A school club needs to raise at least $5,000 to fund a trip. Before selling any tickets, a local sponsor...
GMAT Algebra : (Alg) Questions
A school club needs to raise at least \(\$5,000\) to fund a trip. Before selling any tickets, a local sponsor contributes \(\$300\). The club sells standard tickets for \(\$12\) each and VIP tickets for \(\$35\) each. If \(\mathrm{x}\) is the number of standard tickets sold and \(\mathrm{y}\) is the number of VIP tickets sold, which inequality represents the possible combinations of \(\mathrm{x}\) and \(\mathrm{y}\) that will meet the fundraising goal?
1. TRANSLATE the problem information
- Given information:
- Club needs to raise at least $5,000 total
- Sponsor contributes $300 before any ticket sales
- Standard tickets: $12 each (x tickets sold)
- VIP tickets: $35 each (y tickets sold)
- Need inequality for combinations that meet the goal
- What "at least $5,000" means: \(\mathrm{Total\ funds \geq \$5,000}\)
2. INFER the total funds equation
- \(\mathrm{Total\ funds = Sponsor\ contribution + Ticket\ sales}\)
- \(\mathrm{Total\ funds = 300 + 12x + 35y}\)
- Since we need at least $5,000: \(\mathrm{300 + 12x + 35y \geq 5,000}\)
3. SIMPLIFY to isolate ticket sales
- The question asks for the inequality representing ticket sales combinations
- Subtract the sponsor contribution from both sides:
\(\mathrm{12x + 35y \geq 5,000 - 300}\)
\(\mathrm{12x + 35y \geq 4,700}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand "at least" and use \(\mathrm{\leq}\) instead of \(\mathrm{\geq}\).
They think "at least $5,000" means they can't exceed $5,000, when it actually means they need $5,000 or more. This leads them to write \(\mathrm{12x + 35y \leq 4,700}\).
This may lead them to select Choice B (\(\mathrm{12x + 35y \leq 4,700}\))
Second Most Common Error:
Poor INFER reasoning: Students forget to account for the sponsor contribution when setting up the inequality.
They see the $5,000 goal and immediately write \(\mathrm{12x + 35y \geq 5,000}\), forgetting that the sponsor already contributed $300. Or they might also mix up which ticket type costs which amount.
This may lead them to select Choice C (\(\mathrm{35x + 12y \geq 5,000}\))
The Bottom Line:
This problem tests your ability to carefully translate word problems into mathematical inequalities while keeping track of all components that contribute to the total. The key insight is recognizing that the sponsor contribution reduces what the ticket sales need to achieve.