Say a = length of side of cube. Then you get that the cube has 6 sides of area A = a^2, so the total area of the cube will be 6a^2.

Now think of the rectangular solid: the cube's length has been modified, so you get basically 4 sides that have been changed and 2 that haven't. Your surface area will be 4*2a^2 + 2*a^2 = 10 a^2.

So the ratio will be 6a^2/10a^2 = 6/10 = 3/5.

Please forgive my bad drawing....