A solid triangular prism is made up of 27 identical smaller solid triangular prisms, as shown. The length of every edge of each of the smaller prisms is 1 . If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted?

![](https://cdn.mathpix.com/cropped/2024_04_30_d1e45f8438f865d01c92g-2.jpg?height=285&width=266&top_left_y=1256&top_left_x=1447)
Each of the 27 smaller triangular prisms has 3 faces that are squares with side length 1 and 2 faces that are equilateral triangles with side length 1.

Combined, these prisms have $27 \cdot 3=81$ square faces and $27 \cdot 2=54$ triangular faces.

Each of the 3 square faces of the larger triangular prism is made up of 9 of the smaller square faces, which means that $3 \cdot 9=27$ of these smaller square faces are painted.

Each of the 2 triangular faces of the larger triangular prism is made up of 9 of the smaller square faces, which means that $2 \cdot 9=18$ of these smaller triangular faces are painted.

Therefore, 27 of the 81 smaller square faces are painted and 18 of the 54 smaller triangular faces are painted. In other words, $\frac{1}{3}$ of each type of face is painted, so $\frac{1}{3}$ of the total surface area of the smaller prisms is painted.

ANSWER: $\frac{1}{3}$