What is doubling time?

Doubling time is the period it takes for a quantity growing at a constant percentage rate to double in size. It applies to any exponential growth: money compounding at a fixed interest rate, a population increasing at a fixed annual rate, or a business revenue growing at a steady percentage. The exact formula is T₂ = ln(2) / ln(1 + r/100), where r is the growth rate in percent.

What is the Rule of 70?

The Rule of 70 is a quick mental math shortcut: divide 70 by the annual growth rate (in percent) to get an approximate doubling time. For example, at 7% growth, the doubling time is roughly 70 / 7 = 10 years. The rule works best for rates between 1% and 10% where the approximation is accurate to within a few months of the exact answer.

What is the Rule of 72 and when should I use it?

The Rule of 72 divides 72 by the interest rate to estimate doubling time, and is most commonly used in finance because 72 is divisible by more integers (1, 2, 3, 4, 6, 8, 9, 12) than 70, making mental arithmetic easier. At rates around 8%, the Rule of 72 is more accurate than the Rule of 70; at rates around 2%, the Rule of 70 is slightly better. For a precise answer, always use the exact logarithmic formula.

How long does it take to double money at 6% interest?

At a 6% annual compound interest rate, money doubles in approximately 11.90 years using the exact formula (ln(2) / ln(1.06)). The Rule of 72 gives 72 / 6 = 12 years — very close. The Rule of 70 gives 70 / 6 ≈ 11.67 years. These approximations are useful for quick mental estimates without a calculator.

What growth rate doubles an investment in 10 years?

To double in exactly 10 years, you need an annual growth rate of approximately 7.18% (calculated as (2^(1/10) − 1) × 100 = 7.1773%). This is why a 7% return is often cited as a benchmark for long-term stock market growth — historically, broad market indices have roughly doubled every 10 years over long periods.

Does doubling time apply to continuous or compound growth?

The formula T₂ = ln(2) / ln(1 + r/100) applies to discrete annual compounding — the most common case for finance and economics. For continuous compounding, the formula simplifies to T₂ = ln(2) / r (where r is expressed as a decimal, not a percent). The Rule of 70 and Rule of 72 are approximations valid for both formulations at typical growth rates.

How is doubling time used in biology and population growth?

In biology, doubling time describes how quickly a population or bacterial culture doubles under constant conditions. A healthy E. coli colony can double in as little as 20 minutes. Human population growth at 1% per year gives a doubling time of about 70 years. Epidemiologists use doubling time to measure the early spread of infectious diseases — a shorter doubling time means faster spread.

How does the Rule of 72 help investors?

The Rule of 72 is a quick mental math shortcut for investors: divide 72 by the annual return rate to estimate how many years it takes to double your money. At 6% return: 72 / 6 = 12 years. At 9% return: 72 / 9 = 8 years. At 12% return: 72 / 12 = 6 years. It also works in reverse: to figure out what return you need to double money in 8 years, solve 72 / r = 8, giving r = 9%. This makes compound interest intuitive without a calculator.

How is doubling time used in epidemiology?

During an epidemic, doubling time is one of the most important metrics. A disease doubling every 3 days is far more urgent than one doubling every 14 days. COVID-19 in early 2020 had doubling times of 3–7 days in some regions, indicating very rapid spread. Public health interventions aim to lengthen doubling time — doubling it from 5 days to 10 days gives hospitals twice as long to prepare. Doubling time is directly related to the basic reproduction number R₀.

Doubling Time Calculator — Rule of 70 & 72

Calculates doubling time from a growth rate (and vice versa) — shows the exact formula and Rule of 70 and Rule of 72 approximations.

How to Calculate Doubling Time

This doubling time calculator uses the exact formula for constant compound growth:

T₂ = ln(2) / ln(1 + r/100)

where r is the annual growth rate expressed as a percentage (e.g., 7 for 7%), and ln is the natural logarithm. This formula works for any positive growth rate and gives the precise number of compounding periods needed for a quantity to exactly double. Enter your growth rate above to compute it instantly, or switch to "Find Growth Rate" mode to work backwards from a target doubling time.

Rule of 70 vs. Rule of 72 — Which Should You Use?

Both rules are quick mental-math approximations of the same exact formula:

Rule of 70: T₂ ≈ 70 / r. More accurate at low growth rates (1–5%). Preferred in economics and demography.
Rule of 72: T₂ ≈ 72 / r. More accurate near 8% growth. Preferred in finance because 72 divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12, enabling quick mental calculation for common interest rates.

At 7% annual growth, the exact answer is 10.24 years. The Rule of 70 gives 10 years and the Rule of 72 gives 10.29 years — both excellent approximations. For rates above 15% or below 1%, both rules diverge significantly from the exact answer; always use the exact formula (as this calculator does) for precision.

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Doubling Time in Finance — The Investor's Perspective

Doubling time is one of the most useful mental models in personal finance and investing. At a 7% annual return — roughly the historical long-term inflation-adjusted return of diversified stock index funds — money doubles approximately every 10 years. That means:

$10,000 invested today → ~$20,000 in 10 years
$10,000 invested today → ~$40,000 in 20 years
$10,000 invested today → ~$80,000 in 30 years

This illustrates the power of compound growth: each additional decade doubles the total, regardless of the starting amount. The same principle applies to debt — a credit card at 20% annual interest doubles the balance you owe in about 3.8 years if you make no payments.

Use our proportion calculator to work out ratio-based growth questions, or the standard deviation calculator to analyze the variability of investment returns.

Doubling Time in Biology and Epidemiology

Beyond finance, doubling time is a fundamental concept in any field involving exponential growth:

Population biology: under ideal conditions (unlimited food, no predators), bacterial populations double every 20–30 minutes. Human populations growing at 1% per year have a doubling time of about 70 years.
Epidemiology: during the early, uncontrolled phase of an outbreak, the doubling time of infections indicates transmission speed. A doubling time of 2–3 days is considered fast-spreading; 7–14 days allows more time for intervention.
Oncology: tumor doubling times range from days (aggressive leukemias) to years (some prostate cancers). Radiologists use doubling time on repeat imaging to characterize whether a lesion is likely malignant.

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Doubling Time vs. Half-Life

Doubling time and half-life are exact mathematical inverses: half-life describes how long it takes a quantity to decrease by half under constant negative growth (decay), while doubling time describes how long it takes to increase by a factor of two under constant positive growth. The formulas are structurally identical — only the sign of the rate changes.

Half-life is widely used in nuclear physics (radioactive decay), pharmacology (drug elimination from the body), and environmental science (pollutant degradation). The same logarithmic mathematics governs both processes.

How to Use This Doubling Time Calculator

The calculator has two modes accessible via the toggle at the top:

Find Doubling Time: enter an annual growth rate (as a percentage, e.g. 7 for 7%) and the calculator returns the exact doubling time, the Rule of 70 approximation, and the Rule of 72 approximation — all side by side so you can see how close the shortcuts are.
Find Growth Rate: enter how many years you want the doubling to take, and the calculator returns the exact annual growth rate required.

Both modes use the precise logarithmic formulas with no rounding until display. Results are shown to two decimal places.

Sources & References

Exponential Growth and Decay — Khan Academy
The Rule of 72 — U.S. Securities and Exchange Commission

Doubling Time Calculator

Rule of 70 — Common Examples

Exact Doubling Time Formula

Find Rate from Doubling Time