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Uses and Abuses of Percentages

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					                     Uses and Abuses of Percentages



Wednesday January 5, 2011



Review
  • per cent means “divided by 100”, so for example, 50% = 50/100 = .5 and 125% =
    125/100 = 1.25.

  • To convert a percent to a fraction, replace the percent sign with division by 100 (then
    reduce the fraction). For example,
                                                25   1
                                       25% =        = .
                                                100  4

  • To convert a percent to a decimal, drop the % symbol and move the decimal two
    places to the left (i.e. divide by 100). For example,

                                25% = .25        and 1% = .01

  • To convert a decimal to a percentage, move the decimal two places to the right (i.e.
    multiply by 100) and attach the % symbol. For example,

                              .12 = 12%         and .0013 = 1.3%

  • To convert a fraction to a percentage, first convert the fraction to a decimal (using
    a calculator if necessary) then convert the decimal to a percentage as above. For
    example
                                       1
                                         = .2 = 20%
                                       5

Three Ways of Using Percentages
  1. As Fractions


    Example 1. Express the following as a percent: Of 305 million Americans, 40.4
    million are under the age of 15.


                                            1
  Solution Write as a fraction then convert to percentage:
                          40.4
                               = .13245902 = 13.245902%
                           305
  So approximately 13.25% of Americans are under the age of 15.

2. To Describe Change

    • Absolute change refers to the increase or decrease from a reference value to a
      new value.
    • Relative change is a fraction that describes the relative change in comparison to
      the reference value.

  Example 2. During a 6-month period, Nokia’s stock doubled from $10 a share to
  $20 a share. What were the absolute and relative changes in the stock price?
  Solution The reference value is $10, the starting price of the stock. The new value
  of the stock price is $20. Therefore,

           absolute change = new value − reference value = $20 − $10 = $10

  and
                                         absolute change   $10
                     relative change =                   =     =1
                                         reference value   $10
  Remarks:
  (a) Although the stock price doubled, the relative increase is only %100 (as opposed
  to %200).
  (b) The relative change is unitless. (c) The relative change can be converted to a
  percent by multiplying by 100

  Example 3. The population of the United States increased form 249 million in 1990
  to 308 million in 2010. Find the absolute and percentage change in the the U.S. pop-
  ulation from 1990 to 2010.
  Solution
  reference value = 249 million
  new value = 308 million

                     absolute change (in millions) = 308 − 249 = 59
                                                  59
                             relative change =       = .237
                                                 249
  So the percent change is 23.7%.

3. For Comparisons

    • absolute difference = compered value - reference value
    • relative difference = absolute difference/reference value

                                         2
      Remark As above, we can convert the relative difference to a percent difference by
      multiplying by 100 and suing the % symbol.

      Example 4. Suppose we want to compare the price of a $50,000 Mercedes and a
      $40,000 Lexus. The absolute and percent differences in the prices of the cars will be
      different depending on whether we use the price of the Mercedes or the price of the
      Lexus as our reference value.
      If we use the price of the Lexus as the reference value, we get

                         absolute difference = 50, 000 = 40, 000 = 10, 000

      and as a percentage of the Lexus price, the percent difference in the car prices is
                                        10, 000
                                                = .25 = 25%.
                                        40, 000
      On the other hand if we use the price of the Mercedes as our reference value, we get

                        absolute difference = 40, 000 − 50, 000 = −10, 000.

      and as a percentage of the Mercedes price, the percent difference in the car prices is
                                         10, 000
                                                 = .2 = 20%
                                         50, 000


Friday January 7, 2011


“of” vs. “More Than” or “Less Than”
Consider the rising prices of gas. In 1997, a gallon of gas cost $1.00. Today a gallon of gas
costs $3.00 (if your lucky). The price of gas has tripled since 1997. In terms of percentages,
there are two equivalent ways of saying this; one using “of” and one using “more than”.

   • Using more than: The price of gas today is 200% more than the price of gas in 1997.

   • Using of : the price of gas today is 300% of the price of gas in 1997.

How to convert between of and more than and between of and less than:

   • If the compared value is P % more than the reference value, it is (100 + P )% of the
     reference value.

   • If the compared value is P % less than the reference value, it is (100 − P )% of the
     reference value.

Remark In any particular problem, the compared and reference values should be obvious
(or at least attainable) from the given information. In a real life situation, there is sometimes
a logical reason to pick a particular value to be the reference value.


                                               3
Example 5. Using the gas price example, and taking the price of gas in 1997 as the
reference value, the compared value (i.e. the price of gas today) is 200% more than the
reference value. To convert this to a comparison using of, we take P = 200% in the first bul-
let above. Therefore, the the price today is (100+200)% = 300% of the price of gas in 1997.


Example 6. A store is having a “20% off” sale how does an items sale price compare to
its original price?
Solution 20% off means an items sale price is 20% less than its original price. Therefore,
the sale price is (100 − 20)% = 80% of its original price. If the original price of the item is
$100, the sale price of the item is $80.

Example 7.(continued) Suppose an item that originally cost $100 is marked down to
20% off. Suppose the item is marked down again to 25% off of the sale price. What is the
final price of the the item?
Solution After the first markdown, the item costs $80. After the second markdown, the
item costs (100 − 25)% = 75% of $80, so the final price of the item is

                                 $80 · 75% = $80 · .75 = $60

Notice that the final price of the item is not the same price as marking the item off 45% (in
which case the price of the item would be $55).


Percentages of Percentages
Suppose the interest rate on your student loan gets increased from 10% to 15%. It may
be tempting to say that the interest rate increased by 5%. This statement is not true.
Although the interest rate increased by 5 percentage points, the relative change in interest
rate is
                    new value − old value     15% − 10%       5%
                                           =               =      = 0.5
                       reference value            10%        10%
Converting this to a percent, we see that the interest rate has actually increased by 50%.
   • When you see a change or difference expressed in percentage points, you can assume it
     is an absolute change or difference. If it is expressed with the % symbol, or the word
     percent, it is a relative change.

Example 8. (Tricky Wording) Suppose 40% of the registered voters in Gainesville are
Republicans. Read the following questions carefully and give appropriate answers.
  1. The percentage of voters registered as Republicans in 25% higher in Miami than in
     Gainesville. What percentage of the registered voters in Miami are Republicans?
     Soluteion Here a change is expressed in percent (using the symbol %), so it is a
     relative change (relative to the percent of Republicans in Gainesville). Since 25% of
     40% is 10% (i.e. .25 · 40% = 10%), and the percent of Republicans is 25% higher
     in Miami than in Gainesville, we must add 10% to 40% to get the percentage of
     Republicans in Miami.

                                              4
  2. The percentage of voters registered as Republicans is 25 percentage points higher in
     Miami than in Gainesville. What percentage of the registered voters in Miami are
     Republicans?
     Solution Since the change is given in percentage points, we simply add the percents.
     The percentage of Republicans in Miami is 40% + 25% = 65%.



Solving Percentage Problems
   • If the compared value is P % more than the reference value, then

                        compared value = (100 + P )% · reference value

     and
                                                   compared value
                               reference value =
                                                     (100 + P )%

   • If the compared value is P % less than the reference value, then

                        compared value = (100 − P )% · reference value

     and
                                                   compared value
                               reference value =
                                                     (100 − P )%

Example 9. Suppose you but a shirt with pre-tax price $ 19. If the local sales tax in
7.75%, what is the final price of the shirt?
Solution

Example 10. Suppose you pay $21.00 for a shirt after tax. If the local sales tax is 7.75%,
what is the pre-tax price of the shirt?
Solution

Example 11. Consider the following statement: The rate of smoking in college freshmen
is up 44% to 10.4%. What was the previous rate of smoking for college freshmen?
Solution(you try it)

Monday January 10, 2011


Abuses of Percentages
  1. Shifting Reference Values
     Suppose your employer plans to cut your pay rate by 10% for one month then to raise
     it again by 10%. Will your original pay rate be restored? No. Observe: Suppose your
     original pay rate is $200 per week. A pay cut of 10% reduces your pay rate to

                                 $200 − $20 = $180 per week

                                            5
  A subsequent pay raise of 10% increases your pay rate to

                                $180 + $18 = $198 per week

  Therefore, your employer wins.

  Example 12. Suppose your stock broker tells you that your investment lost 60%
  over his first year on the job, but that on his next year on the job, your investment
  rose 75%. Your stock broker tells you “your investment is now up 15%”. Evaluate
  your brokers intelligence.
  Solution Suppose your initial investment was $1,000. Since 60% of $1,000 is $600,
  after your brokers first year on the job, the value of your investment is

                                   $1, 000 − $600 = $400.

  Moreover, since 75% of $400 is $300, a 75% increase in value of your investment (after
  the 60% decrease in its value) makes your investment worth

                                    $400 + $300 = $700.

  Your actually losing money. Fire your broker. Even if the market changes, he’ll still
  be incompetent.

2. Less Than Nothing

  Example 13. A store manager advertises that it will take “150% off” of the price of
  all merchandise. What should happen when you attempt to purchase a $500 item?
  Solution If the discount wer 100% off, the item would be free. So if the price is 150%
  off, the store should pay you $250 (half the price of the item) to take the item home.
  The manager of the store is clearly FSU alum.

3. Don’t Average Percentages Suppose you take two exams. On the first exam you
   get 70% of the questions correct and on the second exam you get 90% of the questions
   correct. Is it true that between the two exams you got 80% of the questions correct?
   No! The problem is that the exams may have a different number of questions. Consider
   an extreme case for the purpose of illustration. Suppose your first exam had 1,000
   questions and your second exam had 10 questions. Since you scored 70% on your first
   exam you got 700 of the 1,000 questions correct. On your second exam you got 9 of
   10 questions correct. In total, you got 709 of 1,010 questions correct. Converting this
   to a percentage, you got
                                     709
                                         ≈ 0.702 = 70.2%
                                    1010
   of the questions correct.




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