Algebra Ratios and Proportions Study Guide (page 2)

Introduction to Algebra Ratios and Proportions

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction.

—Lev Nikolayevich Tolstoy (1828–1920) Russian Writer

In this lesson, you'll learn how to use algebra to define ratios and proportions and find exact values using these ratios and proportions.

A ratio is a relationship between two or more quantities. If there are 15 girls and 10 boys in your class, we can say that the ratio of girls to boys is "15 to 10." That ratio can be written with a colon, as in 15:10, or as a fraction, as in . Fractions represent division, and we can simplify ratios just as we simplify fractions. Because the greatest common factor of 15 and 10 is 5, the ratio reduces to . For every 3 girls in your class, there are 2 boys.

A proportion is an equation that shows two equal ratios. Because the ratio reduces to , we can write the proportion . We say that these two ratios are in proportion to each other.

Example

The ratio of apples to oranges in a basket is 5:2. If there are 25 apples in the basket, how many oranges are in the basket?

We are given the ratio of apples to oranges, which we can write as the fraction . Because we know that there are 25 apples, the ratio is a reduced form of the ratio of the number of actual apples to the number of actual oranges. We do not know the number of actual oranges, so we will represent that with x. The ratio of the number of actual apples to the number of actual oranges is . Now that we have two equal ratios, we can write a proportion that sets them equal to each other: . To solve a proportion, we cross multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction. Then, set those two products equal to each other:

(5)(x) = (2)(25)

5x = 50

x = 10

The number of actual oranges is 10. We can check our answer by putting 10 into the ratio of actual apples to actual oranges. Because reduces to , 10 must be the correct answer.

We can also use a proportion to find an exact number, given a ratio and a total. If the ratio of apples to oranges is 2:3, and the total number of pieces of fruit is 40, we can write a part-to–total ratio. There are 2 apples for every 3 oranges, which means that there are 2 apples for every 2 + 3 = 5 pieces of fruit. We can find the actual number of apples by comparing the ratio of apples to total pieces of fruit, 2:5, to the ratio of actual apples to actual total number of pieces of fruit, x:40. Now, we can write a proportion and solve for x:

(5)(x) = (2)(40)

5x = 80

x = 16

If there are 40 pieces of fruit and the ratio of apples to oranges is 2:3, then there are 16 apples in the basket.