If 50 people paid a total of $110 for tickets to attend a certain high school play and spent a...

Understanding the Question

Given Information:

• 50 people total attended the play
• Total ticket revenue: \(\mathrm{\$110}\)
• Total refreshment revenue: \(\mathrm{\$100}\)
• Attendees are only adults and children

What We Need to Find:
The exact number of adults who attended

This is a value question - we need a specific number. For a statement to be sufficient, it must allow us to determine exactly one value for the number of adults.

Key Insight: We already know that \(\mathrm{Adults + Children = 50}\). To find the exact number of adults, we need information that creates different spending patterns between adults and children.

Analyzing Statement 1

What Statement 1 Tells Us:

• Adults pay \(\mathrm{\$3}\) per ticket
• Children pay \(\mathrm{\$1}\) per ticket

With this pricing difference and the total revenue of \(\mathrm{\$110}\), let's think about what this means.

Logical Analysis

The key here is that adults pay \(\mathrm{\$2}\) more than children per ticket. Let's use this differential to understand the situation:

• If all 50 attendees were children: \(\mathrm{50 \times \$1 = \$50}\) total
• But we actually have \(\mathrm{\$110}\) total, which is \(\mathrm{\$60}\) more than the all-children scenario
• Since each adult adds \(\mathrm{\$2}\) to the revenue (compared to a child), we need exactly \(\mathrm{\$60 ÷ \$2 = 30}\) adults

To verify our logic: \(\mathrm{30 \text{ adults} \times \$3 + 20 \text{ children} \times \$1 = \$90 + \$20 = \$110}\) ✓

[STOP - Sufficient!] We found exactly one answer: 30 adults.

Conclusion

Statement 1 provides different ticket prices for adults and children, which combined with the total revenue gives us exactly one answer.

Statement 1 is sufficient.

This eliminates answer choices B, C, and E.

Analyzing Statement 2

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

What Statement 2 Tells Us:

• Adults spend \(\mathrm{\$2}\) on refreshments
• Children spend \(\mathrm{\$2}\) on refreshments

Critical Insight

Here's the key realization: If both adults AND children spend the same amount (\(\mathrm{\$2}\)) on refreshments, then the total refreshment revenue simply equals:

\(\mathrm{\$2 \times (\text{total number of people}) = \$2 \times 50 = \$100}\)

But wait - that's exactly what we were told! The \(\mathrm{\$100}\) refreshment revenue just confirms there were 50 people total, which we already knew.

Testing Different Scenarios

Let's verify this insight by testing various adult/child combinations:

• \(\mathrm{40 \text{ adults} + 10 \text{ children} = 50 \text{ people} \rightarrow \$2 \times 50 = \$100}\) refreshments ✓
• \(\mathrm{30 \text{ adults} + 20 \text{ children} = 50 \text{ people} \rightarrow \$2 \times 50 = \$100}\) refreshments ✓
• \(\mathrm{15 \text{ adults} + 35 \text{ children} = 50 \text{ people} \rightarrow \$2 \times 50 = \$100}\) refreshments ✓

Every split that totals 50 people will give us \(\mathrm{\$100}\) in refreshments!

Conclusion

Statement 2 provides no new information about the adult/child split. When both groups spend the same amount per person, the total revenue tells us nothing about how the groups are distributed.

Statement 2 is NOT sufficient.

This eliminates answer choices B and D.

The Answer: A

Statement 1 alone gives us different prices that create a unique solution, while Statement 2's equal spending provides no useful information about the split.

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