One gallon of stain will cover 170 square feet of a surface. A yard has a total fence area of...

1. TRANSLATE the problem information

• Given information:
  • 1 gallon of stain covers 170 square feet
  • Fence area = w square feet
  • Need to stain the fence twice
• Find: Equation for S gallons needed

2. INFER the total area calculation

• Since we're staining the fence twice, we need to cover the fence area two times
• Total area to be stained = 2w square feet
• We need to find how many gallons are required for this total area

3. INFER the rate relationship

• We know: 1 gallon covers 170 square feet
• So: Gallons needed = Total area ÷ Coverage per gallon
• Therefore: \(\mathrm{S = 2w \div 170 = \frac{2w}{170}}\)

4. SIMPLIFY the expression

• \(\mathrm{\frac{2w}{170}}\) can be simplified by finding the GCD of 2 and 170
• Since \(\mathrm{170 = 2 \times 85}\), we can divide both numerator and denominator by 2
• \(\mathrm{S = \frac{2w}{170} = \frac{w}{85}}\)

Answer: \(\mathrm{S = \frac{w}{85}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "stain the fence twice" and think it means they need twice the normal concentration of stain per square foot, rather than covering the fence area twice.

This conceptual confusion leads them to set up incorrect equations or multiply by extra factors unnecessarily, potentially leading to confusion and guessing among the given choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{S = \frac{2w}{170}}\) but then make algebraic errors when simplifying, such as incorrectly canceling terms or making arithmetic mistakes.

This may cause them to arrive at incorrect simplified forms that don't match any of the given choices, leading to random selection.

The Bottom Line:

This problem tests whether students can correctly interpret multiple applications (doing something "twice") as covering twice the area, then properly set up and simplify the resulting rate equation.