QUESTION STEM:Let x and y be nonzero real numbers such that y/x = 3.Consider the expression x^2/m * x *...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
- Let x and y be nonzero real numbers such that \(\frac{\mathrm{y}}{\mathrm{x}} = 3\).
- Consider the expression \(\frac{\mathrm{x}^2}{\mathrm{m} \times \mathrm{x} \times \mathrm{y}}\).
- If \(\frac{\mathrm{x}^2}{\mathrm{m} \times \mathrm{x} \times \mathrm{y}} = \frac{1}{6}\), what is the value of \(\mathrm{m}\)?
Answer Format:
Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- x and y are nonzero real numbers
- \(\mathrm{y/x = 3}\)
- \(\mathrm{x^2/(mxy) = 1/6}\)
- We need to find the value of m
2. INFER a useful relationship
- Since \(\mathrm{y/x = 3}\), we can find the reciprocal: \(\mathrm{x/y = 1/3}\)
- This reciprocal relationship will be key to simplifying our expression
3. SIMPLIFY the complex expression
- Start with: \(\mathrm{x^2/(mxy)}\)
- Factor out one x from numerator and denominator: \(\mathrm{x^2/(mxy) = x/(my)}\)
- Rewrite as: \(\mathrm{x/(my) = (1/m) \times (x/y)}\)
4. INFER the substitution strategy
- We know \(\mathrm{x/y = 1/3}\), so substitute:
- \(\mathrm{(1/m) \times (x/y) = (1/m) \times (1/3) = 1/(3m)}\)
5. SIMPLIFY to solve for m
- Set up the equation: \(\mathrm{1/(3m) = 1/6}\)
- Cross multiply: \(\mathrm{6 = 3m}\)
- Divide both sides by 3: \(\mathrm{m = 2}\)
Answer: 2
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students see \(\mathrm{y/x = 3}\) but don't recognize they need the reciprocal relationship \(\mathrm{x/y = 1/3}\) to make progress. They might try to substitute \(\mathrm{y = 3x}\) directly into the complex fraction without first simplifying, leading to unnecessarily complicated algebra. This often causes them to get stuck and abandon systematic solution, resorting to guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{x/y = 1/3}\), but make algebraic errors when simplifying \(\mathrm{x^2/(mxy)}\). They might incorrectly cancel terms or lose track of the variable m during the manipulation steps. This leads to setting up wrong equations like \(\mathrm{1/m = 1/6}\), giving them the incorrect answer \(\mathrm{m = 6}\).
The Bottom Line:
This problem requires students to see the connection between a given ratio and its reciprocal, then use that insight to transform a complex rational expression into something manageable. The key breakthrough is recognizing that \(\mathrm{x/y = 1/3}\) is the tool that unlocks the solution.