           ## Proportions

Ratios are useful ways to compare two quantities. But how do you compare ratios? For example, in Figure 1 below two out of the three circles are shaded, and in Figure 2 below four out of the six circles are shaded. Although Figure 2 has more circles, the ratio of shaded circles to total circles is the same. That is, . A statement such as this that one ratio is equal to another is called a proportion. Proportional reasoning involves the ability to compare and produce equal ratios. A common use of proportions occurs when making or using maps and scale models.

There are several ways to solve a proportion. One is related to how you find equivalent fractions. To find equivalent fractions, you multiply or divide the numerator and denominator by the same number. Thus, to solve , note that the numerator (2) was multiplied by 3 in order to get 6. Then do the same to the denominator (3) to get n=9. This works well when the numerator and denominator of one fraction are multiples of the other fraction, but it is more difficult to do when they are not, as in the case of . In this case we multiply both sides of the equation by 8n as shown below:  (multiplying both sides by 8n) (divide out the common factors) (divide both sides by 6) (simplify)
The third step above illustrates a rule that can be used when solving proportions, called cross multiplication. That is, when you have a proportion, you can solve it by multiplying the numerator of the first fraction and the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction. Thus, for a proportion . This cross product property only works when solving a proportion. It does not apply when doing operations with fractions, such as multiplying or dividing fractions. Using this idea of cross multiplying inappropriately is a common problem students have. The reason it works is because you are really multiplying both sides of an equation by the product of the two denominators.