The chart shown gives the cost of installing carpet in four rectangular rooms of various sizes. The cost per square metre of installing carpet is always the same.

Width (metres)

Length (metres)

|  | 10 | $y$ |
| :---: | :---: | :---: |
| 15 | $\$ 397.50$ | $\$ 675.75$ |
| $x$ | $\$ 742.00$ | $\$ z$ |

What is the value of $z$ ?
(A) 331.25
(B) 463.75
(C) 1815.25
(D) 476.00
(E) 1261.40
Solution 1

Let $\$ c$ be the cost per square metre of installing carpeting.

Then in each situation, the area of the room times the cost per square metre equals the total price.

From the top left entry in the table, $15 \cdot 10 \cdot \$ c=\$ 397.50$.

From the top right entry in the table, $15 \cdot y \cdot \$ c=\$ 675.75$.

From the bottom left entry in the table, $x \cdot 10 \cdot \$ c=\$ 742.00$.

From the bottom right entry in the table, $x \cdot y \cdot \$ c=\$ z$.

Now,

$$
z=x \cdot y \cdot c=x \cdot y \cdot c \cdot \frac{10 \cdot 15 \cdot c}{10 \cdot 15 \cdot c}=\frac{(x \cdot 10 \cdot c) \cdot(15 \cdot y \cdot c)}{15 \cdot 10 \cdot c}=\frac{(742.00) \cdot(675.75)}{397.50}=1261.40
$$

Therefore, $z=1261.40$.

## Solution 2

Let $\$ c$ be the cost per square metre of installing carpeting.

Then in each situation, the area of the room times the cost per square metre equals the total price.

From the top left entry in the table, $15 \cdot 10 \cdot \$ c=\$ 397.50$.

Thus, $c=\frac{397.50}{15 \cdot 10}=2.65$.

From the top right entry in the table, $15 \cdot y \cdot \$ c=\$ 675.75$.

Thus, $y=\frac{675.75}{15 \cdot 2.65}=17$.

From the bottom left entry in the table, $x \cdot 10 \cdot \$ c=\$ 742.00$.

Thus, $x=\frac{742.00}{10 \cdot 2.65}=28$.

From the bottom right entry in the table, $x \cdot y \cdot \$ c=\$ z$.

Thus, $z=28 \cdot 17 \cdot 2.65=1261.40$.

Therefore, $z=1261.40$.