Population/Resource Doubling Time (Rule of 70) | Online Free Tool | Elysia Tools
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Population/Resource Doubling Time (Rule of 70)
Doubling/halving time from a periodic growth rate: rule-of-70/72/69.3 estimate vs exact discrete and continuous times, with a milestone table.
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Key facts
Category
Health
Input types
number, select
Output type
json
Sample coverage
4
API ready
Yes
Overview
Estimate doubling or half-life time from a compound growth rate per period. Compare a Rule of 70, 72, or 69.3 shortcut with exact discrete compounding and continuous growth, then project ×2/×4/×8/×16 (or matching halves) from an optional starting value.
When to use
When you have a percent growth (or decline) rate per year, generation, or other period and need doubling or half-life time.
When you want to check how far the Rule of 70/72/69.3 sits from exact ln(2)/ln(1+r) and continuous times.
When you need a milestone table of 2×, 4×, 8×, and 16× (or ½, ¼, ⅛, 1/16) with projected amounts.
How it works
1Enter the compound rate per period as a percentage. A negative rate switches the result from doubling time to half-life.
2Choose the rule numerator: 70 (ecology/demography), 72 (finance mental math), or 69.3 (100·ln 2, continuous).
3Optionally set an initial value so milestones can show projected population, capital, or resource amounts.
4The tool reports the rule estimate, exact discrete time, continuous time, percent error, and a ×2–×16 (or ÷2–÷16) milestone table.
Use cases
Estimate how many periods a population, city, or bacterial culture takes to double at a stated growth rate.
Compare Rule of 72 interest doubling with exact discrete and continuous compounding.
Calculate half-life and later remaining fractions for a declining resource, stock, or biomass.
Examples
1. 7% yearly population growth
Demographer
Background
A city of 100 (thousands) is growing 7% per year and planners need doubling time plus later multiples.
Problem
Compare the Rule of 70 with exact compounding and see when the population would reach 4× and 8×.
How to use
Set growth rate per period to 7, keep numerator 70, and enter initial value 100.
ratePct: 7, ruleNumerator: 70, initialValue: 100
Outcome
Rule-of-70 estimate 10 years vs exact 10.24 (−2.39% error); continuous 9.90. Reaches 200 at 10.24, about 400 at 20.49, and about 800 at 30.73.
2. Resource shrinking 3% per year
Resource manager
Background
A stock of 10,000 units is declining 3% each year and managers need half-life and later remaining fractions.
Problem
Get the Rule of 70 half-life, the exact discrete time, and projected amounts at ÷2, ÷4, and ÷8.
How to use
Enter growth rate −3, numerator 70, and initial value 10000.
FAQ
What is the Rule of 70?
It estimates doubling time as 70 divided by the percent rate per period. You can also use 72 or 69.3.
What happens if I enter a negative rate?
The tool computes half-life instead of doubling time, with ÷2, ÷4, ÷8, and ÷16 milestones.
Why is time to 4× not exactly twice the doubling time?
Periodic compounding is discrete, so n-fold times are not strict integer multiples of the doubling (or half-life) time.
Which rule numerator should I choose?
Use 70 for ecology and demography, 72 for finance mental math, and 69.3 for the continuous approximation.
What does the milestone table include?
Times and optional projected values for 2×, 4×, 8×, and 16× growth, or the corresponding halves for a declining rate.