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MAT081 :: Lecture Note :: Week 08
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Overview
Assignments

Percents

The word percent means "per one hundred."

Percent values are suffixed by a % character.

A percent is a fraction where the denominator is 100.

      5%  equals     5/100
     15%  equals    15/100
    120%  equals   120/100
   37.5%  equals  37.5/100

Percent values can be written as a decimal number by multiplying the value by 0.01.

     4%  equals    4(.01)  equals  0.04
    11%  equals   11(.01)  equals  0.11
   110%  equals  110(.01)  equals  1.10

Decimal values can be written as percents by multiplying by 100%.

   0.02  equals  0.02(100)%  equals    2%
   0.33  equals  0.33(100)%  equals   33%
   2.10  equals  2.10(100)%  equals  210%

Percents can be written as fractions by multiplying by 1/100 (or one one-hundreth).

     5%  equals    5(1/100)  equals    5/100
    25%  equals   25(1/100)  equals   25/100
   110%  equals  110(1/100)  equals  110/100

PurpleMath.com:: Converting Between Decimals, Fractions, and Percents [opens new window]

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Converting Fractions Into Percents and Vice Versa

The following algorithm converts a fraction into a percent.

   a)  divide the numerator by the denominator

   b)  multiply the resulting decimal number by 100(%)

Examples.

    3/8 = 0.375;     0.375 * 100(%) = 37.5%
    1/2 = 0.5;       0.5 * 100(%) = 50%
   7/11 = 0.6364;    0.6364 * 100(%) = 63.64%
   11/7 = 1.5714;    1.5714 * 100(%) = 157.14%

The following algorithm converts a percent into a fraction.

   a)  write the percent as a fraction (n% = n/100)

   b)  reduce the fraction, if possible

Examples.

   25% = 25/100;        25/100 reduces to 1/4
   62% = 62/100;        62/100 reduces to 31/50
   202% = 202/100;      202/100 reduces to 2 1/50
   1% = 1/100;          1/100 is reduced
   0.2% = 0.2/100;      0.2/100 reduces to 0.1/50 or 1/500

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Percent Change

The percentage increase between two values is calculated as follows.

   new value - original value
   -------------------------- * 100
          original value

The percentage decrease between two values is calculated as follows.

   original value - new value
   -------------------------- * 100
          original value
Examples

The Maricopa Community Colleges increased 2005-06 in-state tuition rates by $5 per credit-hour. The 2004-05 tuition rate was $55.

            old rate:  $55   [2004-05 rate]
            increase:  $ 5   ["increase" implies addition]
            new rate:  $60   [sum of $55 and $5]

   percentage increase:

      60 - 55            5
      -------     =     ---   =    0.0909
         55              55

      0.0909 * 100 = 9.09%

The Maricopa Community Colleges increased 2006-07 in-state tuition rates from $60 per credit-hour to $65 per credit-hour.

   percentage increase:

      65 - 60            5
      -------     =     ---   =    0.0833
         60              60

      0.0833 * 100 = 8.33%

The Valley Metro Transit is considering decreasing bus fares by $0.25. Current bus rates are $1.25 per two hours of bus riding.

        current rate:  $1.25
   proposed decrease:  $0.25  [the word "decrease" implies subtraction]
            new rate:  $1.00  [difference of $1.25 and $1.00]

   percentage decrease:

   1.25 - 1.00           0.25
   -----------     =     ----   =    0.2000
       1.25              1.25


   0.2000 * 100 = 20%

PurpleMath.com:: "Percent of" Word Problems: General Increase and Decrease [opens new window]

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"Percent of" Problems

Percent problems come in many forms. Three popular forms are often worded as follows.

   What is 'a' percent of 'b'?   

   'a' is what percent of 'b'?   

   'a' is 'b' percent of what?   

Here are some rules to map these type of percent word problems into equations.

   "what" becomes a variable
   "is" becomes an equals operator
   "of" becomes a multiply operator

1) What is 5% of 10?

   n = 5% * 10            ...variable is named 'n'
   n = 5(.01) * 10        ...%5 converted to decimal
   n = 0.05 * 10          ...word "of" became times
   n = 0.5                ...decimal arithmetic

   0.5 is 5% of 10        ...final answer

2) 7 is what percent of 49?

   7 = n * 49
   
   7     n * 49
   -- =  -------
   49       49

   1
   - = n
   7

   .143 = n

   .143 * 100 = 14.3 = n  (final answer:  14.3%)

   7 is 14.3% of 49

3) 4 is 12% of what?

    4 = 12% * n

    4 = 12(.01) * n

     4    .12 * n
    --- = -------
    .12     .12

    33.3 = n

    4 is 12% of 33.3

External Hyperlinks

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Percent Exercises

_______% of the five cells are red.
                   
_______% of the five cells are green.
                   
_______% of the ten cells are not white.
                                       
Shade 45% of the five cells.
                   
Shade 0.5% of the five cells.
                   

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