One gallon of paint will cover 220 square feet of a surface. A room has a total wall area of...

1. TRANSLATE the problem information

• Given information:
  • 1 gallon of paint covers 220 square feet
  • Room has wall area of w square feet
  • Need to paint the walls twice

• What this tells us: We need to find total gallons P for covering more area than just w

2. INFER the total area to cover

• Since we're painting twice, we need to cover the wall area two times
• Total area = \(\mathrm{w + w = 2w}\) square feet

3. INFER the relationship for finding gallons needed

• We know: 1 gallon covers 220 square feet
• To find gallons needed: Divide total area by area per gallon
• Formula: \(\mathrm{P = \frac{total\:area}{area\:per\:gallon}}\)

4. SIMPLIFY the expression

• \(\mathrm{P = 2w ÷ 220}\)
• \(\mathrm{P = \frac{2w}{220}}\)
• Reduce by dividing both numerator and denominator by 2:
• \(\mathrm{P = \frac{w}{110}}\)

Answer: A. \(\mathrm{P = \frac{w}{110}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students miss that "paint twice" means covering 2w total area, not just w area.

They might think the equation is just \(\mathrm{P = \frac{w}{220}}\) (painting once), leading them to select Choice C (\(\mathrm{P = \frac{w}{220}}\)).

Second Most Common Error:

Poor INFER reasoning about the coverage relationship: Students might multiply instead of divide, thinking more area means you multiply by the coverage rate.

This backwards thinking leads to expressions like \(\mathrm{P = 220w}\) or \(\mathrm{P = 440w}\), causing them to select Choice D (\(\mathrm{P = 220w}\)) or Choice B (\(\mathrm{P = 440w}\)).

The Bottom Line:

This problem tests whether students can properly translate a real-world situation into mathematical relationships, specifically recognizing that finding "how many gallons" requires division, not multiplication.

5. Calculator Suggested

Not applicable - all operations involve algebraic simplification rather than numerical computation.