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.
\(\color{blue}{a : b = \frac{a}{b}}\)
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
- Identify which quantity is first (it becomes the numerator) and which is second (the denominator).
- Write the fraction.
- Find the GCF of numerator and denominator.
- Divide both by the GCF to simplify.
- 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
- Write \(\color{blue}{8 : 12}\) as a fraction in simplest form.
- There are 20 boys and 25 girls in a class. Write the ratio of boys to girls as a fraction in simplest form.
- A bag has 15 red and 10 green balls. Write the ratio of green to total as a simplified fraction.
- Write \(\color{blue}{27 : 18}\) as a fraction in simplest form.
- A rectangle is 16 cm wide and 24 cm long. Write the ratio of width to length as a simplified fraction.
- A sports team won 14 games and lost 6. Write the ratio of losses to total games as a simplified fraction.
Answers
- \(\color{blue}{\text{ GCF }(8, 12) = 4}\); \(\color{blue}{\frac{2}{3}}\)
- \(\color{blue}{\text{ GCF }(20, 25) = 5}\); \(\color{blue}{\frac{4}{5}}\)
- \(\color{blue}{\text{ Total } = 25}\); \(\color{blue}{\frac{10}{25}}\); \(\color{blue}{\text{ GCF } = 5}\); \(\color{blue}{\frac{2}{5}}\)
- \(\color{blue}{\text{ GCF }(27, 18) = 9}\); \(\color{blue}{\frac{3}{2}}\)
- \(\color{blue}{\text{ GCF }(16, 24) = 8}\); \(\color{blue}{\frac{2}{3}}\)
- \(\color{blue}{\text{ Total } = 20}\); \(\color{blue}{\frac{6}{20}}\); \(\color{blue}{\text{ GCF } = 2}\); \(\color{blue}{\frac{3}{10}}\)
Frequently Asked Questions
Is a ratio exactly the same thing as a fraction?
They are related but not identical. A fraction always represents a part of a whole (numerator < denominator in most cases), while a ratio compares any two quantities — including part-to-part. When a ratio is written as a fraction, the notation is the same, but the interpretation may differ.
What if the ratio contains decimals or fractions?
If you have a ratio like \(\color{blue}{1.5 : 2}\), multiply both terms by 2 to get \(\color{blue}{3 : 4}\), then write as \(\color{blue}{\frac{3}{4}}\). For fractional ratios like \(\color{blue}{\frac{1}{2} : \frac{3}{4}}\), multiply both terms by the LCD (4) to get \(\color{blue}{2 : 3}\), then write as \(\color{blue}{\frac{2}{3}}\).
Can a ratio as a fraction be greater than 1?
Absolutely. A ratio of 7 : 3 written as a fraction is \(\color{blue}{\frac{7}{3}}\), which is greater than 1. This simply means the first quantity is larger than the second.
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