Ratios are a helpful tool for comparing things to each other in mathematics and real life, so it is important to know what they mean and how to use them. These descriptions and examples will not only help you to understand ratios and how they function but will also make calculating them manageable no matter what the application.

## What Is a Ratio?

In mathematics, a ratio** **is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the *antecedent* and the divisor or number that is dividing termed the *consequent*.

Example: you have polled a group of 20 people and found that 13 of them prefer cake to ice cream and 7 of them prefer ice cream to cake. The ratio to represent this data set would be 13:7, with 13 being the antecedent and 7 the consequent.

A ratio might be formatted as a Part to Part or Part to Whole comparison. A Part to Part comparison looks at two individual quantities within a ratio of greater than two numbers, such as the number of dogs to the number of cats in a poll of pet type in an animal clinic. A Part to Whole comparison measures the number of one quantity against the total, such as the number of dogs to the total number of pets in the clinic. Ratios like these are much more common than you might think.

## Ratios in Daily Life

Ratios occur frequently in daily life and help to simplify many of our interactions by putting numbers into perspective. Ratios allow us to measure and express quantities by making them easier to understand.

Examples of ratios in life:

- The car was traveling 60 miles per hour, or 60 miles in 1 hour.
- You have a 1 in 28,000,000 chance of winning the lottery. Out of every possible scenario, only 1 out of 28,000,000 of them has you winning the lottery.
- There were enough cookies for every student to have two, or 2 cookies per 78 students.
- The children outnumbered the adults 3:1, or there were three times as many children as there were adults.

## How to Write a Ratio

There are several different ways to express a ratio. One of the most common is to write a ratio using a colon as a this-to-that comparison such as the children-to-adults example above. Because ratios are simple division problems, they can also be written as a fraction. Some people prefer to express ratios using only words, as in the cookies example.

In the context of mathematics, the colon and fraction format are preferred. When comparing more than two quantities, opt for the colon format. 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. Consider the context of the comparison when deciding how best to write your ratio.

## Simplifying Ratios

No matter how a ratio is written, it is important that it be simplified down to the smallest whole numbers possible, just as with any fraction. This can be done by finding the greatest common factor between the numbers and dividing them accordingly. With a ratio comparing 12 to 16, for example, you see that both 12 and 16 can be divided by 4. This simplifies your ratio into 3 to 4, or the quotients you get when you divide 12 and 16 by 4. Your ratio can now be written as:

- 3:4
- 3/4
- 3 to 4
- 0.75 (a decimal is sometimes permissible, though less commonly used)

## Practice Calculating Ratios With Two Quantities

Practice identifying real-life opportunities for expressing ratios by finding quantities you want to compare. You can then try calculating these ratios and simplifying them into their smallest whole numbers. Below are a few examples of authentic ratios to practice calculating.

- There are 6 apples in a bowl containing 8 pieces of fruit.
- What is the ratio of apples to the total amount of fruit? (answer: 6:8, simplified to 3:4)
- If the two pieces of fruit that are not apples are oranges, what is the ratio of apples to oranges? (answer: 6:2, simplified to 3:1)

- Dr. Pasture, a rural veterinarian, treats only 2 types of animals—cows and horses. Last week, she treated 12 cows and 16 horses.
- What is the ratio of cows to horses that she treated? (answer: 12:16, simplified to 3:4. For every 3 cows treated, 4 horses were treated)
- What is the ratio of cows to the total number of animals that she treated? (answer: 12 + 16 = 28, the total number of animals treated. The ratio for cows to total is 12:28, simplified to 3:7. For every 7 animals treated, 3 of them were cows)

## Practice Calculating Ratios With Greater Than Two Quantities

Use the following demographic information about a marching band to complete the following exercises using ratios comparing two or more quantities.

**Gender**

- 120 boys
- 180 girls

**Instrument type**

- 160 woodwinds
- 84 percussion
- 56 brass

**Class**

- 127 freshmen
- 63 sophomores
- 55 juniors
- 55 seniors

1. What is the ratio of boys to girls? (answer: 2:3)

2. What is the ratio of freshmen to the total number of band members? (answer: 127:300)

3. What is the ratio of percussion to woodwinds to brass? (answer: 84:160:56, simplified to 21:40:14)

4. What is the ratio of freshmen to seniors to sophomores? (answer: 127:55:63. Note: 127 is a prime number and cannot be reduced in this ratio)

5. If 25 students left the woodwind section to join the percussion section, what would be the ratio for the number of woodwind players to percussion?

(answer: 160 woodwinds – 25 woodwinds = 135 woodwinds;

84 percussionists + 25 percussionists = 109 percussionists. The ratio of the number of players in woodwinds to percussion is 109:135)