Gre math Review 1 Arithmetic

1.6 Ratio

The ratio of one quantity to another is a way to express their relative sizes, often in the form of a fraction, where the first quantity is the numerator and the second quantity is the denominator. Thus, if s and t are positive quantities, then the ratio of s to t can be written as the fraction   s over t. The notation  s followed by the word to, followed by t and the notation   s colon t are also used to express this ratio. For example, if there are 2 apples and 3 oranges in a basket, we can say that the ratio of the number of apples to the number of oranges is   2 over 3, or that it is 2 to 3, or that it is 2 : 3. Like fractions, ratios can be reduced to lowest terms. For example, if there are 8 apples and 12 oranges in a basket, then the ratio of the numbers of apples to oranges is still 2 to 3. Similarly, the ratio 9 to 12 is equivalent to the ratio 3 to 4.

If three or more positive quantities are being considered, say r, s, and t, then their relative sizes can also be expressed as a ratio with the notation, r to s to t. For example, if there are 5 apples, 30 pears, and 20 oranges in a basket, then the ratio of the numbers of apples to pears to oranges is 5 to 30 to 20. This ratio can be reduced to 1 to 6 to 4 by dividing each number by the greatest common divisor of 5, 30, and 20, which is 5.

A proportion is an equation relating two ratios; for example,  9 over 12 = 3 over 4. To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication.

Example 1.6.1: To find a number x so that the ratio of x to 49 is the same as the ratio of 3 to 21 you can write
 
x over 49 = 3 over 21.

Then cross multiply to get   21x = 3 times 49,

and solve for x to get   x = the fraction with numerator 3 times 49, and denominator 21, which is equal to 7.