At a certain banquet, meals were served in only 2 locations. In the main dining room, 40 servings of eggplant...
GMAT Data Sufficiency : (DS) Questions
At a certain banquet, meals were served in only 2 locations. In the main dining room, 40 servings of eggplant parmesan and 60 servings of risotto were served. In the annex, 60 servings of eggplant parmesan and 30 servings of risotto were served. If each guest at the banquet was served exactly 1 serving of eggplant parmesan only, exactly 1 serving of risotto only, or exactly 2 servings, one of eggplant parmesan and the other of risotto, how many guests were at the banquet?
- In the main dining room, each of exactly 5 guests were served both eggplant parmesan and risotto; and in the annex, exactly 5 guests were served risotto only.
- The number of guests that were served eggplant parmesan only was the same in the 2 locations.
Understanding the Question
Let's break down what we're being asked. We have a banquet with two serving locations, and we need to find the exact total number of guests.
Given Information
- Main dining room: 40 eggplant servings, 60 risotto servings
- Annex: 60 eggplant servings, 30 risotto servings
- Guest options: Each guest had exactly one of:
- 1 eggplant only
- 1 risotto only
- 1 eggplant AND 1 risotto (both dishes)
What We Need to Determine
To answer "How many guests were at the banquet?", we need enough information to arrive at one unique value.
Key Insight
The critical insight is that each location has fixed total servings that must be completely distributed among the three guest types. The question becomes: Do we have enough information to uniquely determine how these servings were distributed?
Since guests who had "both" consume one serving from each pile, they affect both the eggplant and risotto counts. This creates an interconnected system where knowing certain values can help us determine all the others.
Analyzing Statement 1
Statement 1 tells us:
- Main dining room: exactly 5 guests had both dishes
- Annex: exactly 5 guests had risotto only
These specific values act as "anchor points" that cascade through our constraints. Let's trace through the logic:
Main dining room analysis:
- 5 guests had both → they consumed 5 eggplant AND 5 risotto servings
- This leaves 35 eggplant servings \((40 - 5)\) → these must go to 35 "eggplant only" guests
- This leaves 55 risotto servings \((60 - 5)\) → these must go to 55 "risotto only" guests
- Main dining room total: \(35 + 55 + 5 = 95\) guests
Annex analysis:
- 5 guests had "risotto only" → they consumed 5 of the 30 risotto servings
- This leaves 25 risotto servings → these must go to guests who had "both" (the only other way to consume risotto)
- So 25 guests had "both" → they also consumed 25 eggplant servings
- This leaves 35 eggplant servings \((60 - 25)\) → these must go to 35 "eggplant only" guests
- Annex total: \(35 + 5 + 25 = 65\) guests
Total guests: \(95 + 65 = 160\) guests
Since Statement 1 provides specific numerical values that completely determine the distribution in both locations, we can find the exact number of guests.
[STOP - Sufficient!] Statement 1 is sufficient.
This eliminates choices B, C, and E.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The number of "eggplant only" guests was the same in both locations.
This creates a relationship between the two locations but doesn't provide any specific numerical anchor points. The key question: Can different values lead to different total guest counts?
Testing Different Scenarios
Scenario 1: Suppose 30 guests had "eggplant only" in each location
- Main: 30 eggplant only → leaves 10 eggplant servings for "both" guests → 10 had both
- Main: 10 had both → leaves 50 risotto servings for "risotto only" → 50 had risotto only
- Annex: 30 eggplant only → leaves 30 eggplant servings for "both" guests → 30 had both
- Annex: 30 had both → leaves 0 risotto servings for "risotto only" → 0 had risotto only
- Total: \((30 + 50 + 10) + (30 + 0 + 30) = 150\) guests
Scenario 2: Suppose 35 guests had "eggplant only" in each location
- Main: 35 eggplant only → leaves 5 eggplant servings for "both" guests → 5 had both
- Main: 5 had both → leaves 55 risotto servings for "risotto only" → 55 had risotto only
- Annex: 35 eggplant only → leaves 25 eggplant servings for "both" guests → 25 had both
- Annex: 25 had both → leaves 5 risotto servings for "risotto only" → 5 had risotto only
- Total: \((35 + 55 + 5) + (35 + 5 + 25) = 160\) guests
Different values for "eggplant only" guests lead to different total guest counts (150 vs 160). Therefore, Statement 2 does NOT provide enough information to determine a unique answer.
This eliminates choices B and D.
The Answer: A
Since Statement 1 alone gives us a unique answer (160 guests) but Statement 2 alone allows for multiple possible totals, only Statement 1 is sufficient.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."