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# Exploring Ratios and Proportions Study Guide

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Updated on Oct 4, 2011

## Introduction to Exploring Ratios and Proportions

Mathematics is a language.

—JOSIAH WILLARD GIBBS, theoretical physicist (1839–1903)

This lesson begins by exploring ratios, using familiar examples to explain the mathematics behind the ratio concept. It concludes with the related notion of proportions, again illustrating the math with everyday examples.

## Ratios

A ratio is a comparison of two numbers. For example, let's say that there are 3 men for every 5 women in a particular club. That means that the ratio of men to women is 3 to 5. It doesn't necessarily mean that there are exactly 3 men and 5 women in the club, but it does mean that for every group of 3 men, there is a corresponding group of 5 women. The table below shows some of the possible sizes of this club.

In other words, the number of men is 3 times the number of groups, and the number of women is 5 times that same number of groups.

A ratio can be expressed in several ways:

• using "to" (3 to 5)
• using "out of" (3 out of 5)
• using a colon (3:5)
• as a fraction
• as a decimal (0.6)

Like a fraction, a ratio should always be reduced to lowest terms. For example, the ratio of 6 to 10 should be reduced to 3 to 5 (because the fraction reduces to ).

Here are some examples of ratios in familiar contexts:

 Last year, it snowed 13 out of 52 weekends in New York City. The ratio 13 out of 52 can be reduced to lowest terms (1 out of 4) and expressed as any of the following: Reducing to lowest terms tells you that it snowed 1 out of 4 weekends, ( of the weekends or 25% of the weekends). Lloyd drove 140 miles on 3.5 gallons of gas, for a ratio (or gas consumption rate) of 40 miles per gallon: miles per gallon The student-teacher ratio at Clarksdale High School is 7 to 1. That means for every 7 students in the school, there is 1 teacher. For example, if Clarksdale has 140 students, then it has 20 teachers. (There are 20 groups, each with 7 students and 1 teacher.) Pearl's Pub has 5 chairs for every table. If it has 100 chairs, then it has 20 tables. The Pirates won 27 games and lost 18, for a ratio of 3 wins to 2 losses. Their win rate was 60%, because they won 60% of the games they played.

In word problems, the word per translates to division. For example, 30 miles per hour is equivalent to . Phrases with the word per are ratios with a bottom number of 1, like these:

 24 miles per gallon \$12 per hour 3 meals per day 4 cups per quart

#### Tip

Ratios can also be used to relate more than two items, but then they are not written as fractions. Example: If the ratio of infants to teens to adults at a school event is 2 to 7 to 5, it is written as 2:7:5.

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