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08 Jan 2025, 03:54
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A library has both fiction and nonfiction books. Of the fiction books, \(\frac{4}{5}\) are hardcover, and \(\frac{3}{4}\) of these hardcover books are authored by local writers. If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?
A. 15
B. 20
C. 40
D. 60
E. 80
A. 15
B. 20
C. 40
D. 60
E. 80
08 Jan 2025, 03:54
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Bookmarks
Official Solution:
A library has both fiction and nonfiction books. Of the fiction books, \(\frac{4}{5}\) are hardcover, and \(\frac{3}{4}\) of these hardcover books are authored by local writers. If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?
A. 15
B. 20
C. 40
D. 60
E. 80
Assuming the number of fiction books is F, then \(F * \frac{4}{5} * \frac{3}{4} = F * \frac{3}{5}\) must be an integer. This implies that F must be a multiple of 5. Additionally, since the number of fiction books is three times the number of nonfiction books, F must also be a multiple of 3. The least common multiple of 5 and 3 is 15, so the least number of fiction books is 15. This makes the least number of nonfiction books 5, and the least total number of books is 15 + 5 = 20.
Answer: B
A library has both fiction and nonfiction books. Of the fiction books, \(\frac{4}{5}\) are hardcover, and \(\frac{3}{4}\) of these hardcover books are authored by local writers. If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?
A. 15
B. 20
C. 40
D. 60
E. 80
Assuming the number of fiction books is F, then \(F * \frac{4}{5} * \frac{3}{4} = F * \frac{3}{5}\) must be an integer. This implies that F must be a multiple of 5. Additionally, since the number of fiction books is three times the number of nonfiction books, F must also be a multiple of 3. The least common multiple of 5 and 3 is 15, so the least number of fiction books is 15. This makes the least number of nonfiction books 5, and the least total number of books is 15 + 5 = 20.
Answer: B
