A rectangle has a perimeter of 52 meters. The length of the rectangle is 6 meters longer than its width....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Perimeter = 52\ meters}\)
  • \(\mathrm{Length = width + 6\ meters}\)
• What we need to find: Area of the rectangle

2. TRANSLATE the relationships into equations

• Let \(\mathrm{W = width}\) and \(\mathrm{L = length}\)
• From perimeter: \(\mathrm{P = 2(L + W) = 52}\), so \(\mathrm{L + W = 26}\)
• From the length relationship: \(\mathrm{L = W + 6}\)

3. INFER the solution strategy

• We have two equations with two unknowns - this calls for substitution
• Substitute the length expression into the perimeter equation

4. SIMPLIFY to find the width

• Substitute \(\mathrm{L = W + 6}\) into \(\mathrm{L + W = 26}\):
  \(\mathrm{(W + 6) + W = 26}\)
  \(\mathrm{2W + 6 = 26}\)
  \(\mathrm{2W = 20}\)
  \(\mathrm{W = 10\ meters}\)

5. INFER the length and calculate area

• Since \(\mathrm{L = W + 6}\): \(\mathrm{L = 10 + 6 = 16\ meters}\)
• Area = \(\mathrm{L \times W = 16 \times 10 = 160\ square\ meters}\)

Answer: D (160)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when solving the linear equation

Many students correctly set up \(\mathrm{(W + 6) + W = 26}\) but then make mistakes like:

• \(\mathrm{2W + 6 = 26}\) becomes \(\mathrm{2W = 26 + 6 = 32}\) (adding instead of subtracting)
• Or getting confused with the arithmetic: \(\mathrm{26 - 6 = 22}\), so \(\mathrm{W = 11}\)

This leads to \(\mathrm{L = 17}\), and \(\mathrm{Area = 17 \times 11 = 187}\), which doesn't match any choice, causing confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "6 meters longer than width"

Some students write \(\mathrm{L = 6 - W}\) instead of \(\mathrm{L = W + 6}\), thinking "longer" means subtraction somehow. This leads to impossible negative dimensions or completely wrong setup, causing them to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can properly translate word relationships into algebra and then execute basic equation solving without computational errors. The setup is straightforward, but small mistakes in translation or arithmetic derail the entire solution.