A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7...

1. TRANSLATE the problem information

• Given information:
  • Area of rectangle = 155 square inches
  • Length = "4 inches less than 7 times the width"
  • Need to find: width

• TRANSLATE the length description: "4 less than 7 times the width" means \(\mathrm{7w - 4}\) (where \(\mathrm{w = width}\))

2. INFER the solution approach

• Since we know area and have expressions for both dimensions, we can use \(\mathrm{Area = width × length}\)
• This will give us one equation with one unknown (the width)

3. Set up the equation

• Area = width × length
• \(\mathrm{155 = w(7w - 4)}\)

4. SIMPLIFY to solve the quadratic

• Expand: \(\mathrm{155 = 7w^2 - 4w}\)
• Rearrange to standard form: \(\mathrm{7w^2 - 4w - 155 = 0}\)
• Factor the quadratic. We need two numbers that multiply to \(\mathrm{(7)(-155) = -1085}\) and add to -4
• Those numbers are 31 and -35: \(\mathrm{(w - 5)(7w + 31) = 0}\)
• Apply zero product property: \(\mathrm{w = 5}\) or \(\mathrm{w = -31/7}\)

5. APPLY CONSTRAINTS to select final answer

• Since width must be positive in real-world context: \(\mathrm{w = 5}\) inches
• Verify: If \(\mathrm{w = 5}\), then \(\mathrm{length = 7(5) - 4 = 31}\), and \(\mathrm{area = 5 × 31 = 155}\) ✓

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "4 less than 7 times the width" as \(\mathrm{4 - 7w}\) instead of \(\mathrm{7w - 4}\)

Students often struggle with the phrase "4 less than [something]" and incorrectly translate it as "4 minus [something]." This leads to setting up the equation as \(\mathrm{w(4 - 7w) = 155}\), which expands to \(\mathrm{4w - 7w^2 = 155}\), or \(\mathrm{-7w^2 + 4w - 155 = 0}\). This quadratic has no real solutions, causing confusion and leading students to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Setting up the correct equation \(\mathrm{w(7w - 4) = 155}\) but making factoring errors

Students may correctly expand to get \(\mathrm{7w^2 - 4w - 155 = 0}\) but struggle with factoring this quadratic. Without proper factoring technique, they might attempt to use the quadratic formula incorrectly or give up entirely, leading to guessing among the answer choices.

The Bottom Line:

The key challenge is accurately translating the verbal description of length into algebra, followed by confident quadratic factoring skills. The constraint application (rejecting negative width) is usually straightforward once students reach that point.