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08 Jan 2025, 01:20
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Of all the students in a certain school, 22 are members of a chess club, 10 are members of a swim team, and 2 are members of both. If all females are members of the swim team and the number of males is twice the number of females, how many students are members of neither the chess club nor the swim team?
A. 0
B. 3
C. 6
D. 10
E. 12
A. 0
B. 3
C. 6
D. 10
E. 12
08 Jan 2025, 01:20
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Official Solution:
Of all the students in a certain school, 22 are members of a chess club, 10 are members of a swim team, and 2 are members of both. If all females are members of the swim team and the number of males is twice the number of females, how many students are members of neither the chess club nor the swim team?
A. 0
B. 3
C. 6
D. 10
E. 12
Total = Chess + Swim - Both + Neither
Total = 22 + 10 - 2 + Neither
Total = 30 + Neither
Since one-third of the students are female and two-thirds are male, the number of females must be at least 10 (\(f ≥ 10\)). This happens when the number of students who are members of neither club is 0. However, the number of females cannot exceed 10 because all females are members of the swim team, which has exactly 10 members (\(f ≤ 10\)).
Therefore, since f must be both at least 10 and at most 10, the number of females is exactly 10. This makes the total number of students 30 (since there are twice as many males as females). Thus, the number of students who are members of neither the chess club nor the swim team is 0 (from 30 = 30 + Neither).
Answer: A
Of all the students in a certain school, 22 are members of a chess club, 10 are members of a swim team, and 2 are members of both. If all females are members of the swim team and the number of males is twice the number of females, how many students are members of neither the chess club nor the swim team?
A. 0
B. 3
C. 6
D. 10
E. 12
Total = Chess + Swim - Both + Neither
Total = 22 + 10 - 2 + Neither
Total = 30 + Neither
Since one-third of the students are female and two-thirds are male, the number of females must be at least 10 (\(f ≥ 10\)). This happens when the number of students who are members of neither club is 0. However, the number of females cannot exceed 10 because all females are members of the swim team, which has exactly 10 members (\(f ≤ 10\)).
Therefore, since f must be both at least 10 and at most 10, the number of females is exactly 10. This makes the total number of students 30 (since there are twice as many males as females). Thus, the number of students who are members of neither the chess club nor the swim team is 0 (from 30 = 30 + Neither).
Answer: A
15 Mar 2025, 03:37
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I like the solution - it’s helpful. And it's a high quality question for sure!
