The 50 participants of a management training seminar ate dinner at a certain restaurant. They had 3 choices for their...

Understanding the Question

We need to find the exact number of participants who ordered blackened catfish.

Given Information

• Total participants: 50
• Three meal choices with fixed prices:
  • Vegetarian lasagna: $12
  • Blackened catfish: $15
  • Stuffed pork chops: $18
• Each participant ordered exactly 1 meal
• Total cost of all meals: $810

What We Need to Determine

To answer "How many participants ordered blackened catfish?", we need a unique value for the number of catfish orders.

Let's define:

• \(\mathrm{L} = \mathrm{number\ who\ ordered\ lasagna}\)
• \(\mathrm{C} = \mathrm{number\ who\ ordered\ catfish}\)
• \(\mathrm{P} = \mathrm{number\ who\ ordered\ pork\ chops}\)

From our given information:

• \(\mathrm{L + C + P = 50}\) (total participants)
• \(\mathrm{12L + 15C + 18P = 810}\) (total cost)

Key Insight

With 2 equations and 3 unknowns, we need one more independent constraint to find a unique value for C. Think of it this way: we know the total headcount and total bill, but there are multiple ways to distribute 50 people among three meals to reach $810.

Analyzing Statement 1

Statement 1: Six more people ordered catfish than lasagna.

What Statement 1 Tells Us

This gives us the relationship: \(\mathrm{C = L + 6}\)

Now we have a third equation! This creates a system of 3 equations with 3 unknowns, which typically has a unique solution.

Solving the System

From \(\mathrm{C = L + 6}\), we get \(\mathrm{L = C - 6}\)

Substituting into our participant equation:

• \(\mathrm{(C - 6) + C + P = 50}\)
• \(\mathrm{2C + P = 56}\)
• \(\mathrm{P = 56 - 2C}\)

Now substituting both expressions into the cost equation:

• \(\mathrm{12(C - 6) + 15C + 18(56 - 2C) = 810}\)
• \(\mathrm{12C - 72 + 15C + 1008 - 36C = 810}\)
• \(\mathrm{-9C + 936 = 810}\)
• \(\mathrm{-9C = -126}\)
• \(\mathrm{C = 14}\)

Verification

Let's verify our answer:

• \(\mathrm{C = 14}\) (catfish orders)
• \(\mathrm{L = 14 - 6 = 8}\) (lasagna orders)
• \(\mathrm{P = 56 - 2(14) = 28}\) (pork chop orders)

Check: \(\mathrm{8 + 14 + 28 = 50}\) ✓
Cost: \(\mathrm{12(8) + 15(14) + 18(28) = 96 + 210 + 504 = 810}\) ✓

Conclusion

Statement 1 provides enough information to determine that exactly 14 participants ordered catfish.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: Twice as many pork chop meals were ordered as catfish meals.

What Statement 2 Provides

This gives us: \(\mathrm{P = 2C}\)

Again, we have a third equation, creating a solvable system.

Solving the System

From \(\mathrm{P = 2C}\) and our participant equation:

• \(\mathrm{L + C + 2C = 50}\)
• \(\mathrm{L + 3C = 50}\)
• \(\mathrm{L = 50 - 3C}\)

Substituting into the cost equation:

• \(\mathrm{12(50 - 3C) + 15C + 18(2C) = 810}\)
• \(\mathrm{600 - 36C + 15C + 36C = 810}\)
• \(\mathrm{600 + 15C = 810}\)
• \(\mathrm{15C = 210}\)
• \(\mathrm{C = 14}\)

Verification

Let's verify this solution:

• \(\mathrm{C = 14}\) (catfish orders)
• \(\mathrm{P = 2(14) = 28}\) (pork chop orders)
• \(\mathrm{L = 50 - 3(14) = 8}\) (lasagna orders)

Check: \(\mathrm{8 + 14 + 28 = 50}\) ✓
Cost: \(\mathrm{12(8) + 15(14) + 18(28) = 96 + 210 + 504 = 810}\) ✓

Conclusion

Statement 2 also provides enough information to determine that exactly 14 participants ordered catfish.

[STOP - Statement 2 is Sufficient!]

The Answer: D

Both statements independently provide enough information to determine that 14 participants ordered blackened catfish.

Answer Choice D: "Each statement alone is sufficient."