Adding Fractions With Different Bottom Numbers: Why 1/2 + 1/3 Is Not 2/5

15 October 2026 · by Larry

A child can add 1/4 and 2/4 without a pause and write 3/4. Then the page turns to 1/2 + 1/3, and the answer comes back as 2/5. Top plus top, bottom plus bottom. It looks tidy, it follows a rule the child has used all their life with numbers, and it is wrong.

This is one of the commonest places for fraction marks to go. It is not carelessness. The child is using an idea that worked everywhere else, and nobody has yet shown them why it stops working here.

Why "add the tops and the bottoms" feels right

With 1/4 + 2/4, the bottom number says what size the pieces are (quarters) and the top number counts them. One quarter and two quarters make three quarters. The bottom number does not change, because the pieces have not changed size.

With 1/2 + 1/3 the pieces are different sizes: halves and thirds. You cannot count them together any more than you can say "one apple and one orange make two apples". The pieces have to be the same size first, and that is the whole idea behind a "common denominator". The bottom number is the name of the piece, and you can only add pieces that share a name.

If your child is shaky on what the two numbers mean in the first place, start with fractions: when they suddenly stop making sense and come back here after.

The sum, step by step

Take 1/2 + 1/3.

• Find a size both pieces can be cut into. Halves and thirds both fit into sixths. (Count up the 2s: 2, 4, 6. Count up the 3s: 3, 6. They first meet at 6.)
• Change each fraction into sixths. 1/2 is 3/6, because 2 × 3 = 6, so the top is also × 3. 1/3 is 2/6, because 3 × 2 = 6, so the top is also × 2.
• Now add. 3/6 + 2/6 = 5/6. The bottom stays 6, because the pieces are all sixths now.

So 1/2 + 1/3 = 5/6. Subtraction works the same way: 2/3 − 1/4 becomes 8/12 − 3/12 = 5/12.

Five slips, and what each one looks like

1. Adding the tops and the bottoms. 1/2 + 1/3 = 2/5. The giveaway: 2/5 is smaller than 1/2, yet the sum started with a half and added something more. An answer smaller than one of the things you added cannot be right.

2. Changing only the bottom. The child knows the pieces must be sixths, so they write 1/6 + 1/6 = 2/6. They changed the name of the piece and left the count alone. A half is three sixths, not one. The top has to change by the same step as the bottom, because the amount has not changed, only how it is cut.

3. Changing one fraction and forgetting the other. In 1/2 + 1/4, the child sees that the bottoms do not match, writes 2/4 for the half, and then carries on as if nothing had changed, adding 2/4 + 1/4 and writing 3/8 or 3/6. When one bottom number already fits into the other, only one fraction needs to change, and the bottom stays at 4. The working is 2/4 + 1/4 = 3/4. This slip is quieter than the first two and shows up as a line of working that starts well and then adds the bottoms after all.

4. Stopping before the answer is in its proper form. 2/5 + 1/10 = 4/10 + 1/10 = 5/10, and 5/10 is the same as 1/2. And 3/4 + 2/3 = 9/12 + 8/12 = 17/12, which is 1 5/12. A child who gets the working right and loses the last mark here is a very common case, so ask whether the answer can be made simpler, and whether it is more than one whole.

5. In mixed numbers, taking the small from the big in each column. For 3 1/4 − 1 1/2, the wholes are fine (3 − 1 = 2). But 1/4 − 1/2 cannot be done, so the child flips it to 1/2 − 1/4 = 1/4 and writes 2 1/4. The correct answer is 1 3/4. Because 1/4 is not big enough to take 2/4 away, one whole is borrowed and cut into quarters: 3 1/4 is 2 5/4, and 2 5/4 − 1 2/4 = 1 3/4. (As a check in decimals: 3.25 − 1.5 = 1.75.) This is the borrowing problem from column sums again, in fraction clothes, and there is more on it in carrying and borrowing: why column sums go wrong.

A five-minute check to do aloud

Ask these with pencil and paper to hand. Each one is chosen to separate the slips above.

1. 1/4 + 2/4 (3/4). If this is wrong, go back to what the two numbers mean before anything else.
2. 1/2 + 1/4 (3/4). Only one fraction needs to change. This shows whether the child can convert at all.
3. 1/2 + 1/3 (5/6). Neither fraction can stay as it is. An answer of 2/5 is slip 1, and 2/6 is slip 2.
4. 3/4 + 2/3 (17/12, which is 1 5/12). The answer is bigger than one whole, so it tests the last step.
5. 3 1/4 − 1 1/2 (1 3/4). This is the mixed-number borrow. An answer of 2 1/4 is slip 5.

The first sum that goes wrong tells you where to start. Do not go back over the ones that went right.

What helps

Draw it before touching the numbers. Two identical rectangles. Cut one into 2 equal parts and shade 1. Cut the other into 3 equal parts and shade 1. Ask: "Can I add these?" The child can see the pieces do not match. Now cut both into sixths (draw the extra lines) and count the shaded pieces: 3 and 2. The picture shows why the tops change as well as the bottoms, which is exactly what slip 2 misses. The same drawing habit is the start of the bar model, covered in the guide to bar models.

Ask "what size is the piece?" before every sum. If both fractions already have the same bottom number, add the tops and keep the bottom. If not, the first job is to make them match. One question, asked every time, stops slip 1 before it starts.

Say the rule as one sentence. "Whatever I do to the bottom, I do to the top." Said while changing 1/2 into 3/6, it keeps the pair together. It also builds the idea that a fraction keeps its value when both numbers are multiplied by the same thing, which is why 1/2 and 3/6 are the same amount.

Estimate first. Is 1/2 + 1/3 more than 1/2? Yes. Is it more than one whole? No, because 1/3 is less than the other half. So the answer sits between 1/2 and 1. The answer 2/5 fails at once; 5/6 passes. It takes five seconds and catches many other sums too.

Keep the times tables sharp. Finding a common size is a times-tables job: counting up the 3s and the 4s until they meet. A child who has to work out 4 × 3 slowly will run out of steam before the fraction is finished. If that is the real problem, times tables: memorising, understanding and the order that works is the place to start.

What does not help

• A shortcut with no picture. There are quick methods that give the right answer without any idea why. They hold until a mixed number, a subtraction or a word problem appears, and then the child has nothing to fall back on.
• Always using the biggest common size. For 1/4 + 1/6, 24 works, but 12 is smaller and easier to handle. Find the first place the counts meet. If a larger common size is the one the child can do reliably, let them use it and simplify at the end.
• Calling the mistake careless. Slip 1 is a logical error, not a lapse in attention. A child told off for it learns that fractions are a game of guessing the teacher's rule. There is more on this in careless mistakes in maths: what they usually are.

Which level is this?

When a school introduces fractions with different bottom numbers, and how far it takes mixed numbers, is set by the school's scheme of work and the MOE syllabus, which you can check at moe.gov.sg. If your child's class is already on it, the five slips above are what to look for in their books. The jump between levels is covered in what changes between P4 and P5.

The short version

• You can only add or subtract pieces of the same size. The bottom number is the size of the piece.
• To make the sizes match, change both numbers of a fraction by the same step. Then add the tops and keep the bottom.
• The five slips: adding the tops and bottoms, changing only the bottom, forgetting to change the other fraction, stopping before the answer is in its proper form, and taking the small from the big in mixed numbers.
• Check with five quick sums and fix the first one that goes wrong.
• Draw it, ask "what size is the piece?", and estimate: is the answer bigger than the first fraction?

Related reading: three quarters of 24: why fraction-of-a-quantity sums go wrong, the other big fraction job at this stage.