The volume of a rectangular prism is $30 \mathrm{~cm}^{3}$. The length of the prism is doubled, the width is tripled, and the height is divided by four. The volume of the new prism is
(A) $31 \mathrm{~cm}^{3}$
(B) $120 \mathrm{~cm}^{3}$
(C) $60 \mathrm{~cm}^{3}$
(D) $90 \mathrm{~cm}^{3}$
(E) $45 \mathrm{~cm}^{3}$
Solution 1

The volume of any rectangular prism is given by the product of the length, the width, and the height of the prism.

When the length of the prism is doubled, the product of the new length, the width, and the height of the prism doubles, and so the volume of the prism doubles.

Since the original prism has a volume of $30 \mathrm{~cm}^{3}$, then doubling the length creates a new prism with volume $30 \times 2=60 \mathrm{~cm}^{3}$.

When the width of this new prism is tripled, the product of the length, the new width, and the height of the prism is tripled, and so the volume of the prism triples.

Since the prism has a volume of $60 \mathrm{~cm}^{3}$, then tripling the width creates a new prism with volume $60 \times 3=180 \mathrm{~cm}^{3}$.

When the height of the prism is divided by four, the product of the length, the width, and the new height of the prism is divided by four, and so the volume of the prism is divided by four. Since the prism has a volume of $180 \mathrm{~cm}^{3}$, then dividing the height by four creates a new prism with volume $180 \div 4=45 \mathrm{~cm}^{3}$.

## Solution 2

The volume of any rectangular prism is given by the product of its length, $l$, its width, $w$, and its height, $h$, which equals $l w h$.

When the length of the prism is doubled, the length of the new prism is $2 l$.

Similarly, when the width is tripled, the new width is $3 w$, and when the height is divided by four, the new height is $\frac{1}{4} h$.

Therefore, the volume of the new prism is the product of its length, $2 l$, its width, $3 w$, and its height, $\frac{1}{4} h$, which equals $(2 l)(3 w)\left(\frac{1}{4} h\right)$ or $\frac{3}{2} l w h$.

That is, the volume of the new prism is $\frac{3}{2}$ times larger than the volume of the original prism. Since the original prism has a volume of $30 \mathrm{~cm}^{3}$, then doubling the length, tripling the width, and dividing the height by four, creates a new prism with volume $30 \times \frac{3}{2}=\frac{90}{2}=45 \mathrm{~cm}^{3}$.

ANSWER: (E)