The equation 21c/7d = sqrt(3s + 48) relates the distinct positive real numbers s, c, and d. Which equation correctly...
GMAT Advanced Math : (Adv_Math) Questions
The equation \(\frac{21\mathrm{c}}{7\mathrm{d}} = \sqrt{3\mathrm{s} + 48}\) relates the distinct positive real numbers s, c, and d. Which equation correctly expresses s in terms of c and d?
\(\mathrm{s = \frac{c^2}{d^2} - 16}\)
\(\mathrm{s = \frac{3c^2}{d^2} - 48}\)
\(\mathrm{s = \frac{9c^2}{d^2} - 48}\)
\(\mathrm{s = \frac{3c^2}{d^2} - 16}\)
\(\mathrm{s = \frac{3c}{d} - 16}\)
1. SIMPLIFY the given equation
- Given: \(\frac{21c}{7d} = \sqrt{3s + 48}\)
- First step: Reduce the fraction on the left side
- \(\frac{21c}{7d} = \frac{3c}{d}\) (since \(21÷7 = 3\))
2. INFER the strategy to eliminate the square root
- We now have: \(\frac{3c}{d} = \sqrt{3s + 48}\)
- To solve for s, we need to eliminate the square root
- Strategy: Square both sides of the equation
3. SIMPLIFY by squaring both sides
- Square the left side: \((\frac{3c}{d})^2 = \frac{9c^2}{d^2}\)
- Square the right side: \((\sqrt{3s + 48})^2 = 3s + 48\)
- Result: \(\frac{9c^2}{d^2} = 3s + 48\)
4. SIMPLIFY to isolate s
- Subtract 48 from both sides: \(\frac{9c^2}{d^2} - 48 = 3s\)
- Divide everything by 3: \(s = \frac{\frac{9c^2}{d^2} - 48}{3}\)
- Distribute the division: \(s = \frac{9c^2}{3d^2} - \frac{48}{3}\)
- Final simplification: \(s = \frac{3c^2}{d^2} - 16\)
Answer: D. \(s = \frac{3c^2}{d^2} - 16\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make arithmetic errors when reducing fractions or performing the final division by 3.
For example, they might incorrectly calculate \(21÷7\) or make errors when dividing \(\frac{9c^2}{d^2}\) by 3, getting \(\frac{9c^2}{3d^2}\) wrong. Some students might also forget to divide the constant term 48 by 3, leading them to select Choice C (\(s = \frac{9c^2}{d^2} - 48\)).
Second Most Common Error:
Poor INFER reasoning about squaring strategy: Students might attempt to solve the equation without properly squaring both sides, or they might square incorrectly.
They might try to bring terms over without eliminating the square root first, leading to confusion and incorrect algebraic manipulation. This causes them to get stuck and guess among the answer choices.
The Bottom Line:
This problem tests systematic algebraic manipulation skills. Success requires careful attention to each step of simplification, particularly fraction operations and the proper technique for eliminating square roots through squaring.
\(\mathrm{s = \frac{c^2}{d^2} - 16}\)
\(\mathrm{s = \frac{3c^2}{d^2} - 48}\)
\(\mathrm{s = \frac{9c^2}{d^2} - 48}\)
\(\mathrm{s = \frac{3c^2}{d^2} - 16}\)
\(\mathrm{s = \frac{3c}{d} - 16}\)