A software company offers bulk licensing packages to businesses. For orders of 12 or more licenses, the company charges $35...
GMAT Algebra : (Alg) Questions
A software company offers bulk licensing packages to businesses. For orders of 12 or more licenses, the company charges $35 per license for the first 12 licenses and $22 per license for each additional license beyond 12. Which function g gives the total cost, in dollars, for an order of n licenses, where \(\mathrm{n \geq 12}\)?
\(\mathrm{g(n) = 22n + 156}\)
\(\mathrm{g(n) = 22n + 420}\)
\(\mathrm{g(n) = 35n + 22}\)
\(\mathrm{g(n) = 57n - 264}\)
1. TRANSLATE the pricing information
- Given information:
- For \(\mathrm{n \geq 12}\) licenses
- First 12 licenses: \(\$35\) each
- Additional licenses beyond 12: \(\$22\) each
- What this tells us: We have two different price tiers that need to be handled separately
2. TRANSLATE each cost component
- Cost for first 12 licenses: \(\mathrm{12 \times \$35 = \$420}\)
- Cost for licenses beyond 12: There are \(\mathrm{(n - 12)}\) additional licenses, each costing \(\$22\)
- So additional cost = \(\mathrm{\$22(n - 12)}\)
3. TRANSLATE the total cost expression
- Total cost = Cost for first 12 + Cost for additional licenses
- \(\mathrm{g(n) = \$420 + \$22(n - 12)}\)
4. SIMPLIFY to standard function form
- \(\mathrm{g(n) = \$420 + \$22(n - 12)}\)
- \(\mathrm{g(n) = \$420 + \$22n - \$264}\) (use calculator for \(\mathrm{22 \times 12 = 264}\))
- \(\mathrm{g(n) = \$22n + (\$420 - \$264)}\) (use calculator for \(\mathrm{420 - 264 = 156}\))
- \(\mathrm{g(n) = \$22n + \$156}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students misunderstand the tiered structure and think the additional licenses also cost \(\$35\), or they add the flat \(\$420\) cost incorrectly.
A common mistake is thinking: "First 12 cost \(\$420\), additional licenses cost \(\$22\) each, so total is \(\mathrm{\$420 + \$22n}\)." This ignores that only (n-12) additional licenses exist, not all n licenses at the additional rate.
This may lead them to select Choice B (\(\mathrm{g(n) = 22n + 420}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students think each license costs the sum of both rates: \(\mathrm{\$35 + \$22 = \$57}\) per license.
They reason: "The problem mentions \(\$35\) and \(\$22\), so maybe each license costs \(\$57\) total." This completely misunderstands the tiered pricing structure.
This may lead them to select Choice C (\(\mathrm{g(n) = 57n - 420}\)) or Choice D (\(\mathrm{g(n) = 57n - 264}\)).
The Bottom Line:
This problem requires careful parsing of tiered pricing language and systematic translation of each pricing component before algebraic combination.
\(\mathrm{g(n) = 22n + 156}\)
\(\mathrm{g(n) = 22n + 420}\)
\(\mathrm{g(n) = 35n + 22}\)
\(\mathrm{g(n) = 57n - 264}\)