Multiply by a one-digit number (formal algorithm) - Involving hundreds
Theory
This Year 5 maths page extends the written algorithm to three- and four-digit numbers with zeros in them, such as \(708 \times 6\) or \(4056 \times 7\). The key idea: a zero digit gives \(0\), but any carry waiting for that column must still be added. It also shows how to estimate first so you can check the answer.
A zero in the top number gives a product of \(0\) in that column, because \(0 \times 6 = 0\).
If a carry is waiting, add it to that \(0\). In \(708 \times 6\), the ones give \(48\), so \(4\) is carried; the tens are \(0 \times 6 + 4 = 4\).
When the last column reaches ten or more, the answer has an extra digit: \(4056 \times 7 = 28\,392\).
In the working, O, T, H, Th, TTh, HTh and M stand for ones, tens, hundreds, thousands, ten thousands, hundred thousands and millions.
Estimate by rounding the large number to its leading digit:
How to multiply a number with zeros
- Estimate first.
- Multiply each digit from the ones, carrying when needed.
- At a zero, the product is \(0\); add any carry and write the result.
- Compare the answer with the estimate.
| \(\text{O: }8 \times 6\) | \(=\) | \(48\ (\text{carry } 4)\) |
| \(\text{T: }0 \times 6 + 4\) | \(=\) | \(4\) |
| \(\text{H: }7 \times 6\) | \(=\) | \(42\) |
The answer is \(4248\).
| \(\text{O: }6 \times 7\) | \(=\) | \(42\ (\text{carry } 4)\) |
| \(\text{T: }5 \times 7 + 4\) | \(=\) | \(39\ (\text{carry } 3)\) |
| \(\text{H: }0 \times 7 + 3\) | \(=\) | \(3\) |
| \(\text{Th: }4 \times 7\) | \(=\) | \(28\) |
The answer is \(28\,392\).
| \(\text{O: }6 \times 7\) | \(=\) | \(42\ (\text{carry } 4)\) |
| \(\text{T: }0 \times 7 + 4\) | \(=\) | \(4\) |
| \(\text{H: }4 \times 7\) | \(=\) | \(28\) |
The tens are \(0 \times 7 + 4 = 4\). The answer is \(2842\).
| \(600 \times 8\) | \(=\) | \(4800\) |
| \(\text{O: }3 \times 8\) | \(=\) | \(24\ (\text{carry } 2)\) |
| \(\text{T: }0 \times 8 + 2\) | \(=\) | \(2\) |
| \(\text{H: }6 \times 8\) | \(=\) | \(48\) |
The estimate is \(4800\) and the exact answer is \(4824\).
| \(\text{O: }7 \times 9\) | \(=\) | \(63\ (\text{carry } 6)\) |
| \(\text{T: }0 \times 9 + 6\) | \(=\) | \(6\) |
| \(\text{H: }0 \times 9\) | \(=\) | \(0\) |
| \(\text{Th: }2 \times 9\) | \(=\) | \(18\) |
The answer is \(18\,063\).
| \(95 \times 4\) | \(=\) | \(380\) |
| \(380 \times 10\) | \(=\) | \(3800\) |
\(950 \times 4 = 3800\).
| \(\text{O: }8 \times 5\) | \(=\) | \(40\ (\text{carry } 4)\) |
| \(\text{T: }0 \times 5 + 4\) | \(=\) | \(4\) |
| \(\text{H: }6 \times 5\) | \(=\) | \(30\) |
\(3040\) trees are planted.
Common pitfalls
Frequently asked questions
What do you do with a zero in the top number?
Multiply it like any digit: the product is \(0\). Then add any carry. In \(708 \times 6\) the tens are \(0 + 4 = 4\).
How do you estimate a product like 603 × 8?
Round the large number: \(600 \times 8 = 4800\). The exact answer should be close.
Why can the answer have five digits?
When the last column reaches ten or more, it adds a new place. \(4056 \times 7 = 28\,392\).
How do you multiply a number ending in zero?
Leave the zero off, multiply, then put it back. \(950 \times 4\): \(95 \times 4 = 380\), so \(3800\).
How do I check my answer?
Compare it with an estimate, and check every column has a digit.