Section 3A - continued Uses and Abuses of Percentages Pages

Uses of Percentages – Reference Values The daily circulation of the USA Today is 1.77 million. The daily circulation of the New York Times is 1.12 million. 67*/151 The daily circulation of the USA Today is 1.77 million. The daily circulation of the New York Times is 1.12 million. Absolute difference = = Relative difference = / =.580 = 58.0% The circulation of USA Today is 58% more than NYT. = = = / = = -36.7% Absolute difference = = Relative difference = / = = -36.7% The circulation of NYT is 36.7% less than USA Today. 3-A

If the new value is P% more than the original value, then the new value is (100+P)% of the original value. If the new value is P% less than the original value, then the new value is (100-P)% of the original value. Percentage Of vs. More (or Less) Than (page 142) 3-A Practice: 73/152 and 75/132

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

Uses of Percentages – Reference Values ex10/145 Your receipt shows that you paid $19.26 for a DVD, tax included. The local sales tax rate is 7%. What is the labeled (pre tax) price? What is reference (original) value? What is compared (new) value? What is P? 87/150 You purchase a bicycle with a labeled (pre-tax) price of $760. The local sales tax rate is 7.6%. What is the final cost? 91/152 Your total cost for dinner after the tip was $ Assuming you left a 20% tip, what was the bill for food and taxes?

Always pay close attention to reference values !

Percentages of Percentages 3-A 81/152 “In the past four decades, the five-year survival rate for all forms of cancer increased from 39% to 60%.” Absolute difference = = 21 Relative difference = 21/39 = The survival rate increased by 21 percentage points. The survival rate today is 53.8% more than it was four decades ago.

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

Practice – Percentages in the News 117/153 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? 119/151 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? 111/153 The percentage of the Chinese population using the Internet increased by 466%, to 7.23%, from 2001 to QUESTION: What percentage of the Chinese population used the Internet in 2001?

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 Abuse of Percentages

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 Abuse of Percentages

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 Abuse of Percentages

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

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? Abuse of Percentages

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 #84, 88, 89, 92, 97, 115, 116, 118