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13 Jan 2026, 10:19
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A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?
(1) \(\frac{3}{5}\) of the students participate in only Math Club.
(2) \(\frac{2}{9}\) of the students participate in both Debate Club and Math Club.
(1) \(\frac{3}{5}\) of the students participate in only Math Club.
(2) \(\frac{2}{9}\) of the students participate in both Debate Club and Math Club.
13 Jan 2026, 10:19
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Official Solution:
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?
(1) \(\frac{3}{5}\) of the students participate in only Math Club.
\(\frac{3}{5}\) of 90 participate in only Math Club, so only Math = 54. Since every student is in at least one club, the remaining 36 students must be those in Debate Club only plus those in both clubs. Therefore, the total number in Debate Club is 36. The total number in Math Club is only Math + both = 54 + both, which is at least 54. So Math Club has more participants than Debate Club for sure. Sufficient.
(2) \(\frac{2}{9}\) of the students participate in both Debate Club and Math Club.
\(\frac{2}{9}\) of 90 participate in both, so both = 20. The remaining 70 are split between only Math and only Debate. We cannot tell which of those two groups is larger. Not sufficient.
Answer: A
A survey of 90 students found that each student participates in at least one of two activities: Debate Club or Math Club. Is the number of students who participate in Debate Club greater than the number who participate in Math Club?
(1) \(\frac{3}{5}\) of the students participate in only Math Club.
\(\frac{3}{5}\) of 90 participate in only Math Club, so only Math = 54. Since every student is in at least one club, the remaining 36 students must be those in Debate Club only plus those in both clubs. Therefore, the total number in Debate Club is 36. The total number in Math Club is only Math + both = 54 + both, which is at least 54. So Math Club has more participants than Debate Club for sure. Sufficient.
(2) \(\frac{2}{9}\) of the students participate in both Debate Club and Math Club.
\(\frac{2}{9}\) of 90 participate in both, so both = 20. The remaining 70 are split between only Math and only Debate. We cannot tell which of those two groups is larger. Not sufficient.
Answer: A
01 Feb 2026, 10:29
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I like the solution - it’s helpful.
