# When to Use the Distributive Property

## Key Questions

• The Distributive Property can help make numbers easier to solve because you are "breaking the numbers into parts".

In Algebra, you can use the Distributive Property if you want to remove parentheses in a problem.

For example: 3(2+5)

What you are essentially doing when you distribute is multiplying the number outside the parentheses by each of the numbers inside the parenthesis. So you would do:

3$\times$2= 6 and 3$\times$5=1 5, now to find the answer just add these numbers up, you will get 21.

• Distributive property of multiplication relative to addition is universal for all numbers - integers, rational, real, complex - and states that
$a \cdot \left(b + c\right) = a \cdot b + a \cdot c$

In particular, if we deal with fractions, when each member of the above formula can be represented in the form $\frac{x}{y}$ where both $x$ and $y$ are integers, the distributive law works in exactly the same way:
$\frac{m}{n} \cdot \left(\frac{p}{q} + \frac{r}{s}\right) = \frac{m}{n} \cdot \frac{p}{q} + \frac{m}{n} \cdot \frac{r}{s}$
where $m , n , p , q , r , s$ are integers and denominators of each fraction $n , q , s$ are not zeros.

If we know the distributive law for integer numbers and understand that a rational number $\frac{x}{y}$ is, by definition, a new number that, if multiplies by $y$, produces $x$, the above formula for fractions can be easily proved by transforming fractions on the left and on the right to a common denominator $n \cdot q \cdot s$:
$\frac{m \cdot \left(p \cdot s + r \cdot q\right)}{n \cdot q \cdot s} = \frac{m \cdot p \cdot s + m \cdot r \cdot q}{n \cdot q \cdot s}$.
In this form the distributive law for fractions is a simple consequence of the distributive law for integers.