A fitness club charges $55 per month for membership. However, new members receive a $15 discount on their first month...

1. TRANSLATE the pricing information

• Given information:
  • Regular monthly cost: \(\$55\)
  • First month discount for new members: \(\$15\)
  • Need equation for total cost y over x months

• What this tells us: First month costs \(\$55 - \$15 = \$40\), all other months cost \(\$55\)

2. INFER the cost pattern

• The discount applies only once (first month)
• For x months: we pay \(\$40\) for month 1, then \(\$55\) for each of the remaining (x-1) months
• Total = \(\$40 + \$55(\mathrm{x}-1)\)

3. SIMPLIFY the expression

• \(\mathrm{y} = \$40 + \$55(\mathrm{x}-1)\)
• \(\mathrm{y} = \$40 + \$55\mathrm{x} - \$55\)
• \(\mathrm{y} = \$55\mathrm{x} - \$15\)

4. Verify with test cases

• For 1 month: \(\mathrm{y} = 55(1) - 15 = \$40\) ✓
• For 2 months: \(\mathrm{y} = 55(2) - 15 = \$95\) ✓

Answer: (A) \(\mathrm{y} = 55\mathrm{x} - 15\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students misunderstand which months get the discount and apply the discounted rate of \(\$40\) to all months, thinking \(\mathrm{y} = 40\mathrm{x}\). Then they might incorrectly try to account for the discount by adding something back, potentially leading to Choice (B) (\(\mathrm{y} = 40\mathrm{x} + 15\)).

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students correctly calculate the first month as \(\$40\) but then think the remaining months are also separate from the x total, writing \(\mathrm{y} = 40 + 55\mathrm{x}\) instead of recognizing that the first month is part of the x months. This leads them to select Choice (C) (\(\mathrm{y} = 55\mathrm{x} + 40\)).

The Bottom Line:

The key insight is recognizing that the \(\$15\) discount creates a one-time reduction from what the total would be at full price (\(\$55\mathrm{x}\)), rather than changing the monthly rate structure.