A number times a bracket

The idea

A number multiplying a bracket multiplies each term inside it (distributive property).

A rectangle in two parts

A rectangle is 4 squares tall. Its width is cut into two parts: x squares and y squares. Its area is 4 × (x + y), written 4(x + y).

How should the 4 multiply the bracket (x + y)?

• Only x: 4x + y
• Both x and y: 4x + 4y
• Neither: 4 + x + y

Count the squares two ways

Change x and y. The coloured parts give 4x + 4y. The whole rectangle gives 4(x + y). Compare the two.

The number multiplies each term

One part has 4x squares and the other has 4y. Counting the whole rectangle at once gives 4(x + y). Both counts are right, so they are equal. This is the distributive property: the multiple of a sum is the sum of the multiples.

Then like terms can be joined: 4(x + y) − y = 4x + 4y − y = 4x + 3y.

A number multiplying a bracket multiplies each term inside it: 4(x + y) = 4x + 4y. This is the distributive property.

• All at once | Part by part
• 4 × (20 + 3) = 92 | 4 × 20 + 4 × 3 = 92
• 4(x + y) | 4x + 4y
• 10(y − 3) | 10y − 30

Notes

A number multiplying a bracket multiplies each term inside it (the distributive property): 4(x + y) = 4x + 4y.

Check yourself

Open the bracket: 5(a + 3).

Open the bracket: 10(y − 3).

Open 3(2p + 5q) to 6p + ?q. What is the missing number?

Answer: 15

3 × 2p = 6p and 3 × 5q = 15q, so 3(2p + 5q) = 6p + 15q.

Simplify 4(x + y) − y to 4x + ?y. What is the missing number?

Answer: 3

4(x + y) − y = 4x + 4y − y = 4x + (4 − 1)y = 4x + 3y.

• 5a + 3. The 5 multiplies the 3 as well: 5 × 3 = 15.
• 5a + 15 — correct. Yes! 5 × a = 5a and 5 × 3 = 15.
• 5 + a + 3. A number written next to a bracket means times, not plus.
• 10y − 3. That is the other expression, 10 × y − 3. In 10(y − 3) the 10 multiplies both y and 3.
• 10y + 30. The term is −3, so the product is −30, not +30.
• 10y − 30 — correct. Yes! 10 × y = 10y and 10 × (−3) = −30.