Multiply by parts

The idea

The distributive property of multiplication over addition states a(b + c) = ab + ac, and by commutativity (a + b)c = ac + bc; it holds for all integers, and a(b + c) means a × (b + c).

Red chairs and blue chairs

A hall has 7 rows of chairs. In every row, 4 chairs are red and 3 are blue.

How could you count all the chairs?

• 7 × (4 + 3)
• 7 × 4 + 7 × 3
• Both ways give the same number

Cut the rectangle

Each square is one chair. The rectangle has a rows and b + c chairs in a row. Drag the purple cut, or use the arrows. Compare the Whole and the Parts.

Multiply each part

The distributive property of multiplication over addition: a(b + c) = ab + ac. By commutativity, (a + b)c = ac + bc. It holds for all integers.

a(b + c) means a × (b + c). We often skip the × next to a bracket, just as 5a means 5 × a. So 23(27 + 1) = 23 × 27 + 23.

The number outside the bracket multiplies every part inside. Not only the first.

Notes

a(b + c) = ab + ac and (a + b)c = ac + bc: the number outside multiplies every part inside the bracket. It holds for all integers.

Check yourself

Which one is equal to 6(10 + 4)?

Use the distributive property to find 7 × 13. Write 13 as 10 + 3.

Answer: 91

7(10 + 3) = 7 × 10 + 7 × 3 = 70 + 21 = 91.

Which one is equal to (x + 4)5?

The property holds for all integers. Find 5(−3 + 7).

Answer: 20

5(−3 + 7) = 5 × (−3) + 5 × 7 = −15 + 35 = 20. Or: 5 × 4 = 20.

• 6 × 10 + 4. That multiplies only the 10 by 6. The 6 must multiply the 4 as well.
• 6 × 10 + 6 × 4 — correct. Yes! 6 × 14 = 84 and 60 + 24 = 84.
• 6 + 10 × 4. That adds 6 instead of multiplying. The 6 multiplies both 10 and 4.
• 5x + 4. The 5 multiplies x, but it must multiply the 4 too.
• x + 20. The 5 multiplies the 4, but it must multiply x too.
• 5x + 20 — correct. Yes! (x + 4)5 = 5x + 5 × 4 = 5x + 20.