Maths
What are equivalent fractions?
Quick answer
Equivalent fractions look different but are the same size, like 1/2, 2/4 and 4/8. You make them by multiplying or dividing the top and bottom by the same number.
In one sentence
Equivalent fractions look different but stand for the same amount, like ½, 2/4 and 4/8. You make one by multiplying or dividing the top and bottom numbers by the same number.
Explained
Cut a watermelon in half and eat one piece: you have eaten ½. If it had been cut into 4 equal slices and you ate 2, that is the same amount, so ½ = 2/4. Cut it into 8 and eat 4: still the same, so 4/8 is equal too.
watermelon- watermelon
pizza
The rule
Whatever you do to the bottom, do to the top. Multiply (or divide) the numerator (top number) and the denominator (bottom number) by the same number, and the fraction keeps its size.
Worked example: write ⅖ with 20 on the bottom.
- What turns 5 into 20? Multiply by 4.
- Do the same to the top: 2 × 4 = 8.
- So ⅖ = 8/20.
Why does it work? Multiplying top and bottom by 4 means cutting every piece into 4 smaller pieces. You have 4 times as many pieces, but each one is a quarter of the size, so the amount stays the same.
Simplifying
Going the other way is called simplifying (also reducing in the US, or cancelling in the UK). Divide top and bottom by a number that goes into both. For 15/20, both divide by 5, giving ¾. A fraction is in simplest form (lowest terms) when only 1 divides into both numbers. Dividing by the greatest common factor (the UK says highest common factor) gets you there in one step. See simplifying fractions and factors and multiples.
Why they matter
Equivalent fractions are the key to comparing and adding fractions. To compare ⅔ and ¾, rewrite both with 12 on the bottom: 8/12 and 9/12. Now it is clear that ¾ is bigger.
Examples
| Start | Change | Equivalent |
|---|---|---|
| ¼ | × 2 | 2/8 |
| ⅗ | × 5 | 15/25 |
| 6/8 | ÷ 2 | ¾ |
| 10/15 | ÷ 5 | ⅔ |
Worked example: ⅓ + ¼. Both 3 and 4 go into 12, so ⅓ = 4/12 and ¼ = 3/12. Then 4/12 + 3/12 = 7/12.
Common mistakes
- Changing only the bottom number, so ½ becomes 1/4. Fix: say the rule aloud: "same to top and bottom".
- Adding instead of multiplying, so ⅔ turns into ¾ by adding 1 to each. Fix: only multiply or divide.
- Stopping too early when simplifying, leaving 6/9. Fix: check again. Both divide by 3, so ⅔.
Quick quiz
Equivalent fractions
Check what you know with 4 quick questions.
Question 1 of 4
Which fraction is equivalent to 1/3?
Practise it
Try it
- Fill the gap: ¼ = ?/12
- Write 18/24 in simplest form.
- Are ⅗ and 9/15 equivalent?
Answers: 1) 3, because 4 × 3 = 12, so 1 × 3 = 3. 2) ¾ (divide both by 6). 3) Yes: multiply the top and bottom of ⅗ by 3 to get 9/15.
Equivalent fraction practice
Practise it step by step in a full course: Grade 4 Maths
For grown-ups
Related topics
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Decimals
Decimals are numbers with a decimal point. Digits after the point show tenths, hundredths and so on, so 2.5 means two and five tenths.
Percentages
A percentage is a number out of 100, written with the sign %. So 25% means 25 out of every 100, which is the same as one quarter.
Factors and multiples
Factors are numbers that divide exactly into a number (the factors of 12 are 1, 2, 3, 4, 6 and 12). Multiples are the numbers in its times table (12, 24, 36…).
Adding and subtracting fractions
To add or subtract fractions, first give them the same denominator. Then add or subtract only the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Comparing fractions
To compare fractions, check whether they share a denominator or a numerator, or rewrite them with a common denominator. The bigger fraction is the one with more of the whole.