# Word Problems Involving the Distributive Property

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

$x(y+z)=xy+xz$ .

This very important property is frequently used in word problems .

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Example 1:
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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.

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

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

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Example 2:
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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 $\text{\$13,\$11and\$8}$ .

$\$13+\$11+\$8$

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.

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

Now Add.

$=384$

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