At a climbing gym, a party package covers admission for up to 6 climbers for a flat fee of $132....

1. TRANSLATE the pricing structure

• Given information:
  • $132 flat fee covers up to 6 climbers
  • $12 for each climber beyond the first 6
  • We need \(\mathrm{C(n)}\) where \(\mathrm{n \geq 6}\)
• What this tells us: We have a two-part cost structure

2. INFER the mathematical relationship

• Since \(\mathrm{n \geq 6}\), every group pays the $132 flat fee
• Additional climbers = \(\mathrm{(n - 6)}\) climbers
• Additional cost = $12 × (number of additional climbers)
• Total cost = Flat fee + Additional cost

3. TRANSLATE into mathematical expression

• \(\mathrm{C(n) = 132 + 12(n - 6)}\)

4. SIMPLIFY to standard form

• \(\mathrm{C(n) = 132 + 12(n - 6)}\)
• \(\mathrm{C(n) = 132 + 12n - 72}\)
• \(\mathrm{C(n) = 12n + (132 - 72)}\)
• \(\mathrm{C(n) = 12n + 60}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "each additional climber costs $12" to mean that ALL climbers cost $12 each, ignoring the flat fee structure.

They think: Total cost = $12 per climber + some base amount, leading them to consider \(\mathrm{C(n) = 12n + 132}\), which matches Choice D.

Second Most Common Error:

Poor INFER skill: Students correctly understand there's a flat fee but miscalculate the number of additional climbers. They might think there are n additional climbers instead of \(\mathrm{(n - 6)}\) additional climbers.

This leads to \(\mathrm{C(n) = 132 + 12n}\), which would be Choice D again, reinforcing the wrong answer.

The Bottom Line:

The key challenge is recognizing that this is a two-tier pricing system where the first 6 climbers are covered by a flat fee, and only climbers beyond that number incur additional charges. Students must carefully parse "additional climbers beyond the first 6" to understand it means \(\mathrm{(n - 6)}\), not n.