h = k/(7g - 4) The given equation relates the distinct positive numbers h, k, and g. Which equation correctly...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{h = \frac{k}{7g - 4}}\)
The given equation relates the distinct positive numbers \(\mathrm{h}\), \(\mathrm{k}\), and \(\mathrm{g}\). Which equation correctly expresses \(\mathrm{7g - 4}\) in terms of \(\mathrm{h}\) and \(\mathrm{k}\)?
- \(\mathrm{7g - 4 = \frac{k}{h}}\)
- \(\mathrm{7g - 4 = hk}\)
- \(\mathrm{7g - 4 = k + h}\)
- \(\mathrm{7g - 4 = \frac{h}{k}}\)
1. TRANSLATE the problem information
- Given equation: \(\mathrm{h = \frac{k}{7g - 4}}\)
- Need to find: An expression for \(\mathrm{7g - 4}\) in terms of h and k
2. INFER the approach
- The expression \(\mathrm{(7g - 4)}\) is currently in the denominator of a fraction
- To isolate it, I need to "undo" the division by treating \(\mathrm{(7g - 4)}\) as a single unit
- Strategy: Use inverse operations to move terms around
3. SIMPLIFY by multiplying both sides by \(\mathrm{(7g - 4)}\)
- Starting equation: \(\mathrm{h = \frac{k}{7g - 4}}\)
- Multiply both sides by \(\mathrm{(7g - 4)}\): \(\mathrm{h(7g - 4) = k}\)
- This eliminates the fraction and puts \(\mathrm{(7g - 4)}\) on the left side
4. SIMPLIFY by dividing both sides by h
- From \(\mathrm{h(7g - 4) = k}\)
- Divide both sides by h: \(\mathrm{(7g - 4) = \frac{k}{h}}\)
- This isolates our target expression
Answer: A. \(\mathrm{7g - 4 = \frac{k}{h}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students attempt to solve for g individually instead of isolating the entire expression \(\mathrm{(7g - 4)}\) as requested. They might try to distribute or separate the terms 7g and -4, leading to unnecessary complexity and confusion about the final form needed.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor INFER reasoning: Students don't recognize that \(\mathrm{(7g - 4)}\) should be treated as a single unit. Instead, they might try to cross-multiply incorrectly or perform operations that don't logically lead to isolating the desired expression.
This may lead them to select Choice B (hk) by incorrectly cross-multiplying or Choice D (h/k) by confusing the relationship.
The Bottom Line:
This problem tests whether students can systematically use inverse operations to isolate a compound expression, rather than getting caught up in trying to solve for individual variables when that's not what's being asked.