Problem: The population density of Cedar County is 230 people per square mile. The county has a population of 85,100...

1. TRANSLATE the problem information

• Given information:
  • Population density of Cedar County = 230 people per square mile
  • Total population = 85,100 people
  • Need to find: Area in square miles

2. INFER the mathematical approach

• This is a population density problem, so we need the relationship: \(\mathrm{Population\ density = \frac{Total\ population}{Area}}\)
• Since we know density and total population, we can solve for area by rearranging this formula

3. Set up the equation

• \(\mathrm{230\ people/square\ mile = \frac{85,100\ people}{Area}}\)

4. SIMPLIFY by solving for area

• Multiply both sides by Area: \(\mathrm{230 \times Area = 85,100}\)
• Divide both sides by 230: \(\mathrm{Area = \frac{85,100}{230}}\)
• \(\mathrm{Area = 370\ square\ miles}\) (use calculator)

Answer: 370

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the relationship and try to multiply density by population instead of dividing population by density.

They might calculate: \(\mathrm{230 \times 85,100 = 19,573,000}\), leading them to think the area is much larger than it actually is. This leads to confusion since such a large number doesn't make sense for a county area, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors in the division.

They might miscalculate \(\mathrm{\frac{85,100}{230}}\), getting answers like 37 or 3,700 instead of 370. This leads to selecting an incorrect answer if similar values appear in multiple choice options.

The Bottom Line:

This problem tests whether students truly understand what population density means as a ratio and can manipulate that relationship algebraically. Many students memorize the density formula but struggle to rearrange it when area (rather than density) is the unknown.