Square P has an area of 16 square feet. The area of square Q is 9 times the area of...

1. TRANSLATE the problem information

• Given information:
  • Square P has area = 16 square feet
  • Square Q has area = 9 times the area of square P
  • Need to find: perimeter of square Q

• What this tells us: We need to work from area information to get perimeter

2. INFER the solution strategy

• To find perimeter, we need the side length of square Q
• To find the side length, we need the area of square Q first
• We can get area of Q from the given relationship with square P

3. Find the side length of square P

• Using \(\mathrm{A = s^2}\): \(\mathrm{16 = s^2}\)
• Taking the square root: \(\mathrm{s = \sqrt{16} = 4}\) feet

4. TRANSLATE to find area of square Q

• Area of Q \(\mathrm{= 9 \times}\) (area of P) \(\mathrm{= 9 \times 16 = 144}\) square feet

5. SIMPLIFY to find the side length of square Q

• Using \(\mathrm{A = s^2}\): \(\mathrm{144 = s^2}\)
• Taking the square root: \(\mathrm{s = \sqrt{144} = 12}\) feet

6. Calculate the perimeter

• Using \(\mathrm{P = 4s}\): \(\mathrm{P = 4 \times 12 = 48}\) feet

Answer: C (48)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to relate perimeters directly without recognizing they need to work through areas and side lengths first.

They might think "if area increases by 9 times, perimeter also increases by 9 times" and calculate \(\mathrm{4 \times 4 \times 9 = 144}\), leading them to select Choice D (144).

Second Most Common Error:

Conceptual confusion about area vs. side length scaling: Students correctly find that area of Q is 144, but then confuse this with the side length or perimeter.

They might select 144 as the perimeter without realizing they need to find the side length first, also leading to Choice D (144).

The Bottom Line:

This problem tests whether students understand the multi-step relationship between area and perimeter through side length, rather than trying to find direct scaling relationships.