How to Solve Ratio Word Problems
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:
3:6
3/6
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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