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

Pages

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**3 Ways of Using Percentages**

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

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**Given: original value and new value**

2. Percents are often used to describe how a quantity changes over time Given: original value and new value Present several examples from contemporary issues in the media that could serve as concrete reminders of the difference between the two changes. Absolute change is POSITIVE if the new value is greater than the reference/original value and NEGATIVE if the new value is less than the original value

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**Given: compared value and reference value:**

3. Percents are often used to compare two values. Given: compared value and reference value:

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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.]

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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.

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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 = = % 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 = ( )% labeled price = % $699 = 1.076×$699 = $752.12

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**Solving Percentage Problems**

The final cost of your new shoes is $ 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 = ( )% labeled price $ = % labeled price $ / = labeled price = $101.40

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**Solving Percentage Problems**

Your dinner bill is $ 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%

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**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

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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%.

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**Abuses of Percentages Beware of Shifting Reference Values**

Less than Nothing Don’t Average Percentages

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**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?

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**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

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**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!

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absolute change is -$400. relative change is - 400/40000 = -.01 = -1%. Your new salary is 1% less than original.

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“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?

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**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

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absolute change is -$300. relative change is -300/1000 = -.3 = % The new value is 30% less than original.

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**A pair of boots was originally marked 20% off**

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?

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**Suppose 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

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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!

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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?

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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!

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**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?

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**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?

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**(8+27) / (10+30) = 35/40 = 87.5% (24+9) / (10+30) = 33/40 = 82.5%**

Not so fast: 10 questions on the midterm 80% correct … 8 correct questions 30 questions on the final 90% correct … correct questions (8+27) / (10+30) = 35/40 = 87.5% 30 questions on the midterm 80% correct … correct questions 10 questions on the final 90% correct … 9 correct questions (24+9) / (10+30) = 33/40 = 82.5%

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**Don’t Average Percentages!**

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3-A Homework Pages # 10, 11, 58, 73, 79, 82, 87, 89, 92, 94, 101, 106, 108

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