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Ratios and Proportions in Word Problems Study Guide

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Updated on Aug 24, 2011

Find practice problems and solutions for these concepts at Ratios and Proportions in Word Problems Practice Problems.

We can use relationships between numbers or pairs of numbers to solve problems. For instance, let's say a deck of cards contains twice as many red cards as black cards. If we know the number of red cards, we can find the number of black cards, because we know the relationship between the number of red cards and the number of black cards. This kind of relationship is called a ratio. A ratio is a comparison, or relationship, between two numbers. Ratios can be shown using a colon or as a fraction. The ratio 3 to 2 can be written as 3:2 or .

Ratios: Subject Review

If there are 16 black pens and 12 blue pens on a desk, then the ratio of black pens to blue pens is 16:12. Ratio can be reduced just like fractions, since ratios are fractions. The greatest common factor of 16 and 12 is 4, so both numbers can be divided by 4. The ratio 16:12 is the same as the ratio 4:3.

Caution

The order of the numbers in a ratio is important. The ratio 3:4 is not the same as the ratio 4:3. Remember, ratios are fractions, and the fraction is not equal to the fraction . If you are describing the ratio of apples to oranges, place the number of apples first in your ratio.

If a word problem simply asks you to find or write a ratio, you may not need the eight-step process to find the answer. However, as with any kind of word problem, the eight-step process will work, and you should use it if you find that you cannot solve a word problem quickly.

Example

If there are 20 red cars and 24 blue cars in a parking lot, what is the ratio of red cars to blue cars?

This problem asks us to find a ratio, and it gives us the number of red cars and the number of blue cars. No computation is needed. The ratio of red cars to blue cars is 20:24. We can reduce the ratio to 5:6.

Example

A parking lot contains only red cards and blue cars. If there are 36 total cars and 16 of them are red, what is the ratio of blue cars to red cars?

Again, we are asked to write a ratio, but this time, we are not given the number of blue cars. We can find the number of blue cars by subtracting the number of red cars from the total number of cars: 36 – 16 = 20. We must be sure to write our ratio in the correct order. The problem asks us to find the ratio of blue cars to red cars, which is 20:16, or 5:4. Determining the question being asked can make the difference between a right or wrong answer.

Pace Yourself

Find the ratio of $1 bills to $5 bills in your wallet or purse. What would that ratio be if you added three $5 bills to your wallet or purse?

Proportions: Subject Review

We can use ratios to solve problems by setting up a proportion. A proportion is a relationship between two equivalent ratios. A proportion usually contains a reduced ratio that represents a relationship between two quantities, and a larger, equivalent ratio that represents the exact values of those two quantities. The ratio 16:12 is equivalent to the ratio 4:3. These equivalent ratios can be expressed as a proportion: .

If the ratio of quantity A to quantity B is 7:2, and the exact value of quantity A is 35, we can set up a proportion to solve for x, the exact value of quantity B. As a fraction, 7:2 is . Set that fraction equal to the exact value of quantity A over the exact value of quantity B: . Cross multiply and set the products equal to each other: 7x = 70, x = 10. If the exact value of quantity A is 35, then exact value of quantity B is 10.

If the ratio of quantity A to quantity B is 7:2, and the exact total of quantity A and quantity B is 27, we can find the exact value of quantity A and quantity B by writing ratios that represent the part-to-whole relationship. If the ratio of quantity A to quantity B is 7:2, then 7 of every 7 + 2 = 9 items are of quantity A. The ratio of quantity A to the whole is 7:9. The whole is 27, and exact number of quantity A is x: . Cross multiply and set the products equal to each other: 9x = 189, x = 21. If the total of quantity A and quantity B is 27, then the total of quantity A is 21. The total of quantity B can be found with the proportion . 9x = 54, x = 6. If the total of quantity A and quantity B is 27, then the total of quantity B is 6.

Now that we remember how to use ratios and proportions, let's look at how to recognize and solve ratio and proportion word problems.

Ratio word problems often contain the word ratio or every. You may also see the ratio expressed in colon form, such as 2:3. Once you've recognized a word problem as a ratio and proportion problem, decide if it is a part-to-part problem or a part-to-whole problem. A part-to-part problem gives you a ratio and the value of one quantity, and asks you for the value of the other quantity. A part-to-whole problem gives you a ratio and the total value of both quantities, and asks you for the value of one quantity.

Inside Track

Although the keyword total can often signal addition, in a ratio problem, the word total can signal a part-to-whole problem rather than a part-to-part problem.

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