What Is the Distributive Property?
The distributive property says: a × (b + c) = a × b + a × c
You can multiply the sum, or multiply each part separately and add the results. Both give the same answer.
Example
6 × 14 = 6 × (10 + 4) = 6 × 10 + 6 × 4 = 60 + 24 = 84
Instead of multiplying 6 × 14 directly, you break 14 into 10 + 4, multiply each part by 6, and add. Same answer, easier computation.
Why it matters
The distributive property is behind:
Mental math: 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686
Multi-digit multiplication: The standard algorithm uses the distributive property at every step.
Area model: Breaking a rectangle into parts uses the distributive property visually.
Algebra: 3(x + 4) = 3x + 12. Distributing is one of the first algebra skills.
Visual understanding
Draw a rectangle 6 units tall and 14 units wide. Split the width into 10 + 4. You get two rectangles:
- 6 × 10 = 60
- 6 × 4 = 24
- Total: 84
The total area is the same whether you compute it as one rectangle or two. That is the distributive property, visualized.
Common confusion
Distributing over multiplication: The distributive property works for multiplication over addition (or subtraction). It does NOT work for multiplication over multiplication: 2 × (3 × 4) ≠ (2 × 3) × (2 × 4).
Related concepts
- Area model: the visual representation
- Multi-digit multiplication: uses the distributive property
- Order of operations: the distributive property is compatible with PEMDAS
- Variables and equations: distributing in algebra