Section 3AUses and Abuses of Percentages

Reprise

Pages 133-147

3 Ways of Using Percentages

1. As fractions – “Percent of”2. To describe change over time3. For comparison

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2. Percents are often used to describe how a quantity changes

over time

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Given: original value and new value

absolute change newvalue original value

absolutechangerelative change

original value

new value original value

original value

3. Percents are often used to compare two values.

Given: compared value and reference value:

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compared vaabsolutedi refference ferencevlue alue

referenceva

absolute differencerelativedifference

c

lue

referencevalueompared value

referencevalue

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.]

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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

Absolute difference = 2,700,000-1,140,000 = 1,560,000

Relative difference = 1,560,000/1,140,000 = 1.37 = 137%

The WSJ has 1,560,000 more readers than the NYT.

The WSJ has 137% more readers than the NYT.

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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

Absolute difference = 1,140,000-2,700,000 = -1,560,000

Relative difference = -1,560,000/2,700,000 = -.577 = -57.8%

The NYW has 1,560,000 fewer readers than the WSJ.

The NYT has 57.8% fewer readers than the WSJ.

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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

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

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Solving Percentage Problems

Your dinner bill is $18.75. You leave $22. What percent tip did you leave?

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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%

Percentages of Percentages of PercentagesPercentages

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

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1%3%

4%-3%3%

=

.33

33%

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%.

Abuses of PercentagesAbuses of Percentages1. Beware of Shifting Reference Values

2. Less than Nothing

3. Don’t Average Percentages

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1. Shifting Reference Values:

Example:If you accept a 10% pay cut nowAnd 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 = $10,000First 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 $100Suppose the boots initially cost $100

To take 20% off means the boots now cost (100-20)% = 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 = $100Final 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 questions30 questions on the final 90% correct … 27 correct questions

(8+27) / (10+30) = 35/40 = 87.5%

30 questions on the midterm80% 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!

Homework

Pages 147-151# 10, 11, 58, 73, 79, 82, 87, 89,

92, 94, 101, 106, 108

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