Proportions: Overview
Students have seen that ratios are useful ways to compare two quantities. The next step is to compare ratios. For example, Figure 1 below shows two out of the three circles shaded, and Figure 2 below shows four out of the six circles 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, stating 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 , you note that the numerator (2) would be multiplied by 3 in order to get 6. So you do the same to the denominator (multiply by 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 you could 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 cross multiplication, a method that can be used when solving proportions. 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 .
The reason cross multiplication works is because you are really multiplying both sides of an equation by the product of the two denominators. 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 cross multiplying inappropriately is a common mistake many students make.