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. \(\color{blue}{\text{ GCF }(45, 60) = 15}\); \(\color{blue}{3 : 4}\)

Frequently Asked Questions

How do you write ratios?

Name the two quantities in the order the problem gives them, then write the first number followed by the second. If a class has 14 girls and 11 boys, the ratio of girls to boys is 14 to 11, or 14 : 11. Order matters: the ratio of boys to girls would be 11 : 14, which is a different comparison.

How do you write ratios in 3 ways?

A ratio can be written in word form, colon form or fraction form. The ratio of 3 red marbles to 5 blue marbles can be written as 3 to 5, 3 : 5 or 3/5. All three mean the same comparison, so choose whichever form the question asks for.

How do you write ratios in simplest form?

Divide both numbers by their greatest common factor (GCF). For 12 : 8, the GCF is 4, so 12 ÷ 4 = 3 and 8 ÷ 4 = 2, giving 3 : 2. If the numbers have units, make sure they are the same unit first: 30 minutes to 2 hours becomes 30 : 120, which simplifies to 1 : 4.

Can you write ratios as fractions?

Yes. Put the first quantity in the numerator and the second in the denominator, so 7 : 4 becomes 7/4. The fraction does not need to be less than 1, because a ratio is a comparison, not always a part of a whole. Just be careful: a part-to-part ratio like 7 : 4 is not the same as the fraction of the total, which would be 7/11.

Can we write a ratio as a decimal?

Yes. Divide the first number by the second. The ratio 3 : 4 equals 3 ÷ 4 = 0.75, and 9 : 2 equals 4.5. A decimal makes it easy to compare two ratios, for example 0.75 is greater than 0.7, so 3 : 4 is a bigger ratio than 7 : 10.

How do you write a ratio in the form 1 : n?

Divide both numbers by the first number. For 4 : 10, dividing both by 4 gives 1 : 2.5. For 5 : 8, dividing by 5 gives 1 : 1.6. In this form, n can be a decimal, and it tells you how many of the second quantity there are for each one of the first.

Related Topics

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