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Word Problems Involving the Distributive Property

The Distributive Property states that, for all real numbers $x, y, and z$ ,

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

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.

$5 x$

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

$7 x$

Add up the numbers of all the tracks.

$x + 5 x + 7 x$

Simplify using the distributive property.

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

So, Rico's MP3 player holds $13 x$ 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 and $8$ .

$$ 13 + $ 11 + $ 8$

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

$12 ($ 13 + $ 11 + $ 8)$

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$ .