A landscaping company's weekly cost is given by 12h + 7m = C, where h represents hours of labor, m...

1. TRANSLATE the problem information

• Given equation: \(12\mathrm{h} + 7\mathrm{m} = \mathrm{C}\)
• Goal: Express \(\mathrm{m}\) in terms of \(\mathrm{h}\) and \(\mathrm{C}\) (isolate \(\mathrm{m}\) on one side)

2. SIMPLIFY through algebraic manipulation

• First, subtract \(12\mathrm{h}\) from both sides:

\(12\mathrm{h} + 7\mathrm{m} = \mathrm{C}\)

\(7\mathrm{m} = \mathrm{C} - 12\mathrm{h}\)

• Next, divide both sides by 7:

\(\mathrm{m} = \frac{\mathrm{C} - 12\mathrm{h}}{7}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly get to \(\mathrm{m} = \frac{\mathrm{C} - 12\mathrm{h}}{7}\) but then select Choice B because they don't recognize the importance of parentheses.

Without parentheses, \(\mathrm{m} = \mathrm{C} - \frac{12\mathrm{h}}{7}\) means \(\mathrm{m} = \mathrm{C} - \left(\frac{12\mathrm{h}}{7}\right)\), which is completely different from \(\mathrm{m} = \frac{\mathrm{C} - 12\mathrm{h}}{7}\). The order of operations makes division happen before subtraction when there are no parentheses.

This leads them to select Choice B (\(\mathrm{m} = \mathrm{C} - \frac{12\mathrm{h}}{7}\))

Second Most Common Error:

Conceptual confusion about inverse operations: Students get confused about which operation to perform first and subtract in the wrong order.

Instead of \(7\mathrm{m} = \mathrm{C} - 12\mathrm{h}\), they write \(7\mathrm{m} = 12\mathrm{h} - \mathrm{C}\), leading to \(\mathrm{m} = \frac{12\mathrm{h} - \mathrm{C}}{7}\).

This may lead them to select Choice C (\(\mathrm{m} = \frac{12\mathrm{h} - \mathrm{C}}{7}\))

The Bottom Line:

This problem tests careful algebraic manipulation and attention to order of operations. The key insight is recognizing that parentheses are essential to show that the entire expression \(\mathrm{C} - 12\mathrm{h}\) must be divided by 7, not just the \(12\mathrm{h}\) term.