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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?
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.
To answer "How many guests were at the banquet?", we need enough information to arrive at one unique value.
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.
Statement 1 tells us:
These specific values act as "anchor points" that cascade through our constraints. Let's trace through the logic:
Main dining room analysis:
Annex analysis:
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.
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?
Scenario 1: Suppose 30 guests had "eggplant only" in each location
Scenario 2: Suppose 35 guests had "eggplant only" in each location
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.
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."