A manufacturing company operates with a linear cost function \(\mathrm{C(n) = an + f}\), where n is the number of...
GMAT Algebra : (Alg) Questions
A manufacturing company operates with a linear cost function \(\mathrm{C(n) = an + f}\), where \(\mathrm{n}\) is the number of units produced, \(\mathrm{a}\) represents the cost per unit, and \(\mathrm{f}\) represents the fixed cost. The table shows the total cost \(\mathrm{C(n)}\) for producing different numbers of units. What is the value of the fixed cost \(\mathrm{f}\)?
| \(\mathrm{n}\) | \(\mathrm{C(n)}\) |
|---|---|
| 3 | 121 |
| 5 | 145 |
| 7 | 169 |
| 9 | 193 |
12
36
85
121
1. TRANSLATE the problem information
- Given information:
- Linear cost function: \(\mathrm{C(n) = an + f}\)
- \(\mathrm{n}\) = number of units produced
- \(\mathrm{a}\) = cost per unit
- \(\mathrm{f}\) = fixed cost (what we need to find)
- Data table with 4 coordinate pairs
2. INFER the solution strategy
- Since this is a linear function, we need to find the slope \(\mathrm{(a)}\) first
- The slope tells us how much cost increases per additional unit
- Once we have the slope, we can use any data point to solve for \(\mathrm{f}\)
3. SIMPLIFY to find the slope
- Look at the pattern in the table:
- When \(\mathrm{n}\) increases by 2 (from 3→5, 5→7, 7→9)
- \(\mathrm{C(n)}\) increases by 24 each time (121→145, 145→169, 169→193)
- Slope = \(\mathrm{\frac{rise}{run} = \frac{24}{2} = 12}\)
- So \(\mathrm{a = 12}\) (cost per unit)
4. SIMPLIFY to find the fixed cost
- Use the equation \(\mathrm{C(n) = an + f}\) with any data point
- Using point (3, 121):
\(\mathrm{121 = 12(3) + f}\)
\(\mathrm{121 = 36 + f}\)
\(\mathrm{f = 121 - 36 = 85}\)
5. INFER verification step
- Check with another point (5, 145):
\(\mathrm{145 = 12(5) + f}\)
\(\mathrm{145 = 60 + f}\)
\(\mathrm{f = 85}\) ✓
Answer: C (85)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students jump directly to trying to solve for \(\mathrm{f}\) without first finding the slope systematically. They might try to use the equation \(\mathrm{C(n) = an + f}\) with unknown values for both \(\mathrm{a}\) and \(\mathrm{f}\), creating confusion with two unknowns. Without the strategic insight to find the slope first, they get stuck in algebraic complexity and may guess randomly.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what the fixed cost \(\mathrm{f}\) represents. They might think \(\mathrm{f}\) is the cost when one unit is produced, leading them to select Choice D (121) since that's the cost when \(\mathrm{n=3}\), the smallest value in the table.
The Bottom Line:
This problem tests whether students can work systematically with linear functions by recognizing that slope must be determined first before solving for the y-intercept. The key insight is treating this as a two-step process rather than trying to solve everything at once.
12
36
85
121