Presentation on theme: "Section 3A Uses and Abuses of Percentages Reprise"— Presentation transcript:
1Section 3A Uses and Abuses of Percentages Reprise Pages
23 Ways of Using Percentages As fractions – “Percent of”To describe change over timeFor comparison
3Given: original value and new value 2. Percents are often used to describe how a quantity changes over timeGiven: original value and new valuePresent several examples from contemporary issues in the media that could serve as concrete reminders of the difference between the two changes.Absolute change is POSITIVE if the new value is greater than the reference/original value and NEGATIVE if the new value is less than the original value
4Given: compared value and reference value: 3. Percents are often used to compare two values.Given: compared value and reference value:
53-AThe daily circulation of the Wall Street Journal is ≈ 2.7 million. The daily circulation of the New York Times is ≈ 1.14 million[Find the absolute and relative difference. Assume that the first quantity is the compared value and the second is the reference value.]
63-AThe daily circulation of the Wall Street Journal is ≈ 2.7 million. The daily circulation of the New York Times is ≈ 1.14 millionAbsolute difference = 2,700,000-1,140,000 = 1,560,000The WSJ has 1,560,000 more readers than the NYT.Relative difference = 1,560,000/1,140,000 = 1.37 = 137%The WSJ has 137% more readers than the NYT.
73-AThe daily circulation of the Wall Street Journal is ≈ 2.7 million. The daily circulation of the New York Times is ≈ 1.14 millionAbsolute difference = 1,140,000-2,700,000 = -1,560,000The NYW has 1,560,000 fewer readers than the WSJ.Relative difference = -1,560,000/2,700,000 = = %The NYT has 57.8% fewer readers than the WSJ.
8Solving Percentage Problems You purchase a bicycle with a labeled (pre-tax) price of $699. The local sales tax rate is 7.6%. What is your final cost?final cost= 100% of labeled price + 7.6% of labeled price= ( )% labeled price= % $699 = 1.076×$699= $752.12
9Solving Percentage Problems The final cost of your new shoes is $ The local sales tax rate is 6.2%. What was the labeled (pre-tax) price.final cost= 100% labeled price + 6.2% of labeled price= ( )% labeled price$ = % labeled price$ / = labeled price= $101.40
10Solving Percentage Problems Your dinner bill is $ You leave $22. What percent tip did you leave?Total bill$22 = dinner bill + tiptip = $22 - $18.75 = $3.25$3.25 is what percent of 18.75?$3.25/18.75 = .1733= 17.33%
11Percentages of Percentages Interest rate increases from 3% to 4%Please DON’T say “my interest rate increased by 1%”Do you mean absolute interest rate? Or relative interest rate?
12Interest rate increased from 3% to 4% Absolute change = 1 percentage pointRelative change
13Example:“The percentage of all bachelor’s degrees awarded to women rose from 44% in 1972 to 58% in 2000.”The percentage of degrees awarded to women rose by 14 percentage points.The percentage of degrees awarded to women rose by 31.8%.
14Abuses of Percentages Beware of Shifting Reference Values Less than NothingDon’t Average Percentages
151. Shifting Reference Values: Example:If you accept a 10% pay cut nowAnd get a 10% pay raise in 6 months . . .In six months – will you be back to your original salary?
16Starting salary = $40,000/year If you take a 10% pay cut – your salary will become (100-10)% $40,000/year= 90% $40,000/year= .9 $40,000/year= $36,000/year
17Six months later, salary = $36,000/year You get a 10% pay raise – your salary will become (100+10)% $36,000/year= 110% $36,000/year= 1.10 $36,000/year= $39,600/yearWhich is not as much ($40,000/year) as you started with!
18absolute change is -$400.relative change is - 400/40000 =-.01 = -1%.Your new salary is 1% less than original.
19“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?
20Starting investment = $10,000 First year – lost 60% (retained 40%)40% $10,000= .4 $10,000 = $4,000Second year – gained 75%(of $4,000)175% $4,000 = 1.75 $4,000= $7,000
21absolute change is -$300.relative change is -300/1000 = -.3 = %The new value is 30% less than original.
22A pair of boots was originally marked 20% off 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?
23Suppose the boots initially cost $100 To take 20% off means the boots now cost (100-20)% = 80% of their original priceSo, they cost 80% $100 = .8 $100 = $80Now take another 30% off.So the boots will cost (100-30)% = 70% of the $80 sale price.That is, 70% $80= .7 $80 = $56
24Original Price = $100Final sale price = $56absolute change is -$44.relative change is -44/100 = -.44 = -44%.The final price is 44% less than original.Saleslady said the boots would be 50% off (i.e. $50).She was wrong!Percentages don’t add!
252. 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?
26Less than Nothing:120% of 15.97= 1.2 × $15. 97= $19.16You should get $19.16 OFF the $15.97 price.The store should pay you $3.19!
27Less than Nothing: Can an athlete give a 110% effort? Can a glass of juice have 125% of the minimum daily requirement of vitamin C?Can Mary be 100% shorter than her older sister Vivian?Can Vivian be 110% taller than her younger sister Mary?
283. 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?
29(8+27) / (10+30) = 35/40 = 87.5% (24+9) / (10+30) = 33/40 = 82.5% Not so fast:10 questions on the midterm80% correct … 8 correct questions30 questions on the final90% correct … correct questions(8+27) / (10+30) = 35/40 = 87.5%30 questions on the midterm80% correct … correct questions10 questions on the final90% correct … 9 correct questions(24+9) / (10+30) = 33/40 = 82.5%