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All courses Geometry · G-MG.2 53 of 57
Apply density concepts based on area and volume in modeling situations.

Find density by dividing a quantity by area

Problem
What is the population density for \(120000~\text{people}\) over an area of \(30~\text{square}~\text{miles}\)?
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Hint

Use the density formula \(\text{density} = \frac{\text{population}}{\text{area}}\).

Population density tells how many people there are for each unit of area, so divide people by square miles or square kilometers.

Solution walkthrough

01

Define population density

\[\text{density}=\text{population}/\text{area}\]

Population density measures the number of people assigned to each unit of area.

02

Substitute the stated quantities

\[\text{density}=120000~\text{people}/30~\text{mi}^2\]

The numerator is the given population and the denominator is the given 30-square-mile area.

03

Divide and preserve rate units

\[120000/30=4000~->~4000~\text{people}/\text{mi}^2\]

Dividing distributes the population evenly across one square mile.

04

Interpret the result

\[4000~\text{people}~\text{per}~\text{square}~\text{mile}\]

The area has an average of 4000 people for every square mile.

+

Another way

  1. Cancel a factor of 10 first: 12000/3=4000 people per square mile.

!

Common mistake

Do not divide area by population. Density is population per area, so people belong in the numerator and square miles in the denominator.

Solution walkthrough video