A contractor builds a straight railing using d vertical posts placed in a row. Between each pair of adjacent posts,...
GMAT Algebra : (Alg) Questions
A contractor builds a straight railing using \(\mathrm{d}\) vertical posts placed in a row. Between each pair of adjacent posts, exactly one metal connector is required. The contractor pays a one-time inspection fee of $52, and each metal connector costs $26. Which function gives the total cost \(\mathrm{C(d)}\), in dollars, of building the railing with \(\mathrm{d}\) posts, where \(\mathrm{d}\) is an integer greater than or equal to 1?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{d}\) vertical posts placed in a row
- Exactly one metal connector between each pair of adjacent posts
- One-time inspection fee: $52
- Each connector costs: $26
- Find: Total cost function \(\mathrm{C(d)}\)
2. INFER the number of connectors needed
- Key insight: If you have \(\mathrm{d}\) posts in a row, how many gaps are between them?
- Think about it: 2 posts → 1 gap, 3 posts → 2 gaps, 4 posts → 3 gaps
- Pattern: \(\mathrm{d}\) posts create \(\mathrm{(d-1)}\) gaps
- Therefore: Number of connectors needed = \(\mathrm{(d-1)}\)
3. TRANSLATE this into the cost structure
- Total cost = Inspection fee + Connector costs
- \(\mathrm{C(d) = 52 + 26(d-1)}\)
4. SIMPLIFY to match answer choice format
- \(\mathrm{C(d) = 52 + 26(d-1)}\)
- \(\mathrm{C(d) = 52 + 26d - 26}\)
- \(\mathrm{C(d) = 26d + 26}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students incorrectly assume that \(\mathrm{d}\) posts require \(\mathrm{d}\) connectors instead of \(\mathrm{(d-1)}\) connectors.
They think: "\(\mathrm{d}\) posts means \(\mathrm{d}\) connectors needed" without visualizing that connectors go between posts, not at each post location. This leads them to calculate: \(\mathrm{C(d) = 52 + 26d}\).
This may lead them to select Choice B (\(\mathrm{C(d) = 26d + 52}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\mathrm{C(d) = 52 + 26(d-1)}\) but make algebraic errors when expanding.
They might incorrectly distribute or combine like terms, getting confused about whether the final form should have +52 or +26 as the constant term.
This leads to confusion and guessing among the remaining choices.
The Bottom Line:
The key challenge is visualizing the physical setup - recognizing that posts create gaps between them, not that each post needs a connector. Once students grasp this spatial relationship, the algebra flows naturally.