A rectangular garden has a perimeter of 32 feet. The length is 4 feet more than the width. What is...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 32 feet
  • Length is 4 feet more than width
  • Need to find: width
• TRANSLATE the relationship: If \(\mathrm{width = w}\), then \(\mathrm{length = w + 4}\)

2. INFER the approach needed

• We have perimeter and a relationship between length and width
• Strategy: Use the perimeter formula to create an equation with one variable (width)

3. INFER which formula to apply

• For rectangles: \(\mathrm{Perimeter = 2(length + width)}\)
• Substitute our expressions: \(\mathrm{P = 2((w + 4) + w) = 2(2w + 4)}\)

4. SIMPLIFY by setting up and solving the equation

• Set up: \(\mathrm{2(2w + 4) = 32}\)
• Distribute: \(\mathrm{4w + 8 = 32}\)
• Subtract 8: \(\mathrm{4w = 24}\)
• Divide by 4: \(\mathrm{w = 6}\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "length is 4 feet more than width" into the correct mathematical relationship. They might set up \(\mathrm{width = length + 4}\) instead of \(\mathrm{length = width + 4}\), reversing the relationship.

With this error, they would get: \(\mathrm{2(w + (w - 4)) = 32}\), leading to \(\mathrm{4w - 8 = 32}\), so \(\mathrm{4w = 40}\), and \(\mathrm{w = 10}\).

This may lead them to select Choice D (10).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly but make algebraic errors during the solving process. Common mistakes include forgetting to distribute the 2, or making arithmetic errors when combining like terms.

For example, they might incorrectly simplify \(\mathrm{2(2w + 4) = 32}\) as \(\mathrm{2w + 4 = 32}\), leading to \(\mathrm{2w = 28}\) and \(\mathrm{w = 14}\), which isn't among the choices.

This leads to confusion and guessing among the available options.

The Bottom Line:

This problem tests whether students can accurately translate verbal relationships into algebraic expressions and then systematically solve a linear equation. The key challenge is maintaining precision in both the translation and algebraic manipulation phases.