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MAT081 :: Lecture Note :: Week 07
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Ratios

A ratio is the "relationship in quantity, amount, or size between two or more things." [source::m-w.com::Merriam-Webster Online]

The general form of a ratio is as follows is n:m, where n is the quantity of one thing and m is the quantity of another thing. For example, if a recipe calls for 3 parts lemon and 5 parts lime, then the ratio is 3:5.

Sometimes ratios are written using the word to instead of a colon. For example, 3 parts lemon to 5 parts lime.

Fractions can be written as ratios.

    1/7  is   1:7  or   1 to 7
    3/8  is   3:8  or   3 to 8
   15/7  is  15:7  or  15 to 7

Which of the following, if any, are valid ratios?

   8:8   1.5:1    4:2.25    0:55    4:0    2:-1

A rate is a special kind of ratio, indicating a "relationship between two measurements with different units."

PurpleMath.com:: Ratios [opens new window]

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Introduction to Rates

A rate is a ratio that "relate" different "units." Merriam-Webster Online defines rate as "a fixed ratio between two things."

Example rates.

   65 miles per hour (mpg)   [ratio -- 65:1]
   You can travel 65 miles in one hour.

   $0.59 per pound           [ratio -- 0.59:1]
   It will cost you $0.59 to buy one pound of foo.

   60 beats per minute       [ratio -- 60:1]
   A good heart rate for a man at rest is 60 beats in one minute.
   {"beat" is one complete pulsation of the heart}

It is possible for us to manually measure our heart rate. We were given a ratio of 60 beats per minute, but a minute is a long time; therefore, we can approximate our heart rate by counting the heart beats for 10 seconds and multiply the final count by 6. [There are 60 seconds per minute.]

TopEndSports.com:: Heart rate measurement

The term per implies a ratio (i.e. fraction i.e. division).

   Receive 10 coupons per every 3 purchases.
                                                    _
   ratio... 10:3   fraction... 10/3   division... 3.3

There are many "forms" of rates. Birth rates, death rates, tax rates, and numerous other financial rates (e.g. mortgage rates, commission rates, interest rates, inflation rates just to name a few). There are also discount rates, tipping rates, data transfer rates, phone rates, turnover rates, failure rates and so on.

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Proportions

A proportion is a "statement of equality between two ratios in which the first of the four terms divided by the second equals the third divided by the fourth." [source::m-w.com::Merriam-Webster Online]

Given two ratios, n:m and a:b, n is a as m is to b.

Two ratios are equal if the cross-products are equal.

     /----- * ----\
     |            |
   2:4  equals   10:20
   |                 |
   \------- * -------/

   2 * 20 = 40
   4 * 10 = 40

From the book "Zero: The Biography of a Dangerous Idea"...

   "To the Pythagoreans, ratios and proportions controlled
    musical beauty, physical beauty, and mathematical beauty.
    Understanding nature was as simple as understanding the
    mathematics of proportions."

PurpleMath.com:: Proportions: Introduction [opens new window]

YouTube.com::Math Education Professor Dor Abrahamson

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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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