For positive numbers p, q, and r, the sum of p and 280% of q equals r. The number q...

1. TRANSLATE the problem information

• Given information:
  • "The sum of p and 280% of q equals r" → \(\mathrm{p + 2.8q = r}\)
  • "q is 25% of r" → \(\mathrm{q = 0.25r}\)
• What we need to find: What percent of q is p? This means \(\mathrm{(p/q) \times 100\%}\)

2. INFER the solution strategy

• We have two equations but three variables (p, q, r)
• Key insight: Express everything in terms of one variable
• Since we need p as a percent of q, let's express both p and r in terms of q

3. SIMPLIFY to find r in terms of q

From \(\mathrm{q = 0.25r}\):

• Divide both sides by 0.25: \(\mathrm{r = q/0.25 = 4q}\)

4. SIMPLIFY to find p in terms of q

Substitute \(\mathrm{r = 4q}\) into the first equation:

• \(\mathrm{p + 2.8q = r}\)
• \(\mathrm{p + 2.8q = 4q}\)
• \(\mathrm{p = 4q - 2.8q = 1.2q}\)

5. Calculate the final percentage

• We found \(\mathrm{p = 1.2q}\)
• So \(\mathrm{(p/q) \times 100\% = (1.2q/q) \times 100\% = 1.2 \times 100\% = 120\%}\)

Answer: C) 120%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse the direction of the percentage relationships. They might interpret "q is 25% of r" as "r is 25% of q" and write \(\mathrm{r = 0.25q}\) instead of \(\mathrm{q = 0.25r}\).

This reversal leads to \(\mathrm{r = q/0.25}\) becoming \(\mathrm{r = 0.25q}\), which when substituted gives \(\mathrm{p + 2.8q = 0.25q}\), resulting in \(\mathrm{p = -2.55q}\) (impossible since p must be positive). This causes confusion and typically leads to guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the equations but make algebraic errors. For example, when solving \(\mathrm{p + 2.8q = 4q}\), they might subtract incorrectly and get \(\mathrm{p = 6.8q}\) instead of \(\mathrm{p = 1.2q}\).

This leads to \(\mathrm{(p/q) \times 100\% = 680\%}\), which doesn't match any answer choice. Students then often pick the largest available percentage, which would be Choice D (280%).

The Bottom Line:

This problem requires careful attention to the direction of percentage relationships and systematic algebraic substitution. Students who rush through the translation step or make careless algebraic errors will struggle to reach the correct answer.