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 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 * 31x = 6x = 6 / 1x = 6To 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.
Solving Ratio Word Problems
To use proportions to solve ratio word problems, we need to follow these steps:
- Identify the known ratio and the unknown ratio.
- Set up the proportion.
- Cross-multiply and solve.
- Check the answer by plugging the result into the unknown ratio.
Your favorite store says it will donate to your soccer team $3 for every $50 that anyone wearing a soccer shirt spends at the store. Your team needs at least $1,200 donated to be able to travel to a tournament. How much money needs to be spent at the store by people wearing soccer shirts?Our known ratio is $3 donated / $50 spent, and the unknown ratio is $1,200 donated / ? spent. The proportion would look like this:
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