How to do long subtraction
Long subtraction, or column subtraction, is the written way to take one big number from another. You work one column at a time, from the ones on the right, so every step is a small take-away you can do in your head.
The rule
Start with the ones and work left. In each column, take the bottom digit from the top digit. If the top digit is too small, exchange one from the column on its left first.
How to do it
- Write the bigger number on top and the smaller one underneath, lined up on the right, ones under ones and tens under tens. Draw a line under them.
- Start with the ones column. If the top digit is as big as the bottom digit or bigger, take the bottom one away and write the answer under the line.
- If the top digit is too small, exchange: cross out the digit in the next column to the left and write it one smaller above it. Then cross out the small digit and write it with ten added, so a 2 becomes 12.
- If the next column to the left is 0, it has nothing to give. Move further left to the first digit that is not 0, take one from it, and every 0 you passed becomes 9.
- Move one column to the left and do the same, using the new digits above any you crossed out. Carry on until every column is done.
Worked examples
What is 586 − 243?
| 5 | 8 | 6 | |
| − | 2 | 4 | 3 |
| 3 | 4 | 3 |
Every top digit is at least as big as the one under it, so nothing needs exchanging. 6 − 3 = 3, 8 − 4 = 4, 5 − 2 = 3, and 586 − 243 = 343.
What is 352 − 127?
| 4 | 12 | ||
| 3 | 5 | 2 | |
| − | 1 | 2 | 7 |
| 2 | 2 | 5 |
You cannot take 7 from 2, so exchange one ten: the 5 tens become 4, and the 2 ones become 12. 12 − 7 = 5, 4 − 2 = 2, 3 − 1 = 2, so 352 − 127 = 225.
What is 412 − 138?
| 3 | 10 | 12 | |
| 4 | 1 | 2 | |
| − | 1 | 3 | 8 |
| 2 | 7 | 4 |
Two exchanges in a row. 2 is less than 8, so the 1 ten goes to the ones: 12 − 8 = 4. That leaves 0 tens, less than 3, so a hundred comes across: the 4 becomes 3 and the tens become 10. 10 − 3 = 7, 3 − 1 = 2, and 412 − 138 = 274.
What is 503 − 276?
| 4 | 9 | 13 | |
| 5 | 0 | 3 | |
| − | 2 | 7 | 6 |
| 2 | 2 | 7 |
The ones need an exchange, but there are no tens to give. Go to the hundreds: 5 becomes 4, the 0 tens become 9, and the 3 ones become 13. 13 − 6 = 7, 9 − 7 = 2, 4 − 2 = 2, so 503 − 276 = 227.
Exchanging, borrowing and regrouping
British schools call this step exchanging. In America it is borrowing, or more often now regrouping, and you will hear all three from teachers and in books. They all mean the same thing, and the written working looks the same.
Counting on, the shopkeeper's method
Instead of taking away, count up from the smaller number to the bigger one, in easy jumps. For 300 − 128: from 128 to 130 is 2, from 130 to 200 is 70, from 200 to 300 is 100. The jumps add up to 172, so 300 − 128 = 172.
It is how a shopkeeper counts out change, and it is often easier than exchanging when the top number is full of zeros. Many British schools teach it on a number line before column subtraction.
Equal addition
Some countries, and some older British teachers, use equal addition instead. When the top digit is too small, add 10 to it as usual, but add 1 to the bottom digit of the next column instead of taking 1 from the top. For 352 − 127: 12 − 7 = 5, the 2 tens underneath become 3, 5 − 3 = 2, then 3 − 1 = 2. The same 225.
It works because adding 10 to both numbers does not change the gap between them. It is a fine method, but mixing it with exchanging in one sum gives wrong answers, so stick to the one your child's school teaches.
Why it works
An exchange swaps one ten for ten ones, and that changes nothing about the number. 352 is 3 hundreds, 5 tens and 2 ones. Write it as 3 hundreds, 4 tens and 12 ones and it is still 352: 300 + 40 + 12. You have only moved some of it into the column that needs it.
Exchanging across a zero is the same idea twice. 503 has no tens, so one hundred is broken into 10 tens, and one of those tens into 10 ones. That leaves 4 hundreds, 9 tens and 13 ones, which is still 400 + 90 + 13 = 503.
To check, add the answer to the number you took away. You should get back to the top number: 274 + 138 = 412.
Where mistakes happen
- Taking the smaller digit from the bigger, whichever is on top. In 52 − 18, the ones are 2 − 8, not 8 − 2. Doing it backwards gives 46; exchanging gives the right answer, 34.
- Forgetting to make the lending digit smaller. If the 5 tens in 352 − 127 stay as 5 after giving one away, the answer comes out as 235 instead of 225. Cross the digit out and write its new value above it every time.
- Getting stuck at a zero. In 503 − 276 the tens have nothing to lend, so the exchange comes from the hundreds, and the 0 becomes 9 on the way through, not 10. Leaving it as 0, or making it 10, gives the wrong tens digit.
- Lining the numbers up on the left. They line up on the right, so that every column holds one place value.
Questions people ask
How do I check my answer?
Add it back. The answer plus the number you took away should give the number you started with, and long addition does that in columns the same way. For 412 − 138 = 274, check that 274 + 138 = 412.
What if the bottom number is bigger than the top?
Then the answer is less than zero. Swap the numbers round, subtract as usual, and put a minus sign in front of the answer, so 138 − 412 = −274. At primary school the bigger number nearly always goes on top.
What if the numbers have different lengths?
Line them up on the right as always. An empty space in the bottom number counts as 0, so in 503 − 76 the hundreds column is 5 − 0.
Why start with the ones and not the hundreds?
Because an exchange takes from the column on the left. Starting on the right means you never have to go back and change a digit you have already written in the answer.