On a certain map, 6 inches represents 15 miles of actual distance. What is the number of inches on the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{6\ inches\ on\ map = 15\ miles\ in\ reality}\)
• What we need to find: How many inches = 1 mile

2. INFER the approach

• This is asking for a unit rate: inches per mile
• We need to find how many inches represent exactly 1 mile
• Since 6 inches represent 15 miles, we divide both by 15 to find what represents 1 mile

3. Set up the calculation

• Inches per mile = \(\mathrm{6\ inches ÷ 15\ miles}\) = \(\mathrm{\frac{6}{15}}\)

4. SIMPLIFY the fraction

• Find the greatest common divisor of 6 and 15: \(\mathrm{GCD(6,15) = 3}\)
• \(\mathrm{\frac{6}{15} = \frac{6÷3}{15÷3} = \frac{2}{5}}\)

5. Convert to decimal (if needed)

• \(\mathrm{\frac{2}{5} = 0.4}\)

Answer: \(\mathrm{\frac{2}{5}}\) or \(\mathrm{0.4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students may set up the proportion backwards, thinking "if 6 inches is 15 miles, then 1 mile should be 15/6 inches." This reversed thinking comes from not clearly identifying what the question is asking for.

This may lead them to calculate \(\mathrm{\frac{15}{6} = 2.5}\) and select an incorrect answer.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly find \(\mathrm{\frac{6}{15}}\) but fail to reduce it to lowest terms, leaving their answer as \(\mathrm{\frac{6}{15}}\) instead of \(\mathrm{\frac{2}{5}}\).

While \(\mathrm{\frac{6}{15}}\) is mathematically correct, the problem specifically asks for the fraction in lowest terms, so this would be marked incorrect.

The Bottom Line:

Map scale problems require clear thinking about what represents what. The key insight is recognizing that you're finding a unit rate - how much of one unit corresponds to exactly one unit of another.