Question:An engineering team uses the formula below to calculate power efficiency in electrical systems, where k represents the efficiency coefficient...

1. TRANSLATE the problem information

• Given equation: \(5k(n - 4) = w\)
• Goal: Express \(k\) in terms of \(w\) and \(n\) (isolate \(k\))

2. INFER the solution strategy

• Since \(k\) is multiplied by \(5(n - 4)\), we need to "undo" this multiplication
• The opposite of multiplication is division
• Divide both sides by \(5(n - 4)\) to isolate \(k\)

3. SIMPLIFY by applying division property of equality

• Divide both sides: \(\frac{5k(n - 4)}{5(n - 4)} = \frac{w}{5(n - 4)}\)
• Left side simplifies: \(k = \frac{w}{5(n - 4)}\)
• This matches answer choice (B)

4. Verify the result

• Substitute back: \(5 \times \frac{w}{5(n - 4)} \times (n - 4) = w\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students attempt to separate the division incorrectly, thinking they can distribute the division operation. They might reason: "w divided by 5(n - 4) is the same as w/5 minus (n - 4)."

This leads them to select Choice A (\(k = \frac{w}{5} - (n - 4)\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to isolate \(k\) but choose the wrong algebraic operation. Instead of dividing to "undo" the multiplication, they might think they need to multiply both sides or flip the entire equation structure.

This may lead them to select Choice C (\(k = \frac{5(n - 4)}{w}\)) or Choice D (\(k = 5w(n - 4)\)).

The Bottom Line:

The key challenge is correctly applying the division property of equality while maintaining proper fraction notation. Students must recognize that when dividing by a compound expression like \(5(n - 4)\), the entire expression stays in the denominator as one unit.