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How to Solve Ratio Word Problems

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  • 0:04 Where Are Ratios?
  • 0:50 What is a Ratio?
  • 1:23 What is a…
  • 2:33 Solving Proportions…
  • 3:11 Solving Ratio Word Problems
  • 4:46 Lesson Summary
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Lesson Transcript
Instructor
Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

We see ratios all around us every day. From the grocery store, to forecasting into the future, to enlarging or shrinking pictures, a command of ratios can be a powerful math tool. In this lesson, learn how to solve word problems with ratios in them.

Where Are Ratios?

Ratios are everywhere around us. Try these on for size:

  • A 5 oz. bag of gummy bears is $1.49. Is it a better deal to get the 144 oz. bag for $15.99?
  • You've got 60 homework problems to do and it took you 10 minutes to do eight of them. At that rate, how long will it take?
  • Your favorite painting in the museum is 5 feet by 8 feet. How big will the eyes in that painting be on your smart phone's 4.3-inch screen?

We could go on and on; and while each of these appear to be different problems - dealing with money, time, and size - they are, at their core, the same. They all involve ratios.

Let's break down ratios a little more and see how they can help us solve these types of problems.

What Is a Ratio?

A ratio is a comparison between two numbers. To keep it simple, we'll ignore the units (e.g., cost in dollars or weight in ounces) and focus just on the number part for a bit. For example, how does 3 compare to 6? Well, three is half of six. We can write ratios in one of three ways:

  1. 3:6
  2. 3/6
  3. 3 to 6

Because we'll be using ratios mathematically, we'll use the format '/' for the rest of the lesson.

What Is a Proportion?

By itself, a ratio is limited to how useful it is. However, when two ratios are set equal to each other, they are called a proportion. For example, 1/2 is a ratio and 3/6 is also a ratio. If we write 1/2 = 3/6, we have written a proportion. We can also say that 1/2 is proportional to 3/6. In math, a ratio without a proportion is a little like peanut butter without jelly or bread.

How Proportions Can Help

In math problems and in real life, if we have a known ratio comparing two quantities, we can use that ratio to predict another ratio, if given one half of that second ratio. In the example 1/2 = 3/?, the known ratio is 1/2. We know both terms of the known ratio. The unknown ratio is 3/?, since we know one term, but not the other (thus, it's not yet a comparison between two ratios). We only know one of the two terms in the unknown ratio. However, if we set them as a proportion, we can use that proportion to find the missing number.

Solving Proportions with an Unknown Ratio

There are a few different methods we can use to solve proportions with an unknown ratio. However, the easiest and most fail-safe method is to cross-multiply and solve the resulting equation. For the last example, we would have:

ratio

1 * x = 2 * 3
1x = 6
x = 6 / 1
x = 6

To check the accuracy of our answer, simply divide the two sides of the equation and compare the decimal that results. In the example, 1/2 = 0.5 and 3/6 = 0.5. That was the correct result.

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Additional Activities

Ratio Word Problems in Real Life

Imagine you are going to host a large dinner party for 20 people, including yourself. You have a lot of favorite recipes you want to make, but none of them are written to serve 20 people. Can you use your skills in solving ratio word problems to figure out the quantities you would need for the ingredients listed below?

Problems

1) Your recipe for chocolate chip cookies uses 1 cup of flour for 12 cookies. How much flour will you need in order to make 20 cookies?

2) You recipe for beef and barley soup serves 6 people and requires 4 cups of beef broth. How many cups of beef broth will you need to serve 20 people?

3) Your spaghetti sauce recipe for your world-famous spaghetti and meatballs requires 3.5 pounds of tomatoes. The sauce recipe makes enough sauce for 8 servings. How many pounds of tomatoes will you need to make enough sauce for 20 servings?

4) Your Caesar salad recipe uses 4 pounds of romaine lettuce for 15 servings. How much romaine lettuce do you need to serve all 20 people?

Solutions

1) 1 cup of flour is used for 12 cookies, and we can write that ratio as 1/12. We need to find how many cups of flour are needed for 20 cookies. We can set up a proportion and solve by cross multiplying.

1/12 = x/20

1 * 20 = 12x

20/12 = x

5/3 = x

You will need 5/3 cups of flour - or 1 2/3 cups of flour.


2) 4 cups of beef broth are needed for 6 people, which gives a ratio of 4/6. To find how many cups of beef broth are needed to make enough soup for 20 people, set up a proportion and solve by cross multiplying.

4/6 = x/20

4 * 20 = 6x

80 = 6x

80/6 = x

40/3 = x

You will need 40/3, or 13 1/3, cups of beef broth.

3) 3.5 pounds of tomatoes makes enough sauce for 8 servings. This is the ratio 3.5/8. To find how many pounds of tomatoes are needed for 20 servings, set up a proportion and solve with cross multiplication.

3.5/8 = x/20

3.5 * 20 = 8x

70 = 8x

70/8 = x

8.75 = x

You will need 8.75 pounds of tomatoes.

4) 4 pounds of romaine lettuce for 15 servings will result in the ratio 4/15. To find how many pounds are needed for 20 servings, set up a proportion and cross multiply.

4/15 = x/20

4 * 20 = 15x

80 = 15x

80/15 = x

16/3 = x

You will need 16/3 pounds, or 5 1/3 pounds of romaine lettuce.

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