Question

How many girls are members of both the Diving Team and the Swim Team?
(1) At a joint meeting of the Diving and Swim Teams, no members were absent and 18 girls were present.
(2) The Diving Team has 27 members, one-third of whom are girls, and the Swim Team has 24 members, half of whom are girls.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient

Hint:

Set theory problems on the GMAT often use the Inclusion-Exclusion Principle. Remember the formula: Total = Group1 + Group2 - Both + Neither. For overlapping groups, the key is often to combine information about the total with information about the individual groups.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem involving sets. We need to find the number of elements in the intersection of three sets: Girls, Diving Team members, and Swim Team members.
Step 2: Key Formula or Approach:
We can use the Principle of Inclusion-Exclusion for sets. For two groups (in this case, girls on the Diving Team and girls on the Swim Team), the formula is:
Total Girls = (Girls on Diving Team) + (Girls on Swim Team) - (Girls on BOTH teams)
Let \(G_D\) be the number of girls on the Diving Team.
Let \(G_S\) be the number of girls on the Swim Team.
Let \(G_{Both}\) be the number of girls on both teams (this is what we need to find).
Let \(G_{Total}\) be the total number of distinct girls across both teams.
The formula is: \(G_{Total} = G_D + G_S - G_{Both}\).
Step 3: Detailed Explanation:
Analyzing Statement (1):
"At a joint meeting of the Diving and Swim Teams, no members were absent and 18 girls were present."
This tells us the total number of unique girls in the union of the two teams. So, \(G_{Total} = 18\).
Our formula becomes: \(18 = G_D + G_S - G_{Both}\).
We have one equation with three unknowns (\(G_D\), \(G_S\), and \(G_{Both}\)). We cannot solve for \(G_{Both}\).
Statement (1) is not sufficient.
Analyzing Statement (2):
"The Diving Team has 27 members, one-third of whom are girls, and the Swim Team has 24 members, half of whom are girls."
From this, we can calculate \(G_D\) and \(G_S\).
Number of girls on Diving Team: \(G_D = \frac{1}{3} \times 27 = 9\).
Number of girls on Swim Team: \(G_S = \frac{1}{2} \times 24 = 12\).
This gives us the number of girls on each team, but we don't know the total number of girls or how many are on both teams. For example, all 9 girls from the Diving Team could also be on the Swim Team, or none of them could be.
Statement (2) is not sufficient.
Analyzing Both Statements Together:
From statement (1), we have the equation: \(G_{Total} = G_D + G_S - G_{Both}\), with \(G_{Total} = 18\).
From statement (2), we have the values: \(G_D = 9\) and \(G_S = 12\).
Now we can substitute these values into the equation:
\[ 18 = 9 + 12 - G_{Both} \] \[ 18 = 21 - G_{Both} \] \[ G_{Both} = 21 - 18 \] \[ G_{Both} = 3 \] Together, the statements allow us to find a unique value for the number of girls on both teams.
Therefore, the statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient.