Subtract whole numbers (written strategies) - Across zeros
Theory
Subtracting from a number with zeros, like \(600\) or \(1000\), needs trading across the zeros: you cannot take from a zero, so you go left to the first non-zero digit and trade down through each zero, which becomes \(9\). This Year 5 maths page shows the method, a shortcut for numbers like \(1000\) and when a mental strategy is quicker.
You cannot trade from a column that holds \(0\). Instead, go left to the first non-zero digit and trade from there.
Trading down through zeros: \(600\) is \(6\) hundreds, which can be renamed as \(5\) hundreds, \(9\) tens and \(10\) ones. Each zero you pass becomes a \(9\), and the column you need gets \(10\).
A shortcut for round numbers: take one off both numbers first. \(1000 - 368\) has the same answer as \(999 - 367\), which needs no trading.
In the working, O, T, H, Th, TTh, HTh and M stand for ones, tens, hundreds, thousands, ten thousands, hundred thousands and millions.
Renaming numbers with zeros:
How to subtract across zeros
- Start with the ones. If the top digit is too small, look left.
- Find the first non-zero digit and take \(1\) from it.
- Change each zero you pass into \(9\), and add ten to the column you need.
- Subtract every column and check by adding.
Shortcut for 100, 1000, 10 000
- Take one off both numbers: \(1000 - 368\) becomes \(999 - 367\).
- Subtract; no trading is needed because every top digit is \(9\).
| \(\text{O: }10 - 8\) | \(=\) | \(2\) |
| \(\text{T: }9 - 4\) | \(=\) | \(5\) |
| \(\text{H: }5 - 2\) | \(=\) | \(3\) |
The answer is \(352\).
| \(\text{O: }10 - 8\) | \(=\) | \(2\) |
| \(\text{T: }9 - 6\) | \(=\) | \(3\) |
| \(\text{H: }9 - 3\) | \(=\) | \(6\) |
The answer is \(632\). Shortcut check: \(999 - 367 = 632\).
| \(\text{O: }10 - 7\) | \(=\) | \(3\) |
| \(\text{T: }9 - 2\) | \(=\) | \(7\) |
| \(\text{H: }9 - 5\) | \(=\) | \(4\) |
| \(\text{Th: }3 - 1\) | \(=\) | \(2\) |
The answer is \(2473\).
| \(\text{O: }14 - 8\) | \(=\) | \(6\) |
| \(\text{T: }9 - 5\) | \(=\) | \(4\) |
| \(\text{H: }9 - 6\) | \(=\) | \(3\) |
| \(\text{Th: }6 - 2\) | \(=\) | \(4\) |
The answer is \(4346\).
| \(\text{O: }10 - 5\) | \(=\) | \(5\) |
| \(\text{T: }9 - 2\) | \(=\) | \(7\) |
| \(\text{H: }9 - 7\) | \(=\) | \(2\) |
| \(\text{Th: }9 - 4\) | \(=\) | \(5\) |
| \(\text{TTh: }2 - 1\) | \(=\) | \(1\) |
The answer is \(15\,275\).
(i) \(6000 - 4\)
(ii) \(3000 - 1999\)
| \(6000 - 4\) | \(=\) | \(5996\) |
| \(3000 - 2000 + 1\) | \(=\) | \(1001\) |
(i) \(5996\) by counting back.
(ii) \(1001\): take away \(2000\), then add back the extra \(1\).
| \(\text{O: }10 - 5\) | \(=\) | \(5\) |
| \(\text{T: }9 - 6\) | \(=\) | \(3\) |
| \(\text{H: }7 - 3\) | \(=\) | \(4\) |
The trade must come from the hundreds: \(800\) becomes \(7\) hundreds, \(9\) tens and \(10\) ones. The answer is \(435\).
Common pitfalls
Frequently asked questions
How do you subtract across zeros?
Go left to the first non-zero digit and take one from it. Each zero you pass becomes \(9\), and the column you need gets ten more. \(600 - 248 = 352\).
Why do the zeros become nines?
Renaming \(600\) as \(5\) hundreds, \(9\) tens and \(10\) ones keeps the same value, so each zero turns into \(9\).
Is there a shortcut for subtracting from 1000?
Take one off both numbers: \(1000 - 368\) gives the same answer as \(999 - 367\), which is \(632\) with no trading.
When should I use a mental strategy?
When the numbers are friendly, such as \(6000 - 4\) or \(3000 - 1999\). Use the written method for harder numbers.
How do you check the answer?
Add the answer to the number you subtracted. \(352 + 248 = 600\).