Question: A rectangle has a perimeter of 170 centimeters. Its length measures 40 centimeters. What is the width of the...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 170 cm
  • Length = 40 cm
  • Width = ? (unknown)

• What this tells us: We have a rectangle where we know the perimeter and one dimension, but need to find the other dimension.

2. INFER the approach

• Since we know the perimeter and need to find a missing dimension, we should use the rectangle perimeter formula
• The perimeter formula \(\mathrm{P = 2(L + W)}\) will let us set up an equation with width as the unknown
• We can substitute our known values and solve algebraically

3. TRANSLATE the formula and substitute

• Rectangle perimeter formula: \(\mathrm{P = 2(L + W)}\)
• Substitute known values: \(\mathrm{170 = 2(40 + W)}\)

4. SIMPLIFY to solve for width

• Distribute the 2: \(\mathrm{170 = 80 + 2W}\)
• Subtract 80 from both sides: \(\mathrm{170 - 80 = 2W}\) → \(\mathrm{90 = 2W}\)
• Divide both sides by 2: \(\mathrm{W = 45}\)

5. Verify the answer

• Check: \(\mathrm{P = 2(40 + 45) = 2(85) = 170}\) ✓

Answer: 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about perimeter vs. area: Students may mix up the perimeter formula \(\mathrm{P = 2(L + W)}\) with the area formula \(\mathrm{A = L \times W}\), leading them to set up an incorrect equation like \(\mathrm{170 = 40 \times W}\), which gives \(\mathrm{W = 4.25}\).

This leads to confusion since 4.25 doesn't make sense as a reasonable width compared to the length of 40, causing them to guess or abandon the systematic approach.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{170 = 2(40 + W)}\) but make arithmetic errors during algebraic manipulation. For example, they might incorrectly distribute to get \(\mathrm{170 = 40 + 2W}\) (forgetting to multiply 40 by 2), leading to \(\mathrm{130 = 2W}\) and \(\mathrm{W = 65}\).

This creates an unreasonable rectangle where width > length, but students may not catch this error and submit the incorrect answer.

The Bottom Line:

This problem tests whether students can distinguish between perimeter and area concepts, and whether they can accurately perform multi-step algebraic solving. The key insight is recognizing that perimeter involves adding all sides, not multiplying dimensions.