Eight unit cubes are used to form a larger 2 by 2 by 2 cube. The six faces of this larger cube are then painted red. When the paint is dry, the larger cube is taken apart. What fraction of the total surface area of the unit cubes is red?
(A) $\frac{1}{6}$
(B) $\frac{2}{3}$
(C) $\frac{1}{2}$
(D) $\frac{1}{4}$
(E) $\frac{1}{3}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-111.jpg?height=323&width=296&top_left_y=1020&top_left_x=1294)
Solution 1

Each unit cube has three exposed faces on the larger cube, and three faces hidden.

Therefore, when the faces of the larger cube are painted, 3 of the 6 faces (or $\frac{1}{2}$ of the surface area) of each small cube are painted, so overall exactly $\frac{1}{2}$ of the surface of area of the small cubes is painted.

## Solution 2

Since the bigger cube is 2 by 2 by 2 , then it has six 2 by 2 faces, for a total surface area of $6 \times 2 \times 2=24$ square units. All of this surface area will be painted.

Each of the unit cubes has a surface area of 6 square units (since each has six 1 by 1 faces), so the total surface area of the 8 unit cubes is $8 \times 6=48$ square units.

Of this 48 square units, 24 square units is red, for a fraction of $\frac{24}{48}=\frac{1}{2}$.

ANSWER: (C)