Skills Beta

Growth Rates, Doubling Time and the Rule of 70

A population growing by a fixed percent each year doubles in a fixed time.

Practice 6: Mathematical Routines

Part 1 · Hook

Why this matters

A city growing "only" 3.5% a year doubles in 20 years: twice the schools, water and roads within one generation. Growth at a steady percent speeds up as the base gets bigger. The rule of 70 lets you turn any yearly growth rate into a doubling time in your head, and the exam asks for it every year.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is the percent change from 50 to 100?

100%
50%
200%
2%

Show the answer

(100 - 50) / 50 × 100 = 100%.

Correct: 100%:
50%:
200%:
2%:

2. What does "per capita" mean?

Per person
Per year
Per city
Per area

Show the answer

Per capita is per person.

Correct: Per person:
Per year:
Per city:
Per area:

Part 4 · See it

See it first

A curve from invented data: a population of 1 million in 2000 grows 2 percent a year. By the rule of 70 the doubling time is 70 divided by 2 = 35 years, so it reaches about 2 million in 2035 and about 4 million in 2070. The curve gets steeper as the population grows. — Growing 2% a year, a population doubles about every 35 years. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

A population grows by a fixed percent each yeareach year’s increase is larger than the last
Divide 70 by the percent growth rateyou get the doubling time in years
Add the doubling time to the start yearyou get the year the population doubles

Part 6 · Key ideas

Key ideas

A growth rate is the percent change per year: 2% a year means 2 more for every 100 each year.
Rule of 70: doubling time ≈ 70 ÷ growth rate (in %, not as a decimal).
Doubling time depends on the rate, not on the starting size.
Each doubling multiplies by 2: after n doublings, multiply by 2 n times (2, 4, 8, 16, ...).

Part 7 · Misconception

A common mistake

The wrong idea: A question asking "in what year" wants the doubling time.

What actually happens: The doubling time is a number of years. "In what year" asks you to add it to the starting year: 70 ÷ 2.5 = 28 years, so 2024 + 28 = 2052.

Part 8 · Check yourself

Check yourself

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

1. A population grows 2% a year. Write the setup for its doubling time by the rule of 70, then give the answer.

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

Show the answer

$\frac{70}{2} = 35$ years.

Setup: $\frac{70}{2}$ ; answer 35 yr

2. A town of 50,000 people in 2024 grows 2.5% a year. In what year will it reach about 100,000 people?

Type a number. Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

Doubling time: $\frac{70}{2.5} = 28$ years. Year: $2024 + 28 = 2052$ .

Answer: 2052

3. A species has 400 individuals and a doubling time of 10 years. If the growth continues, how many will there be after 30 years?

Type a number. Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

30 years is $\frac{30}{10} = 3$ doublings: $400 \times 2 \times 2 \times 2 = 3{,}200$ .

Answer: 3200

Data

Growth rates of four cities

Population in 2025 and yearly growth rate of four cities (invented data for practice)
City	Population in 2025	Growth rate (% per year)
North Port	400,000	3.5
Ridgeview	1.2 million	1.4
Elm Falls	90,000	7.0
Stonebridge	2.0 million	0.5

4. What is the doubling time of North Port?

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

Show the answer

$\frac{70}{3.5} = 20$ years.

Answer: 20 yr

5. In what year will Ridgeview reach about 2.4 million people?

Type a number. Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

2.4 million is double 1.2 million. $\frac{70}{1.4} = 50$ years, so $2025 + 50 = 2075$ .

Answer: 2075

6. Which city will double its population soonest?

Elm Falls
North Port
Ridgeview
Stonebridge

Show the answer

The highest growth rate gives the shortest doubling time: Elm Falls $\frac{70}{7.0} = 10$ years.

Correct: Elm Falls: Right: 10 years.
North Port: North Port: 20 years.
Ridgeview: Ridgeview: 50 years.
Stonebridge: Stonebridge: 140 years, even though it is the largest.

7. If Elm Falls keeps its growth rate, about how many people will it have in 2045?

Type a number. Scale words (2.5 million) and powers of ten (2.5 × 10^6) are read.

Show the answer

Doubling time 10 years; 2025 to 2045 is 20 years = 2 doublings: $90{,}000 \times 2 \times 2 = 360{,}000$ .

Answer: 360,000

Part 9 · Summary

Summary

A population growing by a fixed percent each year doubles in a fixed time. By the rule of 70, doubling time ≈ 70 ÷ growth rate in percent. Add the doubling time to the start year for "in what year"; multiply by 2 for each doubling to project the size. Doubling time depends only on the rate: twice the rate, half the time.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic