A wooden rectangular prism has dimensions 4 by 5 by 6 . This solid is painted green and then cut into 1 by 1 by 1 cubes. The ratio of the number of cubes with exactly two green faces to the number of cubes with three green faces is
(A) $9: 2$
(B) 9:4
(C) 6:1
(D) 3:1
(E) $5: 2$
A wooden rectangular prism has dimensions 4 by 5 by 6 . This solid is painted green and then cut into 1 by 1 by 1 cubes. The ratio of the number of cubes with exactly two green faces to the number of cubes with three green faces is
(A) $9: 2$
(B) $9: 4$
(C) 6:1
(D) 3:1
(E) $5: 2$

## Solution

The cubes with two green faces are the cubes along the edges, not counting the corner cubes. In each dimension, we lost two cubes to the corners so we then have four edges with 4 cubes, four with 3 cubes and four with 2 cubes. The total number of cubes with paint on two edges is then $4(4)+4(3)+4(2)=36$. The number of cubes that have paint on three sides are the corner cubes of which there are eight. The required ratio is then $36: 8$ or $9: 2$.

ANSWER: (A)