This question has the following situation: "In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hardcover books and exactly 1/4 of the softcover books are nonfiction. What is the greatest possible number of books in this home library that could be nonfiction?"

(A) 18
(B) 40
(C) 67
(D) 72
(E) 96

The correct answer is (C). The solution represents the situation with the following equation: "(2/3)x + (1/4)(108 - x), where x is the number of hardcover books in the library." The equation is simplified to 27 + (5/12)x. I am fine up to this point, and I become confused when the solution explains the following: "Since there are some softcover books in the library, x cannot equal 108, and the next greatest possible whole number value for the expression, (5/12)x, is 40, if x is 96. The greatest possible number of nonfiction books in the library, then, is 27 + 40 = 67." Can you please explain how we obtained x = 40 from the original equation?

Hey Atomico:

I'd look at this problem this way - your goal is to get as many nonfiction books as possible, right? If that's the case, and 2/3 of hardcovers are nonfiction as opposed to only 1/4 of softcovers, you're going to want to maximize the number of hardcover books.

But there's a catch...in order to have exactly 1/4 of the softcovers as nonfiction, we need to have at least 4 softcover books. But to only have 4 would mean that there are 104 hardcovers, and because we can't divide 104 into 2/3, we need to find a multiple of softcovers that will leave a multiple of 3 hardcovers. The common multiple of 3 and 4 is 12, and since we want to minimize the number of softcovers, we only want to use 12 of them.

Therefore, as nonfiction books we have:

1/4 of the 12 softcovers = 3
2/3 of the 96 remaining hardcovers = 64

Sum them for a total of 67 nonfiction books.

In the equation they give in the solution, they get to 40 by taking 5/12 of the number of hardcover books, and as we did above noting that we can't have 108 as that number so they're taking 5/12 of 96 (96/12 = 8, which multiplied by 5 is 40). That's another interpretation of how to solve, but if the pure algebra is annoying you can do this one by more logically reasoning out the steps with numbers, too.