20/p = 20/q - 20/r - 20/s The given equation relates the positive variables p, q, r, and s. Which...

1. TRANSLATE the problem information

• Given equation: \(\frac{20}{\mathrm{p}} = \frac{20}{\mathrm{q}} - \frac{20}{\mathrm{r}} - \frac{20}{\mathrm{s}}\)
• Need to find: An expression equivalent to q
• All variables are positive

2. INFER the most efficient approach

• Rather than trying to solve for q directly with all those 20s, let's first simplify by dividing everything by 20
• This will give us a cleaner equation to work with

3. SIMPLIFY by dividing both sides by 20

• \(\frac{20}{\mathrm{p}} ÷ 20 = (\frac{20}{\mathrm{q}} - \frac{20}{\mathrm{r}} - \frac{20}{\mathrm{s}}) ÷ 20\)
• \(\frac{1}{\mathrm{p}} = \frac{1}{\mathrm{q}} - \frac{1}{\mathrm{r}} - \frac{1}{\mathrm{s}}\)

4. INFER the next strategic move

• To isolate q, we need to first isolate 1/q
• Add 1/r and 1/s to both sides: \(\frac{1}{\mathrm{p}} + \frac{1}{\mathrm{r}} + \frac{1}{\mathrm{s}} = \frac{1}{\mathrm{q}}\)

5. SIMPLIFY the right side by finding a common denominator

• \(\frac{1}{\mathrm{p}} + \frac{1}{\mathrm{r}} + \frac{1}{\mathrm{s}}\) needs denominator prs
• \(\frac{1}{\mathrm{p}} = \frac{\mathrm{rs}}{\mathrm{prs}}\), \(\frac{1}{\mathrm{r}} = \frac{\mathrm{ps}}{\mathrm{prs}}\), \(\frac{1}{\mathrm{s}} = \frac{\mathrm{pr}}{\mathrm{prs}}\)
• So: \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs} + \mathrm{ps} + \mathrm{pr}}{\mathrm{prs}}\)

6. INFER the final step

• Since \(\frac{1}{\mathrm{q}} = \frac{\mathrm{rs} + \mathrm{ps} + \mathrm{pr}}{\mathrm{prs}}\), we take the reciprocal of both sides
• \(\mathrm{q} = \frac{\mathrm{prs}}{\mathrm{rs} + \mathrm{ps} + \mathrm{pr}} = \frac{\mathrm{prs}}{\mathrm{pr} + \mathrm{ps} + \mathrm{rs}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students often try to solve for q directly without first simplifying the equation by dividing by 20. They get overwhelmed by the multiple 20s and don't recognize the pattern.

This leads to getting stuck in messy algebra and eventually guessing, or attempting to factor out 20 incorrectly and selecting Choice D \(\frac{\mathrm{prs}}{20\mathrm{p}+20\mathrm{r}+20\mathrm{s}}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly get to \(\frac{1}{\mathrm{q}} = \frac{1}{\mathrm{p}} + \frac{1}{\mathrm{r}} + \frac{1}{\mathrm{s}}\) but make errors when finding the common denominator. They might forget one of the terms in the numerator (like writing \(\mathrm{rs} + \mathrm{ps}\) instead of \(\mathrm{rs} + \mathrm{ps} + \mathrm{pr}\)).

This leads to an incorrect final expression and confusion about which answer choice to select.

The Bottom Line:

This problem tests your ability to see through the complexity of multiple terms and recognize that strategic simplification early on makes the entire solution much more manageable. The key insight is working with reciprocals rather than trying to manipulate the original fractions directly.