Each of the dinners served at a banquet was either chicken or beef or fish. The ratio of the number...

Understanding the Question

We need to find the total number of dinners served at a banquet. We're told that dinners were served in a specific ratio of \(7:5:2\) (chicken:beef:fish), and there were more than 5 fish dinners.

Since the dinners follow this exact ratio, we can express them as:

• Chicken dinners = \(7\mathrm{k}\)
• Beef dinners = \(5\mathrm{k}\)
• Fish dinners = \(2\mathrm{k}\)

where k is a positive integer (we can't serve fractional dinners!).

This gives us a total of 14k dinners (\(7\mathrm{k} + 5\mathrm{k} + 2\mathrm{k} = 14\mathrm{k}\)).

From the constraint that fish dinners > 5:

• We have \(2\mathrm{k} > 5\)
• So \(\mathrm{k} > 2.5\)
• Since k must be a positive integer, k ≥ 3

What makes a statement sufficient? We need to find the exact value of k, which will give us the exact total number of dinners.

Analyzing Statement 1

Statement 1 tells us: The total number of beef dinners and fish dinners was less than 30.

This means: beef + fish < 30

• So \(5\mathrm{k} + 2\mathrm{k} < 30\)
• Which gives us \(7\mathrm{k} < 30\)
• Therefore: \(\mathrm{k} < 30/7 \approx 4.29\)

Combined with our constraint \(\mathrm{k} \geq 3\), we now know that k can only be 3 or 4.

Let's check what this means for the total:

• If k = 3: Total dinners = \(14 \times 3 = 42\)
• If k = 4: Total dinners = \(14 \times 4 = 56\)

Since we get two different possible totals (42 or 56), we cannot determine a unique answer.

Statement 1 alone is NOT sufficient.

This eliminates answer choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The number of chicken dinners was less than 25.

This means: chicken < 25

• So \(7\mathrm{k} < 25\)
• Therefore: \(\mathrm{k} < 25/7 \approx 3.57\)

Combined with our constraint \(\mathrm{k} \geq 3\), this means k must equal exactly 3.

Let's verify:

• k = 3 gives chicken = \(7 \times 3 = 21\) dinners (✓ less than 25)
• k = 4 would give chicken = \(7 \times 4 = 28\) dinners (✗ not less than 25)

With k = 3:

• Total dinners = \(14 \times 3 = 42\) dinners

[STOP - Sufficient!] We found exactly one value for the total.

Statement 2 alone IS sufficient.

This eliminates answer choices C and E.

The Answer: B

Statement 2 alone gives us exactly one possible value for the total (42 dinners), while Statement 1 alone allows two different values (42 or 56 dinners).

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."