Subtract across zeros
Theory
Subtracting across zeros is a Year 3 maths skill for take-aways like \(500 - 137\) and \(1000 - 382\). When the ones are too small and the tens are zero, you change one hundred into 10 tens, then one ten into 10 ones, so an empty column can lend. This page shows the column method, the exchange facts, and how to count up on a number line. Curriculum: VC2M3N04 / AC9M3N03.
When you subtract in columns, you sometimes need to regroup (also called borrowing). You take one from the next column so a small digit has enough to subtract from. Subtracting across zeros happens when that next column is a zero and has nothing to lend.
The fix is to move down one more place. Change one hundred into 10 tens first. Now the tens column is not empty, so change one ten into 10 ones. The one hundred trickles down through the zero until the ones have enough.
Take \(500 - 137\). The ones are \(0 - 7\), which cannot be done, and the tens are also \(0\). Change 1 hundred into 10 tens (the 5 hundreds become 4), then 1 ten into 10 ones (the tens become 9). Now subtract each column: the answer is \(363\).
Subtracting across zeros has no formula, but these exchange facts are what you use every time you regroup.
How to subtract across zeros
- Line up the columns. Write the numbers with hundreds under hundreds, tens under tens and ones under ones. Look at the ones first.
- Regroup across the zeros. If the ones are too small and the tens are \(0\), change 1 hundred into 10 tens (the hundreds digit drops by 1), then change 1 ten into 10 ones (the tens become 9).
- Subtract each column. Take away the ones, then the tens, then the hundreds. If the numbers are close, you can count up on a number line instead.
| \(8\text{ hundreds}\) | \(\to\) | \(7,\ \text{tens}\to9,\ \text{ones}\to10\) |
| \(\text{ones}\) | \(:\) | \(10 - 6 = 4\) |
| \(800 - 6\) | \(=\) | \(794\) |
Both zeros lend, so \(800 - 6 = 794\).
| \(\text{ones }3 - 7\) | \(:\) | \(\text{regroup}\) |
| \(\text{hundreds}\) | \(\to\) | \(8,\ \text{tens}\to9\) |
| \(\text{ones}\) | \(:\) | \(13 - 7 = 6\) |
| \(\text{tens}\) | \(:\) | \(9 - 4 = 5\) |
| \(903 - 547\) | \(=\) | \(356\) |
The hundreds are \(8 - 5 = 3\), so the answer is \(356\).
| \(216 \to 300\) | \(:\) | \(+84\) |
| \(300 \to 500\) | \(:\) | \(+200\) |
| \(84 + 200\) | \(=\) | \(284\) |
So \(500 - 216 = 284\).
| \(700 - 265\) | \(:\) | \(\text{regroup 1 hundred, 1 ten}\) |
| \(\text{ones}\) | \(:\) | \(10 - 5 = 5\) |
| \(\text{tens}\) | \(:\) | \(9 - 6 = 3\) |
| \(700 - 265\) | \(=\) | \(435\) |
Mia has \(\$435\) left.
Common pitfalls
Frequently asked questions
How do you subtract across zeros?
When the ones are too small and the tens are \(0\), change 1 hundred into 10 tens first. Then change 1 ten into 10 ones. Now every column has something to lend, so you subtract the ones, then the tens, then the hundreds.
Why can't you borrow from a zero?
A zero in the tens column means there are no tens to lend. So you move down from the hundreds: one hundred becomes 10 tens, and then one of those tens can lend 10 ones.
Does one hundred become 10 or 10 tens when you regroup?
One hundred becomes 10 tens, not 10. One hundred equals \(100\), which is the same as 10 tens. That is why the tens column jumps to \(10\) before you lend a ten to the ones.
How do you work out 500 minus 137?
Change 1 hundred into 10 tens, so the \(5\) hundreds become \(4\). Then change 1 ten into 10 ones, so the tens become \(9\). Ones \(10 - 7 = 3\), tens \(9 - 3 = 6\), hundreds \(4 - 1 = 3\). So \(500 - 137 = 363\).
How do you subtract a number from 1000?
The easiest way is to count up. For \(1000 - 382\), jump from \(382\) to \(400\) (that is \(+18\)), then from \(400\) to \(1000\) (that is \(+600\)). Add the jumps: \(18 + 600 = 618\). This avoids regrouping across three zeros.
Is there an easier way than borrowing across zeros?
Yes. If the two numbers are close, count up on a number line. For \(800 - 796\) just count up \(+4\). You can also subtract from a number one less, like using \(499\) for \(500\), then add \(1\) back at the end.