A manufacturing company has fixed overhead costs of C dollars per month. In addition to the overhead, the company incurs...

1. TRANSLATE the problem information

• Given information:
  • Fixed overhead costs: \(\mathrm{C}\) dollars per month
  • Variable production costs: \(\mathrm{15\%}\) of total value \(\mathrm{V}\) produced
  • Constraint: Total costs should be at least \(\mathrm{1.8C}\) and at most \(\mathrm{2.2C}\)

• What this tells us:
  • Total monthly costs = Fixed costs + Variable costs = \(\mathrm{C + 0.15V}\)
  • We need: \(\mathrm{1.8C \leq Total\ costs \leq 2.2C}\)

2. INFER the approach

• We need to solve a compound inequality for \(\mathrm{V}\)
• Strategy: Substitute the total cost expression into the constraint inequality, then isolate \(\mathrm{V}\)

3. SIMPLIFY by setting up and solving the inequality

• Substitute total costs into the constraint:

\(\mathrm{1.8C \leq C + 0.15V \leq 2.2C}\)

• Subtract \(\mathrm{C}\) from all parts:

\(\mathrm{1.8C - C \leq 0.15V \leq 2.2C - C}\)

\(\mathrm{0.8C \leq 0.15V \leq 1.2C}\)

• Divide all parts by \(\mathrm{0.15}\):

\(\mathrm{\frac{0.8C}{0.15} \leq V \leq \frac{1.2C}{0.15}}\)

\(\mathrm{\frac{0.8}{0.15}C \leq V \leq \frac{1.2}{0.15}C}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the constraint and set up the inequality as \(\mathrm{1.8C \leq 0.15V \leq 2.2C}\), completely omitting the fixed cost \(\mathrm{C}\) from the total cost expression.

Their reasoning: "The total costs should be between \(\mathrm{1.8C}\) and \(\mathrm{2.2C}\), and the variable costs are \(\mathrm{0.15V}\), so \(\mathrm{1.8C \leq 0.15V \leq 2.2C}\)." They forget that total costs include both fixed AND variable costs.

This leads them to solve \(\mathrm{1.8C \leq 0.15V \leq 2.2C}\), giving them \(\mathrm{\frac{1.8}{0.15}C \leq V \leq \frac{2.2}{0.15}C}\), which matches Choice D.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{1.8C \leq C + 0.15V \leq 2.2C}\) but make algebraic errors when manipulating the inequality, particularly when subtracting \(\mathrm{C}\) or dividing by \(\mathrm{0.15}\).

A common mistake is incorrectly handling the subtraction: instead of getting \(\mathrm{0.8C \leq 0.15V \leq 1.2C}\), they might get confused and think the constraint is just about the variable portion being \(\mathrm{0.8C}\) to \(\mathrm{1.2C}\) directly, leading them to select Choice A.

The Bottom Line:

This problem requires careful attention to what "total costs" actually means - it's the sum of fixed AND variable costs, not just the variable portion. Students who rush through the setup often miss this crucial detail.