Distributive property in multiplication
Theory
The distributive property lets you split one factor into parts, multiply each part, then add the results: \(8 \times 36 = 8 \times 30 + 8 \times 6\). It also works with subtraction, which is handy for numbers just below a hundred: \(4 \times 99 = 4 \times 100 - 4 \times 1\). This Year 5 maths page explains both, with area-model pictures, word problems and common mistakes.
The distributive property says that multiplying a sum is the same as multiplying each part and adding: \(8 \times (30 + 6) = 8 \times 30 + 8 \times 6\).
Splitting by place value uses tens and ones: \(58 = 50 + 8\). You can also use any split that is easy, such as \(28 = 14 + 14\).
Numbers just below a round number are split with subtraction: \(98 = 100 - 2\), so \(6 \times 98 = 6 \times 100 - 6 \times 2\).
Every part must be multiplied. \(4 \times 73\) is \(4 \times 70 + 4 \times 3\), not \(4 \times 70 + 3\).
The distributive property with addition and subtraction:
Split and add
- Split the larger factor into tens and ones.
- Multiply each part by the other factor.
- Add the partial products.
Round and subtract
- Round the factor up to a friendly number, such as \(100\).
- Multiply by the friendly number.
- Subtract the other factor times the amount you rounded by.
| \(8 \times 30\) | \(=\) | \(240\) |
| \(8 \times 6\) | \(=\) | \(48\) |
| \(240 + 48\) | \(=\) | \(288\) |
\(8 \times 36 = 288\).
| \(6 \times 50\) | \(=\) | \(300\) |
| \(6 \times 8\) | \(=\) | \(48\) |
| \(300 + 48\) | \(=\) | \(348\) |
The box is \(8\), and \(6 \times 58 = 348\).
| \(4 \times 100\) | \(=\) | \(400\) |
| \(4 \times 1\) | \(=\) | \(4\) |
| \(400 - 4\) | \(=\) | \(396\) |
\(4 \times 99 = 396\).
(i) \(6 \times 98\)
(ii) \(3 \times 199\)
| \(600 - 12\) | \(=\) | \(588\) |
| \(600 - 3\) | \(=\) | \(597\) |
(i) \(6 \times 100 - 6 \times 2 = 588\).
(ii) \(3 \times 200 - 3 \times 1 = 597\).
| \(140 + 56\) | \(=\) | \(196\) |
| \(98 + 98\) | \(=\) | \(196\) |
Both splits give \(196\), because both pairs of parts add to \(28\).
| \(15 \times 20\) | \(=\) | \(300\) |
| \(15 \times 4\) | \(=\) | \(60\) |
| \(300 + 60\) | \(=\) | \(360\) |
The total cost is \(\$360\).
| \(4 \times 70\) | \(=\) | \(280\) |
| \(4 \times 3\) | \(=\) | \(12\) |
| \(280 + 12\) | \(=\) | \(292\) |
Eli did not multiply the \(3\) by \(4\). The answer is \(292\).
Common pitfalls
Frequently asked questions
What is the distributive property?
Multiplying a sum is the same as multiplying each part and adding. \(8 \times 36 = 8 \times 30 + 8 \times 6 = 288\).
How do you multiply by 99 in your head?
Multiply by \(100\), then subtract one lot of the number. \(4 \times 99 = 400 - 4 = 396\).
Does it matter how I split the number?
No. Any parts that add to the number give the same answer. \(7 \times 28\) is \(196\) whether you split \(28\) as \(20 + 8\) or \(14 + 14\).
Can you use subtraction with the distributive property?
Yes. \(6 \times 98 = 6 \times 100 - 6 \times 2 = 588\).
Why is the distributive property useful?
It turns a hard multiplication into easy ones you can do in your head.