Skills Beta

Rates and Per-Capita Values

A rate divides one quantity by another: liters per day, people per km², kilograms per person.

Practice 6: Mathematical Routines

Part 1 · Hook

Why this matters

Which city wastes more water: one that uses 200 million liters a day, or one that uses 20 million? You cannot tell until you know how many people live in each. Dividing a total by the number of people, the area or the time turns raw totals into rates you can compare fairly, and the exam asks for them constantly.

Part 2 · Prerequisite check

Quick check before you start

1. What is 6.0×1072.0×103\frac{6.0 \times 10^{7}}{2.0 \times 10^{3}}?

  1. 3.0×1043.0 \times 10^{4}
  2. 3.0×10103.0 \times 10^{10}
  3. 12×10412 \times 10^{4}
  4. 3.0×10213.0 \times 10^{21}
Show the answer

Divide the fronts (3.0) and subtract the exponents (7 - 3 = 4).

  • Correct: 3.0×1043.0 \times 10^{4}:
  • 3.0×10103.0 \times 10^{10}:
  • 12×10412 \times 10^{4}:
  • 3.0×10213.0 \times 10^{21}:

2. Which reservoir holds most of Earth’s freshwater that is not ice?

  1. Groundwater
  2. Rivers
  3. Clouds
  4. Lakes
Show the answer

Groundwater holds far more than lakes, rivers and the air.

  • Correct: Groundwater:
  • Rivers:
  • Clouds:
  • Lakes:

Part 3 · See it

See it first

Two bar charts from invented data. Total use: Town A uses 4.5 million liters a day and Town B 2 million liters a day, so Town A uses more in total. Use per person: Town A has 30,000 people, so 4,500,000 divided by 30,000 is 150 liters per person per day; Town B has 8,000 people, so 2,000,000 divided by 8,000 is 250 liters per person per day. Town B uses more per person.
The bigger total is not always the bigger amount per person. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

  1. Divide a total by the number of peopleyou get a per-capita value
  2. Divide by an area or a time insteadyou get a density or a rate per unit of time
  3. Keep the units in the setupthe answer’s unit tells you what you found: L per person, people per km²

Part 5 · Key ideas

Key ideas

  • A rate compares two quantities with different units: liters per day, people per km², kg per person.
  • A unit rate is a rate for one unit of the second quantity: 150 L per 1 person.
  • Per capita means per person: the total ÷ the population.
  • The word after "per" goes on the bottom: liters per km = L ÷ km.

Part 6 · Misconception

A common mistake

The wrong idea: The place with the biggest total uses resources the most heavily.

What actually happens: Big totals often just mean more people. Divide by the population (per capita) before you decide who uses more.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. A town of 30,000 people uses 4,500,000 liters of water a day. How many liters is that per person per day?

Type a number, then its unit (L/person/day). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

4,500,000L30,000people\frac{4{,}500{,}000\,\text{L}}{30{,}000\,\text{people}}=150= 150 L per person per day.

  • Answer: 150 L/person/day

2. A family’s car used 1,200 liters of gasoline in 15,000 km of driving. Write the setup for the fuel used per km, with units, then give the answer.

Type the setup as on the exam, with brackets and units, then the answer.

Show the answer

1,200L15,000km\frac{1{,}200\,\text{L}}{15{,}000\,\text{km}}=0.08= 0.08 L per km.

  • Setup: 1200L15000km\frac{1200\,\text{L}}{15000\,\text{km}}; answer 0.08 L/km

3. A county of 2,500 square kilometers has 400,000 people. How many people live there per square kilometer?

Type a number, then its unit (people/km^2). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

400,000people2,500km2\frac{400{,}000\,\text{people}}{2{,}500\,\text{km}^{2}}=160= 160 people per km².

  • Answer: 160 people/km^2

Data

Household trash in three towns

Population and household trash collected in one year in three towns (invented data for practice)
TownPopulationTrash collected per year (metric tons)
Alder12,0006,000
Birch50,00020,000
Cedar4,0002,600

4. How many metric tons of trash per person did Alder collect in the year?

Type a number, then its unit (metric tons/person). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

6,000t12,000people\frac{6{,}000\,\text{t}}{12{,}000\,\text{people}}=0.5= 0.5 metric tons per person.

  • Answer: 0.5 metric tons/person

5. Which town produced the most trash per person?

  1. Cedar
  2. Birch
  3. Alder
  4. Alder and Birch tied
Show the answer

Per person: Alder 0.5, Birch 20,00050,000\frac{20{,}000}{50{,}000}=0.4= 0.4, Cedar 2,6004,000=0.65\frac{2{,}600}{4{,}000} = 0.65 metric tons. Cedar is highest even though its total is lowest.

  • Correct: Cedar: Right: 0.65 metric tons per person.
  • Birch: Birch has the largest total but 0.4 per person.
  • Alder: Alder is 0.5 per person.
  • Alder and Birch tied: Alder (0.5) and Birch (0.4) are not equal.

6. How many kilograms of trash per person did Birch collect? (1 metric ton = 1,000 kg.)

Type a number, then its unit (kg/person). Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

20,000t50,000people\frac{20{,}000\,\text{t}}{50{,}000\,\text{people}}=0.4= 0.4 t per person; 0.4×1,000=4000.4 \times 1{,}000 = 400 kg per person.

  • Answer: 400 kg/person

7. A newspaper headline says "Birch is the most wasteful town." What does the per-person data show?

  1. Birch has the largest total but the smallest per-person amount.
  2. Birch is the most wasteful by both measures.
  3. Birch has the most trash per person.
  4. The towns are too different in size to compare.
Show the answer

Birch’s total (20,000 t) is largest because it has the most people; per person (0.4 t) it is the least wasteful of the three.

  • Correct: Birch has the largest total but the smallest per-person amount.: Right: totals reflect size; per person shows behavior.
  • Birch is the most wasteful by both measures.: Per person, Birch is the lowest.
  • Birch has the most trash per person.: Cedar has the most per person (0.65 t).
  • The towns are too different in size to compare.: Per-person values make a fair comparison across sizes.

Part 8 · Summary

Summary

A rate divides one quantity by another: liters per day, people per km², kilograms per person. A per-capita value divides a total by the number of people, so places of different sizes can be compared fairly. Write the setup with units: the word after "per" goes on the bottom, and the units of the answer tell you what you found.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

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