Doubling Time Calculator
Enter a growth rate and this calculator shows exactly how long any quantity takes to double. Choose continuous (exponential) or discrete (compounded) growth, switch to the "values" mode to derive doubling time from a measured start and end, or check the Rule of 70 and Rule of 72 quick estimates side-by-side. A live chart traces the quantity across several doubling periods so you can see exponential growth in action.
Formula
Worked example
An investment grows at 7% per year (discrete). r = 0.07. Doubling time = ln(2) / ln(1.07) = 0.6931 / 0.0677 = 10.24 years. The Rule of 72 gives 72/7 = 10.29 years - an overestimate of about 0.4%. The Rule of 70 gives 70/7 = 10.00 years - an underestimate of about 2.3%.
What is doubling time?
Doubling time is the number of time periods it takes for a quantity growing at a constant rate to become exactly twice its current size. It applies to any exponentially growing quantity: a savings account earning compound interest, a bacterial culture, a national population, a carbon dioxide concentration, or the number of users on a platform. Because exponential growth is hard to visualise intuitively, doubling time gives you a single, concrete number that makes the speed of growth easy to grasp and compare. A population doubling every 35 years sounds very different from one doubling every 20 years, even though the underlying percentages are only a few points apart.
Continuous vs. discrete growth - which model should you use?
The two models differ in how often growth is applied. Discrete (compounded) growth adds the growth at fixed intervals - once a year for an annual interest rate, once a quarter for quarterly compounding. The formula is td = ln(2) / ln(1 + r), where r is the rate per period as a decimal. Continuous growth applies the rate at every infinitesimal instant, like bacteria reproducing without any fixed breeding cycle or radioactive atoms decaying. The formula simplifies to td = ln(2) / r. For the same stated rate, continuous growth is always slightly faster and gives a slightly shorter doubling time. In finance, use discrete compounding; in biology, chemistry, and physics, use continuous. The difference matters most at high rates but is small at the low rates typical of savings and inflation.
The Rule of 70, Rule of 72, and Rule of 69.3
Because dividing by a logarithm is inconvenient by hand, several shortcut rules are widely used. The Rule of 72 divides 72 by the percentage growth rate to estimate doubling time. It is popular because 72 has many integer divisors (2, 3, 4, 6, 8, 9, 12, 18, 24, 36), making mental arithmetic easy. It slightly overestimates the doubling time for the discrete model and gives its most accurate result for rates near 8%. The Rule of 70 divides 70 by the rate; it is preferred by economists when discussing inflation and demographic growth, where rates tend to be low (under 5%), and where it is slightly more accurate than the Rule of 72. The most mathematically precise rule uses 69.3 (that is, 100 x ln 2), which matches the continuous formula exactly but is harder to divide by hand. For everyday use at rates between 5% and 10%, the Rule of 72 is the standard choice.
Deriving doubling time from observed data
If you have measured a quantity at two points in time and want to know its doubling time, you can use the "values" mode. The calculator first derives the implied continuous growth rate using r = ln(N / N0) / t, where N0 is the initial measurement, N is the later measurement, and t is the time elapsed. It then plugs that rate into the continuous doubling-time formula td = ln(2) / r. This approach works for any quantity that grows roughly exponentially over the measured interval: tumor size on successive scans, server load over weeks, a social-media follower count, or a colony count on serial dilution plates. The accuracy depends entirely on how well a constant exponential rate describes the real process - if the rate is not constant, the derived doubling time is the best-fit estimate over that particular interval, not a reliable long-run prediction.
Real-world applications
Doubling time appears across many fields. In economics, the rule of 72 tells an investor that money growing at 6% per year doubles in 12 years, and at 12% per year it doubles in 6 years. In epidemiology, a disease spreading with a daily growth rate of 10% doubles case counts roughly every 7 days, a number that drives public-health response decisions. In biology, E. coli bacteria can double every 20 to 30 minutes under ideal conditions; slower-growing organisms like yeast double in about 90 minutes. In demography, world population grew from 3 billion to 6 billion between 1960 and 2000 - a doubling time of 40 years. In technology, transistor counts on microchips doubled roughly every two years for decades (a pattern known as Moore's Law). Understanding doubling time helps you compare growth rates across domains that otherwise look very different.
Common doubling times by growth rate
| Growth rate (% per period) | Exact doubling time (periods) | Rule of 72 estimate | Typical context |
|---|---|---|---|
| 0.5 | 138.63 | 144.0 | Low-yield savings account |
| 1 | 69.32 | 72.0 | Low inflation / bond return |
| 2 | 34.66 | 36.0 | Moderate inflation |
| 3 | 23.10 | 24.0 | Stock market average (real) |
| 5 | 13.86 | 14.4 | S&P 500 long-run nominal return |
| 7 | 9.90 | 10.3 | Classic "doubles every 10 years" rule of thumb |
| 10 | 6.93 | 7.2 | High-growth investment / high inflation |
| 20 | 3.47 | 3.6 | High-growth startup / fast-growing population |
| 50 | 1.39 | 1.44 | Aggressive growth / compounding at short intervals |
| 100 | 0.69 | 0.72 | Quantity doubles each period |
Exact continuous-model doubling times for a selection of common growth rates. Finance calculations typically use discrete compounding instead.
Frequently asked questions
What is the doubling time formula?
There are two versions. For continuous (exponential) growth: td = ln(2) / r, where r is the growth rate as a decimal (e.g. 7% = 0.07) and ln is the natural logarithm. For discrete (compounded) growth: td = ln(2) / ln(1 + r). Both give very similar results at low rates; the difference grows larger at high rates. For growth rates below about 5%, either formula gives a result within 1-2% of the other.
What is the Rule of 72?
The Rule of 72 is a quick mental shortcut: divide 72 by the percentage growth rate to estimate how many periods it takes for a quantity to double. For example, at 6% per year: 72 / 6 = 12 years. It is most accurate for discrete compounding at rates between 6% and 10%. It slightly overestimates the doubling time at low rates and underestimates at very high rates. The Rule of 70 is a similar shortcut preferred in economics; 69.3 (= 100 x ln 2) is the most exact continuous version.
How long does it take to double money at 7% interest?
At 7% annual interest compounded yearly (discrete model), the exact doubling time is ln(2) / ln(1.07) = 10.24 years. The Rule of 72 gives 72 / 7 = 10.29 years, a tiny overestimate of about 0.4%. Many personal-finance guides use "doubles every 10 years at 7%" as a useful approximation for long-run stock-market returns after inflation.
How is doubling time used in biology?
In microbiology, the doubling time (also called generation time) is the time for a bacterial population to double under constant conditions. E. coli under ideal lab conditions doubles every 20-25 minutes. Slower-growing organisms or cultures under stress have longer doubling times. Measuring a colony's doubling time is used to assess culture health, antibiotic efficacy, and the effect of nutrient changes. The continuous exponential model is the standard in these contexts because cell division happens continuously rather than at fixed intervals.
What is the difference between doubling time and half-life?
They are mirror images of each other. Doubling time applies to quantities that are growing: td = ln(2) / r with a positive r. Half-life applies to quantities that are decaying: t1/2 = ln(2) / k with a positive decay constant k. Both use the same formula structure with the natural logarithm of 2; the difference is just the sign of the rate. A substance with a half-life of 10 years is losing half its quantity every decade; an investment with a doubling time of 10 years is gaining twice its value every decade.
Can I use doubling time for non-exponential growth?
Technically no: the formula assumes growth at a constant exponential rate. If a quantity grows linearly (by a fixed amount each period rather than a fixed percentage), the time to double decreases as the quantity grows, so a single "doubling time" is not meaningful. If the rate is roughly constant over your observation window, the exponential model gives a useful approximation. When rates change significantly over time, the computed doubling time is only valid for the interval over which it was measured.
Why does the Rule of 70 differ from the Rule of 72?
Both are approximations to 100 x ln(2) = 69.315. The Rule of 70 rounds down (easier to divide by low round numbers like 2 and 5), while the Rule of 72 rounds up (72 is divisible by 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36, giving clean answers for many common rates). Because the discrete compounding formula adds a small correction on top of the continuous formula, 72 tends to give a closer answer for discrete compounding at typical financial rates (6-10%). For continuous growth and for low rates common in inflation or demographic work, 70 (or 69.3) is closer.