What this calculator does
Doubling time is how many periods it takes a quantity growing at a constant percentage rate per period to double in size. It applies to anything that compounds: an investment earning a fixed annual return, a population growing at a steady rate, or a bacterial culture multiplying under stable conditions, and it is calculated the same way regardless of what is actually growing.
There are two common ways to work it out. The exact formula, t = ln(2) / ln(1 + r), gives the precise doubling time for a given growth rate r. The Rule of 72, t ≈ 72 / (r × 100), is a mental-arithmetic shortcut that gets very close for typical growth rates without needing logarithms, which is why it is so widely quoted for compound interest.
The formula
For the exact answer, divide the natural log of 2 by the natural log of (1 plus the growth rate as a decimal). For the Rule of 72 approximation, divide 72 by the growth rate expressed as a plain number (so 8% is just 8, not 0.08). Both results are in the same time unit as the growth rate: if the rate is a monthly rate, the doubling time comes out in months.
| Term | Meaning |
|---|---|
| r | The growth rate per period, as a percentage, assumed constant over time. |
| Exact doubling time | ln(2) ÷ ln(1 + r), the precise number of periods to double at rate r. |
| Rule of 72 | 72 ÷ (r as a plain number), a fast approximation to the exact doubling time. |
The inputs explained
| Field | What to enter |
|---|---|
| Growth rate per period (%) | The constant growth rate per period, as a percentage. Use whatever period the growth rate is quoted for (annual, monthly, daily and so on); the answer comes out in that same period. |
When to use it
Estimating investment growth
The Rule of 72 is the classic mental shortcut for how long an investment takes to double at a given annual return, and this calculator shows exactly how close that shortcut is to the true answer at any rate.
Modelling population or biological growth
Populations, bacterial cultures and other quantities that grow by a fixed percentage each period all double on the same schedule for a given growth rate, regardless of what is actually being counted.
Understanding why the Rule of 72 sometimes misses
The Rule of 72 is a close approximation across a normal range of growth rates but drifts further from the exact answer at very high growth rates, since it is a linear shortcut standing in for a genuinely logarithmic relationship.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Exact doubling time versus the Rule of 72 across growth rates
The exact and Rule of 72 doubling times at a range of constant growth rates.
| Growth rate per period | Doubling time (exact) | Doubling time (Rule of 72) |
|---|---|---|
| 1% | 69.66 periods | 72.00 periods |
| 2% | 35.00 periods | 36.00 periods |
| 5% | 14.21 periods | 14.40 periods |
| 8% | 9.01 periods | 9.00 periods |
| 10% | 7.27 periods | 7.20 periods |
| 20% | 3.80 periods | 3.60 periods |
Questions
Why 72 and not some other number?
ln(2) is about 0.6931. Multiplying both the numerator and denominator of the exact formula by 100 and approximating ln(1+r) as r for small r gives roughly 69.3 ÷ (r×100); 72 is used instead of 69.3 because it divides evenly by more small numbers (2, 3, 4, 6, 8, 9, 12), making the mental arithmetic easier, at a small cost in accuracy.
Does the Rule of 72 work for any growth rate?
It is closest to the exact answer for rates roughly in the single digits to low teens per period. At very high growth rates, the exact formula and the Rule of 72 diverge more noticeably, so the exact result is worth checking whenever precision matters.
Can this be used for halving time as well as doubling time?
The same logic applies to a quantity shrinking at a fixed percentage rate, though the formula and rule are usually stated for growth. For decay, ln(2) ÷ ln(1+r) still gives the exact number of periods to halve when r is expressed as a negative rate.
Is this the same as the hCG doubling time used in early pregnancy?
No. That is a specific clinical calculation using two actual hCG blood test readings and the days between them to check whether hCG is rising at a healthy rate; see the hCG doubling time calculator for that. This calculator is the general compound-growth formula, used for investments, populations and any other quantity growing at a fixed percentage rate.
For the specific clinical hCG doubling time calculation used in early pregnancy, see the hCG doubling time calculator. For working the other direction from a fixed number of periods, the compound interest calculator shows how a quantity grows over a set timeframe.