A custom printing company charges a one-time setup fee of $28 for each printing job. In addition to the setup...
GMAT Algebra : (Alg) Questions
A custom printing company charges a one-time setup fee of \(\$28\) for each printing job. In addition to the setup fee, the company charges \(\$12.75\) for each item printed. If a customer has a budget of \(\$340\), what is the maximum number of items that can be printed without exceeding the budget?
Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Setup fee: $28 (one-time charge)
- Cost per item: $12.75
- Budget limit: $340
- Need: Maximum number of items without exceeding budget
- What this tells us: We need an inequality since we cannot exceed the budget.
2. TRANSLATE to set up the mathematical relationship
- Let \(\mathrm{x}\) = number of items to be printed
- Total cost = Setup fee + (Cost per item × Number of items)
- Total cost = \(\mathrm{28 + 12.75x}\)
- Constraint: Total cost ≤ Budget
- Mathematical inequality: \(\mathrm{28 + 12.75x \leq 340}\)
3. SIMPLIFY by solving the inequality
- Subtract the setup fee from both sides:
\(\mathrm{12.75x \leq 340 - 28}\)
\(\mathrm{12.75x \leq 312}\)
- Divide both sides by 12.75 (use calculator):
\(\mathrm{x \leq 312 \div 12.75}\)
\(\mathrm{x \leq 24.47...}\)
4. APPLY CONSTRAINTS to find the final answer
- Since you can't print a partial item, the number of items must be a whole number
- We must round DOWN to 24 (not up to 25, as that would exceed the budget)
- Verification: \(\mathrm{28 + 12.75(24) = 28 + 306 = \$334 \leq \$340}\) ✓
Answer: 24
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak APPLY CONSTRAINTS reasoning: Students correctly solve to get \(\mathrm{x \leq 24.47...}\) but then round UP to 25 instead of down to 24.
They think "24.47 is closer to 25, so I should round to 25." However, this violates the budget constraint since \(\mathrm{28 + 12.75(25) = \$346.75}\), which exceeds the $340 budget. This leads to an incorrect answer of 25.
Second Most Common Error:
Incomplete TRANSLATE execution: Students forget to include the setup fee and only focus on the per-item cost.
They set up \(\mathrm{12.75x \leq 340}\), which gives \(\mathrm{x \leq 26.67...}\), leading them to answer 26. This completely ignores the $28 setup fee that must be paid regardless of how many items are printed.
The Bottom Line:
This problem tests whether students understand that real-world constraints require careful attention to inequality direction and realistic rounding. The mathematical solution gives a decimal, but the real-world context demands a whole number that doesn't violate the budget limit.