I would use the word "circumscribed" where you used both "described" and "inscribed", but I understand what you mean, I think.
I used a much simpler method and got the same answer you did, 5:1. So maybe you are actually right. (I could be wrong, too!)
What I did was to take the radius of the inner ball as r, and think about a cross-section of the figure through the center of the ball and a vertical side of the prism. This led to finding that the radius of the outer ball is [MATH]r\sqrt{5}[/MATH], and therefore to your answer.
Make sure you looked at the answer to the right problem!
I can see a simple mistake they could have made to get an answer of 4:1. On the other hand, if the outer ball is not circumscribed about the prism (passing through all 6 of its vertices), but something else, then maybe we're solving the wrong problem.