What Is Doubling Time?
Doubling time is the time needed for a quantity to grow to twice its size when it grows at a constant percentage rate. It applies to a population, an investment, a case count or a bacterial culture. The exact formula is Doubling time = ln(2) ÷ ln(1 + g) for a compound growth rate g per period.
The calculator works in three directions: from a growth rate to a doubling time, from two measurements to a doubling time, and from a doubling time back to a growth rate. It also shows the rule of 70, rule of 72 and rule of 69.3 estimates so you can see how close each shortcut is, and how long the quantity takes to triple or grow tenfold.
Further down you will find the doubling time formula, a variables table, worked examples, the limits of the constant-growth assumption and a note on half-life, the mirror image of doubling.
How to Use the Doubling Time Calculator
- Choose what you know: the rate, the continuous rate, two measurements or the doubling time.
- Enter the numbers and pick the time unit.
- Read the doubling time, the equivalent rate and the time to triple or grow tenfold.
- Compare the rule-of-70 and 72 estimates with the exact answer.
Doubling Time Formula
Doubling time from a compound growth rate g
Continuous growth
From two measurements
Rules of thumb
Growth from doubling time
Doubling Time Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| g | Growth rate per period | As a decimal in the formula, e.g. 5 % = 0.05 |
| k | Continuous growth rate | k = ln(1 + g) |
| Td | Doubling time | In the same time unit as the rate |
| A, B | First and second measurement | Used when you give two values |
| t | Time between the two measurements | Td = t × ln(2) ÷ ln(B/A) |
Doubling Time Worked Example
With the example values loaded in the calculator above, the steps are:
- Compound growth rate g = 3.5% per period.
- Doubling time T = ln 2 ÷ ln(1 + g) = 0.693147 ÷ 0.034401427 = 20.1488.
- Rule of 70 gives 70 ÷ 3.5 = 20; rule of 72 gives 20.571.
Result: Doubling time (years) = 20.1488; Growth rate per year = 3.5 %.
Worked Example: Doubling Time From Two Measurements
A quantity measured as 1,000 at the start is 1,800 after 6 years. The growth factor is 1,800 ÷ 1,000 = 1.8, so T = 6 × ln 2 ÷ ln 1.8 = 6 × 0.6931 ÷ 0.5878 = 7.07 years.
The same data give a compound growth rate of 1.8^(1/6) − 1, about 10.3% a year. The rule of 72 would estimate 72 ÷ 10.3 = 7.0 years, close to the exact answer. That agreement is a useful sanity check on any result.
The rules of 70 and 72
Because ln 2 ≈ 0.693, the doubling time at r% growth is about 69.3 divided by r for continuous growth. The rule of 70 is the easiest to apply to continuous or low-rate compound growth. The rule of 72 is slightly more accurate at typical interest rates of 6% to 10%, and has the convenience that 72 divides evenly by many numbers (2, 3, 4, 6, 8, 9, 12).
| Rate | Exact | Rule of 70 | Rule of 72 |
|---|---|---|---|
| 1% | 69.66 | 70.0 | 72.0 |
| 2% | 35.00 | 35.0 | 36.0 |
| 5% | 14.21 | 14.0 | 14.4 |
| 8% | 9.01 | 8.75 | 9.0 |
| 10% | 7.27 | 7.0 | 7.2 |
| 20% | 3.80 | 3.5 | 3.6 |
Where doubling time is used
- Finance: how many years until an investment doubles at a given annual return.
- Demography: how fast populations grow; 1% growth means doubling in about 70 years.
- Epidemiology: how quickly case counts double at the start of an outbreak.
- Microbiology: generation time of bacteria, often 20 to 60 minutes.
- Technology: capacity and data growth, as in Moore’s law.
Limits of the model
Doubling time assumes the growth rate stays constant. Real systems slow down as they meet limits (food, space, immunity, market size), so a logistic curve often fits better in the long run. Use the non-linear regression calculator to fit a logistic or exponential-with-offset model to measured data.
Half-life is the mirror image
For decay at a rate d, the half-life is ln 2 ÷ −ln(1 − d). The same formula and rules of thumb apply with the rate of decline in place of the rate of growth.
Where Doubling Time Is Used
Growth rate to doubling time
Find how long a quantity takes to double at a steady growth rate.
From two measurements
Work out the doubling time from a start and end value.
Rules of thumb
Compare the exact answer with the rule of 70, 72 and 69.3.
Time to triple or ten times
See other growth milestones for the same rate.
Common Doubling Time Mistakes to Avoid
- Use the same time unit for the rate and the answer.
- For percentage rates use the compound formula; for continuous rates use ln 2 ÷ k.
- The rule of 72 is a quick check, not a replacement for the exact formula.
Doubling Time Terms and Definitions
- Exponential growth
- Growth by a constant percentage per period, so the amount multiplies by the same factor in equal times.
- Continuous growth rate
- The k in N = N₀eᵏᵗ.
Doubling Time FAQs
How do you calculate doubling time?
Divide ln 2 by ln(1 + growth rate). A quick estimate is 70 or 72 divided by the percentage growth rate.
What is the rule of 72?
A shortcut: divide 72 by the annual percentage rate to estimate the years to double. At 6% it gives 12 years.
What is the difference between the rule of 70 and the rule of 72?
Both approximate ln 2 × 100. 70 is more accurate for low or continuous rates; 72 is more accurate for typical interest rates around 8%.
How do I find the growth rate from the doubling time?
Use g = 2^(1/T) − 1. A doubling time of 12 years means about 5.95% growth per year.
How do I find doubling time from two data points?
Use T = t × ln 2 ÷ ln(final ÷ initial) where t is the time between the measurements.
Does doubling time stay constant?
Only for pure exponential growth. In real systems the growth rate usually changes, so doubling time changes too.
How long does it take to double at 2% per year?
Exactly ln 2 ÷ ln 1.02 = 35.00 years. The rule of 70 gives 35 and the rule of 72 gives 36.
How do I find a bacterial doubling time?
Use the two-measurements option. If a culture grows from 1,000 to 8,000 cells in 90 minutes, T = 90 × ln 2 ÷ ln 8 = 30 minutes, because 8 is three doublings.
Is doubling time the same as half-life?
They are mirror images. Doubling time is for growth and half-life is for decay at a constant percentage rate. The same logarithm formula applies to both, with the rate of decline in place of the rate of growth.