9m = 4 - 27k/3n The given equation relates the distinct positive real numbers k, m, and n. Which equation...
GMAT Advanced Math : (Adv_Math) Questions
\(9\mathrm{m} = 4 - \frac{27\mathrm{k}}{3\mathrm{n}}\)
The given equation relates the distinct positive real numbers k, m, and n. Which equation correctly expresses n in terms of k and m?
1. SIMPLIFY the given equation
- Start with: \(\mathrm{9m = 4 - \frac{27k}{3n}}\)
- Simplify the fraction: \(\mathrm{\frac{27k}{3n} = \frac{9k}{n}}\)
- Rewrite as: \(\mathrm{9m = 4 - \frac{9k}{n}}\)
2. INFER the isolation strategy
- Goal: Get n by itself on one side
- Strategy: Move terms systematically to isolate the fraction containing n, then solve
3. SIMPLIFY through algebraic manipulation
- Subtract 4 from both sides: \(\mathrm{9m - 4 = -\frac{9k}{n}}\)
- Multiply both sides by -1 to clean up the negative: \(\mathrm{4 - 9m = \frac{9k}{n}}\)
- Multiply both sides by n: \(\mathrm{n(4 - 9m) = 9k}\)
- Divide both sides by (4 - 9m): \(\mathrm{n = \frac{9k}{4 - 9m}}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when manipulating the equation, especially when dealing with the negative fraction or when multiplying by -1.
For example, when moving from \(\mathrm{9m - 4 = -\frac{9k}{n}}\) to the next step, students might incorrectly write \(\mathrm{9m - 4 = \frac{9k}{n}}\) (forgetting the negative sign) or make errors when multiplying both sides by -1. These algebraic mistakes lead to incorrect final expressions that don't match any of the given choices, causing confusion and guessing.
Second Most Common Error:
Inadequate SIMPLIFY reasoning: Students fail to simplify the fraction \(\mathrm{\frac{27k}{3n}}\) to \(\mathrm{\frac{9k}{n}}\) at the beginning, making the subsequent algebra much more complicated and error-prone.
Working with \(\mathrm{\frac{27k}{3n}}\) throughout the problem creates unnecessary complexity and increases the likelihood of computational errors. This may lead them to select Choice A (\(\mathrm{n = \frac{3k}{4-9m}}\)) if they partially simplify but make errors in the final steps.
The Bottom Line:
This problem tests systematic algebraic manipulation skills. Success requires careful attention to signs, methodical step-by-step work, and the discipline to simplify fractions early in the process to avoid unnecessary complexity.