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.
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.
The order of a ratio is very important. The ratio 5:2 is not the same as the ratio 2:5. If we say that there are five apples for every two oranges, we must write 5:2. The ratio 2:5 would mean that there are two apples for every five oranges, which is a different relationship.
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.
Algebraic Expressions in Ratios
The values in a ratio can also be algebraic expressions. We can use ratios and proportions to solve for the value of the variable in these expressions.
The ratio of sopranos to altos in a choir is 5:7. If there are 6 more altos than sopranos in the choir, how many sopranos are in the choir?
The number of sopranos is unknown, so we will let x represent that number. There are 6 more altos than sopranos, which means that the number of altosis equal to x + 6. The ratio of sopranos to altos is 5:7, or . We can write the ratio of actual sopranos, x, to actual altos, x + 6, as a ratio, too: . Since these ratios are equal, we can write a proportion and solve for x:
|7x = 5x + 30|
|2x = 30|
|x = 15|
|There are 15 sopranos in the choir.|
There is more than one way to write a ratio that contains algebraic expressions. In the previous example, if we had wanted to find the number of altos in the choir, we could have let x equal the number of altos, and the number of sopranos would have been x – 6, since there are 6 more altos than sopranos. We would have set equal to , and found that x, the number of altos, is equal to 21. As always, let x represent the value that you are looking to find.
Find practice problems and solutions for these concepts at Algebra Ratios and Proportions Practice Questions.
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