Equivalent fractions: what they are and how to find them

What are equivalent fractions?

Equivalent fractions are fractions that look different but have the same value. ¹⁄₂ and ²⁄₄ are equivalent: cut a bar in half, or cut the same bar into quarters and take two, and you shade exactly the same amount.

The shaded parts are the same length every time, so ¹⁄₂ = ²⁄₄. The same goes for ⁴⁄₈. They're also at the same point on a number line: halfway between 0 and 1. "Equivalent" just means "equal in value".

Why one-half is the same as two-quarters

Splitting each part into smaller equal parts doesn't change how much is shaded. Take a bar showing ¹⁄₂. Cut each half into 2 and there are 4 parts, with 2 of them shaded. The amount shaded hasn't changed, so ¹⁄₂ = ²⁄₄. Cut each quarter in half again and you get ⁴⁄₈.

Notice what happens to the numbers. Doubling the number of parts doubles the denominator, and doubles the number of shaded parts, the numerator, too. That's the rule.

How to find equivalent fractions

To find an equivalent fraction, multiply the numerator and the denominator by the same number. Whatever you do to the bottom, do to the top.

Example 1: find three fractions equivalent to one-third.

• Multiply top and bottom by 2: ¹⁄₃ = ²⁄₆.
• Multiply top and bottom by 3: ¹⁄₃ = ³⁄₉.
• Multiply top and bottom by 4: ¹⁄₃ = ⁴⁄₁₂.

So ²⁄₆, ³⁄₉ and ⁴⁄₁₂ are all equivalent to ¹⁄₃. You can keep going for ever: every fraction has endless equivalents.

Example 2: find the equivalent of three-fifths with a denominator of 10.

5 has been multiplied by 2 to make 10, so multiply the numerator by 2 as well: 3 × 2 = 6. That gives ³⁄₅ = ⁶⁄₁₀.

Example 3: fill in the missing number in three-quarters = ? twenty-fourths.

4 × 6 = 24, so multiply the 3 by 6 too: 3 × 6 = 18. So ³⁄₄ = ¹⁸⁄₂₄.

You can also go the other way and divide. 9 and 12 can both be divided by 3, so ⁹⁄₁₂ = ³⁄₄. Dividing to get the smallest numbers possible is called simplifying fractions, which children learn properly in Year 6.

Using the fraction wall to find equivalent fractions

A fraction wall shows one whole in every row, split into different numbers of equal parts. Any bricks that line up exactly are equivalent:

Run a ruler down from the end of the ¹⁄₂ brick and it lines up with ²⁄₄, ³⁄₆, ⁴⁄₈, ⁵⁄₁₀ and ⁶⁄₁₂. From the end of the first ¹⁄₃ brick, it lines up with ²⁄₆ and ⁴⁄₁₂. There's more about the wall on the fractions page.

The times tables connection

Every equivalent fraction comes from a times table. The fractions equivalent to ¹⁄₄ are ²⁄₈, ³⁄₁₂, ⁴⁄₁₆ and so on: the numerators count in ones and the denominators go up the 4 times table.

Government guidance for teachers puts it like this: ¹⁄₄ and ³⁄₁₂ are equivalent because 1 is the same part of 4 as 3 is of 12. In both, the denominator is 4 times the numerator. So children who know their times tables and the matching division facts find equivalent fractions much easier.

Equivalent fractions in Year 3, Year 4 and Year 5

Children build this up over several years:

• Year 2: knowing that ²⁄₄ and ¹⁄₂ are the same.
• Year 3: recognising and showing equivalent fractions with small denominators, using diagrams such as bars and fraction walls.
• Year 4: families of common equivalent fractions, such as ¹⁄₄ = ²⁄₈, and spotting that ⁶⁄₉ can be written as ²⁄₃.
• Year 5: naming and writing equivalent fractions of any given fraction, including tenths and hundredths.
• Year 6: using equivalent fractions to compare fractions and add and subtract fractions with different denominators.

The Year 3, Year 4 and Year 5 guides cover the rest of each year.

Equivalent fractions and decimals

Tenths and hundredths link equivalent fractions to decimals. ³⁄₁₀ = ³⁰⁄₁₀₀, and both are 0.3 as a decimal. In Year 4, children learn that ¹⁄₂ is the same as 0.5, ¹⁄₄ is 0.25 and ³⁄₄ is 0.75. That works because ¹⁄₂ = ⁵⁰⁄₁₀₀, which is 50 hundredths. In Year 5 they write decimals such as 0.71 as ⁷¹⁄₁₀₀. The page on fractions to decimals goes further.

Common mistakes to watch for

• Adding instead of multiplying. Adding 1 to the top and bottom of ¹⁄₂ gives ²⁄₃, which isn't equivalent. Always multiply or divide.
• Changing only one number. Doubling the denominator of ¹⁄₃ to get ¹⁄₆ halves the fraction. Change both, by the same number.
• Thinking bigger numbers mean a bigger fraction. ⁴⁄₈ looks bigger than ¹⁄₂, but it's the same amount.
• Different-sized wholes. When comparing with drawings, the bars must be the same length, or the picture won't show equivalence.

How to help at home

• Fold paper. Fold a strip in half and shade one half. Fold again and count: 2 quarters are shaded. Fold again: 4 eighths.
• Share food. Half a pizza is the same whether it's cut into 2, 4 or 8 slices.
• Make chains. Start with ¹⁄₅ and take turns to say the next equivalent fraction: ²⁄₁₀, ³⁄₁₅, ⁴⁄₂₀.
• Practise the times tables and division facts. They do most of the work here.

Equivalent fractions questions to try

1. Write two fractions equivalent to ¹⁄₂.
2. Complete: ²⁄₃ = ? sixths.
3. Complete: ³⁄₅ = ? twentieths.
4. Is ²⁄₅ equivalent to ⁴⁄₁₀?
5. Is ²⁄₃ equivalent to ³⁄₄?
6. Complete: ¹²⁄₁₆ = ? quarters.

Answers

1. Any two of ²⁄₄, ³⁄₆, ⁴⁄₈, ⁵⁄₁₀ and so on
2. 4: ²⁄₃ = ⁴⁄₆ (multiply top and bottom by 2)
3. 12: ³⁄₅ = ¹²⁄₂₀ (multiply top and bottom by 4)
4. Yes (multiply top and bottom by 2)
5. No: in twelfths, ²⁄₃ = ⁸⁄₁₂ and ³⁄₄ = ⁹⁄₁₂, so ³⁄₄ is a little bigger
6. 3: ¹²⁄₁₆ = ³⁄₄ (divide top and bottom by 4)

For everything else children learn in these years, see the KS2 maths guide.