Column subtraction with exchanging — an explanation for Year 3

Subtraction with exchanging (also called 'borrowing' or 'regrouping') is column subtraction where the ones digit of the top number is smaller than the ones digit of the bottom number, so you need to 'exchange' a whole ten and move it into the ones column. For example in 52 − 27, you cannot work out 2 − 7 directly, so you exchange one ten from the 5 (which becomes 4), add it to the 2 (which becomes 12), and work out 12 − 7 = 5 and 4 − 2 = 2, giving 25. The method is taught in Year 3 and is the basis of every future subtraction with bigger numbers.

What is exchanging in subtraction and why do we need it?

When you subtract two numbers in columns, you always start with the ones column (on the right). But sometimes the top digit is smaller than the bottom digit — for example in 52 − 27, the ones column has 2 on top and 7 underneath. You cannot take 7 from 2 and get a positive answer without exchanging.

The fix is to 'exchange' one ten — that is, take a whole ten from the tens column and change it into 10 ones, which join the ones column. It is exactly like changing a £10 note into £1 coins so you have enough small change to pay. This step is also called 'borrowing' or 'regrouping' — all the names describe exactly the same idea. Most schools in England now say 'exchange', because nothing is ever given back.

Subtraction with exchanging step by step: 52 − 27

1. Step 1 — set it out in columns: write 52 above 27, with the ones column lined up with the ones column and the tens column with the tens column.
2. Step 2 — check the ones column: 2 (top) take away 7 (bottom) — not possible, because 2 is smaller than 7. We need to exchange.
3. Step 3 — the exchange: go to the tens column, take one ten from the 5 (which becomes 4), and move it to the ones column: the 2 becomes 12 (2 plus 10).
4. Step 4 — subtract the ones column: 12 − 7 = 5. Write 5 in the answer.
5. Step 5 — subtract the tens column: 4 (after the exchange) take away 2 = 2. Write 2 in the answer.
6. Step 6 — the final answer: 25.

What happens when you need to exchange twice? — 502 − 267

Sometimes the tens column is not enough either, and then you exchange from there too — including the special case where there is a 0 in the tens column, and you have to 'exchange through the zero' from the hundreds. This is the moment that confuses Year 3 pupils most, so let's go through it slowly.

1. Start with the ones: 2 take away 7 — not possible. We need to exchange from the tens. But the tens has 0 — we cannot exchange from there directly.
2. Exchange from the hundreds first: take 1 from the 5 (the hundreds go from 5 to 4), and move it to the tens — now the tens are 10 (instead of 0).
3. Now we can exchange from the tens to the ones: take 1 from the 10 (the tens become 9), and move it to the ones — now the ones are 12 (2 + 10).
4. Subtract the ones: 12 − 7 = 5.
5. Subtract the tens: 9 − 6 = 3.
6. Subtract the hundreds: 4 − 2 = 2.
7. The answer: 235.

Common mistakes in subtraction with exchanging

• Forgetting to take 1 off the column you exchanged from — the child adds 10 to the ones column but forgets to reduce the tens column by 1, and gets an answer that is too big.
• Subtracting the wrong way round — when the top digit is smaller than the bottom digit, some children subtract 'backwards' (7 − 2 instead of 2 − 7) instead of exchanging. This is a very common mistake and gives a completely wrong answer.
• Confusion when exchanging through zero — skipping the first step (exchanging from the hundreds) and trying to exchange directly from a tens column that has 0 in it.
• Columns not lined up — writing the digits out of line, so a ones column gets subtracted from a tens column.
• Forgetting the check — not checking the answer with the inverse addition, and missing small mistakes.

Tips for parents

The best way to get the idea across is with real objects before moving on to digits. Use lolly sticks in bundles of 10 (or Dienes/base-ten blocks if the school sends them home) — to exchange, you open a whole bundle into 10 single sticks, exactly like changing a note. Once your child has seen this with their own eyes a few times, the move to digits on paper becomes much clearer, because they are no longer remembering an arbitrary rule but an idea they have understood.

It is also worth practising subtraction without any exchanging first, making sure the idea of 'columns' and 'place value' is clear, and only then adding the layer of exchanging. Jumping straight to questions with a double exchange before the basics are secure causes unnecessary confusion.

Practice

Start with questions that need only one exchange (like 52 − 27), and only when your child is secure with those move on to questions with a double exchange or exchanging through zero (like 502 − 267). The Year 3 addition and subtraction worksheet includes column addition and subtraction questions graded by difficulty, with full solutions for self-checking.

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