Write the answer under the line and move one column left.
When a column cannot pay
Before this page: the column layout from column addition, and what each digit is worth in integers and decimals.
Column subtraction takes one place value at a time. In 876 − 342 the ones ask 6 − 2 = 4, the tens ask 7 − 4 = 3 and the hundreds ask 8 − 3 = 5. Each answer is written under the column it came from, and reading them together gives 534. The number as a whole is never handled: the layout turns one large subtraction into three small ones.
That works because every column here holds enough to pay what is asked of it. 872 − 346 is set out in exactly the same way, but the ones column asks 2 − 6, and there are not six ones there to take.
100s
10s
1s
8
7
2
−
3
4
6
?
Nothing has gone wrong. 872 is the larger number, so there is an answer — the ones column simply cannot pay for it on its own. What it needs is sitting one place to the left, in the 7 tens. Fetching it is called exchanging, and it is the only thing subtraction adds to the column method.
Exchanging
872 means 800 + 70 + 2. It also means 800 + 60 + 12: the same number, with one ten swapped for ten ones.
872 = 800 + 70 + 2
872 = 800 + 60 + 12
Swapping a £10 note for ten £1 coins does not change how much money you have. It changes what you can pay with. An exchange does exactly that, and it is available in every column: one hundred for ten tens, one tenth for ten hundredths.
The ones column now holds 12, and 12 − 6 = 6. The other two columns are straightforward: 6 − 4 = 2 and 8 − 3 = 5.
Two marks record the swap. Cross out the 7 and write 6 above it, because the tens gave one away. Write a small 1 in front of the 2, because the ones received it — that column now reads 12.
100s
10s
1s
6
8
7
12
−
3
4
6
5
2
6
Both marks matter. Take the ten without paying for it and the top number has grown by ten; pay for it without writing the 1 and you have thrown ten away.
Every column asks the same question
Is the top digit big enough to take the digit below it?
Take one unit from the column on the left, lower that column by one, add ten here, then subtract.
Setting it out
Two things to settle before any subtracting starts.
The larger number goes on top. Addition can be written either way round; subtraction cannot, because 8 − 3 and 3 − 8 are different questions. If the number underneath is the larger one, the answer is below zero, which is adding and subtracting negatives.
Every column needs a digit. Decimals of different lengths do not finish in the same column, so line the points up and fill the gaps with zeros. An empty column has nothing to subtract from and nothing to exchange, so 14.6 − 8.35 cannot begin until the 14.6 is written as 14.60.
14.60
Subtracting whole numbers and decimals
Watch the hundreds column in the first of these: it gives its last hundred away, and then has to borrow one itself.
Example: calculate 6,145 − 2,978
6,145 − 2,978
Align equal place values
Begin with equal place values directly beneath one another. The ones sit under the ones, the tens under the tens, and so on.
6,145 − 2,978 = 3,167
Once the points are lined up, the method takes no further notice of them. The columns carry on past the point, and the answer takes its point from the points above.
Example: calculate 14.6 − 8.35
14.6 − 8.35
Align equal place values
Begin with the decimal points directly beneath one another. Write 14.6 as 14.60 so every occupied place has a digit to work with.
14.60 − 8.35 = 6.25
Where it turns up
Exam questions rarely use the word subtract. They describe one of two situations, and both are answered by the same columns.
What is left
A roll holds 14.6 m of cable and 8.35 m is cut off it. What remains is 14.60 − 8.35 = 6.25 m.
Change from a payment is the same shape of question: the money handed over is the starting amount, and the price comes out of it.
How far apart
One shelf is 2.4 m long and another is 1.75 m. Nothing is cut; the question is the gap between them, which is 2.40 − 1.75 = 0.65 m.
0.65 m
1.75 m
2.4 m
Difference, change, fewer, further, colder, how much more: each of these is one of the two, and in both of them the larger quantity goes on top.
Subtract two chosen numbers
The second number is the larger one, so the answer falls below zero. Lower it, or read adding and subtracting negatives.
Align equal place values
Begin with the decimal points directly beneath one another. Write 4351.2 as 4351.20 so every occupied place has a digit to work with.
Common mistakes
Almost every lost mark comes from one of four places, and none of them is the arithmetic.
- Taking the small digit from the large oneFaced with 2 above 6, it is tempting to write 4. That answers 6 − 2 instead, and the total comes out ten too big. The order within a column is fixed.
- Half an exchangeThe ten arrives, but the column it came from is left as it was, so the top number has quietly grown by ten. Cross out and lower the giving column first.
- Using the old digitOnce a column has been lowered, the new digit is the one to read. Using the crossed-out 4 instead of the 3 above it undoes the exchange you have just made.
- Starting before the setting out is finishedAn unpadded decimal, or the two numbers written the wrong way round, is a wrong answer already, however neatly the columns are then worked.
Summary
- The larger number goes on top, points beneath points, with empty decimal places filled by zeros before anything else.
- Work from the right, one column at a time, writing each result under the column it came from.
- If the top digit is too small, exchange: one unit from the left becomes ten here. The number is rewritten, not changed.
- Record both halves — cross out and lower the column that gave, and write the 1 in front of the digit that received.
- Check by adding: the answer plus the number taken away must return the number you started with.