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11 Jul 2022, 08:00
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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
(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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19 Aug 2022, 03:33
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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 6731 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 6731 times ]
General Discussion
11 Jul 2022, 08:36
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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
