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Venn diagram & overlapping sets

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by piti » Wed Aug 15, 2012 4:39 am
The following question appeared in https://www.beatthegmat.com/venn-diagram-t84951.html:

"In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football.7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football.If 18 students does not play any of these given sports, ho many students play exactly two of these sports?"

Two solutions were provided - one using a Venn diagram which gave an answer of 10, and another using an equation, which gave an answer of 13. Can someone please explain which is the correct solution? If the second one (equation) is correct, can you please show the Venn diagram of this solution and how it all adds up? When I created a Venn for the second solution it didn't seem to add up.
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Source: — Quantitative Reasoning |
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by theCEO » Wed Aug 15, 2012 4:16 pm
I am getting 10 for both answers

Eq 1-3
Intersection of H and C = HC + HCF = 7 ---- HC = 7 - HCF
Intersection of C and F = CF + HCF = 4 ---- CF = 4 - HCF
Intersection of H and F = HF + HCF = 5 ---- HF = 5 - HCF

Question wants us to find HC+CF+HF

Total = H + C + F - (HC + HF + CF) -2HCF + neither = 50
H + C + F - (HC + HF + CF) -2HCF = 32
46 - 32 - 2HCF = HC + HF + CF
14 - 2HCF = HC + HF + CF
14 - 2HCF = (7-HCF) + (5-HCF) + (4-HCF)
14 = (7-HCF) + (5-HCF) + (4-HCF)+ 2HCF
14 = 16 - HCF
HCF = 2

Substituting 2 for HCF in Eq1-3, we have the answer to be (7-2)+(4-2)+(5-2)=10
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