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19 Aug 2022, 03:50
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B
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30% (02:11) correct
70%
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wrong
based on 156
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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.
(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.
30 Sep 2024, 07:28
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There’s a formula in set theory that might be helpful to solve this question very quickly:
Total = #in set 1 + #in set 2 + #in set 3 -#in exactly 2 sets -(2)(#in 3 sets) + #in none of the sets.
Google can probably explain in detail why this formula works, but the basic principle is subtracting elements that are counted more than once.
For this question, the solution becomes:
100+ 200 + 150 -150 + 2X - 2X, where X is the number who root for none of the teams. If you subtract the terms, you’re left with 300.
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Total = #in set 1 + #in set 2 + #in set 3 -#in exactly 2 sets -(2)(#in 3 sets) + #in none of the sets.
Google can probably explain in detail why this formula works, but the basic principle is subtracting elements that are counted more than once.
For this question, the solution becomes:
100+ 200 + 150 -150 + 2X - 2X, where X is the number who root for none of the teams. If you subtract the terms, you’re left with 300.
Posted from my mobile device
General Discussion
19 Aug 2022, 03:50
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Official Solution:
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) 150 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.
For more check ADVANCED OVERLAPPING SETS PROBLEMS
Answer: B
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) 150 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.
For more check ADVANCED OVERLAPPING SETS PROBLEMS
Answer: B
