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A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 150 of them root for exactly two of the three teams. How many members does the fan club have ?
(1) Equal number of members root for Portugal only and for Argentina only.
(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
M38-06
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(1) Equal number of members root for Portugal only and for Argentina only.
(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
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M38-06
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GMAT CLUB Official Explanation:
A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 150 of them root for exactly two of the three teams. How many members does the fan club have ?
Check the diagram below:
Given:
(i) 100 people root for Portugal: \(a + d + f + g = 100\);
(ii) 200 people root for Argentina: \(b + e + d + g = 200\).
(iii) 150 people root for France: \(c + e + f + g = 150\);
(iiii) 120 people root for exactly two of the three teams: \(d + e + f = 150\).
The question asks to find \(total = a + b + c + d + e + f + g + N = ?\)
Sum (i), (ii), and (iii):
\((a + d + f + g) + (b + e + d + g) + (c + e + f + g) = 450\);
\(a + b + c + 2(d + f+ e) + 3g = 450\).
Since given that \(d + e + f = 150\) (iiii), then:
\(a + b + c + 2*150+ 3g = 450.\);
\(a + b + c = 150 - 3g\)
Thus:
\(total = (a + b + c) + (d + e + f) + g + N = (150 - 3g) + 150 + g + N = 300 -2g + N=?\)
(1) Equal number of members root for Portugal only and for Argentina only.
This means that a = b, which is not sufficient to get the value of total = 300 -2g + N.
(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
This means that \(N = 2g\). Thus, \(total = 300 - 2g + N = 300 -2g + 2g = 300\). Sufficient.
Answer: B.
For more check ADVANCED OVERLAPPING SETS PROBLEMS
WC2022.png [ 30.89 KiB | Viewed 9007 times ]
_________________
A poll conducted among the members of a football fan club, revealed that 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina. Also, 150 of them root for exactly two of the three teams. How many members does the fan club have ?
Check the diagram below:
Given:
(i) 100 people root for Portugal: \(a + d + f + g = 100\);
(ii) 200 people root for Argentina: \(b + e + d + g = 200\).
(iii) 150 people root for France: \(c + e + f + g = 150\);
(iiii) 120 people root for exactly two of the three teams: \(d + e + f = 150\).
The question asks to find \(total = a + b + c + d + e + f + g + N = ?\)
Sum (i), (ii), and (iii):
\((a + d + f + g) + (b + e + d + g) + (c + e + f + g) = 450\);
\(a + b + c + 2(d + f+ e) + 3g = 450\).
Since given that \(d + e + f = 150\) (iiii), then:
\(a + b + c + 2*150+ 3g = 450.\);
\(a + b + c = 150 - 3g\)
Thus:
\(total = (a + b + c) + (d + e + f) + g + N = (150 - 3g) + 150 + g + N = 300 -2g + N=?\)
(1) Equal number of members root for Portugal only and for Argentina only.
This means that a = b, which is not sufficient to get the value of total = 300 -2g + N.
(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
This means that \(N = 2g\). Thus, \(total = 300 - 2g + N = 300 -2g + 2g = 300\). Sufficient.
Answer: B.
For more check ADVANCED OVERLAPPING SETS PROBLEMS
Attachment:
WC2022.png [ 30.89 KiB | Viewed 9007 times ]
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General Discussion
Posts: 18
First make the Venn diagram, and consider the following regions in the overlapping circles.
a, b , c - Only Portugal(P), France(F) and Argentina(A) respectively
d, e, f - Both P&F, Both P&A, Both F&A respectively
g - All 3
T - Total
N - Neither
(i) T - N = a+b+c+d+e+f+g
(ii) a+d+e+g = 100 (Portugal)
(iii) b+d+g+f = 150(France)
(iv) e+g+f+c = 200(Argentina)
(v) i + ii + iii + iv = a + b + c + 2(d+e+f) + 3g = 450 ----- (d+e+f = 120, given)
This will be simplified to a+b+c +3g = 210, hence a+b+c = 210-3g (vi)
Putting (d+e+f = 120) & (vi) in eqn (i) we get
T = 330 - 2g - N, hence we need value of g & N to get the total
St 1 - Tells a = c, Not sufficient
St 2 - For every 1 g, there are 2 N's hence N/G is in ratio of 2:1, hence N = 2g
Put this in equation , T = 330 - 2g + 2g, T =330 Answer Hence: Sufficient
a, b , c - Only Portugal(P), France(F) and Argentina(A) respectively
d, e, f - Both P&F, Both P&A, Both F&A respectively
g - All 3
T - Total
N - Neither
(i) T - N = a+b+c+d+e+f+g
(ii) a+d+e+g = 100 (Portugal)
(iii) b+d+g+f = 150(France)
(iv) e+g+f+c = 200(Argentina)
(v) i + ii + iii + iv = a + b + c + 2(d+e+f) + 3g = 450 ----- (d+e+f = 120, given)
This will be simplified to a+b+c +3g = 210, hence a+b+c = 210-3g (vi)
Putting (d+e+f = 120) & (vi) in eqn (i) we get
T = 330 - 2g - N, hence we need value of g & N to get the total
St 1 - Tells a = c, Not sufficient
St 2 - For every 1 g, there are 2 N's hence N/G is in ratio of 2:1, hence N = 2g
Put this in equation , T = 330 - 2g + 2g, T =330 Answer Hence: Sufficient
wadhwakaran
Joined: 31 Mar 2022
Last visit: 22 Jul 2025
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Concentration: General Management, International Business
Schools: INSEAD Aug'24 intake Oxford Said'25 Judge '25 Imperial'25 HEC Sep'24 Intake LBS '26 ISB '27
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Schools: INSEAD Aug'24 intake Oxford Said'25 Judge '25 Imperial'25 HEC Sep'24 Intake LBS '26 ISB '27
GMAT 1: 700 Q50 V35
Posts: 257
I--- Only P+ Only A + Only F
II--- P&F together but not A + P&A together but not F + A&F together but not P
III--- all P, F, A together
O--- None of the P,F&A
II=120
I+2II+3III=450
I+3III=210
Total Football Fanclub members, T=O+I+II+III
S(1) Equal number of members root for Portugal only and for Argentina only.
Only P= Only A
does not help us to identify T
S(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
If III=x, O will be 2x
T= I+II+x+2x
T=I+II+3x or I+II+3III
T=120+I+3III=120+210=330
S2 is sufficient
Therefore, B
II--- P&F together but not A + P&A together but not F + A&F together but not P
III--- all P, F, A together
O--- None of the P,F&A
II=120
I+2II+3III=450
I+3III=210
Total Football Fanclub members, T=O+I+II+III
S(1) Equal number of members root for Portugal only and for Argentina only.
Only P= Only A
does not help us to identify T
S(2) For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
If III=x, O will be 2x
T= I+II+x+2x
T=I+II+3x or I+II+3III
T=120+I+3III=120+210=330
S2 is sufficient
Therefore, B
Posts: 73
Bunuel
Attachments
WhatsApp Image 2022-07-11 at 11.27.18 PM.jpeg [ 232.22 KiB | Viewed 8400 times ]
Posts: 152
Portugal (P) - 100
France (F) - 150
Argentina (A) - 200
PF - ?
FA - ?
PA - ?
PFA - ?
P only - ?
F only - ?
A only - ?
None - ?
PF+FA+PA = 120
Important question to keep in mind -
Total number of members = P + F + A - (PF+FA+PA) - 2 (PFA) + None
Statement 1 -
P only = A only
Insufficient as it doesn't give us information that can lead us to answer.
Statement 2-
None = 2 (PFA)
Total = 100+150+200-(120)-2(PFA)+None
Now , replacing the value of None = 2(PFA)
Total = 330 - 2PFA + 2PFA
Total = 330
Hence, statement 2 is sufficient.
B is the correct choice.
France (F) - 150
Argentina (A) - 200
PF - ?
FA - ?
PA - ?
PFA - ?
P only - ?
F only - ?
A only - ?
None - ?
PF+FA+PA = 120
Important question to keep in mind -
Total number of members = P + F + A - (PF+FA+PA) - 2 (PFA) + None
Statement 1 -
P only = A only
Insufficient as it doesn't give us information that can lead us to answer.
Statement 2-
None = 2 (PFA)
Total = 100+150+200-(120)-2(PFA)+None
Now , replacing the value of None = 2(PFA)
Total = 330 - 2PFA + 2PFA
Total = 330
Hence, statement 2 is sufficient.
B is the correct choice.
From the given data - 100 of them root for Portugal, 150 of them root for France, and 200 of them root for Argentina.
Lets assume E1 for Exactly 1 similarly for E2 for 2 and E3 for 3.
\(E1+2E2+3E3 = 100+150+200 = 450\)
also \(E2 = 120\)
To find - How many members does the fan club have
\(E1 + E2 + E3 + none = Total\)
1st - Equal number of members root for Portugal only and for Argentina only.
Not useful.
2nd - For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
none = 2E3
Therefore, \(Total = E1 + E2 + 3E3\)
This we can calculate from given data
\(E1+2E2+3E3 = 450\)
\(E1+E2+3E3 = 450-E2\)
But \(E2 = 120\)
\(E1+E2+3E3 = 450-120 = 330\)
Sufficient
Answer is B.
Lets assume E1 for Exactly 1 similarly for E2 for 2 and E3 for 3.
\(E1+2E2+3E3 = 100+150+200 = 450\)
also \(E2 = 120\)
To find - How many members does the fan club have
\(E1 + E2 + E3 + none = Total\)
1st - Equal number of members root for Portugal only and for Argentina only.
Not useful.
2nd - For every 2 members of the club who root for none of the three teams, there is 1 member who roots for all three of the teams.
none = 2E3
Therefore, \(Total = E1 + E2 + 3E3\)
This we can calculate from given data
\(E1+2E2+3E3 = 450\)
\(E1+E2+3E3 = 450-E2\)
But \(E2 = 120\)
\(E1+E2+3E3 = 450-120 = 330\)
Sufficient
Answer is B.
Posts: 63
Hello How did you arrive at d+e+f=150?
Bunuel
NoeticImbecile
120 was a typo, it should have been: 150 of them root for exactly two of the three teams.
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