An event planner is planning a party. It costs the event planner a one-time fee of $35 to rent the...

1. TRANSLATE the problem information

• Given information:
  • One-time venue rental fee: $35
  • Cost per attendee: $10.25
  • Total budget available: $300
  • Need to find: greatest number of attendees without exceeding budget

• This tells us we need an inequality since we want to stay "within" the budget limit

2. TRANSLATE into mathematical form

• Let \(\mathrm{x}\) = number of attendees
• Total cost = fixed cost + variable cost
• Total cost = \(35 + 10.25\mathrm{x}\)
• Since we cannot exceed the budget: \(35 + 10.25\mathrm{x} \leq 300\)

3. SIMPLIFY the inequality step by step

• Start with: \(35 + 10.25\mathrm{x} \leq 300\)
• Subtract 35 from both sides: \(10.25\mathrm{x} \leq 265\)
• Divide both sides by 10.25: \(\mathrm{x} \leq 25.853...\) (use calculator)

4. APPLY CONSTRAINTS to select final answer

• Since the number of attendees must be a whole number in real-world context
• We must round down from 25.853 to 25 attendees
• Verification: \(35 + 10.25(25) = 35 + 256.25 = \$291.25 \leq \$300\) ✓

Answer: 25 attendees

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle with setting up the correct inequality direction and often write \(35 + 10.25\mathrm{x} = 300\) as an equation instead of an inequality, missing that we want the "greatest number possible without exceeding."

This leads them to get \(\mathrm{x} = 25.853\) and then round up to 26, not realizing this exceeds the budget. When they check their work (if at all), they find \(35 + 10.25(26) = \$301.50 \gt \$300\), creating confusion.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the inequality to get \(\mathrm{x} \leq 25.853\) but then round up to 26 instead of down to 25, not recognizing that real-world constraints require staying within the budget limit.

This causes them to select an answer of 26 attendees, which actually violates the budget constraint.

The Bottom Line:

This problem tests whether students can distinguish between "maximum possible" (requiring inequalities) versus "exact amount" (requiring equations), and whether they understand that constraint problems often require rounding in the direction that satisfies the constraint.