A gym offers a membership with a one-time $30 signup fee and a charge of $18 per class attended. A...

1. TRANSLATE the problem information

• Given information:
  • One-time signup fee: \(\$30\)
  • Cost per class: \(\$18\)
  • Number of classes: c (the variable)
  • Coupon discount: \(\$12\) off the total
  • Budget constraint: at most \(\$120\)
• What this tells us: We need to build an expression for total cost and set up an inequality.

2. INFER how costs combine

• The signup fee is paid once regardless of classes attended
• Each class costs \(\$18\), so c classes cost \(\$18\mathrm{c}\)
• The coupon reduces the final total, not the per-class cost
• "At most \(\$120\)" means the total can be \(\$120\) or less (≤)

3. TRANSLATE into mathematical expression

• Total cost before coupon: \(\$30 + \$18\mathrm{c}\)
• Total cost after coupon: \(\$30 + \$18\mathrm{c} - \$12\)
• Set up inequality: \(\$30 + \$18\mathrm{c} - \$12 \leq \$120\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding where the coupon applies

Students often think the \(\$12\) coupon reduces the per-class cost rather than the total bill. They might reason: "The coupon makes each class cost \(\$18 - \$12 = \$6\), so the cost is \(\$30 + \$6\mathrm{c}\)."

This may lead them to select Choice D (\(\$30 + (18 - 12)\mathrm{c} \leq 120\))

Second Most Common Error:

Incomplete TRANSLATE reasoning: Forgetting components of the total cost

Students might focus only on the class costs and forget either the signup fee or the coupon discount. For example, they might only consider the variable costs and ignore the \(\$30\) fee.

This may lead them to select Choice A (\(\$18\mathrm{c} \leq 120\)) or Choice B (\(\$30 + 18\mathrm{c} \leq 120\))

The Bottom Line:

Success requires carefully parsing the problem to understand that some costs are fixed (signup fee), some are variable (per-class), and discounts typically apply to totals rather than unit costs. The key insight is recognizing the coupon as a reduction from the final sum, not a per-class discount.