During a production shift, a factory uses two assembly lines to manufacture products. Line A produces x items per hour...

1. TRANSLATE the problem information

• Given information:
  • Line A: \(\mathrm{x}\) items per hour, operates \(\mathrm{h}\) hours
  • Line B: \(\mathrm{2x}\) items per hour, operates \(\mathrm{k}\) hours
  • Total items produced: 480
  • Line A operates twice as long as Line B: \(\mathrm{h = 2k}\)
• What we need to find: Total items produced by Line A

2. TRANSLATE into mathematical equations

• Production equation: \(\mathrm{xh + 2xk = 480}\)
• Time relationship: \(\mathrm{h = 2k}\)

3. INFER the solution strategy

• We have two equations with three unknowns (\(\mathrm{x}\), \(\mathrm{h}\), \(\mathrm{k}\))
• The key insight: We don't need to find individual values of \(\mathrm{x}\), \(\mathrm{h}\), or \(\mathrm{k}\)
• We can use substitution to find what Line A produced directly

4. SIMPLIFY by substitution

• Substitute \(\mathrm{h = 2k}\) into the production equation:

\(\mathrm{x(2k) + 2xk = 480}\)

• Combine like terms:

\(\mathrm{2xk + 2xk = 4xk = 480}\)

• Solve for the product \(\mathrm{xk}\):

\(\mathrm{xk = 120}\)

5. INFER Line A's total production

• Line A produces \(\mathrm{xh}\) items total
• Since \(\mathrm{h = 2k}\): \(\mathrm{xh = x(2k) = 2xk}\)
• Therefore: Line A's production = \(\mathrm{2xk = 2(120) = 240}\)

Answer: C (240)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to set up the correct equations from the word problem, especially the relationship \(\mathrm{h = 2k}\). They might write \(\mathrm{k = 2h}\) instead, reversing the relationship.

With the wrong time relationship, they would substitute incorrectly and get a different value for \(\mathrm{xk}\), leading them to calculate Line A's production incorrectly. This systematic error in translation cascades through the entire solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when combining terms. They might incorrectly simplify \(\mathrm{2xk + 2xk}\) as \(\mathrm{2xk}\) instead of \(\mathrm{4xk}\), or make arithmetic errors when solving \(\mathrm{4xk = 480}\).

These calculation errors lead to wrong values for \(\mathrm{xk}\), causing them to select Choice A (160) or Choice B (192) instead of the correct answer.

The Bottom Line:

This problem requires careful attention to translating relationships correctly (especially "twice as many hours") and systematic algebraic manipulation. Students who rush through the setup or make careless algebraic errors will likely select an incorrect answer choice.