# Ratios and proportions

A ratio is a comparison between quantities. A proportion is a set of ratios that are equal. Ratios in a proportion are related to one another by multiplication by some constant.

## Ratios

There are a few different ways to express a ratio. For example, the ratio of boys to girls in a class can be expressed as:

• 2 boys for every 3 girls
• 2 to 3
• 2:3
• 2/3

Ratios can also be expressed as part-to-part or part-to-whole ratios. The above example (boys to girls) is an example of a part-to-part ratio. Assuming there are only 5 students in the class, an example of a part-to-whole ratio is the ratio of girls to students in the class, or 3:5. For boys, the ratio would be 2:5.

All ratios can be scaled to form equivalent ratios by multiplying both parts of the ratio by the same constant. This is useful in everyday applications such as cooking, where scaling a recipe up or down may be necessary. For example, a pasta recipe that calls for 2 cups of pasta to 3 cups of water that feeds 2 people could be doubled to an equivalent ratio of 4:6 to feed 4 people.

## Proportions

Proportions are equations made up of two equivalent ratios. The following are all proportions:

• 2:3 = 6:9

A ratio of 2:3 is said to be proportional to a ratio of 4:6 (or 6:9, 8:12, etc.). Proportions indicate that the relative sizes of the objects being compared are the same. This means that given two objects that are proportional, it is possible to determine certain attributes of either object given information about the other; this is done by solving the proportion using cross multiplication.

Example

If the ratio of apples to students is proportional, and there are 2 apples for every 1 student, how many apples are there in a class of 30 students?

This problem can be set up as a proportion. The ratio of apples to students is 2:1. The total number of students is known. To find the number of apples, write the ratios in fraction form and cross multiply:

2 × 30 = 1 × #

# = 60

If there are 2 apples per student, and there are 30 students, then there are 60 apples.