# 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. > Canonical page: https://elysiatools.com/en/tools/population-doubling-time-rule70 - **Category:** Health - **Keywords:** doubling time, rule of 70, rule of 72, half-life growth, exponential growth, population growth, compound growth, demography calculator, generation time, bacterial growth, interest doubling, growth rate ## 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. ## Inputs - **Growth rate per period (%)** (number): Compound rate per period (years, generations, minutes…). Negative rates compute half-life instead of doubling time. - **Rule numerator** (select) - **Initial value (optional)** (number): Starting amount for the milestone table (population, currency, biomass — any unit). ## 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 - Enter the compound rate per period as a percentage. A negative rate switches the result from doubling time to half-life. - Choose the rule numerator: 70 (ecology/demography), 72 (finance mental math), or 69.3 (100·ln 2, continuous). - Optionally set an initial value so milestones can show projected population, capital, or resource amounts. - The 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. ## Frequently asked questions ### 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. ## Related tools - [A-a Gradient (Alveolar-Arterial O₂)](https://elysiatools.com/en/tools/a-a-gradient-calculator): Calculate the alveolar-arterial O₂ gradient to distinguish hypoxemia causes. PAO₂ = FiO₂×(P_atm−47) − PaCO₂/0.8; A-a = PAO₂ − PaO₂. Normal ≈ (Age+10)/4. Elevated → V/Q mismatch, shunt, diffusion impairment. Derived from StatPearls, LITFL, Cornell PICU, and Martin 1999. 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