Section 3A - More Uses and Abuses of Percentages Pages

Warm-up problems for 3-A On Tuesday I bought regular unleaded gas for $2.399/gallon and today unleaded gas at the same station is $ per gal. What is the absolute change and relative change? What will I use to compare the price of a gallon of Regular unleaded gas ($2.339/gallon) at Greenway Market to the price of a gallon of Premium Unleaded gas( $2.539) at Greenway Market? Why? A Lazboy Rocker Recliner is advertised as on sale at a savings of 44%. The regular price is $539. What is the sale price of the chair before taxes?

Percentages of Percentages   Interest rate increases from 3% to 4%   Please DON’T say “my interest rate increased by 1%”  absolute relative  Do you mean absolute interest rate? relative interest rate? 3-A

  Interest rate increased from 3% to 4%  Absolute change = 1 percentage point  Relative change 3-A

Example: “ “The percentage of all bachelor’s degrees awarded to women rose from 44% in 1972 to 58% in 2000.” 14 percentage points. The percentage of degrees awarded to women rose by 14 percentage points. 31.8%. The percentage of degrees awarded to women rose by 31.8%.

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

1. Shifting Reference Values: Example: Accept a 10% pay cut now Get 10% pay raise in 6 months In six months – will you be back to your original salary?

Starting salary = $40,000/year (100-10)% If you take a 10% pay cut – your salary will become (100-10)%  $40,000/year 90% = 90%  $40,000/year.9 =.9  $40,000/year $36,000/year = $36,000/year

Six months later, salary = $36,000/year (100+10)% You get a 10% pay raise – your salary will become (100+10)%  $36,000/year 110% = 110%  $36,000/year 1.10 = 1.10  $36,000/year $39,600/year = $39,600/year Which is not as much ($40,000/year) as you started with!

“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 the stock broker correct?

Starting investment = $10,000 First year First year – lost 60% (retained 40%) 40% 40%  $10,000.4 $4,000 =.4  $10,000 = $4,000 Second year – gained 75% (of $4,000) (of $4,000) 175% %  $4,000 = 1.75  $4,000 = $7,000

A pair of boots was originally marked 20% off. Then they were marked down an additional 30%. The sales clerk tells you this means the boots are now 50% off the original price. Is she correct?

Suppose the boots initially cost $100 To take 20% off means the boots now cost ( )% = 80% of their original price So, they cost 80%  $100 =.8  $100 = $80 Now take another 30% off. So the boots will cost (100-30)% = 70% of the $80 sale price. That is, 70%  $80 =.7  $80 = $56

Original Price = $100 Final sale price = $56 This is a total of 44% off. Saleslady said the boots would be 50% off (i.e. $50). She was wrong! Percentages don’t add!

2. Less than Nothing: Example: A store advertises that it will take “120% off” all red-tagged items. You take a red-tag blouse marked $15.97 to the counter. How much should it cost you?

Less than Nothing: 120% of = 1.2 × $ = $19.16 You should get $19.16 OFF the $15.97 price. The store should pay you $3.19!

Less than Nothing: Can an athlete give a 110% effort?

3. Don’t Average Percentages: Example: You answered 80% of the midterm questions correctly. You answered 90% of the final exam questions correctly. Conclusion: You answered (80%+90%)/2 = 85% of the test questions correctly.Right?

Not so fast: 10 questions on the midterm 80% correct … 8 correct questions 30 questions on the final 90% correct … 27 correct questions 90% correct … 27 correct questions (8+27) / (10+30) = 35/40 = 87.5% of all the questions correct!

Don’t Average Percentages!

Homework for Friday: Pages # 72, 88, 89, 92, 96, 98,102, 108