A rectangle has length x and width y, where both dimensions are positive. The area of the rectangle is 18...

1. TRANSLATE the problem information

• Given information:
  • Rectangle has length x and width y (both positive)
  • Area = 18 square units
  • Perimeter = 3 times the length

• What this tells us:
  • We need two equations: one for area, one for perimeter

2. TRANSLATE the conditions into equations

• Area condition: \(\mathrm{xy = 18}\)
• Perimeter condition: \(\mathrm{2x + 2y = 3x}\)

3. INFER the solution strategy

• We have two equations with two unknowns - this suggests using substitution
• The perimeter equation looks easier to solve for one variable first

4. SIMPLIFY the perimeter equation

• Start with: \(\mathrm{2x + 2y = 3x}\)
• Subtract 2x from both sides: \(\mathrm{2y = x}\)
• Divide by 2: \(\mathrm{y = x/2}\)

5. INFER the next step and SIMPLIFY

• Substitute \(\mathrm{y = x/2}\) into the area equation:
• \(\mathrm{x(x/2) = 18}\)
• \(\mathrm{x²/2 = 18}\)
• \(\mathrm{x² = 36}\)
• \(\mathrm{x = 6}\) (taking the positive root since length must be positive)

6. APPLY CONSTRAINTS to verify

• Check our solution: If \(\mathrm{x = 6}\), then \(\mathrm{y = x/2 = 3}\)
• Area check: \(\mathrm{6 × 3 = 18}\) ✓
• Perimeter check: \(\mathrm{2(6 + 3) = 18}\), and \(\mathrm{3 × 6 = 18}\) ✓

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "perimeter is 3 times the length" as meaning the perimeter formula should be \(\mathrm{P = 3x}\), rather than setting the perimeter equal to 3x.

This leads them to write \(\mathrm{P = 3x}\) instead of \(\mathrm{2x + 2y = 3x}\), creating an incorrect equation. Without the proper relationship between x and y, they cannot solve the system and end up guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving, such as:

• Getting \(\mathrm{y = 2x}\) instead of \(\mathrm{y = x/2}\) from the perimeter equation
• Making sign errors or calculation mistakes when solving \(\mathrm{x² = 36}\)

These errors lead to incorrect values that don't match any of the given choices, causing confusion and potentially selecting Choice A (4) or Choice D (9) through incorrect calculations.

The Bottom Line:

This problem requires careful translation of the perimeter condition and systematic algebraic manipulation. The key insight is recognizing that "perimeter equals 3 times the length" means \(\mathrm{2x + 2y = 3x}\), not that the perimeter formula itself is 3x.