Write a Ratio

A ratio is one of the most useful tools in everyday math — from cooking to shopping to mixing paint. On the GED Math test, you need to recognize ratios, write them in the correct form, and simplify them. This lesson covers exactly what a ratio is, the three ways to write one, and how to simplify.

What Is a Ratio?

A ratio compares two quantities of the same kind. It tells you how many times larger (or smaller) one quantity is relative to another. For example, if there are 3 red marbles and 5 blue marbles, the ratio of red to blue is 3 to 5.

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Important: the order of a ratio matters. “3 to 5” is not the same as “5 to 3.”

Three Ways to Write a Ratio

1. Word form

Write the two numbers separated by the word “to.”

Example: 3 to 5

2. Colon form

Write the two numbers separated by a colon.

Example: \(\color{blue}{3 : 5}\)

3. Fraction form

Write the first number as the numerator and the second as the denominator.

Example: \(\color{blue}{\frac{3}{5}}\)

All three forms — 3 to 5, 3 : 5, and \(\color{blue}{\frac{3}{5}}\) — represent the exact same ratio.

Simplifying a Ratio

A ratio is in simplest form (or lowest terms) when the two numbers share no common factor other than 1. To simplify, divide both numbers by their Greatest Common Factor (GCF).

Example: Simplify \(\color{blue}{12 : 8}\).

\(\color{blue}{\text{ GCF }(12, 8) = 4}\). Divide both: \(\color{blue}{12 \div 4 = 3}\) and \(\color{blue}{8 \div 4 = 2}\). Simplified ratio: \(\color{blue}{3 : 2}\).

Step-by-Step Summary

1. Identify the two quantities being compared and their correct order.
2. Write the ratio in the form asked (word, colon, or fraction).
3. Find the GCF of the two numbers.
4. Divide both numbers by the GCF to write the ratio in simplest form.

Watch: Introduction to Ratios (Video Lesson)

Khan Academy introduces ratios with clear real-world examples including apples and oranges:

Worked Examples

Example 1: A class has 12 boys and 16 girls. Write the ratio of boys to girls in simplest form.

Ratio = \(\color{blue}{12 : 16}\). \(\color{blue}{\text{ GCF }(12, 16) = 4}\). Simplified: \(\color{blue}{3 : 4}\).

Example 2: Write the ratio of 9 to 15 as a fraction in simplest form.

\(\color{blue}{\frac{9}{15}}\). \(\color{blue}{\text{ GCF }(9, 15) = 3}\). Simplified: \(\color{blue}{\frac{3}{5}}\).

Example 3: A bag has 5 apples and 3 oranges. Write the ratio of oranges to total fruit.

\(\color{blue}{\text{ Total } = 5 + 3 = 8}\). Ratio of oranges to total = \(\color{blue}{3 : 8}\) (already in simplest form since \(\color{blue}{\text{ GCF } = 1}\)).

Example 4: Simplify the ratio \(\color{blue}{36 : 24}\).

\(\color{blue}{\text{ GCF }(36, 24) = 12}\). \(\color{blue}{36 \div 12 = 3}\), \(\color{blue}{24 \div 12 = 2}\). Simplified: \(\color{blue}{3 : 2}\).

More Practice: Ratios and Rates Video

Math Antics covers ratios and rates with visual explanations and real-world examples:

Exercises

1. Write the ratio of 4 to 10 in all three forms and simplify.
2. A recipe uses 6 cups of flour and 2 cups of sugar. Write the ratio of flour to sugar in simplest form.
3. In a parking lot there are 20 cars and 8 trucks. Write the ratio of trucks to cars in simplest form.
4. Simplify: \(\color{blue}{18 : 30}\)
5. A team won 9 games and lost 3 games. Write the ratio of wins to total games in simplest form.
6. Write the ratio \(\color{blue}{45 : 60}\) in simplest form.

Answers

1. \(\color{blue}{4 \text{ to } 10}\) = \(\color{blue}{4 : 10}\) = \(\color{blue}{\frac{4}{10}}\); simplified: \(\color{blue}{2 : 5}\)
2. \(\color{blue}{6 : 2}\); \(\color{blue}{\text{ GCF } = 2}\); simplified: \(\color{blue}{3 : 1}\)
3. \(\color{blue}{8 : 20}\); \(\color{blue}{\text{ GCF } = 4}\); simplified: \(\color{blue}{2 : 5}\)
4. \(\color{blue}{\text{ GCF }(18, 30) = 6}\); \(\color{blue}{3 : 5}\)
5. Total \(\color{blue}{\text{ games } = 9 + 3 = 12}\); ratio = \(\color{blue}{9 : 12}\); \(\color{blue}{\text{ GCF } = 3}\); simplified: \(\color{blue}{3 : 4}\)
6. \(\color{blue}{\text{ GCF }(45, 60) = 15}\); \(\color{blue}{3 : 4}\)

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Frequently Asked Questions

Is a ratio the same as a fraction?

They are closely related. A ratio compares two quantities and can be written as a fraction, but a fraction usually represents part of a whole while a ratio can compare any two quantities — including part-to-part and part-to-whole. The math of simplifying them is identical.

Does the order of numbers in a ratio matter?

Yes, very much. “Boys to girls” and “girls to boys” are different ratios. Always match the order of the numbers to the order of the words or the order specified in the problem.

Can a ratio have a decimal or fraction in it?

You can write one, but ratios are usually simplified to whole numbers. If you end up with a decimal ratio like \(\color{blue}{1.5 : 2}\), multiply both sides by 2 to get \(\color{blue}{3 : 4}\).

Related Topics

• Using a Fraction to Write Down a Ratio
• Ratio Tables
• Finding Equivalent Ratio
• Word Problems Involving Writing a Ratio
• Simplifying Fractions