Section 3A - continued Uses and Abuses of Percentages Pages

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 = = Relative difference = / =.654 = 65.4% The circulation of WSJ is 65.4% more than NYT. = = = / = = -39.6% Absolute difference = = Relative difference = / = = -39.6% The circulation of NYT is 39.6% less than WSJ. 3-A

Uses of Percentages – Reference Values 78/150 The final cost of your new shoes is $ The local sales tax rate is 6.2%. What was the labeled (pre-tax) price? Final Price = Labeled Price + 6.2% of Labeled Price Final Price = 100% of Labeled Price + 6.2% of Labeled Price Final Price = 106.2% * Labeled Price $ = * Labeled Price $ = Labeled Price and Tax is.062* = $6.29 Is the labeled price 6.2% less than final price? Final Price – 6.2% of Final Price * = – 6.68 = $101.01

Always pay close attention to reference values !

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 = = 14 Relative difference = 14/44 = The percentage of degrees awarded to women rose by 14 percentage points. The percentage of degrees awarded to women rose by 31.8%.

When describing change in percentages: Absolute change uses percentage points. Relative change uses percentage.

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?

QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ 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?

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

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.

Value of investment $1000 with 60% drop: new value is 60% less than previous value. new value = 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 = x 400 = $700 OR final value is 175% of new value final value = 1.75  400 = $700 #2

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.

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

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.

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

QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ 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/$0/-$ A glass of orange juice has 110% of the minimum daily requirement for vitamin C. Do this make sense? Yes/No/Depends

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

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

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 QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ

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

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

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

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