Column subtraction: how to do it, with exchanging

How to do column subtraction

To do column subtraction, write the bigger number on top, the number you're taking away underneath, ones under ones, and subtract one column at a time from right to left. In each column, take the bottom digit away from the top digit and write the answer under the line.

The steps are:

1. Write the larger number on top and line up the ones on the right.
2. Start with the ones: top digit take away bottom digit.
3. If the top digit is smaller than the bottom digit, exchange: take 1 from the next column to the left and turn it into 10 in this column.
4. Move left to the tens, then the hundreds, and repeat.

Children usually learn column addition first, then column subtraction, and the two work in the same way. Our guide to column addition covers the adding side.

Column subtraction without exchanging

The simplest column subtraction has no exchanging: in every column, the top digit is bigger than or the same as the bottom digit. Here is 786 − 254:

1. Ones: 6 − 4 = 2.
2. Tens: 8 − 5 = 3.
3. Hundreds: 7 − 2 = 5.

Schools encourage children to say what each column is worth: "6 ones minus 4 ones is equal to 2 ones, 8 tens minus 5 tens is equal to 3 tens". That keeps the value of each digit in view.

Column subtraction with exchanging

Exchanging is what you do when a column's top digit is smaller than the one you're taking away. You take 1 from the next column to the left, which is worth 10 in the column you're working on. It works because 1 ten is the same as 10 ones, and 1 hundred is the same as 10 tens. Here is 452 − 127, with one exchange:

1. Ones: you can't take 7 from 2, so exchange 1 ten for 10 ones (the tens digit becomes 4). Now there are 12 ones: 12 − 7 = 5.
2. Tens: 4 − 2 = 2.
3. Hundreds: 4 − 1 = 3.

You can't take 7 ones from 2 ones, so one of the 5 tens is exchanged for 10 ones. The 5 is crossed out and becomes 4, and the 2 ones become 12 ones. The number on top is still worth 452: it's just been split differently, as 400, 40 and 12. That's flexible partitioning, and it's why exchanging makes sense rather than being a trick.

Some adults learned this as "borrowing". English schools call it exchanging, which describes what really happens: 1 ten is swapped for 10 ones, and the number keeps its value.

Two exchanges

Sometimes more than one column needs an exchange. Here is 532 − 278, where both the ones and the tens need one:

1. Ones: you can't take 8 from 2, so exchange 1 ten for 10 ones (the tens digit becomes 2). Now there are 12 ones: 12 − 8 = 4.
2. Tens: you can't take 7 from 2, so exchange 1 hundred for 10 tens (the hundreds digit becomes 4). Now there are 12 tens: 12 − 7 = 5.
3. Hundreds: 4 − 2 = 2.

After the first exchange, there are only 2 tens left on top, which isn't enough to take away 7 tens. So 1 hundred is exchanged for 10 tens as well.

Exchanging with zeros in the top number

When the next column along is a zero, there's nothing to exchange from it, so you go one more column to the left first. In 503 − 278, there are no tens to exchange, so 1 hundred becomes 10 tens, then 1 of those tens becomes 10 ones:

1. Ones: you can't take 8 from 3, so exchange 1 hundred for 10 tens (the hundreds digit becomes 4), then exchange 1 of those tens for 10 ones, leaving 9 tens. Now there are 13 ones: 13 − 8 = 5.
2. Tens: 9 − 7 = 2.
3. Hundreds: 4 − 2 = 2.

With more zeros, the exchanges ripple along. In 7,001 − 2,468, 1 thousand is exchanged for 10 hundreds, 1 hundred for 10 tens and 1 ten for 10 ones before the ones column can be done:

1. Ones: you can't take 8 from 1, so exchange 1 thousand for 10 hundreds (the thousands digit becomes 6), then exchange 1 of those hundreds for 10 tens, leaving 9 hundreds, then exchange 1 of those tens for 10 ones, leaving 9 tens. Now there are 11 ones: 11 − 8 = 3.
2. Tens: 9 − 6 = 3.
3. Hundreds: 9 − 4 = 5.
4. Thousands: 6 − 2 = 4.

These are the questions children find hardest, so it's worth taking them slowly. In Year 4, children use column subtraction with four-digit numbers like this, and in Year 5 with more than four digits.

Subtracting decimals in columns

From Year 5, children subtract decimals in columns, lining up the decimal points so tenths sit under tenths. If the numbers have different numbers of decimal places, fill the gap with a zero: for 6.5 − 2.38, write 6.5 as 6.50. Then subtract as usual, exchanging 1 tenth for 10 hundredths in the first column. The answer is 4.12. We don't have a drawn diagram for decimals yet, so this one is in words.

Estimating and checking with addition

Before working it out, estimate by rounding. For 532 − 278, round to 530 − 280 = 250. The answer, 254, is close, so it's probably right.

After working it out, check by adding. Addition is the inverse of subtraction, so the answer plus the number you took away should give the number you started with: 254 + 278 = 532. If it doesn't, look for a slip in one of the columns.

When is column subtraction taught?

Column subtraction is taught in Year 3, with numbers up to three digits, after children are confident with column addition. It moves to four digits in Year 4 and more than four digits in Year 5, and children keep practising it in Year 6.

Children need quick recall of subtraction facts within 20, like 12 − 7 = 5, before they start, because every column with an exchange uses one. Number bonds are the foundation.

Column subtraction isn't always the best choice. For 400 − 35, counting on from 35 to 400 on a number line is quicker and avoids exchanging across two zeros.

Common column subtraction mistakes

• Taking the smaller digit from the bigger one. This is the most common mistake. In 532 − 278, doing 8 take away 2 in the ones because 2 take away 8 "doesn't work" leads to 346 instead of 254. The bottom number is always taken from the top, so when the top digit is smaller, exchange.
• Forgetting to reduce the digit exchanged from. In 532 − 278, adding 10 to the ones but leaving the 3 tens as 3 gives 364 instead of 254. When you exchange, cross out the digit and write the new, smaller one above it.
• Getting stuck on zeros. In 503 − 278, trying to exchange from the 0 tens. Go to the next column that has something in it.
• Putting the smaller number on top. The larger number always goes on top.
• Misaligned digits. Line up the ones on the right, especially when the numbers have different numbers of digits.

How to help at home

• Use money. Make 52p with five 10p coins and two 1p coins, then take away 27p. To take 7p, you have to change a 10p for ten 1p coins: that's exchanging.
• Always check with addition. Make it a habit: answer plus the number taken away should equal the top number.
• Ask "is there enough?" Before each column, ask whether the top digit is big enough to take the bottom one from.
• Practise the facts. Quick subtraction facts within 20 make every column easier.
• Follow the school's layout. Ask your child to show you how they write the exchange in class.

Column subtraction questions to try

1. 865 − 432
2. 643 − 218
3. 724 − 386
4. 600 − 147
5. 5,003 − 1,856
6. The class wants to raise £472 for sports equipment and has raised £158 so far. How much more does it need?

Answers

1. 433 (no exchanging)
2. 425
3. 338
4. 453
5. 3,147
6. £314

For the rest of these years, see the Year 3 maths, Year 4 maths and Year 5 maths guides, or the KS2 maths overview.