Multiply by two-digit numbers - 3-digit x 2-digit
Theory
Long multiplication works the same way for a three-digit number: multiply by the ones digit, then by the tens (with a zero placeholder), and add the two partial products. This Year 5 maths page shows the method, what happens when the multiplier ends in zero, how to estimate first, and how to use it in word problems.
\(437 \times 26 = 437 \times 6 + 437 \times 20\). The first row is \(2622\), the second is \(8740\), and the total is \(11\,362\).
The second row starts with a zero in the ones place because it multiplies by tens.
If the multiplier ends in zero, such as \(40\), the first row would be all zeros. It is quicker to multiply by \(4\) and write one zero: \(385 \times 40 = 15\,400\).
Long multiplication with a three-digit number:
Long multiplication (three-digit by two-digit)
- Estimate first.
- Row 1: multiply the three-digit number by the ones digit.
- Row 2: write \(0\) in the ones, then multiply by the tens digit.
- Add the rows and compare with the estimate.
| \(213 \times 4\) | \(=\) | \(852\) |
| \(213 \times 10\) | \(=\) | \(2130\) |
| \(852 + 2130\) | \(=\) | \(2982\) |
The answer is \(2982\).
| \(437 \times 6\) | \(=\) | \(2622\) |
| \(437 \times 20\) | \(=\) | \(8740\) |
| \(2622 + 8740\) | \(=\) | \(11\,362\) |
The answer is \(11\,362\).
| \(385 \times 4\) | \(=\) | \(1540\) |
| \(1540 \times 10\) | \(=\) | \(15\,400\) |
\(385 \times 40 = 15\,400\). The ones row of zeros is not needed.
| \(500 \times 30\) | \(=\) | \(15\,000\) |
| \(512 \times 8\) | \(=\) | \(4096\) |
| \(512 \times 20\) | \(=\) | \(10\,240\) |
| \(4096 + 10\,240\) | \(=\) | \(14\,336\) |
The estimate is \(15\,000\) and the exact answer is \(14\,336\).
| \(609 \times 3\) | \(=\) | \(1827\) |
| \(609 \times 20\) | \(=\) | \(12\,180\) |
| \(1827 + 12\,180\) | \(=\) | \(14\,007\) |
The answer is \(14\,007\).
| \(148 \times 6\) | \(=\) | \(888\) |
| \(148 \times 30\) | \(=\) | \(4440\) |
| \(888 + 4440\) | \(=\) | \(5328\) |
It carries \(5328\) cartons.
| \(256 \times 4\) | \(=\) | \(1024\) |
| \(256 \times 30\) | \(=\) | \(7680\) |
| \(1024 + 7680\) | \(=\) | \(8704\) |
The \(3\) means \(3\) tens, so the second row is \(256 \times 30 = 7680\). The answer is \(8704\).
Common pitfalls
Frequently asked questions
How do you multiply a three-digit number by a two-digit number?
Multiply by the ones digit, then by the tens digit (with a zero placeholder), and add. \(437 \times 26 = 2622 + 8740 = 11\,362\).
What happens if the two-digit number ends in zero?
Multiply by the tens digit and write one zero. \(385 \times 40 = 1540 \times 10 = 15\,400\).
How do you estimate a three-digit by two-digit product?
Round the first number to the nearest hundred and the second to the nearest ten. \(512 \times 28 \approx 500 \times 30 = 15\,000\).
Why is there a zero at the start of the second row?
The second row multiplies by tens, so its answer is a number of tens. The zero keeps the digits in the right places.
How many digits can the answer have?
A three-digit number times a two-digit number has four or five digits, for example \(213 \times 14 = 2982\) and \(437 \times 26 = 11\,362\).