Exactly three faces of a $2 \times 2 \times 2$ cube are partially shaded, as shown. (Each of the three faces not shown in the diagram is not shaded.) What fraction of the total surface area of the cube is shaded?
(A) $\frac{1}{3}$
(B) $\frac{1}{4}$
(C) $\frac{1}{6}$
(D) $\frac{3}{8}$
(E) $\frac{2}{3}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-045.jpg?height=219&width=227&top_left_y=709&top_left_x=1342)

## Part B: Each correct answer is worth 6.
Solution 1

In the given diagram, we can see 3 of the 6 faces, or $\frac{1}{2}$ of the cube.

The remaining 3 faces (also $\frac{1}{2}$ of the cube) is unshaded.

Of the visible faces, $\frac{1}{2}$ of the area is shaded.

Therefore, the fraction of the total surface area that is shaded is $\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$.

## Solution 2

Since the cube is $2 \times 2 \times 2$, the area of each face is $2 \times 2=4$.

Since a cube has six faces, the total surface area of the cube is $6 \times 4=24$.

Each of the three faces that is partially shaded is one-half shaded, since each face is cut into two identical pieces by its diagonal.

Thus, the shaded area on each of these three faces is 2 , and so the total shaded area is $3 \times 2=6$. Therefore, the fraction of the total surface area that is shaded is $\frac{6}{24}=\frac{1}{4}$.

ANSWER: (B)