In a home library consisting of 108 books, some hardcover

BTGmoderatorDC wrote:In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hard cover 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

Goal:
To MAXIMIZE the number of nonfiction books.
Since the fraction of hardcover books that are nonfiction (2/3) is greater than the fraction of softcover books that are nonfiction (1/4), the number of nonfiction books will be maximized if we must MAXIMIZE THE NUMBER OF HARDCOVER BOOKS.

Since 2/3 of the hardcover books are nonfiction, the total number of hardcover books must be a MULTIPLE OF 3.
Since 1/4 of the softcover books are nonfiction, the total number of softcover books must be a MULTIPLE OF 4.
Some some of the books must be softcover, not all 108 books may be hardcover.
The greatest possible multiple of 3 that is less than 108 is 105.
Starting with 105, test decreasing multiples of 3 for the hardcover books until a multiple of 4 is yielded for the number of softcover books.

Options:
Hard = 105, soft = 108-105 = 3.
Hard = 102, soft = 108-102 = 6.
Hard = 99, soft = 108-99 = 9.
Hard = 96, soft = 108-96 = 12.

The option in blue yields a multiple of 4 for the number of softcover books.
Implication:
Greatest number of hardcover books that could be nonfiction = (2/3)(96) = 64.
Resulting number of softcover books that are nonfiction = (1/4)(12) = 3.
Thus:
Greater number of books that could be nonfiction = 64+3 = 67.

The correct answer is C.

To get the greatest number of non-fiction books, hardcover needs to be maximized while softcover minimized.

S = 12
H = 96

1/4*S = 3
2/3*H = 64

3 + 64 = 67.

Hence, C is the correct answer. Regards!

BTGmoderatorDC wrote:In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hard cover 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. Since we want the greatest possible number of nonfiction books in this home library and the fraction of hardcover that is nonfiction is greater than that of softcover, we want as many hardcover books as possible and as few softcover books as possible.

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:

⅔(96) + ¼(12) = 64 + 3 = 67

Answer: C