Dividing fractions by whole numbers: a guide for parents

How to divide fractions by a whole number

To divide a fraction by a whole number, there are two ways, and which is easier depends on the numbers. The national curriculum's Year 6 example is ¹⁄₃ ÷ 2 = ¹⁄₆, and both ways give the same answer when they both work.

Way 1: divide the numerator

If the numerator divides exactly by the whole number, divide it and keep the denominator.

Take ⁶⁄₇ ÷ 3. Six sevenths is 6 one-sevenths, and 6 of anything shared into 3 equal groups gives 2 in each. So ⁶⁄₇ ÷ 3 = ²⁄₇. The parts are still sevenths: only the number of them has changed.

This is like 6 ÷ 3 = 2, with sevenths instead of ones.

Way 2: multiply the denominator

If the numerator doesn't divide exactly, keep the numerator and multiply the denominator by the whole number. This makes the parts smaller.

Take ¹⁄₃ ÷ 2. One third can't be split into 2 whole thirds, so split it into 2 smaller pieces instead. Halve a third and each piece is a sixth of the whole, so ¹⁄₃ ÷ 2 = ¹⁄₆.

The bar for ¹⁄₆ is half the length of the bar for ¹⁄₃, which is what dividing by 2 means.

Here's another: ³⁄₄ ÷ 2 = ³⁄₈. Each quarter is cut in half to make eighths, and there are still 3 of them.

Why multiplying the denominator works

Multiplying the denominator works because the denominator tells you how many parts the whole is split into. The more parts, the smaller each one is. Splitting a whole into 6 parts instead of 3 makes each part half the size, so sixths are half of thirds.

So dividing a fraction by 2 is the same as finding half of it, and dividing by 3 is the same as finding a third of it. The curriculum links multiplying by a fraction with dividing from Year 5, and you can see it here: ¹⁄₂ of ¹⁄₃ = ¹⁄₆, the same answer as ¹⁄₃ ÷ 2. There's more on that in multiplying fractions.

How to divide a fraction in half

To divide a fraction in half, divide it by 2. If the numerator is even, halve the numerator: ⁴⁄₅ ÷ 2 = ²⁄₅. If it's odd, double the denominator: ³⁄₅ ÷ 2 = ³⁄₁₀.

Both answers can be checked by doubling them back. ²⁄₅ × 2 = ⁴⁄₅, and ³⁄₁₀ × 2 = ⁶⁄₁₀, which simplifies to ³⁄₅.

Checking a division with multiplication

Division is the inverse of multiplication, so multiplying the answer by the whole number should take you back to where you started. To check ¹⁄₃ ÷ 2 = ¹⁄₆, work out ¹⁄₆ × 2 = ²⁄₆. Then simplify: ²⁄₆ = ¹⁄₃, which is the fraction you started with.

Year 6 children are expected to give answers in their simplest form, and simplifying fractions is often the last step. For example, ⁴⁄₉ ÷ 2 = ²⁄₉, which can't be simplified, but ²⁄₃ ÷ 4 = ²⁄₁₂, which simplifies to ¹⁄₆.

Dividing whole numbers to make fractions

A fraction is also a division. ³⁄₄ means 3 divided by 4, so 3 ÷ 4 = ³⁄₄. If 3 pizzas are shared equally between 4 people, each person gets ³⁄₄ of a pizza.

Year 6 children use this to change fractions into decimals: the curriculum's example is 3 ÷ 8 = 0.375, so ³⁄₈ is 0.375 as a decimal. The fractions to decimals guide covers this, and the bus stop method is the written division children use for it.

What about dividing a fraction by a fraction?

Dividing a fraction by a fraction, such as ¹⁄₂ ÷ ¹⁄₄, isn't in the primary curriculum. The Year 6 curriculum only asks children to divide proper fractions by whole numbers. Dividing by a fraction, and dividing mixed numbers, come at secondary school, where children learn to use all four operations with every kind of fraction.

If your child brings home a question like this, it's worth asking the school how they teach it rather than showing a shortcut that their teacher hasn't introduced yet.

Common mistakes to watch for

• Dividing the denominator. ⁴⁄₆ ÷ 2 is ²⁄₆, not ⁴⁄₃. Dividing the denominator makes the parts bigger, so the answer grows, which can't be right.
• Multiplying the numerator. ¹⁄₃ ÷ 2 is ¹⁄₆, not ²⁄₃. That's multiplying by 2, not dividing.
• Forgetting the parts get smaller. When the denominator is multiplied, the answer is smaller, even though the bottom number got bigger.
• Not checking. Multiplying back by the whole number is quick and catches most mistakes.
• Not simplifying. ²⁄₁₂ is right, but in Year 6 it's usually expected as ¹⁄₆.

How to help at home

• Share real things. Half a pizza shared between 2 people is a quarter each. Ask your child to say it as a division.
• Draw a bar. Shade the fraction, then split each part into equal pieces to see the new, smaller parts.
• Ask "is it smaller?" Dividing a fraction by a whole number always gives a smaller fraction. If the answer is bigger, something's gone wrong.
• Keep the times tables sharp. Multiplying denominators and simplifying both use times tables facts.

Dividing fractions questions to try

1. ⁸⁄₉ ÷ 4
2. ¹⁄₄ ÷ 3
3. ²⁄₅ ÷ 3
4. Divide ⁶⁄₁₁ in half
5. ³⁄₄ ÷ 6 (simplify the answer)
6. 5 cakes are shared equally between 8 children. What fraction of a cake does each child get?

Answers

1. ⁸⁄₉ ÷ 4 = ²⁄₉
2. ¹⁄₄ ÷ 3 = ¹⁄₁₂
3. ²⁄₅ ÷ 3 = ²⁄₁₅
4. ⁶⁄₁₁ ÷ 2 = ³⁄₁₁
5. ³⁄₄ ÷ 6 = ³⁄₂₄, and ³⁄₂₄ = ¹⁄₈
6. 5 ÷ 8 = ⁵⁄₈, so each child gets ⁵⁄₈ of a cake

Dividing fractions builds on adding and subtracting fractions and equivalent fractions. For everything else children learn in the last year of primary school, see the Year 6 maths guide and KS2 maths. Fraction calculations like these come up on the Year 6 SATs arithmetic paper.