The equation k/x = k/m - 2k/n relates the positive variables k, x, m, and n. Which of the following...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{k/x = k/m - 2k/n}\)
• Goal: Express x in terms of m and n only
• All variables are positive

2. INFER the solution strategy

• Since k appears in every term and \(\mathrm{k \gt 0}\), we can divide it out completely
• This will eliminate k and leave us with an equation involving only x, m, and n
• To solve for x when it appears in a denominator, we'll need to use reciprocals

3. SIMPLIFY by dividing both sides by k

Since k is positive, \(\mathrm{k ≠ 0}\), so we can safely divide:

\(\mathrm{k/x ÷ k = (k/m - 2k/n) ÷ k}\)

This gives us: \(\mathrm{1/x = 1/m - 2/n}\)

4. SIMPLIFY the right side by finding a common denominator

The common denominator for m and n is mn:

• \(\mathrm{1/m = n/(mn)}\)
• \(\mathrm{2/n = 2m/(mn)}\)

So: \(\mathrm{1/x = n/(mn) - 2m/(mn) = (n - 2m)/(mn)}\)

5. SIMPLIFY by taking reciprocals to solve for x

Since \(\mathrm{1/x = (n - 2m)/(mn)}\), we have:

\(\mathrm{x = mn/(n - 2m)}\)

Answer: C. \(\mathrm{mn/(n - 2m)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that k can be eliminated by division since it's positive.

Students might try to work with the equation as-is, getting tangled up trying to isolate x while k is still present. They may attempt to cross-multiply or use other complex manipulations instead of the elegant approach of dividing by k first. This leads to unnecessarily complicated algebra and often errors.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when combining fractions.

Students correctly divide by k to get \(\mathrm{1/x = 1/m - 2/n}\), but when finding the common denominator, they might write \(\mathrm{(n + 2m)/(mn)}\) instead of \(\mathrm{(n - 2m)/(mn)}\), forgetting that \(\mathrm{2/n}\) becomes \(\mathrm{2m/(mn)}\), so the subtraction gives \(\mathrm{n - 2m}\), not \(\mathrm{n + 2m}\).

This may lead them to select Choice B (\(\mathrm{mn/(n + 2m)}\)).

The Bottom Line:

This problem tests whether students can recognize when a variable can be eliminated to simplify the problem, combined with careful fraction manipulation. The key insight is that dividing by k transforms a complex-looking rational equation into a much simpler one.