The area of a rectangle is 240 square inches. The length of the rectangle is 4 inches greater than twice...

1. TRANSLATE the problem information

• Given information:
  • Area = 240 square inches
  • Length = 4 inches greater than twice the width

• TRANSLATE the length relationship: If width = \(\mathrm{w}\), then length = \(\mathrm{2w + 4}\)

2. INFER the approach

• We have area and expressions for both dimensions, so we can use the area formula
• Set up the equation: Area = length × width

3. Set up and expand the equation

• \(\mathrm{240 = (2w + 4) \times w}\)
• \(\mathrm{240 = 2w^2 + 4w}\)

4. SIMPLIFY to standard quadratic form

• \(\mathrm{2w^2 + 4w - 240 = 0}\)
• Divide by 2: \(\mathrm{w^2 + 2w - 120 = 0}\)

5. SIMPLIFY by solving the quadratic

• Factor: Look for two numbers that multiply to -120 and add to 2
• Those numbers are 12 and -10: \(\mathrm{(w + 12)(w - 10) = 0}\)
• Solutions: \(\mathrm{w = -12}\) or \(\mathrm{w = 10}\)

6. APPLY CONSTRAINTS to select final answer

• Since width must be positive in real-world context: \(\mathrm{w = 10}\)

Answer: B. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle with "4 inches greater than twice the width" and might write length = \(\mathrm{2w - 4}\) or length = \(\mathrm{4w + 2}\), misunderstanding the phrase structure.

If they write length = \(\mathrm{2w - 4}\), their equation becomes \(\mathrm{240 = (2w - 4) \times w = 2w^2 - 4w}\), leading to \(\mathrm{2w^2 - 4w - 240 = 0}\), or \(\mathrm{w^2 - 2w - 120 = 0}\). This factors to \(\mathrm{(w - 12)(w + 10) = 0}\), giving \(\mathrm{w = 12}\), which would lead them to select Choice C (12).

Second Most Common Error:

Poor SIMPLIFY execution: Students may set up the correct equation but make algebraic errors when expanding or solving the quadratic, such as sign errors or factoring mistakes.

This leads to confusion and incorrect solutions, causing them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests whether students can accurately translate verbal relationships into algebraic expressions and then work systematically through multi-step algebra. The key challenge is the initial translation - once that's correct, the algebra follows standard procedures.