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Lena keeps all of her books on 7 wooden shelves. The average (arithmetic mean) number of pages per book in her collection is exactly 252.08. How many books does she have?
(1) Each shelf contains between 2 and 7 books, inclusive.
(2) The total number of pages across all of her books is less than 10,000.
(1) Each shelf contains between 2 and 7 books, inclusive.
(2) The total number of pages across all of her books is less than 10,000.
13 Jan 2026, 11:11
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Official Solution:
Lena keeps all of her books on 7 wooden shelves. The average (arithmetic mean) number of pages per book in her collection is exactly 252.08. How many books does she have?
An average of 252.08 pages per book means:
\(\frac{\text{total pages}}{\text{total books}} = 252.08 = \frac{25208}{100} = \frac{6302}{25}\).
Since 6302/25 is in lowest terms, this implies that the total number of pages is a multiple of 6302 and the total number of books is a multiple of 25.
(1) Each shelf contains between 2 and 7 books, inclusive.
With 7 shelves, the total number of books is between 2 * 7 = 14 and 7 * 7 = 49, inclusive. The only multiple of 25 in this range is 25. So the number of books is 25. Sufficient.
(2) The total number of pages across all of her books is less than 10,000.
There is only one multiple of 6,302 less than 10,000, 6,302 itself. Thus, the total number of pages is 6,302 and the number of books is 25. Sufficient.
Answer: D
Lena keeps all of her books on 7 wooden shelves. The average (arithmetic mean) number of pages per book in her collection is exactly 252.08. How many books does she have?
An average of 252.08 pages per book means:
\(\frac{\text{total pages}}{\text{total books}} = 252.08 = \frac{25208}{100} = \frac{6302}{25}\).
Since 6302/25 is in lowest terms, this implies that the total number of pages is a multiple of 6302 and the total number of books is a multiple of 25.
(1) Each shelf contains between 2 and 7 books, inclusive.
With 7 shelves, the total number of books is between 2 * 7 = 14 and 7 * 7 = 49, inclusive. The only multiple of 25 in this range is 25. So the number of books is 25. Sufficient.
(2) The total number of pages across all of her books is less than 10,000.
There is only one multiple of 6,302 less than 10,000, 6,302 itself. Thus, the total number of pages is 6,302 and the number of books is 25. Sufficient.
Answer: D
