Ross MackellUK-qualified primary teacher · 10 min read

What's a half plus a third?

If you said two fifths, you've just made the most common fractions mistake there is. It's the exact mistake I see children make in Year 5 and Year 6 every single year, and it isn't because they're bad at maths. It's because nobody showed them why it's wrong.

I'm Ross, a UK-qualified primary teacher with over ten years in the classroom. In this guide I'll show you how your child is taught to add fractions with different denominators: why you can't just add the bottoms, the one skill it all depends on, the easy case and the harder case, what to do when the answer is bigger than one, and the three mistakes children make most.

The language your child uses at school

Before we start, two words. Your child will hear these every lesson, so it helps if you use them too.

• Numerator: the top number. It tells you how many pieces you have.
• Denominator: the bottom number. It tells you how many equal pieces the whole has been cut into.

That second one is the key to everything on this page. The denominator isn't really a number you count. It's the size of the pieces.

Why a half plus a third isn't two fifths

Here's a half and a third, drawn as bars. This is exactly how I draw it on the board in class.

The problem is that the pieces are different sizes. Halves are big. Thirds are smaller. You can't add pieces of different sizes, any more than you can add 1 pound and 1 euro and call it 2.

So we cut them until the pieces match:

• Cut each half into three. A half is now 3 sixths.
• Cut each third into two. A third is now 2 sixths.

That's the whole idea in one picture. Make the pieces the same size, then add. Everything else on this page is just a way of doing that quickly with numbers instead of drawings.

If you've ever wondered why your child draws bars and diagrams instead of just "doing the sum", this is why. I explain more about that in why maths is taught differently now.

Start here: fractions with the same denominator

When the denominators are already the same, adding fractions is easy.

27 + 37 = 57

You add the tops. The bottom stays the same, because the size of the pieces hasn't changed. 2 sevenths and 3 sevenths is 5 sevenths, in the same way 2 apples and 3 apples is 5 apples.

Children do this in Year 3 and Year 4. The tricky step comes in Year 5 and Year 6, when the denominators are different. You can see this progression in the national curriculum for maths. And to do that tricky step, your child needs one skill first.

The one skill it all depends on: equivalent fractions

Equivalent fractions are fractions that look different but are worth the same amount. Look at this fraction wall:

The rule your child needs is this: whatever you multiply the bottom by, multiply the top by the same.

12 = 1 × 42 × 4 = 48

If your child is shaky on this, adding fractions will always feel hard, however many worksheets they do. Equivalent fractions is the bit to fix first. Five minutes with a chocolate bar or a pizza ("if I cut every slice in half, how many slices have I got now?") does more than any amount of explaining.

The easy case: one denominator goes into the other

Here's 14 + 38. Teach your child to ask one question first: can I turn one of these into the other?

Does 4 go into 8? Yes. So we only need to change the quarter.

1. 4 × 2 = 8, so multiply the top and bottom of 14 by 2. That gives 28.
2. Now both fractions are in eighths: 28 + 38 = 58.

Only one fraction had to change. The 38 was already in eighths, so we leave it alone. This is the version of the question your child meets in Year 5, and it's a great place to build confidence before the harder case.

The harder case: finding a common denominator

Now what if neither denominator goes into the other? We need a number that both denominators go into. That's called a common multiple, and this is where times tables come in.

Example: 2/3 + 1/5

1. Thirds and fifths. 3 doesn't go into 5. So ask: what's a number in the 3 times table and the 5 times table? 15.
2. 23: 3 × 5 = 15, so multiply the top by 5 too. That gives 1015.
3. 15: 5 × 3 = 15, so multiply the top by 3 too. That gives 315.
4. Add the tops: 1015 + 315 = 1315.

A tip for when your child gets stuck: multiplying the two denominators together always works. 3 × 5 is 15. It won't always give the smallest number, but it will always give a number that works.

Example: 3/4 + 1/6

Multiplying the denominators gives 24, and that works:

1824 + 424 = 2224 = 1112

But there's a smaller number that both 4 and 6 go into: 12.

34 + 16 = 912 + 212 = 1112

Same answer, smaller numbers, less to simplify at the end. This is why times tables matter so much here. A child who knows them well spots the 12 straight away. A child who doesn't is stuck working with big numbers, and big numbers mean more chances to slip up. If that's your child, my guide on how to learn times tables fast is the place to start.

When the answer is bigger than one

Here's 34 + 23.

1. The common denominator is 12: 912 + 812 = 1712.
2. 17 twelfths is more than 12 twelfths, and 12 twelfths is one whole.
3. So we take out one whole and have 5 twelfths left over: 1512.

A fraction like 1712, where the top is bigger than the bottom, is called an improper fraction. Children are usually expected to turn it into a mixed number: a whole number and a fraction.

Adding mixed numbers: 2 1/2 + 1 1/3

1. Add the wholes: 2 + 1 = 3.
2. Add the fractions: 12 + 13 = 36 + 26 = 56.
3. Put them together: 356.

If the fractions add up to more than one, carry that whole over to the whole numbers. It's exactly like carrying in column addition, and children who are confident carrying tens usually find this part easy.

Quick bonus: subtracting fractions

Subtracting works exactly the same way. Find a common denominator, change both fractions, then subtract the tops.

56 − 14 = 1012 − 312 = 712

3 mistakes children make when adding fractions

Mistake 1: Adding the bottoms

12 + 13 = 25

It's the big one. Give your child this quick check: if the answer is smaller than one of the fractions you started with, something's gone wrong. Two fifths is smaller than a half, so it can't be right.

When you see it, don't just correct the answer. Draw the two bars and ask "are these pieces the same size?" Let them find the problem.

Mistake 2: Changing the bottom but not the top

Your child turns 14 into eighths and writes 18. The bottom got multiplied, the top didn't. And 18 is half the size of 14, so the whole question falls apart from there.

The fix is one sentence, said every time: whatever you do to the bottom, do to the top.

Mistake 3: Losing the whole

With mixed numbers, the fractions add up to more than one and the extra whole gets forgotten. 134 + 123 isn't 2 and 1712. It's 3512, because 1712 contains another whole.

Ask your child: "Is the top of your fraction bigger than the bottom? Then there's a whole hiding in there."

How to practise at home (5 minutes)

1. Pick two fractions where one denominator goes into the other, like 13 + 29.
2. Do one together, out loud, using the three steps.
3. Then let your child do two on their own while you watch.
4. Once that's easy, move to pairs where neither denominator goes into the other, like 12 + 15.

Pizza and chocolate bars are brilliant for showing why the pieces need to match. Little and often wins: five minutes three times a week does more than an hour on a Sunday.

If your child is in Year 6, it's also worth looking at the fractions questions on the Year 6 SATs arithmetic paper. Adding and subtracting fractions comes up every year, and you can find real examples in the official KS2 past papers. Once fractions are secure, percentages become much easier too, because a percentage is really just a fraction out of 100.

Frequently asked questions

How do you add fractions with different denominators?

Find a common denominator (a number both denominators go into), change both fractions into equivalent fractions with that denominator, then add the numerators. The denominator stays the same. For example, 2/3 + 1/5 becomes 10/15 + 3/15, which is 13/15.

Why can't you just add the denominators?

The denominator tells you the size of the pieces, not how many you have. Adding the denominators changes the size of the pieces and gives the wrong answer: 1/2 + 1/3 is 5/6, not 2/5. A quick check is that the answer must be bigger than each fraction you started with.

What year do children learn to add fractions?

In England, children add fractions with the same denominator in Year 3 and Year 4. In Year 5 they add fractions where one denominator is a multiple of the other, and in Year 6 they add any fractions with different denominators, including mixed numbers.

What is the easiest way to find a common denominator?

First check if the smaller denominator goes into the larger one. If it does, use the larger one. If not, multiplying the two denominators together always works. Children who know their times tables well will often spot a smaller common multiple, which means less simplifying at the end.

How do you add mixed numbers?

Add the whole numbers, then add the fractions using a common denominator, then put them together. If the fractions add up to more than one whole, carry that whole over to the whole number part. So 2 1/2 + 1 1/3 is 3 + 5/6, which is 3 5/6.

Does subtracting fractions work the same way?

Yes. Find a common denominator, change both fractions, then subtract the numerators. For example, 5/6 − 1/4 becomes 10/12 − 3/12, which is 7/12.

And if your child gets upset or shuts down as soon as fractions come out, you're not alone. I wrote about that in my child hates maths: what to do.