Ratios and Proportions
      Ratios and proportion properties
      Direct proportion
      Cross product, means and extremes
A ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. 
A ratio can be written as,  a : b  or as a fraction  a/b  or by the phrase  ' a to b '. 
A ratio is a way in which quantities can be divided or shared. It shows how much bigger one thing is than another.
Make the numbers in ratios smaller so that they are easier to compare. Do this by dividing each side of the ratio by the same number, the highest common factor.
A proportion is a statement of the equality of two ratios. If the fractions both reduce to the same value, the proportion is true.
Example: 9 : 6 = 3/2   and,   21 : 14 = 3/2,  follows,  9 : 6 = 21 : 14.
Ratios and proportion properties
A ratio's value is not altered if both the first and second term are multiplied or divided by the same number different from zero.
              a : b = (a c) : (b c)    
Two quantities are in direct proportion when they increase or decrease in the same ratio.
When two ratios are equal, then the cross products of the ratios are equal. 
That is, a : b = c : or
the product of the means is equal to the product of the extremes.
The first and last term in a proportion are called the extremes, the second and third terms are called the means.
If any three terms in a proportion are given, the fourth may be found.
  the extremes   the means  
Examples:  a)  2 : x = 6 : 15   b)  8 : (x - 1) = 2 : 7   c)  (7 - x) : 3 = 4x : 9
           9 (7 - x) = 3 4x
               63 - 9x = 12x
            x = 5                    x = 29                 21x = 63 | 21
                            x = 3
The first and third terms, and the second and the fourth terms in a proportion are the corresponding terms.
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