If 50 people paid a total of $110 for tickets to attend a certain high school play and spent a...
GMAT Data Sufficiency : (DS) Questions
If 50 people paid a total of $110 for tickets to attend a certain high school play and spent a total of $100 for refreshments at intermission, how many adults attended the play?
- Each adult who attended the play paid $3 for tickets, and each child who attended paid $1 for tickets.
- Each adult who attended the play spent $2 for refreshments at intermission, and each child spent $2 for refreshments at intermission.
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."