Rectangle P has a perimeter of 20 centimeters and a length of 6 centimeters. Square Q has a side length...

1. TRANSLATE the problem information

• Given information:
  • Rectangle P: perimeter = 20 cm, length = 6 cm
  • Square Q: side length = 5 cm
  • Need to find: total area of both shapes

2. INFER the approach

• We can't find rectangle P's area yet because we only know length, not width
• We need to use the perimeter to find the missing width first
• Then calculate each area separately and add them together

3. SIMPLIFY to find rectangle P's width

• Use perimeter formula: \(\mathrm{P = 2(length + width)}\)
• Substitute known values: \(\mathrm{20 = 2(6 + width)}\)
• Distribute: \(\mathrm{20 = 12 + 2(width)}\)
• Subtract 12: \(\mathrm{8 = 2(width)}\)
• Divide by 2: \(\mathrm{width = 4\,cm}\)

4. Calculate rectangle P's area

• Area = length × width = 6 × 4 = 24 square cm

5. Calculate square Q's area

• Area = side² = 5² = 25 square cm

6. SIMPLIFY to find total area

• Total area = 24 + 25 = 49 square cm

Answer: C (49)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to calculate rectangle area immediately without realizing they need to find the width first.

They might see "length = 6 cm" and think they can calculate area, but area requires both length AND width. Without recognizing this multi-step strategy, they get stuck and either guess or try to use the perimeter value incorrectly in area calculations. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the perimeter equation correctly but make algebraic errors.

For example: \(\mathrm{20 = 2(6 + width)}\) becomes \(\mathrm{20 = 12 + width}\) (forgetting to distribute the 2), leading to \(\mathrm{width = 8}\). This gives rectangle area = 6 × 8 = 48, and total area = 48 + 25 = 73, which isn't even an answer choice, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students can recognize that missing information (rectangle width) can be found using given information (perimeter), then systematically work through multiple calculations without losing track of the overall goal.