A company uses the formula C = nh + F to calculate the total cost, C, in dollars, to produce...
GMAT Advanced Math : (Adv_Math) Questions
A company uses the formula \(\mathrm{C = nh + F}\) to calculate the total cost, \(\mathrm{C}\), in dollars, to produce a batch of a certain product. In the formula, \(\mathrm{n}\) is the number of units in the batch, \(\mathrm{h}\) is the cost for labor per unit in dollars, and \(\mathrm{F}\) is a fixed setup fee in dollars. Which of the following correctly expresses the number of units, \(\mathrm{n}\), in terms of \(\mathrm{C}\), \(\mathrm{h}\), and \(\mathrm{F}\)?
1. TRANSLATE the problem information
- Given: The cost formula \(\mathrm{C = nh + F}\) where C is total cost, n is number of units, h is cost per unit, and F is fixed setup fee
- Find: Express n in terms of the other variables
2. INFER the solution strategy
- This is a 'solve for the variable' problem
- I need to isolate n on one side of the equation using inverse operations
- Since n is currently multiplied by h and then has F added to it, I need to undo these operations in reverse order
3. SIMPLIFY by removing the constant term first
- Start with: \(\mathrm{C = nh + F}\)
- Subtract F from both sides: \(\mathrm{C - F = nh}\)
- This eliminates F from the right side
4. SIMPLIFY by isolating the variable
- Now I have: \(\mathrm{C - F = nh}\)
- Divide both sides by h: \(\mathrm{\frac{C - F}{h} = n}\)
- This gives me: \(\mathrm{n = \frac{C - F}{h}}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize the proper order of operations for isolating variables. They might try to divide by h first, leading to something like \(\mathrm{\frac{C}{h} = n + \frac{F}{h}}\), then incorrectly attempt to subtract F instead of \(\mathrm{\frac{F}{h}}\).
This type of confusion about operation order can lead them to select Choice C \(\mathrm{(\frac{C}{h} - F)}\) or cause them to get stuck and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students make sign errors when moving terms across the equals sign. They might incorrectly add F instead of subtracting it when trying to isolate the nh term.
This leads them to write \(\mathrm{C + F = nh}\), and then \(\mathrm{n = \frac{C + F}{h}}\), causing them to select Choice B \(\mathrm{(\frac{C + F}{h})}\).
The Bottom Line:
Success on this problem requires systematic thinking about inverse operations and careful algebraic manipulation. Students who rush or don't have a clear strategy for variable isolation are most likely to make errors.