A fitness center charges new members $120 for the first month of membership and $60 for each additional month. Which...
GMAT Algebra : (Alg) Questions
A fitness center charges new members \(\$120\) for the first month of membership and \(\$60\) for each additional month. Which of the following equations gives the total cost \(\mathrm{y}\), in dollars, for \(\mathrm{x}\) months of membership, where \(\mathrm{x}\) is a positive integer?
1. TRANSLATE the pricing structure into mathematical expressions
- Given information:
- First month: \(\$120\)
- Each additional month: \(\$60\)
- Need equation for total cost y after x months
- What this tells us: We have a two-part pricing system, not a simple per-month rate
2. INFER the relationship between total months and additional months
- Key insight: If someone has x months total, they have:
- 1 first month (at \(\$120\))
- \(\mathrm{(x-1)}\) additional months (at \(\$60\) each)
- This is the foundation for building our equation
3. TRANSLATE this insight into a mathematical equation
- Total cost = Cost of first month + Cost of additional months
- Total cost = \(\$120 + \$60(\mathrm{x}-1)\)
4. SIMPLIFY through algebraic manipulation
- \(\mathrm{y} = 120 + 60(\mathrm{x}-1)\)
- \(\mathrm{y} = 120 + 60\mathrm{x} - 60\)
- \(\mathrm{y} = 60\mathrm{x} + 60\)
5. Verify with test cases
- 1 month: \(\mathrm{y} = 60(1) + 60 = \$120\) ✓
- 2 months: \(\mathrm{y} = 60(2) + 60 = \$180 = \$120 + \$60\) ✓
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students misinterpret the pricing structure as "\(\$60\) per month plus a \(\$120\) setup fee" rather than understanding that the first month has a special price.
They set up the equation as: Total cost = \(\$120 + \$60\mathrm{x}\) (thinking it's a flat setup fee plus monthly charges)
This leads them to select Choice C (\(\mathrm{y} = 60\mathrm{x} + 120\))
Second Most Common Error:
Poor INFER skill: Students recognize the two-part pricing but fail to realize that x months means \(\mathrm{(x-1)}\) additional months beyond the first month.
They might think: "First month is \(\$120\), and there are x more months at \(\$60\) each"
Setting up: \(\mathrm{y} = 120 + 60\mathrm{x}\)
This also leads them to select Choice C (\(\mathrm{y} = 60\mathrm{x} + 120\))
The Bottom Line:
This problem requires students to carefully parse a two-tier pricing system and translate it into the correct mathematical relationship. The key insight is recognizing that "x months total" includes the specially-priced first month, leaving only \(\mathrm{(x-1)}\) additional months at the regular rate.