Section 3A - continued Uses and Abuses of Percentages

Section 3A - continuedUses and Abuses of Percentages Pages 133-147

3-A Uses of Percentages – Reference Values 67*/151The daily circulation of the USA Today is 1.77 million. The daily circulation of the New York Times is 1.12 million. Absolute difference= 1770000-1120000=650000Relative difference= 650000/1120000 = .580 = 58.0% The circulation of USA Today is 58% more than NYT. Absolute difference= 1120000 - 1770000= -650000Relative difference= -65000/1770000 = -.367 = -36.7% The circulation of NYT is 36.7% less than USA Today.

3-A Percentage Of vs. More (or Less) Than(page 142) If the new value is P% more than the original value, then the new value is (100+P)% ofthe original value. If the new value is P% less than the original value, then the new value is (100-P)%of the original value. 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 + .40*(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 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? 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? 91/152 Your total cost for dinner after the tip was $76.40. Assuming you left a 20% tip, what was the bill for food and taxes? What is reference (original) value? What is compared (new) value? What is P?

Always pay close attention to reference values !

3-A Percentages of Percentages 81/152 “In the past four decades, the five-year survival rate for all forms of cancer increased from 39% to 60%.” Absolute difference = 60-39 = 21 Relative difference = 21/39 = 0.5384 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 111/153 The percentage of the Chinese population using the Internet increased by 466%, to 7.23%, from 2001 to 2004.QUESTION: What percentage of the Chinese population used the Internet in 2001? 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?

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

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 #1 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

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? 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 yourinvestments fell 60% during my first year on the job.Thisyear, however, their value has increased by 75%, so youare now 15% ahead!”Is he correct?Yes/No/Depends

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 #2 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. A stockbroker says “I admit that the value of yourinvestments fell 60% during my first year on the job.Thisyear, however, their value has increased by 75%, so youare now 15% ahead!”Is he correct?Yes/No/Depends Value before fall and rise: $1000 Value after fall and rise: $700

Abuse of Percentages 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 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 #3 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.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.

3-A Abuses of Percentages • Beware of Shifting Reference Values

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 6. A glass of orange juice has 110% of the minimum daily requirement for vitamin C.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

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

3-A Abuses of Percentages • Beware of Shifting Reference Values • Less than Nothing

QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ 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

#8 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. 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

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

Homework Pages 151-154 #84, 88, 89, 92, 97, 115, 116, 118

Section 3A - continued Uses and Abuses of Percentages