Ratios and Proportions Study Guide: GED Math (page 3)
Practice problems for these concepts can be found at:
Ratios are numbers that are used to compare things. Ratios play an important role in mathematics because they quantify all of the items that you compare on a day-to-day basis. There are several different ways to write ratios. Here are some examples of ways to write ratios.
- with the word to: 1 to 2
- using a colon (:) to separate the numbers: 1:2
- using the phrase for every: 1 for every 2
- separated by a division sign or fraction bar: 1/2 or
Let's look at an example. In the community gardening group, there are 24 women and 16 men. If you want to compare the number of women to the number of men, you can show this comparison in several different ways:
24 to 16
Regardless of which form is used, the meaning is the same: "There were 24 women for every 16 men." Notice that is a fractional form of a ratio. The fractional form of a ratio is often a convenient way to represent a ratio when solving problems.
In addition to comparing women to men, a comparison could also be made between women and total members. The total membership is 24 + 16 = 40 people. This ratio is , or 24 to 40, or 24:40.
Ratios are usually shown in lowest terms and can be simplified in the same way that fractions are simplified. For example, in the gardening group, there are 24 women and 16 men. This ratio can be expressed as 3:2, because . In this group there are three women for every two men. You can also express the ratio of men to total group members. This ratio is 2:5, because .
Write the following ratio as a fraction: 10 wins to 5 losses.
This ratio is .
Even though looks like an improper fraction, it's not, here—it's a ratio comparing the number of wins to the number of losses. You can, however, reduce the ratio to lowest terms: .
Solving Ratio Problems
There are several kinds of ratio problems. The examples that follow show how to solve different kinds of ratio problems.
- A painter mixes two quarts of red paint to three quarts of white paint. What is the ratio of red paint to white paint?
There are several ways you could write this ratio:
- 2 quarts of red paint to 3 quarts of white paint, or 2 to 3
- 2 quarts red paint : 3 quarts white paint, or 2:3
- 2 quarts red paint/3 quarts white paint, or
What is the ratio of games won to games lost, and what is the ratio of games won to games played?
Write your answers as fractions.
The first part of the question asks for the ratio of games won to games lost. So, you would write . You could reduce the ratio to .
The second part of the question asks for the ratio of games won to games played. First, you need to calculate the total number of games played. Because the Tigers won 30 games, lost 6 games, and tied no games, they must have played a total of 36 games. The ratio of games won to games played is 30 games won to 36 total games, or . You could reduce to .
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.
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.
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.
- Harry earns $6 per hour at his job. If he works nine hours this week, how much will Harry earn?
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.
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:
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