The Distributive Law

What is the distributive law?

The distributive law is one of the most useful rules in all of mathematics. It describes how multiplication "spreads out," or distributes, over addition. In symbols:

a × (b + c) = a × b + a × c

In words: to multiply a number by a sum, you can multiply it by each part of the sum separately, then add the results. The same works for subtraction:

a × (b − c) = a × b − a × c

Seeing why it works

Picture a rectangle of chocolate squares. It is 3 rows tall and split into two blocks: one 4 squares wide and one 2 squares wide.

• Counting the whole thing: 3 × (4 + 2) = 3 × 6 = 18 squares.
• Counting block by block: 3 × 4 + 3 × 2 = 12 + 6 = 18 squares.

Both give 18, because the 3 rows reach across both blocks. That is the why behind the law: the multiplier touches every part inside the bracket.

Using it for mental math

The distributive law turns awkward multiplications into easy ones. Split one number into friendly parts, multiply each, then add.

Example 1 — Work out 7 × 23.

1. Split 23 into 20 + 3.
2. Distribute the 7: 7 × 20 + 7 × 3.
3. Calculate each: 140 + 21.
4. Add: 161.

Example 2 — Work out 6 × 98.

1. Write 98 as (100 − 2) — closer to a round number.
2. Distribute: 6 × 100 − 6 × 2.
3. Calculate: 600 − 12 = 588.

Calculation	Split as	Distributed	Answer
4 × 26	4 × (20 + 6)	80 + 24	104
9 × 31	9 × (30 + 1)	270 + 9	279
5 × 97	5 × (100 − 3)	500 − 15	485
8 × 45	8 × (40 + 5)	320 + 40	360

These are the same shortcuts behind many mental multiplication tricks.

Using it in algebra

The real power of the distributive law shows up in algebra, where it is used to expand brackets. The outside term multiplies every term inside.

Example 3 — Expand 3(x + 4).

1. Multiply 3 by the first term: 3 × x = 3x.
2. Multiply 3 by the second term: 3 × 4 = 12.
3. Write the result: 3x + 12.

Example 4 — Expand 5(2y − 3).

1. 5 × 2y = 10y.
2. 5 × 3 = 15, and the sign is minus.
3. Result: 10y − 15.

The commonest mistake is forgetting to multiply the second term — writing 3x + 4 instead of 3x + 12. Always check that the outside number has reached every term.

A practice activity

Try a "two ways" challenge:

1. Pick a multiplication like 6 × 34.
2. Solve it the normal way (e.g. column multiplication).
3. Now solve it again using the distributive law: 6 × 30 + 6 × 4.
4. Check both answers match.
5. Challenge: write three algebra brackets such as 4(n + 7), expand them, then ask a partner to check that every term was multiplied.

Where this leads

The distributive law is the engine behind expanding brackets, factorising and simplifying expressions. Once it feels natural, head to expanding brackets to practise the algebra version, and you will have unlocked a core skill used throughout secondary maths.