Question

A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

Answer 1

The volume of the larger cube is 3*3*3, which is 27.
The volume of the smaller cubes is 1*1*1, which is 1.
27/1=27
That means all cubes will be used, so only the placement of the cubes matter.
The center of the cube will show no sides,
The center of the face will show 1 side, which has a surface area of 1sq.in.
Since there are 6 white cubes and 6 faces with 6 face centers, and only one cube center, there will be 5 cubes on the center of the faces and 1 in the cube center.

This means there will be 0 faces of 1 cube and 1 faces shown for the rest of the cubes. Since each face is 1sq.un., the rest is math

(0*1)+((1*1)5)
(0)+(1*5)
0+5
5sq.un. of the cube is white.

a=The fraction of surface area showing
b=Surface area of large cube

b=3*3*number of sides on cube(6)

5*a=b
5*a=54
a=54/5

Your answer is 54/5.