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

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Uses of Percentages – Reference Values from Friday: Jack weighs 40% more than Jill. Does Jill weigh 40% less than Jack? Suppose Jill weighs 100 pounds. Then Jack weighs 100 +.40*(100) = 140 pounds. What is 40% less than Jack? 140 –.40*(140) = 84 pounds This does not match Jill’s weight.

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Uses of Percentages – Reference Values 78/150 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? What is reference value? What is compared value? What is P? 77/150 You purchase a bicycle with a labeled (pre-tax) price of $699. The local sales tax rate is 7.6%. What is the final cost?

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Always pay close attention to reference values !

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Percentages of Percentages 3-A 74/149 “The percentage of all bachelor’s degrees awarded to women rose from 44% in 1972 to 58% in 2000.” Absolute difference = 58-44 = 14 Relative difference = 14/44 = 0.318 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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When describing change in percentages: Absolute change uses percentage points. Relative change uses percentage.

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Abuse of Percentages 1. Because of losses by your employer, you are asked to accept a temporary 10% pay cut. Your employer promises to give you a 10% raise after 6 months? Will your salary be the same after the pay cut and pay raise? Yes/No/Depends 2. A stockbroker says “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 he correct? Yes/No/Depends 3. A pair of boots was originally marked 20% off. Then they were marked down an additional 30%. The sales clerk says the boots are now 50% off the original price. Is she correct? Yes/No/Depends Is she correct?

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Starting salary is $40,000 with 10% pay cut: new salary is 10% less than previous salary. new salary = 40000 -.10 x 40000 = $36,000 OR new salary is 90% of previous salary new salary =.90 x 40000 = $36,000 New salary is $36,000 with 10% pay raise: final salary is 10% more than new salary. final salary = 36000 +.10 x 36000 = $39,600 OR final salary is 110% of new salary final salary = 1.10 36000 = $39,600 #1

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1. Because of losses by your employer, you are asked to accept a temporary 10% pay cut. Your employer promises to give you a 10% raise after 6 months? Is your salary the same after the pay cut and pay raise? Yes/No/Depends Salary before pay cut and pay raise: $40,000 Salary after pay cut and pay raise: $39,600 absolute change is -$400. relative change is -400/40000 = -.01 = -1%. Your new salary is 1% less than original.

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Value of investment $1000 with 60% drop: new value is 60% less than previous value. new value = 1000 -.60 x 1000 = $400 OR new value is 40% of previous value new salary =.40 x 1000 = $400 New value of investment is $400 with 75% rise: final value is 75% more than new value. final value = 400 +.75 x 400 = $700 OR final value is 175% of new value final value = 1.75 400 = $700 #2

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2. A stockbroker says “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 he correct? Yes/No/Depends Value before fall and rise: $1000 Value after fall and rise: $700 absolute change is -$300. relative change is -300/1000 = -.3 = -30%. The new value is 30% less than original.

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Original price of boots $100 with 20% discount: new price is 20% less than original price. new price = 100 -.20 x 100 = $80 OR new price is 80% of original price new price =.80 x 1000 = $80 New price of boots is $80 with 30% additional discount: final price is 30% less than new price. final price = 80 -.30 x 80 = $56 OR final price is 70% of new price final price =.70 80 = $56 #3

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3. A pair of boots was originally marked 20% off. Then they were marked down an additional 30%. The sales clerk says the boots are now 50% off the original price. Is she correct? Yes/No/Depends Is she correct? Original Price before discounts: $100 Final Price after discount: $56 absolute change is -$44. relative change is -44/100 = -.44 = -44%. The final price is 44% less than original.

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Abuses of Percentages 1. Beware of Shifting Reference Values 3-A

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Abuse of Percentages 4. Mary is 100% shorter than her older sister Vivian. Do this make sense? Yes/No/Depends 5. Vivian is 100% taller than her younger sister Mary. Do this make sense? Yes/No/Depends 7. A store advertises that it will take 150% off the price of all merchandise. How much will you pay for a $500 item? $350/$250/$0/-$250 6. A glass of orange juice has 110% of the minimum daily requirement for vitamin C. Do this make sense? Yes/No/Depends

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Original price = $500 new price is 150% less than previous price new price = 500-1.50 x 500 = 500 - 750 = - $250 #7 WOW! The store pays you $250 to buy the $500 item!

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Abuses of Percentages 1. Beware of Shifting Reference Values 2. Less than Nothing 3-A

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8. On an exam, you answer 80% of the questions correctly in Part I and 90% of the questions in Part II. What is your score on the exam? 82/85/87/depends Abuse of Percentages

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10 questions in Part I 80% correct … 8 correct questions 30 questions in Part II 90% correct … 27 correct questions Final Score = (8+27) / (10+30) = 35/40 = 87.5% 30 questions in Part I 80% correct … 24 correct questions 10 questions in Part II 90% correct … 9 correct questions Final Score = (24+9) / (10+30) = 33/40 = 82.5% #8

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8. On an exam, you answer 80% of the questions correctly in Part I and 90% of the questions in Part II. What is your score on the exam? 82/85/87/depends

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Abuses of Percentages 1. Beware of Shifting Reference Values 2. Less than Nothing 3. Don’t Average Percentages 3-A

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Homework Pages 147-151 #108,106,105, 95, 94, 93, 88, 87

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Uses of Percentages – Reference Values The daily circulation of the Wall Street Journal is 1.77 million. The daily circulation of the New York Times is 1.07 million. 57/149 The daily circulation of the Wall Street Journal is 1.77 million. The daily circulation of the New York Times is 1.07 million. Absolute difference = 1770000-1070000=700000 Relative difference = 700000/1070000 =.654 = 65.4% The circulation of WSJ is 65.4% more than NYT. = 1070000 - 1770000= -700000 = -700000/1770000 = -.396 = -39.6% Absolute difference = 1070000 - 1770000= -700000 Relative difference = -700000/1770000 = -.396 = -39.6% The circulation of NYT is 39.6% less than WSJ. 3-A

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Practice – Percentages in the News 102/151 In 1991, just 12% of the U.S. population was considered obese; by 1998 it was 20%. QUESTION: What is the percentage increase in the proportion of the U.S population considered obese? 107/151 In 2003, 625,000 U.S. prisoners were released. Within three years, 60% of those former inmates will be rearrested. QUESTION: Approximately how many U.S. prisoners will be rearrested within three years? 109/151 Oracles’ stock rose [to] $13.48 at the end of the week, up 3% over the previous week’s close. QUESTION: What was the price of Oracle’s stock at the previous week’s close?

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