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Please explain how to simplify

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Please explain how to simplify

by oquiella » Sat Sep 05, 2015 10:44 am
In a certain sport, teams receive 3 points for each win, 1 point for each draw, and no points for losses. In a five-team tournament in this sport, in which each of teams G, H, J, K, and L played each other team exactly once, did team L finish the tournament with the highest point total?

(1) Team L finished with 8 points.

(2) The sum of all five teams' point totals for the tournament was 23 points.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.
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Source: — Data Sufficiency |

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by [email protected] » Sun Sep 06, 2015 1:24 pm
Hi oquiella,

This is a 'layered' DS question that requires some serious work to get to the correct answer.

We're told that there are 5 teams and that they each play one another once in a tournament (wins are worth 3 points each, ties are worth 1 point each and losses are with 0 points each). Using the Combination Formula, we can figure out how many total games are played:

5!/(2!3!) = 10 total games played.

So, each team will play in 4 games, but there are only 10 total games played. This is important for a couple of reasons:
1) It establishes the maximum number of points that any team can score (12 points, if a team won ALL its games)
2) It establishes the minimum and maximum points that could be scored by ALL teams (if ALL the games were ties, then there would be 20 points scored; if there are NO ties, then there will be 10 wins and 10 losses, for a total of 30 points).

We're asked if Team L finished with the HIGHEST point total? This is a YES/NO question.

Fact 1: Team L finished with 8 points.

To get 8 points, Team L would have had to have won 2 games and tied 2 games. We have to determine if that COULD have been the highest total and if that might not have been the highest...

IF...
All of the other teams tied one another (except for the two that lost to Team L), then each of the other 4 teams would have scored 4 or fewer points, so Team L COULD have had the highest total and the answer to the question is YES.

IF...
One team tied Team L, but won its other 3 games, then that team would have scored 10 points. In that situation, Team L would NOT have had the highest point total and the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: The Total Points scored in the tournament was 23 points.

This tells us NOTHING about how Team L performed in the tournament. Maybe it was highest (a YES answer), maybe it was lowest (a NO answer), maybe it was somewhere 'in between' (also a NO answer).
Fact 2 is INSUFFICIENT

Combined, we know....
Team L had 8 total points (2 wins and 2 ties).
The Total points for the tournament was 23 points (which is relatively low, implying lots of ties happened).

From here, we have to play around some more with the possibilities; since Team L scored 8 points, that means the other 4 teams scored just 15 points in total. So could another team have scored MORE than Team L?

Since no team won against Team L, for another Team to score MORE than Team L, that team would need to have tied with Team L and WON ALL 3 of its other games...
Team L = 8 points
Team G = 10 points
7 games played so far

This leaves 5 points and 3 remaining games for the other 3 teams. Since each game will generate either 2 points (if the teams tie) or 3 points (if one wins and one loses), 3 remaining games will generate AT LEAST 6 points. Here, there are only 5 points to spare, so this situation is IMPOSSIBLE - It cannot happen - thus, NO TEAM could have scored higher than Team L under these conditions, so Team L's total IS the highest and the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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by Max@Math Revolution » Sun Sep 06, 2015 7:37 pm
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


In a certain sport, teams receive 3 points for each win, 1 point for each draw, and no points for losses. In a five-team tournament in this sport, in which each of teams G, H, J, K, and L played each other team exactly once, did team L finish the tournament with the highest point total?

(1) Team L finished with 8 points.

(2) The sum of all five teams' point totals for the tournament was 23 points.

Transforming the original condition by variable approach method, we have multiple variables (since we need to know the scores of all 5 people), thus we need multiple equations as well. Therefore E is likely the answer. Using both 1) and 2) together, L has 8 points and in order to have 23 points total, L=8, G=5, H=2, J=4, K=4.

Therefore L>G, L>H, L=J, L=K and G>H, G=J, G=K, H=J, H=K, J=K. Here, > stands for victory while = stands for a draw.
Since L has the best points, the conditions are sufficient together, but not separately. In case of 1) all three G>H, G>J, G>K is possible therefore it is not sufficient and in case of 2), L is not a unique answer. Therefore C is the answer.


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