Section 3A - continuedUses and Abuses of Percentages

Pages 133-147

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.

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?

Always pay close attention to reference values !

Percentages of Percentages3-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 = 14Relative 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%.

When describing change in percentages:

Absolute change uses percentage points.

Relative change uses percentage.

Abuse of Percentages1. 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/Depends2. 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

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 $40,000with 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 $36,000with 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

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/DependsSalary before pay cut and pay raise: $40,000Salary 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 $1000with 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 $400with 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

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/DependsValue before fall and rise: $1000Value 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 $100with 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 $80with 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

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: $100Final 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 Percentages1. Beware of Shifting Reference Values

3-A

Abuse of Percentages4. Mary is 100% shorter than her older sister Vivian. Do this make sense? Yes/No/Depends5. 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

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!

Abuses of Percentages1. 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

Abuse of Percentages

10 questions in Part I80% correct … 8 correct questions

30 questions in Part II 90% correct … 27 correct questionsFinal Score = (8+27) / (10+30) = 35/40 = 87.5%

30 questions in Part I80% correct … 24 correct questions

10 questions in Part II 90% correct … 9 correct questionsFinal 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 Percentages1. Beware of Shifting Reference Values

2. Less than Nothing

3. Don’t Average Percentages

3-A

Homework

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

Uses of Percentages – Reference Values57/149 The daily circulation of the The daily circulation of the Wall Street Journal Wall Street Journal is 1.77 is 1.77 million. The daily circulation of the million. The daily circulation of the New York TimesNew York Times is is 1.07 million.1.07 million.

Absolute difference = 1770000-1070000=700000Relative difference = 700000/1070000 = .654 = 65.4%The circulation of WSJ is 65.4% more than NYT.

Absolute difference = 1070000 - 1770000= -700000= 1070000 - 1770000= -700000Relative difference = -700000/1770000 = -.396 = -= -700000/1770000 = -.396 = -39.6%39.6%The circulation of NYT is 39.6% less than WSJ.The circulation of NYT is 39.6% less than WSJ.

3-A

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?