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# Ratio and Rate Word Problems Study Guide

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Updated on Oct 3, 2011

## Introduction to Ratio and Rate Word Problems

There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.

—NICOLAI LOBACHEVSKY (1792–1856)

Whether we enjoy them or not, ratios and rates can be found everywhere in our daily lives. Ratios help us to compare quantities of different things. When traveling between locations, we travel at a rate of speed. When we are shopping at a grocery store, the unit price tells us if the cost is reasonable. The information in this lesson will review information about ratios and rate and will give practice solving word problems on these topics.

### Ratio

A ratio is a comparison of two numbers by division. There are three different ways to write a ratio: as a fraction, using a colon, and using the word to.

For example, if the ratio of cats to dogs is 1 to 3, this can also be written as 1:3 or

The order of the terms in the ratio is very important. For instance, in the example above, the word "cats" was mentioned first and the word "dogs" was mentioned second. This means that the first number in the ratio stands for cats and the second number stands for dogs. Knowing this concept is essential as you move forward into more ratios, rates, and eventually, proportions.

## Simplifying Ratios

Simplifying a ratio is basically the same process as simplifying a fraction. To do this, find the greatest common factor between the two numbers in the ratio and divide each by this common factor. For example, if the fans at a football game number 300 for the home team and 200 for the visiting team, this can be written as the ratio 300:200. The greatest common factor of each of these numbers is 100, so the ratio can be simplified to 3:2, 3 to 2, or . This means there are 3 home team fans for every 2 visiting team fans.

#### Tip:

Although they appear very similar to fractions, ratios are different from fractions. Fractions compare a part to a whole. In a ratio, the values could be comparing a part of a group to another part of a group, or part of a group to the entire group. For example, if there are 14 girls and 15 boys in a class, the ratio of girls to boys is but the ratio of girls to total students is . Be aware of the labels of the ratio to help with this concept.

## Continued Ratios

A continued ratio is a ratio that compares more than two numbers. For example, the ratio 2:3:4 is a continued ratio. In order to simplify a continued ratio, divide each term by the greatest common factor of all the terms.

To simplify the continued ratio 15:30:45, divide by the greatest common factor, 15. The ratio simplifies to 1:2:3.

## Rate

A rate is similar to a ratio, but it is specifically a comparison of two units, not only two numbers. Common rates used in everyday life are miles per hour, feet per second, gallons per minute, and basically any time when you are comparing two units.

#### Tip:

A unit rate is a rate where the second number is always one. For instance, traveling 100 miles in 2 hours is a rate, but simplifying this to traveling 50 miles in 1 hour is a unit rate.

To find a unit rate, simply divide the quantity of the first unit by the quantity of the second unit. For example, if someone can type 200 words on a keyboard in 4 minutes, the unit rate is because 200 ÷ 4 = 50.

Find practice problems and solutions for these concepts at Ratio and Rate Word Problems Practice Questions.

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