Squares — Perimeter Comparison to Area Factor: The perimeters of two squares are 12 cm and 24 cm. The area of the larger square is how many times the area of the smaller square?

Introduction / Context:
Square perimeter is proportional to side length, and area scales with the square of side length. Doubling the perimeter doubles the side; the area therefore quadruples. This item reinforces the linear–quadratic linkage between perimeter and area.

Given Data / Assumptions:

• P1 = 12 cm, P2 = 24 cm
• Perimeter P = 4s ⇒ s = P/4
• Area A = s^2

Concept / Approach:
Compute sides from perimeters, then compare areas. Because s2/s1 equals P2/P1, area ratio equals (P2/P1)^2. Here P2/P1 = 2, so area ratio = 4 : 1 (larger : smaller).

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 2 or 3 times ignores quadratic scaling.
• 5 or 6 times overestimates the effect.

Common Pitfalls:

• Mistaking a linear doubling for an area doubling.

Final Answer:
4 times.