In a circuit, three resistors with positive resistances r, q, and s (in ohms) are connected in parallel, producing an...

1. TRANSLATE the problem information

• Given: \(\frac{1}{\mathrm{p}} = \frac{1}{\mathrm{r}} + \frac{1}{\mathrm{q}} + \frac{1}{\mathrm{s}}\) (parallel resistance formula)
• Find: \(\mathrm{q}\) in terms of \(\mathrm{p}\), \(\mathrm{r}\), and \(\mathrm{s}\)
• What this means: Rearrange the equation to get \(\mathrm{q}\) by itself on one side

2. INFER the solution strategy

• To solve for \(\mathrm{q}\), we first need to isolate the term containing \(\mathrm{q}\) (which is \(\frac{1}{\mathrm{q}}\))
• Strategy: Move all other terms to the left side, then invert both sides

3. SIMPLIFY by isolating \(\frac{1}{\mathrm{q}}\)

• Start with: \(\frac{1}{\mathrm{p}} = \frac{1}{\mathrm{r}} + \frac{1}{\mathrm{q}} + \frac{1}{\mathrm{s}}\)
• Subtract \(\frac{1}{\mathrm{r}}\) and \(\frac{1}{\mathrm{s}}\) from both sides:
  \(\frac{1}{\mathrm{q}} = \frac{1}{\mathrm{p}} - \frac{1}{\mathrm{r}} - \frac{1}{\mathrm{s}}\)

4. SIMPLIFY by combining fractions

• Need common denominator for the right side: \(\mathrm{prs}\)
• Convert each fraction:
  • \(\frac{1}{\mathrm{p}} = \frac{\mathrm{rs}}{\mathrm{prs}}\)
  • \(\frac{1}{\mathrm{r}} = \frac{\mathrm{ps}}{\mathrm{prs}}\)
  • \(\frac{1}{\mathrm{s}} = \frac{\mathrm{pr}}{\mathrm{prs}}\)
• Substitute: \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs}}{\mathrm{prs}} - \frac{\mathrm{ps}}{\mathrm{prs}} - \frac{\mathrm{pr}}{\mathrm{prs}}\)
• Combine: \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs - ps - pr}}{\mathrm{prs}}\)

5. SIMPLIFY by inverting to solve for \(\mathrm{q}\)

• If \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs - ps - pr}}{\mathrm{prs}}\), then \(\mathrm{q} = \frac{\mathrm{prs}}{\mathrm{rs - ps - pr}}\)

Answer: C. \(\frac{\mathrm{prs}}{\mathrm{rs - ps - pr}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students try to solve directly for \(\mathrm{q}\) instead of first isolating \(\frac{1}{\mathrm{q}}\).

They might attempt to multiply through by \(\mathrm{pqrs}\) or try other complex manipulations, getting lost in algebraic complexity. This leads to confusion and often results in guessing among the answer choices or selecting a plausible-looking but incorrect option.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly isolate \(\frac{1}{\mathrm{q}}\) but make sign errors when combining the fractions.

The most frequent mistake is writing \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs + ps + pr}}{\mathrm{prs}}\) instead of \(\frac{\mathrm{rs - ps - pr}}{\mathrm{prs}}\), forgetting that subtracting fractions changes signs. This leads them to select Choice B (\(\frac{\mathrm{prs}}{\mathrm{pr + ps + rs}}\)) instead of the correct answer.

The Bottom Line:

This problem tests whether students can work systematically with reciprocals and fraction operations. The key insight is recognizing that solving for \(\mathrm{q}\) requires first solving for \(\frac{1}{\mathrm{q}}\), then inverting - a strategy that isn't immediately obvious but makes the algebra much cleaner.