If a population grows 2 percent a year, how long until it doubles? Because growth builds on growth, like interest on savings, the answer is shorter than 50 years. A simple rule, using the rate of natural increase, gives a good estimate.
Key idea
The rule of 70: doubling time ≈ 70 ÷ growth rate, with the rate written as a percent (2, not 0.02). At 2% a year, doubling takes about 70 ÷ 2 = 35 years.
Why it works
Each year the population is multiplied by (1 + r). Doubling means (1 + r)n = 2. Taking natural logarithms, n = ln 2 ÷ ln(1 + r). For small r, ln(1 + r) is very close to r, and ln 2 ≈ 0.693. So n ≈ 0.693 ÷ r = 69.3 ÷ (r as a percent). Rounding 69.3 up to 70 makes the arithmetic easy.
Worked examples
Worked example: Doubling time from a rate
A country's RNI is 2.5% a year. About how long will it take to double?
- Use the percent: 2.5.
- 70 ÷ 2.5 = 28.
Answer: About 28 years.
Worked example: From birth and death rates
CBR 31 per 1,000, CDR 7 per 1,000. Find the doubling time.
- 31 − 7 = 24 per 1,000.
- 24 ÷ 10 = 2.4%.
- 70 ÷ 2.4 ≈ 29.2.
Answer: About 29 years.
Worked example: Working backward
A city doubled in about 20 years. What was its yearly growth rate?
- Rule of 70: rate ≈ 70 ÷ doubling time.
- 70 ÷ 20 = 3.5.
Answer: About 3.5% a year.
Common mistake
Students divide 70 by the decimal (70 ÷ 0.02 = 3,500 years). Use the percent: 70 ÷ 2 = 35 years. Check the size: at 2% a year, simple addition would take 50 years to add 100%, and compounding makes it a bit faster, so the answer must be somewhat under 50. Remember too that the rule assumes the rate stays the same, which real populations rarely do.
| Growth rate | 1% | 2% | 3% | 3.5% |
|---|---|---|---|---|
| Doubling time (rule of 70) | 70 years | 35 years | about 23 years | 20 years |