A sports team played 100 games last season. Did this team

Hi Bunuel,

I do not understand the combined portion.

How are we getting at 30+9+9 part?

The overlap of the first 65 games and the last 65 games (out of 100) is middle 30 games: 100 = 65 + 65 - 30. The minimum number of games the team could have won is 30 (overlap) + 9 + 9 = 48.

Hi ashikaverma13

Here we need max & min values of games won. Max possible value is 39+39=78

Minimum value could be under a situation where there are 30 overlaps. So team won 78 matches by playing 65+65=130 games but in reality only 100 games were played. so 30 wins have been counted twice and hence need to be removed from the maximum value

i.e. 78-30=48 could be our minimum value.

Target question: Did this team win at least half of the games it played last season?

Rephrased target question: Did this team win more than 49 games?

Statement 1: The team won 60% of its first 65 games
In other words, the team won 39 of its first 65 games
Since we don't know the results of the last 35 games, we can't answer the target question with certainty. So, statement 1 is NOT SUFFICIENT

Statement 2: The team won 60% of its last 65 games
In other words, the team won 39 of its last 65 games
Since we don't know the results of the first 35 games, we can't answer the target question with certainty. So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Let's first see if the team could have won more than 49 games. To check this out, we'll MAXIMIZE the number of wins. So, let's say the team won the first 39 games (which would account for statement 1) and say the team won the last 39 games (which would account for statement 2)
So, in total, the team won 78 games. So, it is possible that the team won more than 49 games.

Now let's see if it's possible for the team to win fewer than 49 games. To do this, we'll MINIMIZE the number of wins by overlapping the shared wins for statements 1 and 2.
So, for statement 1, let's say the team lost games #1 to #26, and then won games #27 to #65 (39 wins)
For statement 2, let's say the team won games #36 to #74 (39 wins), and then lost games #75 to #100
So, in total, the team won games #27 to #74, which means it won 48 games altogether.

Since we still cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

Assume a, b, c and d are numbers of games the team won from the first 35 games, the next 15 games, the next 15 games and the last 35 games, respectively.
The question asks if a + b + c + d ≥ 50.

Since we have 4 variables and 0 equations, E is most likely to be the answer. So, we should consider 1) & 2) first, since we can save time by first checking whether conditions 1) and 2) are sufficient, when taken together.

Conditions 1) & 2)
a + b + c = 38 ( = 65 * 0.6)
b + c + d = 38 ( = 65 * 0.6)
a = 37, b + c = 1, d = 37 ⇒ a + b + c + d = 75 : Yes
a = 8, b + c = 30, d = 8 ⇒ a + b + c + d = 46 : No

Both conditions together are not sufficient.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

\(\# \,{\rm{wins}}\,\,\,\mathop \ge \limits^? \,\,\,50\)

\(\left( {1 + 2} \right)\,\,\left\{ \matrix{\\
\,39\,\,{\rm{wins}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{first}}\,65\,\,{\rm{games}}\,\,\,\,\left( {1{\rm{st}}\,\,{\rm{till}}\,\,65{\rm{th}}} \right) \hfill \cr \\
\,39\,\,{\rm{wins}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{last}}\,65\,\,{\rm{games}}\,\,\left( {36{\rm{th}}\,{\rm{till}}\,\,100{\rm{th}}} \right) \hfill \cr} \right.\)

\({\rm{If}}\,\,{\rm{wins}}\,\,{\rm{games}}\,\,\# \,\,\left\{ \matrix{\\
\,1,2,3, \ldots ,39\,{\rm{and}}\,\,62,63, \ldots ,100\,\,\,\,\,:\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\left( {78\,\,{\rm{games}}} \right) \hfill \cr \\
\,27,48, \ldots ,65\,\,{\rm{and}}\,\,36, \ldots ,74\,\,\, = \,\,\,27, \ldots ,74\,\,\,\,\,:\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\left( {48\,\,{\rm{games}}} \right) \hfill \cr} \right.\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Thanks for the great explanations and correct answer.
Just Bumping this interesting question and adding a tiny comment on it:
the minimum is 48 (at max. overlap of 30), but the maximum is 74 (at min. overlap of 4)

(1) INSUFFICIENT: The team won 60%, or 39, of its first 65 games last season. What about the last 35 games? Test the extreme cases: winning all of the remaining games and winning none of them.

If the team won all 35 remaining games, then it won 74 of 100 total games—more than half of the season total. If, on the other hand, the team lost all 35 remaining games, then it only won 39 of 100 total games—less than half of the season total.

(2) INSUFFICIENT: The team won 60%, or 39, of its last 65 games last season. What about the first 35 games? Test the extreme cases: winning all of the games and winning none of them.

If the team won the first 35 games, then it won 74 of 100 total games—more than half of the season total. If, on the other hand, the team lost the first 35 games, then it only won 39 of 100 total games—less than half of the season total.

(1) AND (2) INSUFFICIENT: The key to treating the statements together is to observe the overlap between “the first 65 games” and “the last 65 games.” Sketch a quick number line, with the first game labeled #1 and the last labeled #100. In this case, the first 65 games are #1 to #65, and the last 65 games are #36 to #100. Therefore, games #36 to #65—a total of thirty games—are considered in both statements.



Test the extreme cases: as many as possible of the overlapping games were won or as many as possible of the overlapping games were lost.

Maximize overlapping wins: All 30 of the overlapping games could represent winners. In that case, those games account for 30 of the team’s 39 wins in the first 65 games, and for 30 of its wins in the last 65 games. Statement 1 indicates that the team won 39 of the first 65, so the team must also have won an additional 9 games during matches #1 to #35. Likewise, the team must also have won an additional 9 games during matches #66 to #100.



The team would win a total of 9 + 30 + 9 = 48 games, less than half the season total.

Minimize overlapping wins: If the team won games #1 through #35, then it would need to win only 4 games in the overlapping group in order to satisfy statement 1. Similarly, if the team won games #66 through #100 (a total of 35 games), then it would need to win only 4 games in the overlapping group in order to satisfy statement 2.



The team would win a total of 35 + 4 + 35 = 74 games, more than half the season total.

The correct answer is E.

Hi, could someone please clarify if the total number of maximum wins will be 78 or 74? I think it should be 74 and not 78. Bunuel and BrentGMATPrepNow, could you please comment.

Thank you.

We know from 1/2 that the team won 39 of its first 65 games and 39 of its last 65 games, but we do not know how those wins are distributed.

Case 1: No overlap - the team is like the Golden State Warriors and steam-rolls game after game without losing.
Assume the team won games 1 - 39 (inclusive) = 39 wins
and that it also won games 40 - 78 (inclusive) = 39 wins

Total wins = 39 + 39 = 78 wins >= 50 wins, thus Yes

Case 2: Overlap
Assume the team won games 27 - 65 (inclusive) = 39 wins
and that we are told it also won games, 35 - 73 (inclusive) = 39 wins

How many games did it actually win?

Notice the overlap from games 35 to 65; we are double counting the wins, and we have to subtract the overlapping wins; therefore,
Total wins = 39 + 39 - 30 (the overlap) = 48 wins <= 50 wins, thus No