Using a Fraction to Write down a Ratio

Writing a ratio as a fraction is a natural and useful skill because it connects two major areas of math you already know, ratios and fractions, and makes simplification and comparison straightforward. On the GED, many ratio questions expect you to work with the fraction form.

What Does It Mean to Write a Ratio as a Fraction?

When you write a ratio as a fraction, the first quantity becomes the numerator and the second quantity becomes the denominator.

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\(\color{blue}{a : b = \frac{a}{b}}\)

For example, “the ratio of 3 cats to 7 dogs” written as a fraction is \(\color{blue}{\frac{3}{7}}\). This means: for every 3 cats, there are 7 dogs.

Important note: The fraction form of a ratio does not have to be a proper fraction (numerator < denominator). A ratio of \(\color{blue}{8 : 5}\) becomes \(\color{blue}{\frac{8}{5}}\), an improper fraction, and that is perfectly correct.

How to Write a Ratio as a Fraction and Simplify

Step 1: Write the ratio in fraction form

Put the first quantity (the one mentioned first) in the numerator and the second quantity in the denominator.

Step 2: Simplify

Find the GCF of numerator and denominator, then divide both by the GCF.

Part-to-part vs. Part-to-whole

A ratio can compare a part to another part, or a part to the whole.

• Part-to-part: 3 red to 4 blue → \(\color{blue}{\frac{3}{4}}\)
• Part-to-whole: 3 red out of 7 total → \(\color{blue}{\frac{3}{7}}\)

Always check what the question is asking before you write the fraction.

Step-by-Step Summary

1. Identify which quantity is first (it becomes the numerator) and which is second (the denominator).
2. Write the fraction.
3. Find the GCF of numerator and denominator.
4. Divide both by the GCF to simplify.
5. Check: is the answer in simplest form? (\(\color{blue}{\text{ GCF } = 1}\))

Watch: Ratios as Fractions (Video Lesson)

Khan Academy explores whether a ratio like \(\color{blue}{2 : 5}\) is the same as the fraction \(\color{blue}{\frac{2}{5}}\):

Worked Examples

Example 1: Write the ratio 10: 4 as a fraction in simplest form.

\(\color{blue}{\frac{10}{4}}\). \(\color{blue}{\text{ GCF }(10, 4) = 2}\). Simplified: \(\color{blue}{\frac{5}{2}}\).

Example 2: A basket contains 6 apples and 9 oranges. Write the ratio of apples to oranges as a fraction in simplest form.

\(\color{blue}{\frac{6}{9}}\). \(\color{blue}{\text{ GCF }(6, 9) = 3}\). Simplified: \(\color{blue}{\frac{2}{3}}\).

Example 3: Write the ratio of girls to total students as a fraction, given 14 girls and 21 boys.

Total \(\color{blue}{\text{ students } = 14 + 21 = 35}\). Fraction: \(\color{blue}{\frac{14}{35}}\). \(\color{blue}{\text{ GCF }(14, 35) = 7}\). Simplified: \(\color{blue}{\frac{2}{5}}\).

Example 4: A wall is 15 feet long and 9 feet tall. Write the ratio of height to length as a simplified fraction.

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

More Practice: Ratios as Fractions in Simplest Form

This Khan Academy video works through multiple examples of writing ratios as fractions and simplifying:

Exercises

1. Write \(\color{blue}{8 : 12}\) as a fraction in simplest form.
2. There are 20 boys and 25 girls in a class. Write the ratio of boys to girls as a fraction in simplest form.
3. A bag has 15 red and 10 green balls. Write the ratio of green to total as a simplified fraction.
4. Write \(\color{blue}{27 : 18}\) as a fraction in simplest form.
5. A rectangle is 16 cm wide and 24 cm long. Write the ratio of width to length as a simplified fraction.
6. A sports team won 14 games and lost 6. Write the ratio of losses to total games as a simplified fraction.

Answers

1. \(\color{blue}{\text{ GCF }(8, 12) = 4}\); \(\color{blue}{\frac{2}{3}}\)
2. \(\color{blue}{\text{ GCF }(20, 25) = 5}\); \(\color{blue}{\frac{4}{5}}\)
3. \(\color{blue}{\text{ Total } = 25}\); \(\color{blue}{\frac{10}{25}}\); \(\color{blue}{\text{ GCF } = 5}\); \(\color{blue}{\frac{2}{5}}\)
4. \(\color{blue}{\text{ GCF }(27, 18) = 9}\); \(\color{blue}{\frac{3}{2}}\)
5. \(\color{blue}{\text{ GCF }(16, 24) = 8}\); \(\color{blue}{\frac{2}{3}}\)
6. \(\color{blue}{\text{ Total } = 20}\); \(\color{blue}{\frac{6}{20}}\); \(\color{blue}{\text{ GCF } = 2}\); \(\color{blue}{\frac{3}{10}}\)

Frequently Asked Questions

Can you write ratios as fractions?

Yes. A ratio a : b can be written as the fraction a/b, with the first quantity on top and the second on the bottom. The ratio of 3 cats to 7 dogs becomes 3/7. The fraction does not have to be proper, so a ratio of 8 : 5 is correctly written as 8/5.

How do you write a ratio as a fraction in simplest form?

Write the first quantity as the numerator and the second as the denominator, then divide both by their greatest common factor. For a ratio of 18 to 24, write 18/24. The GCF is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. Keep improper results such as 5/2 as fractions rather than mixed numbers.

How do you write a rate as a fraction?

Put the first quantity with its unit in the numerator and the second quantity with its unit in the denominator. A rate of 120 miles in 2 hours becomes 120 miles / 2 hours. Simplifying gives 60 miles / 1 hour, which is the unit rate of 60 miles per hour. Unlike a plain ratio, a rate keeps its units.

How do you write a ratio in fractional notation?

Fractional notation simply means writing a : b as a/b. The ratio of 5 apples to 9 oranges in fractional notation is 5/9. Order matters: the ratio of oranges to apples would be 9/5. Read the wording carefully, because the quantity named first always goes in the numerator.

How do you write the ratio 3 to 4 as a fraction?

The ratio 3 : 4 is written as the fraction 3/4. It is already in simplest form because 3 and 4 share no common factor other than 1. If 3 red beads and 4 blue beads are in a bag, the part-to-part fraction is 3/4, while the part-to-whole fraction for red beads is 3/7.

Related Topics

• Write a Ratio
• Simplifying Fractions
• Finding Equivalent Ratio
• Word Problems Involving Writing a Ratio
• Ratio Tables