Distributive Law
The distributive law says that to multiply a bracket you multiply every part inside it, then add or subtract the results. In NSW Year 7 maths (NESA outcome MA4‑ALG‑C‑01) it is used first with numbers and then with pronumerals, so \(2(x + 5)\) becomes \(2x + 10\).
You will split awkward numbers into friendly sums or differences, for example \(18 \times 12 = 18 \times (10 + 2) = 216\) or \(72 \times (100 - 1)\), and link the law to a rectangle’s area when it is cut into two parts. Then you start expanding brackets. It is handy for pricing 12 items at $49 each without a calculator. This subtopic is part of Year 7 Algebra and Equations.
Every question with a fully worked solution.
- Distributive Law - Video - The Distributive Property Watch
Theory
The distributive law says: to multiply a group in brackets, multiply every part inside the brackets, then add (or subtract) the results. \(a \times (b + c) = a \times b + a \times c\). This Year 7 maths page shows the law as an area model, uses it for quick mental multiplication and expands brackets with pronumerals.
Brackets hold a group together. \(3 \times (20 + 7)\) means \(3\) lots of the whole group \(20 + 7\).
With a sum: \(a \times (b + c) = a \times b + a \times c\)
With a difference: \(a \times (b - c) = a \times b - a \times c\)
Mental maths. Split the harder number into a friendly sum or difference, then multiply each part: \(18 \times 12 = 18 \times (10 + 2) = 180 + 36 = 216\) and \(17 \times 9 = 17 \times (10 - 1) = 170 - 17 = 153\).
The split number can come first too: \((10 + 5) \times 6 = 60 + 30 = 90\).
With pronumerals the law is written without \(\times\) signs: \(a(b + c) = ab + ac\). Removing the brackets is called expanding: \(2(x + 5) = 2x + 10\).
Quick rules from the distributive law:
| To multiply by | Example |
|---|---|
| \(9 = 10 - 1\): times \(10\), take away the number | \(13 \times 9 = 130 - 13 = 117\) |
| \(11 = 10 + 1\): times \(10\), add the number | \(13 \times 11 = 130 + 13 = 143\) |
| \(8 = 10 - 2\): times \(10\), take away double | \(13 \times 8 = 130 - 26 = 104\) |
| \(12 = 10 + 2\): times \(10\), add double | \(13 \times 12 = 130 + 26 = 156\) |
| \(101 = 100 + 1\): times \(100\), add the number | \(13 \times 101 = 1300 + 13 = 1313\) |
| \(99 = 100 - 1\): times \(100\), take away the number | \(13 \times 99 = 1300 - 13 = 1287\) |
How to use the distributive law
- Split one number into a friendly sum or difference (\(99 = 100 - 1\)).
- Multiply every part by the other number.
- Add (or subtract) the results.
- Check the size with an estimate.
| \(6 \times 125\) | \(=\) | \(6 \times (100 + 20 + 5)\) |
| \(\) | \(=\) | \(600 + 120 + 30\) |
| \(\) | \(=\) | \(750\) |
The area is \(750\) m².
(i) \(23 \times 11\)
(ii) \(32 \times 101\)
| \(23 \times 11\) | \(=\) | \(23 \times (10 + 1)\) |
| \(\) | \(=\) | \(230 + 23 = 253\) |
| \(32 \times 101\) | \(=\) | \(32 \times (100 + 1)\) |
| \(\) | \(=\) | \(3200 + 32 = 3232\) |
(i) \(253\)
(ii) \(3232\)
(i) \(7 \times 99\)
(ii) \(16 \times 99\)
| \(7 \times 99\) | \(=\) | \(7 \times (100 - 1)\) |
| \(\) | \(=\) | \(700 - 7 = 693\) |
| \(16 \times 99\) | \(=\) | \(16 \times (100 - 1)\) |
| \(\) | \(=\) | \(1600 - 16 = 1584\) |
(i) \(693\)
(ii) \(1584\)
(i) \(23 \times 8\)
(ii) \(14 \times 12\)
| \(23 \times 8\) | \(=\) | \(23 \times (10 - 2)\) |
| \(\) | \(=\) | \(230 - 46 = 184\) |
| \(14 \times 12\) | \(=\) | \(14 \times (10 + 2)\) |
| \(\) | \(=\) | \(140 + 28 = 168\) |
(i) \(184\)
(ii) \(168\)
| \(15 \times 39\) | \(=\) | \(15 \times (40 - 1)\) |
| \(\) | \(=\) | \(600 - 15\) |
| \(\) | \(=\) | \(585\) |
The tickets cost \(\$585\).
(i) \(26 \times 19\)
(ii) \(34 \times 102\)
| \(26 \times 19\) | \(=\) | \(26 \times (20 - 1)\) |
| \(\) | \(=\) | \(520 - 26 = 494\) |
| \(34 \times 102\) | \(=\) | \(34 \times (100 + 2)\) |
| \(\) | \(=\) | \(3400 + 68 = 3468\) |
(i) \(494\)
(ii) \(3468\)
| \(5(2k - 3)\) | \(=\) | \(5 \times 2k - 5 \times 3\) |
| \(\) | \(=\) | \(10k - 15\) |
\(10k\) and \(15\) are not like terms, so stop here. Check with \(k = 4\): \(5 \times (8 - 3) = 25\) and \(40 - 15 = 25\). So \(5(2k - 3) = 10k - 15\).
Common pitfalls
Frequently asked questions
What is the distributive law?
\(a(b + c) = ab + ac\): multiply every part inside the brackets, then add.
How do you multiply by 99 in your head?
Multiply by \(100\), then take away the number once. \(16 \times 99 = 1600 - 16 = 1584\).
How do you multiply by 11 in your head?
Multiply by \(10\), then add the number. \(23 \times 11 = 230 + 23 = 253\).
What does expanding mean?
Removing brackets by multiplying every term inside. \(2(x + 5) = 2x + 10\).
Why does the area model show the distributive law?
The area of the whole rectangle equals the areas of its parts added together.