A rectangle whose side lengths are whole numbers has area $48 \mathrm{~cm}^{2}$. The perimeter of this rectangle is $32 \mathrm{~cm}$. Measured in $\mathrm{cm}$, the positive difference between the length and the width of the rectangle is
(A) 47
(B) 2
(C) 22
(D) 8
(E) 13

Grade 8
The area of the rectangle, $48 \mathrm{~cm}^{2}$, is given by the product of the rectangle's length and width, and so we must consider all possible pairs of whole numbers whose product is 48 .

In the table below, we systematically examine all possible factors of 48 in order to determine the possible whole number side lengths of the rectangle and its perimeter.

| Factors of 48 | Side Lengths of Rectangle | Perimeter of Rectangle |
| :---: | :---: | :---: |
| $48=1 \times 48$ | 1 and 48 | $2 \times(1+48)=2 \times 49=98$ |
| $48=2 \times 24$ | 2 and 24 | $2 \times(2+24)=2 \times 26=52$ |
| $48=3 \times 16$ | 3 and 16 | $2 \times(3+16)=2 \times 19=38$ |
| $48=4 \times 12$ | 4 and 12 | $2 \times(4+12)=2 \times 16=32$ |
| $48=6 \times 8$ | 6 and 8 | $2 \times(6+8)=2 \times 14=28$ |

The side lengths of the rectangle having area $48 \mathrm{~cm}^{2}$ and perimeter $32 \mathrm{~cm}$, are 4 and 12 . In $\mathrm{cm}$, the positive difference between the length and width of the rectangle is $12-4=8$.