The positive number a is 2,241% of the sum of the positive numbers b and c, and b is 83%...

1. TRANSLATE the problem information

• Given relationships:
  • a is 2,241% of the sum of b and c → \(\mathrm{a = 22.41(b + c)}\)
  • b is 83% of c → \(\mathrm{b = 0.83c}\)
• What we need: Express a as a percentage of b

2. INFER the solution strategy

• We have a in terms of (b + c), but we need a in terms of b only
• Since we know b in terms of c, we can solve for c in terms of b
• Then substitute to eliminate c from our first equation

3. SIMPLIFY to express c in terms of b

From \(\mathrm{b = 0.83c}\), we get:

\(\mathrm{c = \frac{b}{0.83}}\)

4. SIMPLIFY by substitution

Substitute \(\mathrm{c = \frac{b}{0.83}}\) into \(\mathrm{a = 22.41(b + c)}\):

\(\mathrm{a = 22.41(b + \frac{b}{0.83})}\)
\(\mathrm{a = 22.41b(1 + \frac{1}{0.83})}\)

Calculate \(\mathrm{1 + \frac{1}{0.83}}\):

\(\mathrm{\frac{1}{0.83} ≈ 1.2048}\) (use calculator)
\(\mathrm{1 + 1.2048 = 2.2048}\)

5. SIMPLIFY the final calculation

\(\mathrm{a = 22.41b × 2.2048}\)
\(\mathrm{a = 49.41b}\) (use calculator)

Since \(\mathrm{a = 49.41b}\), this means a is 4,941% of b.

Answer: D. 4,941%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often try to work with both variables b and c throughout the problem instead of recognizing they need to eliminate one variable through substitution. They might attempt to set up complex systems or get confused about which variable to eliminate.

This leads to confusion and abandoning systematic solution, causing them to guess among the answer choices.

Second Most Common Error Path:

Poor SIMPLIFY execution: Students correctly set up the equations but make arithmetic errors when calculating \(\mathrm{1 + \frac{1}{0.83}}\) or in the final multiplication step. Common mistakes include treating \(\mathrm{\frac{1}{0.83}}\) as 0.83 or making decimal conversion errors.

This may lead them to select Choice A (23.24%) or Choice B (49.41%) if they drop the percentage conversion or make calculation mistakes.

The Bottom Line:

This problem tests whether students can systematically work through multi-step percentage relationships using substitution. The key insight is recognizing that you need to eliminate one variable to express the final relationship clearly, combined with careful arithmetic execution.