Multiplying fractions: how to multiply fractions step by step

How to multiply fractions by a whole number

To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator. The denominator tells you the size of the parts, and that doesn't change. Only the number of parts does.

Take ²⁄₅ × 3. That's 3 lots of 2 fifths, which is 6 fifths. So ²⁄₅ × 3 = ⁶⁄₅.

The answer is more than one whole, so it can be written as a mixed number: ⁶⁄₅ = 1 ¹⁄₅. The improper fractions and mixed numbers guide shows how.

Because multiplication can be done in either order, 3 × ²⁄₅ gives the same answer as ²⁄₅ × 3. Some children find it easier to read the whole number first: "3 lots of two fifths".

Multiplying fractions times whole numbers: another example

³⁄₈ × 4 is 4 lots of 3 eighths, which is 12 eighths. So ³⁄₈ × 4 = ¹²⁄₈.

Twelve eighths is one whole (8 eighths) and 4 eighths more, so ¹²⁄₈ = 1 ⁴⁄₈. Then ⁴⁄₈ simplifies to ¹⁄₂, giving 1 ¹⁄₂.

Multiplying mixed numbers by whole numbers

To multiply a mixed number by a whole number, multiply the whole number part and the fraction part separately, then add them. Year 5 children learn this with diagrams to help.

For 2 ¹⁄₄ × 3:

1. Multiply the wholes: 2 × 3 = 6.
2. Multiply the fraction: ¹⁄₄ × 3 = ³⁄₄.
3. Add them together: 6 and ³⁄₄ make 6 ³⁄₄.

So 2 ¹⁄₄ × 3 = 6 ³⁄₄. If the fraction part comes to more than one whole, change it to a mixed number and add the extra whole on.

Multiplying by a fraction is finding a fraction of an amount

Multiplying a whole number by a fraction gives the same answer as finding that fraction of the number. ³⁄₄ × 12 is the same as ³⁄₄ of 12. Find one quarter by dividing by 4, then take 3 of those quarters:

So ³⁄₄ of 12 = 9, and ³⁄₄ × 12 = 9 too. Children learn fractions of amounts from Year 3, so this link often makes multiplying fractions feel familiar.

How to multiply two fractions

To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. This is Year 6 work. The curriculum's own example is ¹⁄₄ × ¹⁄₂ = ¹⁄₈.

It helps to read × as "of". ¹⁄₄ × ¹⁄₂ is a quarter of a half. Cut a half into 4 equal pieces and each piece is an eighth of the whole:

A fraction times a fraction, then simplify

The Year 6 curriculum asks for answers in their simplest form. Take ²⁄₃ × ³⁄₄:

1. Multiply the numerators: 2 × 3 = 6.
2. Multiply the denominators: 3 × 4 = 12.
3. So ²⁄₃ × ³⁄₄ = ⁶⁄₁₂.
4. Simplify: 6 and 12 can both be divided by 6, so ⁶⁄₁₂ = ¹⁄₂.

Here's one more: ³⁄₅ × ²⁄₃ = ⁶⁄₁₅. Both 6 and 15 divide by 3, so ⁶⁄₁₅ = ²⁄₅. The simplifying fractions guide explains how to find the number to divide by.

How do I multiply fractions with different denominators?

You multiply fractions with different denominators in exactly the same way: numerators together, denominators together. Unlike adding and subtracting fractions, you don't need to make the denominators the same first. ²⁄₃ × ³⁄₄ above has different denominators, and the method still works.

Why the answer can be smaller

Multiplying by a fraction smaller than one makes a number smaller, which surprises a lot of children who think multiplying always makes things bigger. ¹⁄₂ × 10 = 5, because half of 10 is 5. The curriculum calls this scaling: multiplying by ¹⁄₂ scales a number down to half its size.

A good check is to ask whether the answer makes sense. ²⁄₃ × ³⁄₄ should be less than ³⁄₄, because it's only two thirds of it, and ¹⁄₂ is less than ³⁄₄.

Common mistakes to watch for

• Multiplying the denominator by the whole number too. ²⁄₅ × 3 is ⁶⁄₅, not ⁶⁄₁₅. The parts are still fifths.
• Finding a common denominator first. It isn't wrong, but it's extra work that multiplying doesn't need, and it makes the numbers bigger.
• Forgetting to simplify. In Year 6, ⁶⁄₁₂ is usually expected as ¹⁄₂.
• Multiplying only the whole number in a mixed number. In 2 ¹⁄₄ × 3, both the 2 and the ¹⁄₄ are multiplied.
• Expecting a bigger answer. Multiplying two proper fractions always gives an answer smaller than both of them.

How to help at home

• Use "lots of". Say ³⁄₈ × 4 as "4 lots of three eighths". It makes the answer easier to picture.
• Use "of" in the kitchen. Half of a quarter of a cake, or two thirds of a 12-piece pizza, are real multiplying fractions questions.
• Fold paper. Fold a strip in half, then in quarters the other way, and count the pieces to see that a quarter of a half is an eighth.
• Keep up the times tables. Multiplying and simplifying fractions both lean on times tables facts.

Multiplying fractions questions to try

1. ²⁄₇ × 3
2. ³⁄₄ × 5 (write the answer as a mixed number)
3. 1 ²⁄₅ × 2
4. ¹⁄₃ × ¹⁄₂
5. ³⁄₄ × ²⁄₅ (simplify the answer)
6. ²⁄₃ of 18

Answers

1. ²⁄₇ × 3 = ⁶⁄₇
2. ³⁄₄ × 5 = ¹⁵⁄₄, and ¹⁵⁄₄ = 3 ³⁄₄
3. 1 × 2 = 2 and ²⁄₅ × 2 = ⁴⁄₅, so 1 ²⁄₅ × 2 = 2 ⁴⁄₅
4. ¹⁄₃ × ¹⁄₂ = ¹⁄₆
5. ³⁄₄ × ²⁄₅ = ⁶⁄₂₀, and ⁶⁄₂₀ = ³⁄₁₀
6. One third of 18 is 6, so ²⁄₃ of 18 = 12

Next comes dividing fractions by whole numbers, also in Year 6. For a start on what fractions are, see the fractions guide, and for everything else children learn in these years, see the Year 5 and Year 6 maths guides. Fraction calculations like these come up on the Year 6 SATs arithmetic paper.