The total area of a coastal city is 92.1 square miles, of which 11.3 square miles is water. If the...

1. TRANSLATE the problem information

• Given information:
  • Total city area: 92.1 square miles
  • Water area within city: 11.3 square miles
  • Population: 621,000 people
  • Need: Population density in people per square mile of land area

• Key insight: The problem asks for density per square mile of land area, not total area

2. INFER what calculation approach to use

• Population density requires: Population ÷ Area
• Since we need land area density, we must find land area first
• Land area = Total area - Water area (since water isn't livable land)

3. SIMPLIFY to find the land area

• Land area = \(92.1 - 11.3 = 80.8\) square miles

4. SIMPLIFY to calculate population density

• Population density = \(621{,}000 \div 80.8 = 7{,}685.6...\) (use calculator)
• This rounds to approximately 7,686 people per square mile

5. APPLY CONSTRAINTS to select from answer choices

• Looking at choices: 6,740, 7,690, 55,000, 76,000
• 7,686 is closest to 7,690

Answer: B. 7,690

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students use total area instead of land area for the calculation.

They think: "Population density = \(621{,}000 \div 92.1 = 6{,}743\)" and select the closest answer. This reasoning seems logical because they're using the "total area" given in the problem, but it ignores that people don't live on water. The problem specifically asks for density per square mile of land area.

This may lead them to select Choice A (6,740).

Second Most Common Error Path:

Poor TRANSLATE skills: Students misunderstand what the problem is asking and use water area instead of land area.

They might calculate: "Population density = \(621{,}000 \div 11.3 = 54{,}956\)" thinking the problem wants density relative to water area. This shows confusion about what population density means in a practical context.

This may lead them to select Choice C (55,000).

The Bottom Line:

The key challenge is recognizing that "land area" must be calculated first by subtracting water area from total area. Students who rush through the problem often grab the most obvious number (total area) without carefully reading what type of area is actually needed for a meaningful population density calculation.