cut now
And get a 10% pay raise in 6 months . . .
In six months – will you be back to your
original salary?
Starting salary = $40,000/year
If you take a 10% pay cut – your salary
will become (100-10)% 
$40,000/year
= 90%  $40,000/year
= .9  $40,000/year
= $36,000/year
Six months later, salary = $36,000/year
You get a 10% pay raise – your salary will
become (100+10)%  $36,000/year
= 110%  $36,000/year
= 1.10  $36,000/year
= $39,600/year
Which is not as much ($40,000/year) as
you started with!
absolute change is -$400.
 relative change is - 400/40000 =
-.01 = -1%.


Your new salary is 1% less than
original.
“I admit that the value of your
investments fell 60% during my first
year on the job. This year, however,
their value has increased by 75%, so you
are now 15% ahead!”
Is the stock broker correct?
Starting investment = $10,000
First year – lost 60% (retained 40%)
40%  $10,000
= .4  $10,000 = $4,000
Second year – gained 75%
(of $4,000)
175%  $4,000 = 1.75  $4,000
= $7,000
absolute change is -$300.
 relative change is -300/1000 = -.3 =
-30%


The new value is 30% less than original.
A pair of boots was originally marked 20%
off. Then they were marked down an
additional 30%. The sales clerk tells
you this means the boots are now 50%
off the original price.
Is she correct?
Suppose the boots initially cost $100
To take 20% off means the boots now cost (10020)% = 80% of their original price
So, they cost 80%  $100 = .8  $100 = $80
Now take another 30% off.
So the boots will cost (100-30)% = 70% of the
$80 sale price.
That is, 70%  $80
= .7  $80 = $56
Original Price = $100
Final sale price = $56
absolute change is -$44.
 relative change is -44/100 = -.44 = -44%.
 The final price is 44% less than original.

Saleslady said the boots would be 50% off (i.e.
$50).
She was wrong!
Percentages don’t add!
2. Less than Nothing:
Example:
A store advertises that it will take “120%
off” all red-tagged items.
You take a red-tag blouse marked $15.97
to the counter. How much should it cost
you?
Less than Nothing:
120% of 15.97
= 1.2 × $15. 97
= $19.16
You should get $19.16 OFF the $15.97
price.
The store should pay you $3.19!
Less than Nothing:
Can an athlete give a 110% effort?
Can a glass of juice have 125% of the minimum
daily requirement of vitamin C?
Can Mary be 100% shorter than her older sister
Vivian?
Can Vivian be 110% taller than her younger
sister Mary?
3. Don’t Average Percentages:
Example:
You answered 80% of the midterm
questions correctly.
You answered 90% of the final exam
questions correctly.
Conclusion: You answered (80%+90%)/2
= 85% of the test questions correctly.
Right?
Not so fast:
10 questions on the midterm
80% correct … 8 correct questions
30 questions on the final
90% correct … 27 correct questions
(8+27) / (10+30) = 35/40 = 87.5%
30 questions on the midterm
80% correct … 24 correct questions
10 questions on the final
90% correct … 9 correct questions
(24+9) / (10+30) = 33/40 = 82.5%
Don’t Average Percentages!
3-A
Homework
Pages 147-151
# 10, 11, 58, 73, 79, 82, 87, 89, 92,
94, 101, 106, 108