Question:A rectangular garden has a perimeter P given by P = 2l + 2w, where l is the length and...

1. TRANSLATE the problem information

• Given information:
  • Perimeter formula: \(\mathrm{P = 2l + 2w}\)
  • Perimeter = 42 meters
  • Length = 16 meters
  • We need to find \(\mathrm{3w}\) (not just \(\mathrm{w}\)!)

2. TRANSLATE by substituting known values

• Replace P with 42 and l with 16 in the formula:
  \(\mathrm{42 = 2(16) + 2w}\)

3. SIMPLIFY through algebraic steps

• Calculate \(\mathrm{2(16) = 32}\):
  \(\mathrm{42 = 32 + 2w}\)
• Subtract 32 from both sides:
  \(\mathrm{42 - 32 = 2w}\)
  \(\mathrm{10 = 2w}\)
• Divide both sides by 2:
  \(\mathrm{w = 5}\)

4. TRANSLATE to answer the actual question

• The question asks for \(\mathrm{3w}\), not \(\mathrm{w}\):
  \(\mathrm{3w = 3(5) = 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students solve correctly for \(\mathrm{w = 5}\) but forget that the question asks for \(\mathrm{3w}\), not \(\mathrm{w}\).

They complete all the algebra correctly but stop at \(\mathrm{w = 5}\), thinking they're done. This leads them to enter 5 instead of 15 as their final answer.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors during the algebraic manipulation steps.

Common mistakes include calculating \(\mathrm{2(16) = 26}\) instead of 32, or making subtraction errors like \(\mathrm{42 - 32 = 12}\). These calculation errors propagate through to give incorrect values for \(\mathrm{w}\) and subsequently for \(\mathrm{3w}\).

The Bottom Line:

This problem tests careful reading (what exactly is being asked?) combined with systematic algebraic manipulation. The perimeter formula itself is straightforward, but students must pay attention to the final question and execute clean arithmetic throughout the solution process.