12/n - 2/t = -2/w The given equation relates the variables n, t, and w, where n gt 0, t...

1. TRANSLATE the problem information

• Given equation: \(\frac{12}{n} - \frac{2}{t} = -\frac{2}{w}\)
• Need to find: Expression equivalent to n
• Constraints: \(n \gt 0, t \gt 0, \text{ and } w \gt t\)

2. INFER the solution strategy

• Since we need n by itself, we should first isolate the term containing n (which is \(\frac{12}{n}\))
• This means moving the \(\frac{2}{t}\) term to the right side

3. SIMPLIFY by isolating the term with n

• Add \(\frac{2}{t}\) to both sides:

\(\frac{12}{n} = -\frac{2}{w} + \frac{2}{t}\)

4. SIMPLIFY by combining fractions on the right side

• Find common denominator tw:

\(-\frac{2}{w} = -\frac{2t}{tw}\)

\(\frac{2}{t} = \frac{2w}{tw}\)

• Combine: \(\frac{12}{n} = \frac{-2t + 2w}{tw} = \frac{2(w-t)}{tw}\)

5. INFER that taking reciprocals will help solve for n

• Since we have \(\frac{12}{n} = \frac{2(w-t)}{tw}\), taking reciprocals gives us \(\frac{n}{12}\)

6. SIMPLIFY using reciprocals

• Take reciprocal of both sides: \(\frac{n}{12} = \frac{tw}{2(w-t)}\)
• Multiply both sides by 12: \(n = \frac{12tw}{2(w-t)} = \frac{6tw}{w-t}\)

Answer: D. \(\frac{6tw}{w-t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make sign errors when moving terms or combining fractions.

When adding \(\frac{2}{t}\) to both sides, they might write \(\frac{12}{n} = -\frac{2}{w} - \frac{2}{t}\) instead of \(\frac{12}{n} = -\frac{2}{w} + \frac{2}{t}\). This sign error propagates through the solution, leading to incorrect expressions with negative terms that don't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about reciprocals: Students attempt to solve for n without recognizing they need to use reciprocals strategically.

Some students try to multiply both sides by n too early, creating more complex expressions with n in multiple places. They might get stuck with equations like \(12 = n\left[-\frac{2}{w} + \frac{2}{t}\right]\) and struggle to isolate n effectively. This may lead them to select Choice A (\(12tw\)) by incorrectly thinking they can just multiply the denominators together.

The Bottom Line:

This problem requires careful fraction manipulation and strategic thinking about when to use reciprocals. Success depends on systematic algebraic steps rather than trying shortcuts.