Circle A has a radius of 3n and circle B has a radius of 129n, where n is a positive...

1. TRANSLATE the problem information

• Given information:
  • Circle A has radius = 3n
  • Circle B has radius = 129n
  • n is a positive constant
  • Need to find: Area of B ÷ Area of A

2. INFER the approach needed

• To compare areas, we need to find each circle's area first
• Then divide B's area by A's area to get the ratio
• We'll use the circle area formula: \(\mathrm{A = πr^2}\)

3. SIMPLIFY to find Circle A's area

• \(\mathrm{A_A = π(3n)^2}\)
• \(\mathrm{A_A = π(9n^2) = 9πn^2}\)

4. SIMPLIFY to find Circle B's area

• \(\mathrm{A_B = π(129n)^2}\)
• First compute: \(\mathrm{129^2 = 16,641}\) (use calculator)
• \(\mathrm{A_B = π(16,641n^2) = 16,641πn^2}\)

5. SIMPLIFY to find the ratio

• Ratio = \(\mathrm{A_B / A_A = (16,641πn^2) / (9πn^2)}\)
• The \(\mathrm{πn^2}\) terms cancel: \(\mathrm{= 16,641 / 9}\)
• Compute: \(\mathrm{16,641 ÷ 9 = 1,849}\) (use calculator)

Answer: D. 1,849

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make calculation errors when squaring 129, often getting confused by the large number and miscalculating \(\mathrm{129^2}\) as something other than 16,641.

For example, they might approximate \(\mathrm{129 ≈ 130}\) and compute \(\mathrm{130^2 = 16,900}\), leading to a ratio of \(\mathrm{16,900/9 ≈ 1,878}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse what the question is asking for and compute the ratio of radii instead of areas, getting \(\mathrm{129n ÷ 3n = 129 ÷ 3 = 43}\).

This may lead them to select Choice A (43).

The Bottom Line:

This problem tests whether students can systematically work through area comparisons involving large numbers while maintaining accuracy in both setup and calculation.