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06 Aug 2026, 11:17
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At the end of Monday, a bookstore display had only hardcover books and paperback books, and \(\frac{7}{12}\) of the books on the display were hardcovers. On Tuesday, the store added hardcover books and paperback books to the display and removed none of the books that were already there. After the additions, was the fraction of paperback books on the display less than it had been at the end of Monday?
(1) Less than \(\frac{1}{2}\) of the books added on Tuesday were paperbacks.
(2) After the additions on Tuesday, the total number of books on the display was \(\frac{1}{3}\) greater than it had been at the end of Monday.
(1) Less than \(\frac{1}{2}\) of the books added on Tuesday were paperbacks.
(2) After the additions on Tuesday, the total number of books on the display was \(\frac{1}{3}\) greater than it had been at the end of Monday.
06 Aug 2026, 11:17
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Official Solution:
At the end of Monday, a bookstore display had only hardcover books and paperback books, and \(\frac{7}{12}\) of the books on the display were hardcovers. On Tuesday, the store added hardcover books and paperback books to the display and removed none of the books that were already there. After the additions, was the fraction of paperback books on the display less than it had been at the end of Monday?
At the end of Monday, \(\frac{5}{12}\) of the books were paperbacks. The paperback fraction will decrease only if the fraction of paperbacks among the books added on Tuesday is less than \(\frac{5}{12}\).
(1) Less than \(\frac{1}{2}\) of the books added on Tuesday were paperbacks.
This is not enough, because a fraction less than \(\frac{1}{2}\) could be either less than or greater than \(\frac{5}{12}\).
Not sufficient.
(2) After the additions on Tuesday, the total number of books on the display was \(\frac{1}{3}\) greater than it had been at the end of Monday.
This tells us how many books were added relative to the Monday total, but it does not tell us what fraction of the added books were paperbacks.
Not sufficient.
(1)+(2) Let the Monday total be 12 books. Then there were 5 paperbacks.
Statement (2) says 4 books were added. By statement (1), fewer than 2 of those 4 books were paperbacks, so at most 1 paperback was added.
Then the Tuesday paperback fraction is at most:
\(\frac{5 + 1}{12 + 4} = \frac{6}{16} = \frac{3}{8}\)
This is less than \(\frac{5}{12}\), so the answer is YES.
But let the Monday total be 36 books. Then there were 15 paperbacks.
Statement (2) says 12 books were added. By statement (1), fewer than 6 of those 12 books were paperbacks, so 5 paperbacks could have been added.
Then the Tuesday paperback fraction is:
\(\frac{15 + 5}{36 + 12} = \frac{20}{48} = \frac{5}{12}\)
This is not less than the Monday fraction, so the answer is NO.
Thus, even together, the statements do not give a definite answer.
Not sufficient.
Answer: E
At the end of Monday, a bookstore display had only hardcover books and paperback books, and \(\frac{7}{12}\) of the books on the display were hardcovers. On Tuesday, the store added hardcover books and paperback books to the display and removed none of the books that were already there. After the additions, was the fraction of paperback books on the display less than it had been at the end of Monday?
At the end of Monday, \(\frac{5}{12}\) of the books were paperbacks. The paperback fraction will decrease only if the fraction of paperbacks among the books added on Tuesday is less than \(\frac{5}{12}\).
(1) Less than \(\frac{1}{2}\) of the books added on Tuesday were paperbacks.
This is not enough, because a fraction less than \(\frac{1}{2}\) could be either less than or greater than \(\frac{5}{12}\).
Not sufficient.
(2) After the additions on Tuesday, the total number of books on the display was \(\frac{1}{3}\) greater than it had been at the end of Monday.
This tells us how many books were added relative to the Monday total, but it does not tell us what fraction of the added books were paperbacks.
Not sufficient.
(1)+(2) Let the Monday total be 12 books. Then there were 5 paperbacks.
Statement (2) says 4 books were added. By statement (1), fewer than 2 of those 4 books were paperbacks, so at most 1 paperback was added.
Then the Tuesday paperback fraction is at most:
\(\frac{5 + 1}{12 + 4} = \frac{6}{16} = \frac{3}{8}\)
This is less than \(\frac{5}{12}\), so the answer is YES.
But let the Monday total be 36 books. Then there were 15 paperbacks.
Statement (2) says 12 books were added. By statement (1), fewer than 6 of those 12 books were paperbacks, so 5 paperbacks could have been added.
Then the Tuesday paperback fraction is:
\(\frac{15 + 5}{36 + 12} = \frac{20}{48} = \frac{5}{12}\)
This is not less than the Monday fraction, so the answer is NO.
Thus, even together, the statements do not give a definite answer.
Not sufficient.
Answer: E
