In a home library consisting of 108 books, some hardcover...

swerve wrote:In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hardcover and exactly 1/4 of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

A. 18
B. 40
C. 67
D. 72
E. 96

Since 2/3 of the hardcover are nonfiction as opposed to only 1/4 of the softcover, we want as many hardcover books as possible in the library if we want to determine the greatest possible number of nonfiction books.

However, we can we can’t have all 108 books as hardcover books since it’s stated that there is a least one softcover book. In fact, the number of softcover books must be a multiple of 4 since exactly 1/4 of them are nonfiction. Similarly, the number of hardcover books must be a multiple of 3 since exactly 2/3 of the hardcover books are nonfiction.

Let’s say there are 4 softcover books in the library; then there would be 104 hardcover books. However, 104 is not a multiple of 3.

Now let’s try 8 softcover books; then there would be 100 hardcover books. However, 100 is not a multiple of 3, either.

Finally, let’s try 12 softcover books; then there would be 96 hardcover books, and 96 is a multiple of 3. Thus, the greatest possible number of nonfiction books in the library is:

2/3(96) + 1/4(12) = 64 + 3 = 67

Answer: C