A rectangular garden has a perimeter of 90 feet. The length of the garden is 5 feet longer than its...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 90 feet
  • Length = width + 5 feet
  • Need to find: Area
• What this tells us: We have a relationship between length and width, plus a constraint from the perimeter.

2. INFER the approach

• We need both length and width to find area, but we only have relationships between them
• Strategy: Use the perimeter formula with substitution to find actual dimensions first

3. TRANSLATE relationships into equations

• Let \(\mathrm{w}\) = width in feet
• Then length = \(\mathrm{w + 5}\) (from "5 feet longer than width")
• Perimeter formula: \(\mathrm{P = 2(length + width) = 2(l + w)}\)

4. SIMPLIFY by substituting and solving

• Substitute into perimeter equation: \(\mathrm{90 = 2((w + 5) + w)}\)
• Distribute: \(\mathrm{90 = 2(2w + 5)}\)
  \(\mathrm{90 = 4w + 10}\)
• Solve: \(\mathrm{4w = 80}\), so \(\mathrm{w = 20}\) feet
• Find length: \(\mathrm{l = w + 5 = 20 + 5 = 25}\) feet

5. Calculate the final answer

• Area = length × width = \(\mathrm{25 \times 20 = 500}\) square feet

Answer: (D) 500

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "length is 5 feet longer than width" as meaning the length IS 5 feet, rather than understanding it's a relationship. They might set length = 5 and try to work from there.

This conceptual misunderstanding leads them to incorrect dimensions and subsequently wrong area calculations. They may end up selecting Choice (A) 25 if they confuse dimensions or Choice (B) 45 if they use half the perimeter incorrectly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving \(\mathrm{4w + 10 = 90}\). For example, they might subtract 10 incorrectly or divide by 4 wrong, leading to \(\mathrm{w = 15}\) instead of \(\mathrm{w = 20}\).

With \(\mathrm{w = 15}\), they get length = 20 and area = 300, which isn't among the choices, causing confusion and potentially leading to guessing or selecting Choice (C) 400 as the "closest" answer.

The Bottom Line:

This problem requires careful reading to establish the correct relationship between dimensions, then systematic algebraic work to find actual values. Students who rush through the translation step or make computational errors in the algebra will struggle to reach the correct answer.