# 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
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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.

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:

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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