The positive number a is 24,705% of the difference between the positive numbers b and c, and b is 125%...

1. TRANSLATE the problem information

• Given relationships:
  • a is 24,705% of the difference between b and c
  • b is 125% of c
  • All numbers are positive
• Need to find: What percent of b is a?

2. INFER the solution strategy

• Since everything relates back to c, let's use c as our base variable
• Express b in terms of c, then find the difference, then express a in terms of c
• Finally, calculate the ratio a/b to get our percentage

3. TRANSLATE percentages to work with them mathematically

• \(\mathrm{b = 125\% \text{ of } c = 1.25c}\)
• \(\mathrm{24,705\% \text{ as a decimal } = 247.05}\)

4. SIMPLIFY to find the difference between b and c

• \(\mathrm{Difference = b - c = 1.25c - c = 0.25c}\)

5. SIMPLIFY to express a in terms of c

• \(\mathrm{a = 247.05 × (difference)}\)
  \(\mathrm{a = 247.05 × 0.25c}\)
  \(\mathrm{a = 61.7625c}\)

6. SIMPLIFY to find what percent of b is a

• We need the ratio: \(\mathrm{a/b = 61.7625c / 1.25c = 61.7625/1.25}\)
• Calculate: \(\mathrm{61.7625 ÷ 1.25 = 49.41}\) (use calculator)
• Convert to percentage: \(\mathrm{49.41 × 100\% = 4,941\%}\)

Answer: C (4,941%)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students struggle to convert the complex percentage language into clear mathematical relationships, especially getting confused about which quantity is the base for each percentage calculation.

Many students might set up incorrect equations like \(\mathrm{a = 24,705\% × b}\) instead of recognizing that a equals 24,705% of the difference between b and c. This fundamental misreading derails the entire solution and leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the relationships but make arithmetic errors in the final calculation, particularly in dividing 61.7625 by 1.25, or they forget to convert their decimal answer back to percentage form.

A student might correctly get to \(\mathrm{a/b = 49.41}\) but then report this as Choice A (49.41%) instead of recognizing they need to express this as 4,941%.

The Bottom Line:

This problem tests whether students can navigate multiple percentage relationships while keeping track of which quantity serves as the base for each calculation. The key insight is recognizing that complex percentage problems often become manageable when you choose one variable as your foundation and express everything else in terms of that variable.