# How do you use the distributive property with fractions?

Oct 26, 2014

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.