8/p + 3/q = 5/r The given equation relates the variables p, q, and r, where p gt 0, q...

1. TRANSLATE the problem information

• Given equation: \(\frac{8}{\mathrm{p}} + \frac{3}{\mathrm{q}} = \frac{5}{\mathrm{r}}\)
• Find: An expression equivalent to p
• What this tells us: We need to isolate p on one side of the equation

2. INFER the solution approach

• Since we want p by itself, we need to isolate the term containing p first
• The term \(\frac{8}{\mathrm{p}}\) is currently being added to \(\frac{3}{\mathrm{q}}\), so we'll subtract \(\frac{3}{\mathrm{q}}\) from both sides
• Then we'll need to manipulate the resulting equation to get p alone

3. SIMPLIFY by isolating the p-term

Subtract \(\frac{3}{\mathrm{q}}\) from both sides:

\(\frac{8}{\mathrm{p}} = \frac{5}{\mathrm{r}} - \frac{3}{\mathrm{q}}\)

4. SIMPLIFY the right side using common denominators

To subtract the fractions on the right, find a common denominator of qr:

\(\frac{8}{\mathrm{p}} = \frac{5\mathrm{q} - 3\mathrm{r}}{\mathrm{qr}}\)

5. SIMPLIFY using cross multiplication

Cross multiply to eliminate the fractions:

\(8\mathrm{qr} = \mathrm{p}(5\mathrm{q} - 3\mathrm{r})\)

6. SIMPLIFY to solve for p

Divide both sides by \((5\mathrm{q} - 3\mathrm{r})\):

\(\mathrm{p} = \frac{8\mathrm{qr}}{5\mathrm{q} - 3\mathrm{r}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making a sign error when finding the common denominator

Students correctly isolate \(\frac{8}{\mathrm{p}} = \frac{5}{\mathrm{r}} - \frac{3}{\mathrm{q}}\), but when finding the common denominator, they write:

\(\frac{8}{\mathrm{p}} = \frac{5\mathrm{q} + 3\mathrm{r}}{\mathrm{qr}}\) instead of \(\frac{5\mathrm{q} - 3\mathrm{r}}{\mathrm{qr}}\)

This leads them to get \(\mathrm{p} = \frac{8\mathrm{qr}}{5\mathrm{q} + 3\mathrm{r}}\), causing them to select Choice B \(\left(\frac{8\mathrm{qr}}{5\mathrm{q} + 3\mathrm{r}}\right)\)

Second Most Common Error:

Poor INFER reasoning: Attempting to cross multiply the original equation immediately

Students see the fractions and immediately try cross multiplication without isolating terms first. This creates a complex equation that's much harder to solve and often leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically work through multi-step algebraic manipulation while maintaining accuracy with signs and fractions. The key insight is recognizing that isolation must happen before cross multiplication can be effective.