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Section 3A Uses and Abuses of Percentages Reprise Pages 133-147

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3 Ways of Using Percentages 1. As fractions – Percent of 2. To describe change over time 3. For comparison 3-A

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2. Percents are often used to describe how a quantity changes over time 3-A Given: original value and new value

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

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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

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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. 3-A

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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. 3-A

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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 3-A

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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 3-A

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Solving Percentage Problems Your dinner bill is $18.75. You leave $22. What percent tip did you leave? 3-A 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 DONT say my interest rate increased by 1% Do you mean absolute interest rate? Or relative interest rate? 3-A

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

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Example: The percentage of all bachelors 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 1. Beware of Shifting Reference Values 2. Less than Nothing 3. Dont Average Percentages 3-A

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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 = -30% The new value is 30% less than original.

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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 dont 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. Dont 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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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%

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Dont Average Percentages!

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Homework Pages 147-151 # 10, 11, 58, 73, 79, 82, 87, 89, 92, 94, 101, 106, 108 3-A

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