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Difficulty:
95%
(hard)
Question Stats:
38% (02:48) correct
62%
(02:41)
wrong
based on 68
sessions
History
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The members of a chess club took part in a round robin competition in which each plays every one else once. All members scored the same number of point, except four juniors whose total score were 17.5. How many members were there in the club? Assume that for each win a player scores 1 point, for draw ½ point and zero for losing.
A. 9
B. 15
C. 18
D. 27
E. 36
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A. 9
B. 15
C. 18
D. 27
E. 36
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Bunuel
Let the total members be n. So, the total number of matches that can be played is nC2. And, since each match carries 1 point, the total number of points will also be nC2.
Let the total points of the members other than the 4 juniors be x.
Therefore, nC2-17.5 = (n-4)x
\frac{n(n-1)}{2} - 17.5 = (n-4)x
\frac{n(n-1)-35 }{ (n-4)} = 2x
Since x is a multiple of 0.5, 2x will be an integer.
Hence, the expression \frac{n(n-1) - 35}{(n-4)} will be an integer.
\frac{n^2-4n+3n-12-23}{(n-4)} = 2x
\frac{(n+3)(n-4)-23}{(n-4)} = 2x
(n+3) - \frac{23}{(n-4)} = 2x
Therefore, \frac{23}{(n-4)} must be an integer. So, n must be 27.
Answer will be D.
General Discussion
GMAT Focus 1: 735 Q87 V90 DI82 
Posts: 15
We can immediately eliminate answer choices A, B and C, as 17.5 points for each of 4 juniors requires a minimum of 18 games (17 wins + 1 draw). With 18 members in total, each member would play only 17 games, so there is no way to get to 17.5 points.
Regarding answer choice E, this doesn't work either. The 4 juniors got 17.5 points each, which means 70 points in total for all 4. If we have 36 members in total, that would mean a total of (36*35)/2 games played in the tournament, so a total of 630 games which also implies 630 points in total (as each game brings 1 point in total regardless of the outcome). This means that the remaining 32 participants would have 630-70=560 points to share between them, and we know that each participant got equal number of points except for juniors who got 17.5 each. However, if we divide 560/32, this also brings us to the same 17.5 points per participant - which we can't have per the question. Based on this, D is the only answer option that works.
Regarding answer choice E, this doesn't work either. The 4 juniors got 17.5 points each, which means 70 points in total for all 4. If we have 36 members in total, that would mean a total of (36*35)/2 games played in the tournament, so a total of 630 games which also implies 630 points in total (as each game brings 1 point in total regardless of the outcome). This means that the remaining 32 participants would have 630-70=560 points to share between them, and we know that each participant got equal number of points except for juniors who got 17.5 each. However, if we divide 560/32, this also brings us to the same 17.5 points per participant - which we can't have per the question. Based on this, D is the only answer option that works.
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