A rectangular garden has a perimeter of 36 feet. The length of the garden is 6 feet more than twice...

1. TRANSLATE the problem information into mathematical equations

• Given information:
  • Perimeter of rectangular garden = 36 feet
  • Length = 6 feet more than twice the width
  • Variables: \(\mathrm{w = width, l = length}\)

• This gives us two equations:
  • From perimeter: \(\mathrm{2l + 2w = 36}\), which simplifies to \(\mathrm{l + w = 18}\)
  • From length relationship: \(\mathrm{l = 2w + 6}\)

2. INFER the solving strategy

• We have a system of two linear equations with two unknowns
• Since one equation already expresses l in terms of w, substitution is the most efficient method
• We'll substitute the expression for l into the perimeter equation

3. SIMPLIFY using substitution method

• Substitute \(\mathrm{l = 2w + 6}\) into \(\mathrm{l + w = 18}\):
  \(\mathrm{(2w + 6) + w = 18}\)

• Combine like terms:
  \(\mathrm{3w + 6 = 18}\)

• Solve for w:
  \(\mathrm{3w = 12}\)
  \(\mathrm{w = 4}\)

4. Verify the solution

• If \(\mathrm{w = 4}\), then \(\mathrm{l = 2(4) + 6 = 14}\)
• Check perimeter: \(\mathrm{2(14) + 2(4) = 28 + 8 = 36}\) ✓

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to correctly interpret "6 feet more than twice the width" and write \(\mathrm{l = 2w}\) instead of \(\mathrm{l = 2w + 6}\), missing the "+ 6" part.

When they solve the system with \(\mathrm{l = 2w}\) and \(\mathrm{l + w = 18}\), they get:
\(\mathrm{2w + w = 18}\)
\(\mathrm{3w = 18}\)
\(\mathrm{w = 6}\)

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

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make arithmetic errors when combining terms or solving the final equation.

For example, they might incorrectly get \(\mathrm{3w = 6}\) (instead of \(\mathrm{3w = 12}\)) or \(\mathrm{3w = 24}\), leading to \(\mathrm{w = 2}\) or \(\mathrm{w = 8}\) respectively.

This may lead them to select Choice A (2) or Choice D (8).

The Bottom Line:

This problem tests the crucial skill of translating complex English relationships into precise mathematical equations. Success requires careful attention to every word in the problem statement, especially phrases like "more than" that indicate addition operations.