Square A has side lengths that are 166 times the side lengths of square B. The area of square A...

1. TRANSLATE the problem information

• Given information:
  • Square A has side lengths that are 166 times the side lengths of square B
  • The area of square A is k times the area of square B
  • Need to find k

2. TRANSLATE to mathematical expressions

• Let \(\mathrm{x}\) = side length of square B
• Then side length of square A = \(\mathrm{166x}\)
• Area of square B = \(\mathrm{x^2}\)
• Area of square A = \(\mathrm{(166x)^2}\)

3. INFER the scaling relationship

• Since area of square A = k × area of square B, we can write:
• \(\mathrm{(166x)^2 = k \times x^2}\)
• This gives us an equation to solve for k

4. SIMPLIFY the left side

• \(\mathrm{(166x)^2 = 166^2 \times x^2 = 27,556x^2}\) (use calculator for \(\mathrm{166^2}\))
• So our equation becomes: \(\mathrm{27,556x^2 = k \times x^2}\)

5. SIMPLIFY to find k

• Divide both sides by \(\mathrm{x^2}\) (valid since \(\mathrm{x \gt 0}\)):
• \(\mathrm{k = 27,556}\)

Answer: 27,556

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think that if the side lengths scale by factor 166, then the areas also scale by factor 166.

They reason: "If sides are 166 times bigger, then area is 166 times bigger, so k = 166."

This fails to recognize that area scales by the square of the linear scaling factor. This may lead them to select an answer choice of 166 if available, or causes confusion when 166 isn't an option.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors when computing \(\mathrm{166^2}\).

They might calculate \(\mathrm{166^2}\) incorrectly (perhaps getting 27,556 wrong by a factor of 10, or making multiplication errors), leading to an incorrect value of k.

The Bottom Line:

This problem tests whether students understand that area scaling follows a quadratic relationship with linear scaling, not a linear relationship. The key insight is that doubling the sides quadruples the area, tripling the sides increases area by 9 times, and so forth.