A rectangle has a perimeter of 28 units. The length of the rectangle is 2 less than 3 times the...

1. TRANSLATE the problem information

• Given information:
  • Rectangle has perimeter of 28 units
  • Length = 3 × width - 2
  • Need to find: sum of length and width
• Set up variables: Let \(\mathrm{w}\) = width, \(\mathrm{l}\) = length

2. TRANSLATE each condition into equations

• From perimeter: \(\mathrm{2w + 2l = 28}\)
• Simplify by dividing by 2: \(\mathrm{w + l = 14}\)
• From length constraint: \(\mathrm{l = 3w - 2}\)

3. INFER the solution strategy

• We have two equations with two unknowns - this suggests using substitution
• Since we already have \(\mathrm{l}\) expressed in terms of \(\mathrm{w}\), substitute this into the perimeter equation

4. SIMPLIFY through substitution

• Substitute \(\mathrm{l = 3w - 2}\) into \(\mathrm{w + l = 14}\):
  \(\mathrm{w + (3w - 2) = 14}\)
• Combine like terms: \(\mathrm{4w - 2 = 14}\)
• Add 2 to both sides: \(\mathrm{4w = 16}\)
• Divide by 4: \(\mathrm{w = 4}\)

5. Find the length and calculate the sum

• \(\mathrm{l = 3(4) - 2 = 12 - 2 = 10}\)
• Sum = \(\mathrm{w + l = 4 + 10 = 14}\)

Answer: B (14)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret "2 less than 3 times the width" and write \(\mathrm{l = 3w + 2}\) instead of \(\mathrm{l = 3w - 2}\).

This mistake changes the constraint equation, leading to different algebra:

\(\mathrm{w + (3w + 2) = 14}\)
\(\mathrm{4w + 2 = 14}\)
\(\mathrm{w = 3}\), \(\mathrm{l = 11}\), \(\mathrm{sum = 14}\)

Interestingly, this still gives the correct final answer due to the mathematical structure, but represents flawed reasoning that could fail in similar problems.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors while solving \(\mathrm{4w - 2 = 14}\), such as getting \(\mathrm{w = 3}\) instead of \(\mathrm{w = 4}\).

With \(\mathrm{w = 3}\), they calculate \(\mathrm{l = 3(3) - 2 = 7}\), giving \(\mathrm{sum = 10}\). This may lead them to select Choice A (10).

The Bottom Line:

This problem tests whether students can systematically translate word constraints into algebra and then execute multi-step equation solving without computational errors. The key insight is recognizing that "2 less than" means subtraction, not addition.