Trevor has recently purchased a classical music compendium, a science book and a fiction novel from his neighborhood bookstore for a total of $100. It is known that:

(I) The music compendium costs more than twice the science book.

(II) Three copies of the science book cost more than four copies of the fiction novel.

(III) Three copies of the fiction novel cost more than the music compendium.

Given that all prices are whole numbers of dollars, how much does each item cost?

Calling the books M, S, and N, M>2S and 3S>4N, so 3M>6S>8N. If 3M=6S=8N, a solution could be M=24, S=12, and N=9. Since they do not sum to 100, we scale the solution (working with integers) up to M=53, S=27, and N=20, which doesn't work; however, a bit of fiddling around changes it to M=55, S=26, N=19 and that does it.

Posted by Federico Kereki on 2007-08-11 22:33:22