A factory must produce at least 4{,}200 units of a product. One machine produces a fixed 300 units, and there...

1. TRANSLATE the problem information

• Given information:
  • Total requirement: at least 4,200 units
  • Fixed machine: produces 300 units (constant)
  • Type 1 machines: 125 units per cycle, x cycles
  • Type 2 machines: 240 units per cycle, y cycles

• What 'at least 4,200' means: \(\mathrm{Total\ production \geq 4,200}\)

2. TRANSLATE the total production

• Total production = Fixed output + Variable outputs
• Total production = \(\mathrm{300 + 125x + 240y}\)

3. INFER the constraint setup

• Since total production must be at least 4,200:
• \(\mathrm{300 + 125x + 240y \geq 4,200}\)

4. SIMPLIFY to match answer format

• Subtract 300 from both sides:
• \(\mathrm{125x + 240y \geq 3,900}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students forget to account for the fixed 300 units and set up the inequality as just \(\mathrm{125x + 240y \geq 4,200}\), thinking the variable machines alone must meet the full requirement.

This may lead them to select Choice C (\(\mathrm{240x + 125y \geq 4,200}\)) if they also mix up the coefficients.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly set up \(\mathrm{300 + 125x + 240y \geq 4,200}\) but misunderstand the inequality direction, thinking 'at least' means the production should be limited rather than meeting a minimum.

This may lead them to select Choice B (\(\mathrm{125x + 240y \leq 3,900}\)).

The Bottom Line:

This problem tests whether students can properly translate real-world constraints into mathematical inequalities while keeping track of all components in the system. The key insight is recognizing that the variable machines only need to produce what's left after accounting for the fixed machine's contribution.