The distributive property of multiplication over addition (or subtraction) states that the product of a sum (or difference) is equal to the sum (or difference) of the products.

### Examples

7 × 36 = 7 × (30 + 6) = 7 × 30 + 7 × 6 = 210 + 42 = 252

7 × 36 = 7 × (40 – 4) = 7 × 40 – 7 × 4 = 280 – 28 = 252

### Property

The distributive property can simplify calculations.

It is especially helpful in simplifying mental math.

An operation denoted by ⊗ distributes over an operation denoted by ⊕ if, regardless of the numbers

*a*,

*b*and

*c*, we have :

*a*⊗ (

*b*⊕

*c*) = (

*a*⊗

*b*) ⊕ (

*a*⊗

*c*). This property is called the distributive property.

Specifically, the operation is left-distributive if : *a* ⊗ (*b* ⊕ *c*) = (*a* ⊗ *b*) ⊕ (*a* ⊗ *c*).

It is right-distributive if : (*a* ⊕ *b*) ⊗ *c* = (*a* ⊗ *c*) ⊕ (*a *⊗ *c*).

### Examples

- For the set of real numbers, multiplication distributes over addition :

12 × (3 + 10) = (12 × 3) + (12 × 10) = 36 + 120 = 156 - For the set of real numbers, multiplication distributes over subtraction :

25 × (20 − 5) = (25 × 20) − (25 × 5) = 500 − 125 = 375 - In set theory, the following equalities are obtained for the subsets A, B and C of U :

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

this means that the intersection of sets is left-distributive over the union of sets. It can also be shown that the union of sets distributes over the intersection of sets :

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)