The given cube is cut into four pieces by two planes. The first plane is parallel to face $A B C D$ and passes through the midpoint of edge $B G$. The second plane passes through the midpoints of edges $A B, A D, H E$, and $G H$. Determine the ratio of the volumes of the smallest and largest of the four pieces.
(A) $3: 8$
(B) 7:24
(C) 7:25
(D) $7: 17$
(E) 5:11

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-140.jpg?height=366&width=344&top_left_y=2164&top_left_x=1335)
The given cube is cut into four pieces by two planes. The first plane is parallel to face $A B C D$ and passes through the midpoint of edge $B G$. The second plane passes through the midpoints of edges $A B, A D, H E$, and $G H$. Determine the ratio of the volumes of the smallest and largest of the four pieces.
(A) $3: 8$
(B) 7:24
(C) 7:25
(D) 7:17
(E) 5:11

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-285.jpg?height=420&width=412&top_left_y=679&top_left_x=1298)

## Solution

For convenience, let each edge of the cube have length 2. The plane $P Q R S$ through the mid-point of $B G$ and parallel to face $A B C D$ bisects the volume of the cube. The plane through $K, L, M$, and $N$ also bisects the volume of the cube and contains the line segment $Q S$. Hence the two planes divide the cube into four pieces, two equal 'smaller' pieces and two equal 'larger' pieces. Extend $Q K$ and $S N$ to meet at $T$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-285.jpg?height=537&width=431&top_left_y=1233&top_left_x=1389)

We note that $\triangle T A N$ is similar to $\triangle T P S$. Since $\frac{T A}{T P}=\frac{A N}{P S}$ and $A N=1, P S=2$ and $P A=1$ it is easy to calculate to find $T A=1$.

The volume of the upper 'smallest' piece is equal to the volume of the tetradedron TQSP - volume of the tetradedron TKNA. This volume is, $\frac{1}{3}\left[\left(\frac{1}{2}(2)(2)(2)\right)\right]-\frac{1}{3}\left[\left(\frac{1}{2}\right)(1)(1)(1)\right]$

$$
=\frac{4}{3}-\frac{1}{6}=\frac{7}{6}
$$

The volume of a 'largest' piece is, $2 \times 2 \times 1-\frac{7}{6}=\frac{17}{6}$.

The ratio of the volume of the 'smallest' piece to that of the 'largest' piece is 7:17.

ANSWER: (D)