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: $200
  • Need: Greatest number of attendees without exceeding budget

• What this tells us: We need to find the maximum whole number of attendees where total costs stay within $200.

2. TRANSLATE the relationship into mathematics

• 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: \(35 + 10.25\mathrm{x} \leq 200\)

3. SIMPLIFY to solve the inequality

• Start with: \(35 + 10.25\mathrm{x} \leq 200\)
• Subtract 35 from both sides: \(10.25\mathrm{x} \leq 165\)
• Divide both sides by 10.25: \(\mathrm{x} \leq 165 \div 10.25\)
• Calculate (use calculator): \(\mathrm{x} \leq 16.097...\)

4. APPLY CONSTRAINTS to select the final answer

• Since we need a whole number of attendees, we must round down
• The greatest number of attendees is 16

• Verification: \(35 + 10.25(16) = 35 + 164 = 199 \leq 200\) ✓

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students set up an equation instead of an inequality, writing \(35 + 10.25\mathrm{x} = 200\). They think "use all the budget" instead of "don't exceed the budget."

This leads them to solve for \(\mathrm{x} = 165 \div 10.25 = 16.097...\), then round to 16, coincidentally getting the right answer but with flawed reasoning. However, this error could lead to confusion on similar problems where the answer isn't as forgiving.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly set up and solve the inequality to get \(\mathrm{x} \leq 16.097...\), but then round up to 17 because "16.097 is closer to 17."

They don't recognize that having 17 attendees would cost \(35 + 10.25(17) = \$209.25\), which exceeds the $200 budget. This leads them to an answer of 17, which violates the budget constraint.

The Bottom Line:

This problem tests whether students understand the difference between "using exactly" versus "not exceeding" a limit, and whether they can properly apply real-world constraints to mathematical solutions.