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.
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