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Question

# A rectangular room has length L and width W, where L and W are measured in feet. If carpeting costs x dollars per square yard, express the cost of carpeting this room algebraically.

Solution

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Given:

\begin{align*}\text{Rectangular room} \\ \text{Length room}&=L\text{ feet} \\ \text{Width room}&=W\text{ feet} \\ \text{Price per square yard}&=x\text{ dollars} \end{align*}

Since the room is rectangular, the area of the room is then the product of the length and the width of the room.

\begin{align*}\text{Area room}&= \text{Length room}\times \text{Width room} \\ &=L\text{ feet}\times W\text{ feet} \\ &=LW\text{ square feet} \end{align*}

We have also been given that 1 yard equals 3 feet.

$1\text{ yard}=3\text{ feet}$

1 square yard is basically the product of 1 yard and 1 yard. Similarly, 1 square foot is the product of 1 foot and 1 foot.

\begin{align*} 1\text{ square yard}&=1\text{ yard}\times 1\text{ yard} \\ &=3\text{ feet}\times 3\text{ feet} &\color{#4257b2}1\text{ yard}=3\text{ feet} \\ &=3\times 3\text{ square feet} \\ &=9\text{ square feet} \end{align*}

Thus there are 9 square feet in a square yard.

To re-express the area of the room in square yard, we then need to divide the area in square feet by 9.

\begin{align*}\text{Area room}&= \text{Length room}\times \text{Width room} \\ &=L\text{ feet}\times W\text{ feet} \\ &=LW\text{ square feet} \\ &=\dfrac{LW}{9}\text{ square yard} \end{align*}

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