Distributive property for more terms
A number times a sum of three parts is the sum of three products.
A sum of three parts
We know that (a + b) × n = an + bn. What if the first number is a sum of three parts, like (2 + 3 + 4) × 5?
Is (2 + 3 + 4) × 5 equal to 2 × 5 + 3 × 5 + 4 × 5?
What this lesson covers
The idea
The distributive property holds when one of the numbers has more than two terms: (a + b + c) × n = an + bn + cn.
A sum of three parts
We know that (a + b) × n = an + bn. What if the first number is a sum of three parts, like (2 + 3 + 4) × 5?
Is (2 + 3 + 4) × 5 equal to 2 × 5 + 3 × 5 + 4 × 5?
- Yes, they are equal
- No, the left side is bigger
- No, the right side is bigger
Cut the rectangle in three
The rectangle is n tall and its width is cut into parts a, b and c. Change the numbers. Watch the whole area and the three small areas added up.
Three parts, three products
The distributive property holds when one of the numbers has more than two terms: (a + b + c) × n = an + bn + cn.
(2 + 3 + 4) × 5 = 2×5 + 3×5 + 4×5 = 10 + 15 + 20 = 45
Notes
The distributive property holds when one of the numbers has more than two terms: (a + b + c) × n = an + bn + cn.
Check yourself
Find (3 + 4 + 5) × 6 by working out 3×6 + 4×6 + 5×6.
Answer: 72
3×6 + 4×6 + 5×6 = 18 + 24 + 30 = 72, and (3 + 4 + 5) × 6 = 12 × 6 = 72 too.
Use 123 × 4 = (100 + 20 + 3) × 4. What is the product?
Answer: 492
100×4 + 20×4 + 3×4 = 400 + 80 + 12 = 492.
Which is equal to (a + b + c) × n?
(2 + 3 + 4) × 5 = 10 + 15 + ?. What is the missing number?
Answer: 20
The three products are 2×5 = 10, 3×5 = 15 and 4×5 = 20. Together 45.
- an + bn + cn — correct. Yes! n multiplies each of the three parts.
- a + b + cn. Here n multiplies only c. It must multiply a and b too.
- an + bn + c. Here n multiplies only a and b. It must multiply c too.
- an × bn × cn. The parts are added, not multiplied. The three products are added.