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Section 3A Uses and Abuses of Percentages Reprise

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Section 3AUses and Abuses of PercentagesReprise

Pages 133-147

3-A

- As fractions – “Percent of”
- To describe change over time
- For comparison

3-A

Given: original value and new value

3-A

3. Percents are often used to compare two values.

Given:comparedvalue andreferencevalue:

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

[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.

3-A

- 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

3-A

- 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

3-A

- 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

- 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?

3-A

- Interest rate increased from 3% to 4%
- Absolute change= 1percentage point
- Relative change

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 by14 percentage points.

The percentage of degrees awarded to women rose by31.8%.

3-A

- Beware ofShifting Reference Values
- Less than Nothing
- 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 originalsalary?

Starting salary=$40,000/year

Ifyou take a 10% pay cut – your salary willbecome(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?

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 = $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