Equivalent fractions

Cut a pizza in half and eat one half, or cut it in quarters and eat two quarters: you have eaten the same amount. 1/2 and 2/4 are different numbers for the same amount. Fractions like these are called equivalent fractions, which means equal fractions.

The rule

Multiply or divide the top and the bottom of a fraction by the same number, and you get an equivalent fraction. It is the same amount, written with different numbers.

How to do it

To find an equal fraction with a given bottom number, work out what the old bottom was multiplied by to get there. For 3/4 = ?/12, 4 × 3 = 12, so the number is 3.
Multiply the top by the same number: 3 × 3 = 9, so 3/4 = 9/12.
Going to a smaller bottom number works the same way with dividing. The number has to divide both the top and the bottom exactly.
To simplify a fraction, divide the top and the bottom by a number that goes into both. Keep going until the only number that goes into both is 1. The fraction is then in its simplest form.
For a shortcut, divide once by the highest common factor of the top and the bottom, and you reach the simplest form in one step.

Worked examples

Why are 1/2, 2/4 and 4/8 equal?

1/2 = 2/4 (× 2 on top and bottom)
2/4 = 4/8 (× 2 on top and bottom)

Each bar is cut into twice as many pieces as the one above it, and twice as many are shaded. The shaded part stays exactly half the bar, so 1/2 = 2/4 = 4/8.

What goes in the gap? 3/4 = ?/12

4 × 3 = 12
3 × 3 = 9

The bottom went from 4 to 12, so it was multiplied by 3. Do the same to the top, 3 × 3 = 9, and 3/4 = 9/12. In the picture, each quarter is cut into 3, so the 3 shaded quarters become 9 shaded twelfths.

What goes in the gap? 12/18 = 2/?

12 ÷ 6 = 2
18 ÷ 6 = 3

This time the top is the one you know, and it went down from 12 to 2, so it was divided by 6. Divide the bottom by 6 too, 18 ÷ 6 = 3, and 12/18 = 2/3.

What is 18/24 in its simplest form?

18/24 = 9/12 (÷ 2 on top and bottom)
9/12 = 3/4 (÷ 3 on top and bottom)

Both numbers are even, so halve them: 9/12. 3 goes into 9 and 12, so divide by 3: 3/4. Only 1 goes into both 3 and 4, so 3/4 is the simplest form. Dividing by 6, the highest common factor of 18 and 24, gets there in one step.

Simplify, cancel down, reduce

British schools say simplify, or cancel down, and call the result the simplest form. American schools also say reduce, and lowest terms. They all mean the same thing: divide the top and the bottom by the same number until nothing but 1 goes into both.

Checking two fractions are equal

To check whether two fractions are equal without drawing them, multiply each top number by the other fraction's bottom number. For 2/6 and 3/9: 2 × 9 = 18 and 3 × 6 = 18. The answers match, so the fractions are equal. If they do not match, the fractions are not equal, and comparing fractions shows which is bigger.

A fraction wall

Many classrooms have a fraction wall: rows of bars, one whole long, cut into halves, thirds, quarters and so on, stacked one under another like the pictures on this page. Run a ruler down it and every row that the ruler cuts at the same place shows an equivalent fraction.

Why it works

Multiplying the top and the bottom by 3 is the same as cutting every piece into 3 smaller pieces. There are 3 times as many pieces in the whole, so the bottom is 3 times bigger. You have 3 times as many of them, so the top is 3 times bigger. But no cake has been added or eaten: the amount stays the same.

Dividing the top and the bottom by the same number glues pieces back together in equal groups. 18/24 is 18 small pieces; put them in pairs and it is 9 bigger pieces out of 12. The amount does not change, only the size of the pieces you count in.

Adding does not work the same way. Add 1 to the top and the bottom of 1/2 and you get 2/3, which is more than a half. Only multiplying or dividing both numbers by the same amount keeps the fraction the same.

Where mistakes happen

Adding instead of multiplying. 1/2 and 2/3 are not equal, even though 1 more was put on the top and the bottom. 1/2 = 2/4, because both numbers were multiplied by 2.
Changing only one number. 3/4 is not 3/12: if the bottom is multiplied by 3, the top must be too, so 3/4 = 9/12.
Stopping too soon when simplifying. 18/24 = 9/12 is right, but 9/12 is not the simplest form, because 3 still goes into both. Keep going to 3/4.
Dividing by a number that does not go into both. 9/12 cannot be halved, because 9 is odd. The number has to divide the top and the bottom exactly.
Thinking bigger numbers make a bigger fraction. 9/12 looks bigger than 3/4, but it is the same amount.

Questions people ask

How many equivalent fractions does a fraction have?

As many as you like. Multiply the top and the bottom by 2, 3, 4, 5 and so on, and every one is equal: 1/3 = 2/6 = 3/9 = 4/12 = 5/15, and it never stops.

How do I know a fraction is in its simplest form?

When the only number that divides both the top and the bottom exactly is 1. 3/4, 5/8 and 7/10 are all in their simplest form. Numbers like that are called coprime.

Why do we need equivalent fractions at all?

To add or take away fractions, the bottom numbers have to be the same, and equivalent fractions are how you make them the same. For 1/2 + 1/4, write 1/2 as 2/4 first. See adding fractions.

Does the answer always have to be simplified?

Most schools and tests expect an answer in its simplest form, so it is a good habit. 9/12 is not wrong, it is just not finished.