One gallon of paint will cover 220 square feet of a surface with one coat. If P gallons of paint...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{1}\) gallon covers \(\mathrm{220}\) square feet with one coat
  • \(\mathrm{P}\) gallons total are used
  • Walls get two complete coats
  • Find equation for total wall area \(\mathrm{w}\)

2. INFER the key relationship

• The critical insight: "two complete coats" means we must cover the wall area twice
• So total coverage needed = \(\mathrm{2w}\) square feet
• We need to connect this coverage need to the paint amount used

3. Set up the coverage equation

• Total coverage provided by \(\mathrm{P}\) gallons = \(\mathrm{P \times 220}\) square feet = \(\mathrm{220P}\) square feet
• Total coverage needed for two coats = \(\mathrm{2w}\) square feet
• These must be equal: \(\mathrm{220P = 2w}\)

4. SIMPLIFY to solve for \(\mathrm{w}\)

• From \(\mathrm{220P = 2w}\)
• Divide both sides by \(\mathrm{2}\): \(\mathrm{w = 220P ÷ 2 = 110P}\)

Answer: D) \(\mathrm{w = 110P}\) square feet

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "two complete coats" means mathematically.

Some students think \(\mathrm{P}\) gallons covers \(\mathrm{w}\) square feet directly, forgetting that two coats means covering the area twice. They might set up \(\mathrm{w = 220P}\), leading them to select Choice C (\(\mathrm{w = 220P}\) square feet).

Second Most Common Error:

Poor INFER reasoning: Students recognize two coats but apply it incorrectly to the paint amount rather than the area coverage.

They might think "two coats means double the paint efficiency" and calculate \(\mathrm{w = 220P \times 2 = 440P}\), leading them to select Choice B (\(\mathrm{w = 440P}\) square feet).

The Bottom Line:

This problem tests whether students can correctly model the relationship between paint quantity, coverage rate, and multiple applications. The key insight is distinguishing between what gets doubled (the area coverage needed) versus what stays constant (the coverage rate per gallon).