The distributive property

This is one of the most powerful rules in mathematics. It lets us swap a single multiplication for two , or two for one , depending on which form is easier.

Idea

The distributive property says: $a \times (b + c) = a \times b + a \times c.$ And similarly: $a \times (b - c) = a \times b - a \times c.$

You can picture it as a rectangle of width $a$ and length $b + c$. Total area is $a(b + c)$. Cut the rectangle vertically at $b$, and you have two rectangles of areas $ab$ and $ac$. The two cuts must add up to the whole.

The rule works both ways:

• Expanding , $7 \times 23 = 7 \times (20 + 3) = 140 + 21 = 161$.
• Factoring , $42 + 35 = 7 \times 6 + 7 \times 5 = 7 \times (6 + 5) = 7 \times 11 = 77$.

Use expanding when you have a hard multiplication. Break the second factor into a sum of easy numbers (like $20 + 3$ instead of $23$) and multiply each.

Use factoring when you spot a common factor in a sum. It often turns a tedious problem into a one-step calculation.

The same rule works with three or more terms: $a(b + c + d) = ab + ac + ad$. And mixed signs: $a(b - c + d) = ab - ac + ad$.

Worked examples

Example 1. Use distribution to compute $6 \times 102$ mentally.

$6 \times 102 = 6 \times (100 + 2) = 600 + 12 = 612$.

Example 2. Use distribution backwards: $14 \times 7 + 14 \times 3$.

Common factor is $14$: $14 \times (7 + 3) = 14 \times 10 = 140$.

Example 3. Compute $98 \times 6$ using a difference inside the bracket.

$98 = 100 - 2$, so $98 \times 6 = (100 - 2) \times 6 = 600 - 12 = 588$.

Example 4. Simplify $5 \times (8 + 2 - 3)$ two ways.

By BODMAS: $5 \times (10 - 3) = 5 \times 7 = 35$. By distribution: $5 \times 8 + 5 \times 2 - 5 \times 3 = 40 + 10 - 15 = 35$. Both give $35$. ✓

Try it yourself

1. Expand $4(6 + 9)$.
2. Compute $7 \times 99$ using distribution.
3. Compute $11 \times 25 + 11 \times 75$ using distribution.
4. Expand $9(10 - 2)$.
5. Compute $12 \times 13 + 12 \times 7$.
6. Use the distributive law to simplify $25 \times 16$.
7. Show that $a(b + c + d) = ab + ac + ad$ holds for $a = 3$, $b = 4$, $c = 2$, $d = 5$.
8. Find the missing number: $4 \times \square + 4 \times 8 = 4 \times 12$.

Activity

Draw a $7 \times 13$ grid on graph paper. Count the unit squares in two ways: as one rectangle (= $91$), and as two rectangles $7 \times 10$ and $7 \times 3$ joined side by side (= $70 + 21 = 91$). Convince yourself the distributive law is just rectangle-cutting.