The length of a rectangular garden is 24 feet more than its width. A landscape designer also notes that the...

1. TRANSLATE the problem information

• Given information:
  • "Length is 24 feet more than width" → \(\mathrm{L = W + 24}\)
  • "Length is 3 times width" → \(\mathrm{L = 3W}\)
  • Need to find: Area of rectangle

2. INFER the solving strategy

• Key insight: Both expressions equal the length L
• Strategy: Set the two expressions equal to create one equation with one variable
• This eliminates L and lets us solve for W directly

3. SIMPLIFY to solve for width

Set the expressions equal: \(\mathrm{3W = W + 24}\)

Subtract W from both sides: \(\mathrm{2W = 24}\)

Divide by 2: \(\mathrm{W = 12}\) feet

4. INFER the length using either original equation

Using \(\mathrm{L = 3W}\):

\(\mathrm{L = 3(12) = 36}\) feet

5. Calculate the area

\(\mathrm{Area = L \times W = 36 \times 12 = 432}\) square feet

Answer: C) 432

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly interpret "24 feet more than width" as \(\mathrm{W = L + 24}\) instead of \(\mathrm{L = W + 24}\), reversing the relationship.

If they write \(\mathrm{W = L + 24}\) and \(\mathrm{L = 3W}\), then substituting gives:

\(\mathrm{W = 3W + 24}\), leading to \(\mathrm{-2W = 24}\), so \(\mathrm{W = -12}\)

Since negative width doesn't make sense, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students may try to solve each equation individually rather than recognizing they form a system. They might attempt to find L and W separately without using the constraint that both expressions equal the same length.

This causes them to get stuck early in the problem and resort to guessing among the answer choices.

The Bottom Line:

This problem requires carefully translating English phrases into correct mathematical relationships, then recognizing that having two expressions for the same quantity creates a solvable system of equations.