Question:2/3k = 5m/(n+p)The given equation relates the positive numbers k, m, n, and p. Which equation correctly expresses m in...
GMAT Advanced Math : (Adv_Math) Questions
\(\frac{2}{3\mathrm{k}} = \frac{5\mathrm{m}}{\mathrm{n}+\mathrm{p}}\)
The given equation relates the positive numbers k, m, n, and p. Which equation correctly expresses m in terms of k, n, and p?
- \(\mathrm{m} = \frac{2(\mathrm{n}+\mathrm{p})}{15\mathrm{k}}\)
- \(\mathrm{m} = \frac{15\mathrm{k}(\mathrm{n}+\mathrm{p})}{2}\)
- \(\mathrm{m} = \frac{\mathrm{n}+\mathrm{p}}{15\mathrm{k}}\)
- \(\mathrm{m} = \frac{30\mathrm{k}}{\mathrm{n}+\mathrm{p}}\)
1. TRANSLATE the problem requirements
- Given: \(\frac{2}{3\mathrm{k}} = \frac{5\mathrm{m}}{\mathrm{n}+\mathrm{p}}\)
- Need: Express m in terms of k, n, and p (isolate m)
2. SIMPLIFY by cross-multiplying
Cross-multiplication eliminates the fractions by multiplying each side by the opposite denominator:
- Left side × (n+p): \(2(\mathrm{n}+\mathrm{p})\)
- Right side × (3k): \(5\mathrm{m}(3\mathrm{k}) = 15\mathrm{km}\)
- Result: \(2(\mathrm{n}+\mathrm{p}) = 15\mathrm{km}\)
3. SIMPLIFY by isolating m
Divide both sides by 15k to get m alone:
- \(\frac{2(\mathrm{n}+\mathrm{p})}{15\mathrm{k}} = \frac{15\mathrm{km}}{15\mathrm{k}}\)
- \(\frac{2(\mathrm{n}+\mathrm{p})}{15\mathrm{k}} = \mathrm{m}\)
Answer: A. \(\mathrm{m} = \frac{2(\mathrm{n}+\mathrm{p})}{15\mathrm{k}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make errors during cross-multiplication, often incorrectly distributing terms or forgetting to multiply all components.
For example, they might write \(2(\mathrm{n}+\mathrm{p}) = 5\mathrm{m}(3\mathrm{k})\) but then incorrectly simplify the right side as 15mk instead of 15km, or forget that 3k multiplies the entire numerator 5m. Some students also struggle with the final division step, incorrectly dividing only part of the numerator by 15k.
This leads to selecting incorrect expressions that don't match any of the given choices, causing confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misunderstand what "express m in terms of" means and attempt to solve for a different variable or rearrange incorrectly.
They might try to isolate k or confuse the target variable, leading them to manipulate the equation in ways that don't achieve the goal of isolating m.
This may lead them to select Choice D (\(\frac{30\mathrm{k}}{\mathrm{n}+\mathrm{p}}\)) by incorrectly rearranging terms.
The Bottom Line:
Success requires careful execution of cross-multiplication and systematic algebraic manipulation. The multiple variables can be overwhelming, but the core skill is standard equation-solving technique applied to rational expressions.