12 Days of Christmas GMAT Competition - Day 3: Of all the students in

Question

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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. 9
E. 12

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

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.

siddhantvarma · Answer

• Chess club members: 22
• Swim team members: 10
• Students in both clubs: 2

Number of students in either club is: 22 + 10 - 2 = 30
We're told that all females are on the swim team, and the number of males is twice the number of females.

Let x be the number of females:
• Females: x
• Males: 2x
• Total students: x + 2x = 3x

We know the swim team members (10) are all females, so x = 10

Total number of students:
• Females: 10
• Males: 20
• Total: 30 students

Students in clubs: 30
Students NOT in clubs: 0

The answer is A. 0.

hr1212 · Answer

Member of chess club = 22
Members of swim team = 10
Member of chess and swim = 2

Total members = Members of chess + Members of swim - Members of chess and swim
= 22 + 10 - 2
= 30 --- (1)

Female members = Members of swim team = 10
Male members = 2 * Female members = 20
Total students = male members + female members = 10 + 20 = 30 --- (2)

From (1) and (2), as there are no extra students who are not members of swim team or chess club, answer should be 0.

Answer: A

HarshaBujji · Answer

Let F be the number of females. Let M be the number of males. Given M=2F Total students = M+F=3F. Chess club members: 22. (C) Swim team members: 10. (S) Students in both the chess club and the swim team: 2. We know that C U S = 32-2 => 30. So Total - none = 30. We also know that al female are in S. So max f=10, => f<=10. This implies total <=30. Hence none has to be 0. So #students are members of neither the chess club nor the swim team is 0 (IMO A)

Krunaal · Answer

Total unique members in either club = 22+10−2 = 30......................(1)

All females are member of the swim team, and males is twice the number of females => M = 2F

Total no. of students in school = M + F = 3F

For (1) to satisfy, F has to be 10

Total no. of students in school = 30

Now, Total no. of students in school - Students in neither club = Students in Chess club + Students in Swimming club - Students in both clubs
=> 30 - Students in neither club = 22 + 10 - 2
=> 30 - Students in neither club = 22 + 10 - 2
=>  Students in neither club = 0

Answer A.

Kinshook · Answer

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?

Let the number of students who are neither members of chess club nor the swim team be x

Chess 
 Not Chess 
 Total 
 
  Swim 
 2 
 8 
 10 
 
  Not Swim 
 20 
 x 
 20+x 
 
  Total 
 22 
 8+x 
 30+x

Total students = 30 + x

Let the number of females be f
Number of males = 2f 
Total students = f + 2f = 3f

3f = 30 + x
Since all females are members of the swim team, maximum value of f = 10

30 = 30 + x 
x = 0

The number of students who are members of neither the chess club nor the swim team = 0

IMO A

kingbucky · Answer

Problem Analysis: 
- Total chess club members = 22
- Total swim team members = 10
- Members of both clubs = 2

Calculating Exclusive Memberships: 
- Chess only members = Total chess - Both clubs =  22 - 2 = 20 
- Swim only members = Total swim - Both clubs =  10 - 2 = 8

Total Members in Clubs: 
- Total club members = Chess only + Swim only + Both =  20 + 8 + 2 = 30

Gender-based Team Membership and Ratios: 
- All females (F) are on the swim team.
- Number of males (M) is twice the number of females (M = 2F).

Total Population Breakdown: 
- Let  x  be the number of students neither in chess nor swim.
- Total students = Club members + Neither =  30 + x

Relationship of Total Males and Females to Club Membership: 
- Total swimmers (including both clubs) = 10, hence  F \leq 10 
- Using the ratio of males to females,  M + F = 30 + x  and  M = 2F , we substitute to find:
 -  2F + F = 30 + x \Rightarrow 3F = 30 + x

Determine Maximum Females on Swim Team: 
- Since  3F = 30 + x  and  F \leq 10 , this implies  3F \leq 30 , setting  x + 30 \leq 30 \Rightarrow x \leq 0 .

Concluding the Value of  x : 
- Given  x  must be a non-negative integer and  x \leq 0 , the only possibility is  x = 0 .

Answer: 
- The number of students who are members of neither the chess club nor the swim team is  0 .

Correct Answer: A. 0

nikiki · Answer

Chess 
 Swim 
 Both 
 Neither 
 Total 
 
  Male

2x 
 
  Female 
 0 
 x 
 0 
 0 
 x 
 
  Total 
 22 
 10 
 2 
 N

22+10+2+N = 3x
x = (N + 34)/3

For x to be an integer N+34 must be a multiple of 3

Putting all the values of N from options
When N =
0 x=34/3 not divisible
3 x=37/3 not divisible
6 x=40/3 not divisible
9 x=43/3 not divisible
12 x=46/3 DIVISIBLE

Option E

nishantswaft · Answer

Fairly easy question once we draw the venn diagram and understand the statements in question.

Look at the venn diagram below;

Here, N represents the number of students that are members of neither the chess club nor the swim team.

Now, it is also given to us that;

All females are members of the swim team, and  
 The number of males is twice the number of females or M=2F

Forming a equation, we have;

22+10-2+N= Total, also,

M+F=Total and since M=2F,

3F= Total

Hence, 30+N=3F or N=3F-30.

Now since All females are members of the swim team, 
 0<F≤10

And also since value of N can’t be negative,

N=3(10)-30=30-30= 0

Hence the answer is A

Karanjotsingh · Answer

Given Information

Total Clubs: 
 
  Chess Club:  22 members 
  Swim Team:  10 members 
  Members in Both Clubs:  2 students  
 
  Gender Information: 
 
  All Females  are members of the  Swim Team . 
  The number of Males  is  twice  the number of  Females .  
  
 Step-by-Step Solution:

1. Determine the Number of Females and Males 
 
 Let  F  be the number of  Females . 
 Then, the number of  Males (M)  is  2F . 
  Total Number of Students (N): 
N = F + M = F + 2F = 3F
  
 2. Analyze Club Memberships 
 
  Swim Team Membership: 
 
 All  Females  are in the  Swim Team . 
 Total  Swim Team  members = 10.  
Since all females are on the swim team:
F = 10
Therefore:
M = 2F = 2 × 10 = 20
And:
N = 3F = 3 × 10 = 30  
 3. Break Down Club Memberships 
 
  Members in Both Clubs:  2 students
 
 Since all females are on the swim team and the overlap is with the swim team, these  2 students  are  Females .  
 
  Females Only in Swim Team: 
10 (Total Females) − 2 (Both Clubs) = 8 Females only in Swim Team
 
  Males Only in Chess Club: 
22 (Chess Club) − 2 (Both Clubs) = 20 Males only in Chess Club
  
 4. Summarize Club Memberships 
 
  Chess Club Only:  20 Males 
  Swim Team Only:  8 Females 
  Both Clubs:  2 Females 
  Total Students in At Least One Club: 
20 (Chess Only) + 8 (Swim Only ) + 2 (Both) = 30
  
 5. Calculate Students in Neither Club 
 
  Total Students in School (N):  30 
  Students in At Least One Club:  30 
  Students in Neither Club: 
N − 30 = 30 − 30 = 0
  
 Final Answer: 
 A) 0

seenuvasan · Answer

Given details,

22 members in Chess, 10 members in Swim, and 2 in both. All females are members in Swim.

The below venn diagram help us to understand the scenario. From this we can say that Total members $ = 20+2+8+Neither$. Apart from this we know that all female members are part of Swim club. So the max possible number of females is 10, so max possible number of males is 20. Total max possible is 30 members.

$Total members = 30 + Neither$.

Since total members cannot exceed 30, possible number of neither is 0. Answer is A.

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A_Nishith · Answer

Step 1:  Analyze the given information.
Total members of the chess club: 22
Total members of the swim team: 10
Members of both clubs: 2
All females are on the swim team.
Number of males = 2×number of females.
Let the total number of students be N, and the number of females be F. 
Then the number of males is 2F.

Step 2:  Use the inclusion-exclusion principle.
The total number of students in either the chess club or the swim team (or both) is:

Members in chess club or swim team=(members of chess club)+(members of swim team)−(members of both) =22+10−2=30.
Thus, 30 students are in at least one of the two groups.

Step 3:  Total number of students.
The total number of students, N, is:
N=number of males+number of females=2F+F=3F.

Step 4:  Set up the condition for students not in any group.
The number of students in neither the chess club nor the swim team is:

Neither=N−(members in chess club or swim team).
Neither=3F−30.

Step 5:  Use the condition that all females are on the swim team.
All F females are on the swim team, so the number of swim team members who are female is F. 
The remaining 10−F swim team members are males. Since the total number of males is 2F, the number of males not in the swim team is:

Males not in swim team=2F−(10−F)=3F−10.
Similarly, the number of males not in the chess club can be calculated. However, the overlap calculations ensure we can solve directly for F.

Step 6: Solve for F. 
The total number of students must satisfy 
3F=N, and we know 
3F−30=Neither. Testing integer values quickly shows 
F=10:

N=3×10=30.
Thus, all students are accounted for, and none are in "neither."

Final Answer: A. 0

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