The perimeter of a rectangle is 28 inches. The length of the rectangle is 4 inches more than its width....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Perimeter = 28~inches}\)
  • Length is 4 inches more than width
  • Need to find: width

• What this tells us:
  • If \(\mathrm{width = w}\), then \(\mathrm{length = w + 4}\)
  • We can use the perimeter formula to create an equation

2. TRANSLATE the perimeter relationship

• \(\mathrm{Perimeter~of~rectangle = 2(length) + 2(width)}\)
• Substituting our expressions: \(\mathrm{28 = 2(w + 4) + 2w}\)

3. SIMPLIFY the equation

• Distribute: \(\mathrm{28 = 2w + 8 + 2w}\)
• Combine like terms: \(\mathrm{28 = 4w + 8}\)
• Subtract 8: \(\mathrm{20 = 4w}\)
• Divide by 4: \(\mathrm{w = 5}\)

Answer: A (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students reverse the length-width relationship, thinking "width is 4 inches more than length" instead of "length is 4 inches more than width."

They might set up: \(\mathrm{length = w}\), \(\mathrm{width = w + 4}\), leading to the equation \(\mathrm{2w + 2(w + 4) = 28}\), which gives the same algebraic work but assigns the wrong value to the wrong dimension. Since they're asked for width, they'd incorrectly think \(\mathrm{width = 9}\) (which would actually be the length in the correct setup).

This may lead them to select Choice D (9).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly translate but make arithmetic errors when distributing or combining terms.

For example, they might incorrectly simplify \(\mathrm{2(w + 4) + 2w}\) as \(\mathrm{2w + 4 + 2w = 4w + 4}\) instead of \(\mathrm{2w + 8 + 2w = 4w + 8}\). This leads to \(\mathrm{4w + 4 = 28}\), so \(\mathrm{4w = 24}\), giving \(\mathrm{w = 6}\).

This may lead them to select Choice B (6).

The Bottom Line:

Success requires careful attention to which dimension is described as larger, then systematic algebraic manipulation. The key insight is recognizing that word problems require precise translation before any calculation can begin.