A rectangular garden has a length that is 6 meters more than its width. If the perimeter of the garden...

1. TRANSLATE the problem information

• Given information:
  • Length is 6 meters more than width
  • Perimeter is 68 meters
  • Need to find area

• What this tells us in math notation:
  • If width = \(\mathrm{w}\), then length = \(\mathrm{w + 6}\)
  • \(\mathrm{P = 68}\) meters

2. TRANSLATE the perimeter condition into an equation

• Using the rectangle perimeter formula: \(\mathrm{P = 2(length + width)}\)
• Substitute our expressions: \(\mathrm{68 = 2((w + 6) + w)}\)

3. SIMPLIFY to solve for width

• \(\mathrm{68 = 2(2w + 6)}\)
• \(\mathrm{68 = 4w + 12}\)
• \(\mathrm{56 = 4w}\)
• \(\mathrm{w = 14}\) meters

4. INFER what we need for the area calculation

• We found width = 14 meters
• But area needs both length and width
• So we need to find length first: \(\mathrm{length = 14 + 6 = 20}\) meters

5. Calculate the final area

• \(\mathrm{Area = length \times width = 20 \times 14 = 280}\) square meters

Answer: B. 280

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "length that is 6 meters more than its width" and set up the wrong relationship, such as \(\mathrm{length = 6w}\) instead of \(\mathrm{length = w + 6}\).

This leads to an equation like \(\mathrm{68 = 2(6w + w) = 14w}\), giving \(\mathrm{w \approx 4.86}\). Then they might calculate area as \(\mathrm{6(4.86) \times 4.86 \approx 142}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly find \(\mathrm{w = 14}\) but forget they need to calculate the length before finding area. They might try to use \(\mathrm{area = 14 \times 6 = 84}\) (using width times the "6 more" instead of actual length).

Since 84 isn't an answer choice, this causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can correctly translate a verbal relationship into algebra and then systematically work through a multi-step solution, remembering to find all necessary dimensions before the final calculation.