The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for...
GMAT Algebra : (Alg) Questions
The cost of renting a backhoe for up to 10 days is \(\$270\) for the first day and \(\$135\) for each additional day. Which of the following equations gives the cost \(\mathrm{y}\), in dollars, of renting the backhoe for \(\mathrm{x}\) days, where \(\mathrm{x}\) is a positive integer and \(\mathrm{x \leq 10}\)?
1. TRANSLATE the pricing structure
- Given information:
- $270 for the first day
- $135 for each additional day
- Need cost y for x days total
- What this tells us: We have a fixed cost for day 1, then a per-day rate for the remaining days
2. INFER how many "additional days" there are
- Key insight: If we rent for x days total, then:
- Day 1 costs $270
- Days 2, 3, 4, ..., x are the "additional days"
- Number of additional days = \(\mathrm{x - 1}\)
3. Set up the cost equation
- Total cost = Cost for first day + Cost for additional days
- \(\mathrm{y = 270 + 135(x - 1)}\)
4. SIMPLIFY the equation to match answer choices
- \(\mathrm{y = 270 + 135(x - 1)}\)
- \(\mathrm{y = 270 + 135x - 135}\)
- \(\mathrm{y = 135x + 135}\)
Answer: D. \(\mathrm{y = 135x + 135}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Misunderstanding what "additional days" means
Students often think that if you rent for x days, then all x days are "additional days" after some base period. This leads them to write:
\(\mathrm{y = 270 + 135x}\)
When they try to match this to the answer choices, they might select Choice C (\(\mathrm{y = 135x + 270}\)) since it has the same numbers, just rearranged.
Second Most Common Error:
Poor TRANSLATE reasoning: Mixing up which cost applies to which days
Some students reverse the pricing, thinking the first day costs $135 and additional days cost $270 each. This creates:
\(\mathrm{y = 135 + 270(x - 1)}\)
\(\mathrm{y = 135 + 270x - 270}\)
\(\mathrm{y = 270x - 135}\)
This leads them to select Choice A (\(\mathrm{y = 270x - 135}\)).
The Bottom Line:
The key challenge is recognizing that "x days total" breaks down into "1 first day + (x-1) additional days." Students who miss this distinction will struggle to set up the correct equation, even if they can handle the algebra.