Ratios and Proportions Study Guide: GED Math

Proportions

A proportion is an equation that states that two ratios are equal. Often, in addition to comparing items, it is natural to want to compare ratios.

For example, if you want to choose an amusement park, you might want to compare the number of gift shops to the number of rides. Park A has 10 gift shops and 50 rides. Park B has 17 gift shops and 85 rides. For park A, the ratio is 1 to 5: shops/rides = . For park B, the ratio is also 1 to 5: shops/rides = . For both parks, the ratios are exactly the same. By using a proportion, you can compare these parks on this issue. The proportion is , because both ratios equal .

A proportion is a way of relating two ratios to one another. Let's say that 8 out of 10 students in your study group are expected to take the GED this year. If there are 100 people in your study group, then 80 people are expected to take the test this year. This is an example of a proportion. Proportions can be written as equations. For example, this proportion can be written as = .

Proportions show equivalent fractions. Both and reduce to the same fraction: .

For a proportion to work, the terms in both ratios have to be written in the same order. Notice that the numerator in each ratio in the study group proportion example refers to the number of students expected to take the exam. The denominator refers to the total number of students.

Let's say you didn't see immediately that would be equal to . How could you have figured out the equivalent ratio? Remember in Chapter 2 when you were working with fractions? You learned the following steps to raise a fraction to higher terms.

Step 1     Divide the denominator of the fraction into the new denominator.

Step 2     Multiply the quotient, or the answer to step 1, by the numerator.

Step 3     Write the product, or the answer to step 2, over the new denominator.

Example

Divide the denominator into the new denominator, which is 100: 100 ÷ 10 = 10.

Multiply 10 by the numerator: 8 × 10 = 80.

Write 80 over the new denominator: .

There's another way to solve for the missing term. It's called cross multiplying or finding the cross products. Here's how cross multiplying works.

Step 1     Multiply the numerator of the first ratio by the denominator in the second ratio.

Step 2     Divide the product (the answer to step 1) by the denominator in the first ratio. The answer is the missing numerator in the second ratio.

You can also use cross multiplication to check that two ratios are equal. When a proportion is set up properly, the results of cross multiplication should be equal.

Example

Use cross multiplication to check that the two ratios in this proportion are equal: = .

Multiply the numerator of the first ratio by the denominator in the second ratio: 8 × 100 = 800.

Multiply the denominator of the first ratio by the numerator in the second ratio: 10 × 80 = 800.

The answers are equal, so your proportion is valid: 800 = 800.

Solving Proportion Word Problems

Proportions are common in word problems. Let's look at some examples of proportion word problems.

Examples

Margaret drove 220 miles in five hours. If she maintained the same speed, how far could she drive in seven hours?

Set up a proportion: 220 miles/5 hours = ? miles/7 hours, or .

Solve for the missing number in the second ratio: 220 × 7 ÷ 5 = 308, so .

Check your work by cross multiplying:

220 × 7 = 5 × 308

1,540 = 1,540

Keeping the same speed, Margaret could drive 308 miles in seven hours.

1. Harry earns $6 per hour at his job. If he works nine hours this week, how much will Harry earn?

Set up a proportion: $6/1 hour = $?/9 hours, or .

Solve for the missing number in the second ratio: 6 × 9 ÷ 1 = $54. Therefore, .

Check your work by cross multiplying: 6 × 9 = 54 and 1 × 54 = 54.

Working seven hours, Harry will make $54.

Practice problems for these concepts can be found at:

Measurement Practice Problems: GED Math