### Ratio Definition

In mathematics, a **ratio **is a numerical comparison of 2 or more quantities which indicates their relative sizes. It can be considered as a way of comparing numbers by division. In a ratio of two numbers, the first value is termed the ancedent and the second number is the consequent.

### Ratios in Daily Life

- 60 miles per 1 hour (the "per" word instantly indicates a ratio)
- Cost per text message

- 1 to 28,000,000 chance of winning the lottery
- 3 teachers per 76 students

### How to Write a Ratio

It's fine to write a ratio using a colon, as a this-to-that comparison, or as a fraction. In math, it's usually preferable to simplify the comparison to the smallest whole numbers. So, instead of comparing 12 to 16, you can divide each number by 4 to get a ratio of 3 to 4.

- 3:4
- 3 to 4
- 3/4
- 0.75 (a decimal is also permissible, although it's less commonly used)

If you are asked to provide an answer "as a ratio", the colon format or fraction is usually preferred over the verbal comparison.

The big advantage of using the colon for ratios is apparent when you compare more than two values. For example, if you are preparing a mixture that calls for 1 part oil, 1 part vinegar, and 10 parts water, you could express the ratio of oil to vinegar to water as 1:1:10. It's also useful to express dimension of an object. For example, the ratio of dimensions of a block of wood might be 2:4:10 (a two-by-four that is 10 feet long).

Note that the numbers are not simplified in this context.

### Ratio Example Calculations

A simple example would be comparing numbers of types of fruit in a bowl. If there are 6 apples in a bowl containing 8 pieces of fruit, the ratio of apples to the total amount of fruit would be 6:8, which reduces to 3:4.

If two of the pieces of fruit are oranges, the ratio of apples to oranges is 6:2 or 3:1.

For example: Dr. Pasture, a rural veterinarian, treats only 2 types of animals – cows and horses. Last week, she treated 12 cows and 16 horses.

Part to Part Ratio: What is the ratio of cows to horses that she treated?

Simplify: 12:16 = 3:4

For every 3 cows that Dr. Pasture treated, she treated 4 horses.

Part to Whole Ratio: What is the ratio of cows that she treated to the total number of animals that she treated?

Simplify: 12:30 = 2:5

This can be written as:

- 2:5
- 2 to 5
- 2/5

For every 5 animals that Dr. Pasture treated, 2 of them were cows.

### Sample Ratio Exercises

Use the demographic information about the marching band to complete the following exercises.

Dale Union High School Marching Band

Gender

- 120 boys
- 180 girls

Instrument type

- 160 woodwinds
- 84 percussionists
- 56 brass players

Class

- 127 Freshmen
- 63 Sophomores
- 55 Juniors
- 55 Seniors

1. What is the ratio of boys to girls? 2:3 or 2/3

2. What is the ratio of freshmen to the total number of band members? 127:300 or 127/300

3. What is the ratio of percussionists to the total number of band members? 7:25 or 7/25

4. What is the ratio of juniors to seniors? 1:1 or 1/1

5. What is the ratio of sophomores to juniors?

63:55 or 63/55

6. What is the ratio of freshmen to seniors? 127:55 or 127/55

7. If 25 students left the woodwind section to join the percussion section, what would be the new ratio of woodwinds to percussionists?

160 woodwinds – 25 woodwinds = 135 woodwinds

84 percussionists + 25 percussionists = 109 percussionists

109:135 or 109/135

Edited by Anne Marie Helmenstine, Ph.D.