# Word Problems Involving the Distributive Property

The Distributive Property states that, for all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z$ ,

$x\left(y+z\right)=xy+xz$ .

This very important property is frequently used in word problems .

Example 1:

Rico's MP3 player holds songs of three different genres: grindcore, zydeco, and kuduro. There are $5$ times as many grindcore tracks as there are zydeco tracks, and there are $7$ times as many kuduro tracks as there are zydeco tracks. Let $x$ represent the number of zydeco tracks. Write an expression for the total number of tracks on the MP3 player, and simplify it.

Multiply the number of zydeco tracks by $5$ to get the number of grindcore tracks.

$5x$

Multiply the number of zydeco tracks by $7$ to get the number of kuduro tracks.

$7x$

Add up the numbers of all the tracks.

$x+5x+7x$

Simplify using the distributive property.

$\begin{array}{l}=\left(1+5+7\right)x\\ =13x\end{array}$

So, Rico's MP3 player holds $13x$ tracks.

Example 2:

A volleyball uniform costs $13$ for the shirt, $11$ for pants, and $8$ for socks. Write two equivalent expressions for the total cost of $12$ uniforms. Then find the cost.

Write an expression for the cost of $1$ uniform, add .

$13+11+8$

Write an expression for the cost of $12$ uniforms, multiply $12$ by the cost of $1$ uniform.

$12\left(13+11+8\right)$

Simplify using the distributive property.

$\begin{array}{l}=12\cdot 13+12\cdot 11+12\cdot 8\\ =156+132+96\end{array}$

$=384$
So, the total cost of $12$ uniforms is $384$ .