Ross MackellUK-qualified primary teacher · 9 min read

Your child has finally cracked adding fractions. Common denominators, the lot. Then the next worksheet says 2/3 × 3/4, and they set off hunting for a common denominator again. Ten minutes later they have a page of crossings out and an answer that's wrong.

Here's the good news: multiplying fractions is actually easier than adding them. Dividing a fraction by a whole number is easier still. Most children only struggle because nobody has told them that the rules are different, and why.

I'm Ross, a UK-qualified primary teacher with over ten years in the classroom. In this guide I'll show you how UK schools teach multiplying and dividing fractions step by step, why the answer gets smaller, the mistakes I see most often, and a 10-minute practice routine with answers.

What your child needs first

Multiplying and dividing fractions is where three earlier skills come together. Before you start, check your child has:

• Secure times tables. Every single step is a multiplication or a division fact. If tables are shaky, start with how to learn times tables fast.
• Simplifying. Almost every answer needs simplifying at the end. If 6/12 doesn't instantly look like 1/2, spend a few days on equivalent fractions and simplifying first.
• Improper fractions and mixed numbers. They need to know that 6/5 is the same as 1 1/5.

In the national curriculum, children multiply proper fractions and mixed numbers by whole numbers in Year 5. In Year 6 they multiply pairs of fractions, writing the answer in its simplest form, and divide fractions by whole numbers. Both turn up on the Year 6 SATs arithmetic paper every year.

Why "of" means multiply

This is the idea that makes everything else click. In maths, "of" means multiply. Half of 10 is the same as 1/2 × 10, which is 5.

So 1/2 × 1/3 just means half of a third. Look at this bar. The top row shows 1/3 shaded. The bottom row takes half of that shaded part:

Half of a third is a sixth. And 1 × 1 = 1 on top, 3 × 2 = 6 on the bottom, so the rule gives exactly the same answer: 1/2 × 1/3 = 1/6.

This also explains the thing that confuses so many children: multiplying two fractions makes the answer smaller. If you take half of something, you end up with less of it. That's not a mistake, it's exactly what should happen.

Multiplying a fraction by a whole number

This is the Year 5 skill, and it's the best place to start.

Example 1: 2/5 × 3

1. Multiply the top by the whole number. 2 × 3 = 6.
2. Leave the bottom alone. It stays as 5.
3. Tidy up. 6/5 is more than a whole, so turn it into a mixed number: 1 1/5.

Why does the bottom stay the same? Because 2/5 × 3 is just 2/5 + 2/5 + 2/5. Three lots of two fifths is six fifths. You're counting more pieces, but the pieces are still fifths.

Example 2: 3/4 × 8

1. Multiply the top: 3 × 8 = 24, so 24/4.
2. Simplify: 24 ÷ 4 = 6.

So 3/4 × 8 = 6. There's a shortcut here too. Because "of" means multiply, this is the same as 3/4 of 8. Find a quarter of 8 (that's 2), then three quarters (that's 6). Children who've learned fractions of amounts often find this way easier, and either way is fine.

Multiplying two fractions

This is the Year 6 skill. The rule is short enough to fit on a sticky note: top times top, bottom times bottom.

Example 3: 2/3 × 3/4

1. Multiply the tops. 2 × 3 = 6.
2. Multiply the bottoms. 3 × 4 = 12.
3. Simplify. 6 and 12 are both in the 6 times table, so 6/12 = 1/2.

So 2/3 × 3/4 = 1/2. In SATs, an answer that isn't simplified can lose the mark when the question asks for simplest form, so make simplifying the final step every time.

Example 4: 3/5 × 1/2

1. Tops: 3 × 1 = 3.
2. Bottoms: 5 × 2 = 10.
3. Can 3 and 10 be simplified? Nothing goes into both except 1, so the answer is 3/10.

A tip for confident children: you can simplify before you multiply. In 2/3 × 3/4, there's a 3 on the top and a 3 on the bottom, so they cancel out, leaving 2/4, which is 1/2. It keeps the numbers small. It's a nice trick, but only once the basic rule is solid.

Multiplying mixed numbers

Year 5 children also multiply mixed numbers by whole numbers. There are two good ways.

Example 5: 2 1/2 × 3

Method 1: partition it. Split the mixed number into its whole part and its fraction part, multiply both, then add. This is exactly the idea behind partitioning.

1. 2 × 3 = 6.
2. 1/2 × 3 = 3/2, which is 1 1/2.
3. 6 + 1 1/2 = 7 1/2.

Method 2: make it an improper fraction. 2 1/2 is 5/2. Then 5/2 × 3 = 15/2, which is 7 1/2.

Same answer both ways. Partitioning is usually easier for younger children. The improper fraction method is what they'll need in secondary school.

Dividing a fraction by a whole number

This is the other half of the Year 6 objective, and it's the one children find most surprising.

Example 6: 1/3 ÷ 2

Picture a third of a pizza being shared between two people. Each person gets half of that third, which is a sixth. So 1/3 ÷ 2 = 1/6.

The rule: keep the top the same, and multiply the bottom by the whole number. 3 × 2 = 6, so 1/6.

Notice that dividing by 2 gave the same answer as multiplying by 1/2 earlier. Halving is halving, whichever way you write it.

Example 7: 4/5 ÷ 2

1. Multiply the bottom by 2: 4/10.
2. Simplify: 4/10 = 2/5.

There's a quicker way when the top divides neatly. 4 ÷ 2 = 2, so 4/5 ÷ 2 = 2/5 straight away. Four fifths shared by two is two fifths each. Both ways are correct.

Example 8: 3/4 ÷ 3

1. Multiply the bottom by 3: 3/12.
2. Simplify: 3/12 = 1/4.

Or the quick way: 3 ÷ 3 = 1, so 1/4. Three quarters shared between three people is one quarter each.

What about dividing by a fraction?

Dividing one fraction by another, like 1/2 ÷ 1/4, isn't part of the primary curriculum. Children meet it in secondary school, where they usually learn the "keep, change, flip" method: keep the first fraction, change ÷ to ×, and flip the second fraction upside down. So 1/2 ÷ 1/4 becomes 1/2 × 4/1 = 4/2 = 2.

That makes sense if you think of it as a question: how many quarters fit into a half? Two. If your child is in Year 6 and their homework only ever divides by a whole number, that's normal. You don't need to teach keep, change, flip yet.

Watch: the times tables behind it

Every step of multiplying and dividing fractions is a times table fact, and simplifying is times tables in reverse. If your child is slow at these, tightening up tables is often the quickest fix. This video shows the order I teach them in:

4 common mistakes and how to fix them

Mistake 1: Finding a common denominator

Your child turns 2/3 × 3/4 into 8/12 × 9/12 before they start, because that's what adding taught them.

The fix: make the difference explicit. "Adding and subtracting need the same denominator. Multiplying and dividing don't." Some children like writing "+ − : same bottom" and "× ÷ : no need" at the top of the page.

Mistake 2: Multiplying the top and the bottom by the whole number

They write 2/5 × 3 = 6/15. That's just an equivalent fraction of 2/5, so nothing has been multiplied at all.

The fix: write the whole number as a fraction first: 3 is 3/1. Now top times top (2 × 3 = 6) and bottom times bottom (5 × 1 = 5) gives 6/5, and the mistake disappears.

Mistake 3: Dividing the bottom instead of multiplying it

For 1/3 ÷ 2, they try to halve the 3 and get stuck, or write 1/1.5.

The fix: go back to the pizza. Sharing a piece makes the pieces smaller, and smaller pieces means a bigger number on the bottom. Thirds become sixths.

Mistake 4: Not simplifying the answer

They get 6/12 and stop. It's correct, but on a test that asks for simplest form it can cost the mark.

The fix: make "Can I simplify?" the last step on every single question. A quick check: are both numbers even? Both in the 3 times table? Both end in 0 or 5?

A 10-minute practice routine

Same three steps I use for every skill: watch, copy, practise.

1. Watch: do one example yourself, out loud. "Top times top, 2 times 3 is 6. Bottom times bottom, 3 times 4 is 12. Can I simplify? Yes, 6/12 is 1/2."
2. Copy: your child does a very similar one, saying each step out loud.
3. Practise: three or four on their own, ending each one with "Can I simplify?"

Only move on when they're getting three in a row right. If they keep tripping over the multiplying itself, spend a few days on times tables instead. Ten minutes, a few times a week, is plenty.

Frequently asked questions

How do you multiply fractions?

Multiply the numerators (top numbers) together and the denominators (bottom numbers) together, then simplify. For example, 2/3 × 3/4 = 6/12, which simplifies to 1/2. You do not need a common denominator.

How do you multiply a fraction by a whole number?

Multiply the numerator by the whole number and keep the denominator the same. For example, 2/5 × 3 = 6/5, which is 1 1/5 as a mixed number. It can help to write the whole number as a fraction over 1.

How do you divide a fraction by a whole number?

Keep the numerator the same and multiply the denominator by the whole number, then simplify. For example, 1/3 ÷ 2 = 1/6 and 4/5 ÷ 2 = 4/10 = 2/5. If the numerator divides exactly, you can divide it instead: 4/5 ÷ 2 = 2/5.

Why does multiplying fractions make the answer smaller?

Because multiplying by a fraction less than 1 means taking part of something. In maths, "of" means multiply, so 1/2 × 1/3 means half of a third, which is 1/6. Taking half of anything leaves you with less.

What year do children learn to multiply and divide fractions?

In England, children multiply proper fractions and mixed numbers by whole numbers in Year 5. In Year 6 they multiply pairs of proper fractions, writing the answer in its simplest form, and divide proper fractions by whole numbers. Dividing a fraction by a fraction is taught in secondary school.

Sources and further reading

• Department for Education, National curriculum in England: mathematics programmes of study (fractions in Years 5 and 6)
• Standards and Testing Agency, Key stage 2 mathematics test framework
• Ross Mackell, Teacher's Perspective on YouTube