Factory X charges $8 per hour for machine operation and $12 per hour for skilled labor. Factory Y charges $10...

1. TRANSLATE the problem information

• Given information:
  • Factory X: \(\$8/\mathrm{hour}\) machine + \(\$12/\mathrm{hour}\) skilled labor = \(\$84\) total
  • Factory Y: \(\$10/\mathrm{hour}\) machine + \(\$16/\mathrm{hour}\) skilled labor = \(\$108\) total
  • Same production order at both factories
• Let \(\mathrm{m}\) = hours of machine operation, \(\mathrm{l}\) = hours of skilled labor

2. TRANSLATE into mathematical equations

• Factory X equation: \(8\mathrm{m} + 12\mathrm{l} = 84\)
• Factory Y equation: \(10\mathrm{m} + 16\mathrm{l} = 108\)

3. SIMPLIFY the equations to make solving easier

• Divide Factory X equation by 4: \(2\mathrm{m} + 3\mathrm{l} = 21\)
• Divide Factory Y equation by 2: \(5\mathrm{m} + 8\mathrm{l} = 54\)

4. INFER the best solution strategy

• Use elimination method to remove one variable
• Target eliminating \(\mathrm{l}\) since we want to find \(\mathrm{m}\)

5. SIMPLIFY using elimination method

• Multiply first equation by 8: \(16\mathrm{m} + 24\mathrm{l} = 168\)
• Multiply second equation by 3: \(15\mathrm{m} + 24\mathrm{l} = 162\)
• Subtract second from first: \(\mathrm{m} = 6\)

6. Verify the solution

• If \(\mathrm{m} = 6\), then \(2(6) + 3\mathrm{l} = 21\), so \(\mathrm{l} = 3\)
• Check: Factory Y gives \(10(6) + 16(3) = 108\) ✓

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the initial equations by mixing up the hourly rates between factories or incorrectly interpreting which costs go together.

For example, they might write: \(8\mathrm{m} + 10\mathrm{l} = 84\) and \(12\mathrm{m} + 16\mathrm{l} = 108\), confusing which rates belong to which factory. This completely changes the system and leads to wrong values that don't match any answer choice, causing them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the elimination process, particularly when multiplying equations by constants or when subtracting equations.

Common mistakes include: \(16\mathrm{m} + 24\mathrm{l}\) minus \(15\mathrm{m} + 24\mathrm{l} = 168 - 162\), but incorrectly calculating this as \(\mathrm{m} = 8\) instead of \(\mathrm{m} = 6\). This may lead them to select Choice C (8).

The Bottom Line:

This problem challenges students to correctly translate real-world constraints into mathematical language and then execute multi-step algebraic procedures without computational errors. Success requires both strong translation skills and careful arithmetic execution.