The perimeter P of a rectangle with length L and width W is given by P = 2L + 2W....

1. INFER the solution strategy

• Given: \(\mathrm{P = 2L + 2W}\)
• Goal: Express W in terms of P and L (get W by itself on one side)
• Strategy: Use algebraic manipulation to isolate W

2. SIMPLIFY by removing the 2L term

• Start with: \(\mathrm{P = 2L + 2W}\)
• Subtract 2L from both sides: \(\mathrm{P - 2L = 2W}\)
• This eliminates 2L from the right side

3. SIMPLIFY by removing the coefficient of W

• Current equation: \(\mathrm{P - 2L = 2W}\)
• Divide both sides by 2: \(\mathrm{W = \frac{P - 2L}{2}}\)
• This gives W isolated on the left side

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly subtract 2L from both sides to get \(\mathrm{P - 2L = 2W}\), but then make distribution errors when dividing by 2. They might divide only P by 2, writing \(\mathrm{W = \frac{P}{2} - 2L}\), forgetting that the entire expression \(\mathrm{(P - 2L)}\) must be divided by 2.

This leads them to select Choice B (\(\mathrm{W = \frac{P}{2} - 2L}\)).

Second Most Common Error:

Poor INFER reasoning about the isolation process: Students recognize they need to eliminate terms from the right side but subtract L instead of 2L, getting \(\mathrm{P - L = 2W}\), then \(\mathrm{W = \frac{P - L}{2}}\). However, they forget the division step entirely and think \(\mathrm{W = P - L}\).

This causes them to select Choice A (\(\mathrm{W = P - L}\)).

The Bottom Line:

This problem tests careful algebraic manipulation where students must apply operations to entire expressions, not just individual terms. The key insight is that whatever you do to one side of the equation must be done to the complete other side.