Section 3A Uses and Abuses of Percentages Reprise
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Transcript of Section 3A Uses and Abuses of Percentages Reprise
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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