Population Doubling Time Calculator

What this calculator does

If something grows at a steady rate, how long until it doubles? That is the doubling time, and it is one of the most useful ways to grasp the power of steady growth, whether it is a population, an investment, or an economy. This works it out from the growth rate.

Give it a growth rate per period and it returns the doubling time in periods. It runs right here in the browser.

Using the calculator

1. Enter an initial value (this is just for context, as you will see below).
2. Enter the growth rate per period, as a percentage. A period can be whatever you like, a year, a month, a decade, as long as the rate matches it.
3. Press Calculate.

It returns the doubling time in those same periods. Reset clears the boxes.

What doubling time is

Doubling time is the flip side of growth rate, and often the more meaningful one. A rate like "3 percent a year" is easy to shrug off as small. But "doubles every 23 years" lands differently, because it makes the long run vivid. Anything growing at a fixed percentage doubles over a fixed stretch of time, again and again: it doubles, then doubles again to four times, then to eight, and so on.

It is the exact mirror of a half-life. Where a half-life is the constant time for something to halve, a doubling time is the constant time for something to double. Same idea, opposite direction.

The formula

For growth of r percent per period, the doubling time is:

doubling time = ln(2) ÷ ln(1 + r ÷ 100)

where ln is the natural logarithm and r is the percentage growth rate. The ln(2) on top is there because it is a doubling; if you wanted a tripling time, that would become ln(3). The bottom captures how much each period multiplies the amount. It looks fiddly, which is exactly why the shortcut below exists.

The shortcut: the rule of 70 (and 72)

Almost nobody works that logarithm out in their head, and they do not need to. There is a famous mental shortcut, the rule of 70: to get the doubling time, just divide 70 by the growth rate as a percentage.

doubling time ≈ 70 ÷ (growth rate percent)

So 7 percent growth doubles in roughly 70 ÷ 7 = 10 periods. 2 percent takes about 35. The number 70 is used because it is close to 100 times the natural log of 2, which is about 69.3. In finance you will often see the rule of 72 instead, dividing 72 by the rate, because 72 divides cleanly by 2, 3, 4, 6, 8 and 9, which makes the mental sums easier, and it is a touch more accurate for the 6 to 10 percent range typical of investments. Both are approximations of the exact formula this tool uses, and the rule of 72 is old, traced back to Luca Pacioli in 1494.

A worked example

Take a growth rate of 7 percent per period.

1. By the exact formula: ln(2) ÷ ln(1.07) = 0.6931 ÷ 0.0677 = 10.24 periods.
2. By the rule of 70: 70 ÷ 7 = 10 periods.

The shortcut lands within a quarter of a period of the exact answer, which is why it is trusted for quick estimates.

Why the starting value does not matter

Here is something worth noticing: the doubling time does not depend on the initial value at all. That is why the tool can work it out from the rate alone. A town of 500 and a city of 5 million, both growing at the same rate, take the same time to double. It makes sense when you think about it, since doubling is a proportion, not an amount. The starting value is there only to picture what is growing; it plays no part in the answer.

For the underlying growth itself, term by term, see the exponential growth calculator.

Questions people ask

What is doubling time?

The time it takes for a quantity growing at a fixed rate to double. It stays constant, so the quantity doubles again over each further stretch of that length.

What is the rule of 70?

A mental shortcut for doubling time: divide 70 by the growth rate as a percentage. At 5 percent, that is about 70 ÷ 5 = 14 periods.

Rule of 70 or rule of 72, which should I use?

Both are close. The rule of 70 is common for population and economic growth; the rule of 72 is favoured in finance because 72 divides evenly by more numbers and is slightly more accurate around 6 to 10 percent.

Does the starting amount affect the doubling time?

No. Doubling time depends only on the growth rate, so any starting amount doubles in the same time.

How does this relate to half-life?

It is the mirror image. Doubling time is the constant time to double under growth; half-life is the constant time to halve under decay.

References

A note on where this comes from. Doubling time is derived from the exponential growth model as ln(2) divided by the logarithm of the per-period growth factor. The rule of 70, and the closely related rule of 72 used in finance, are standard approximations of it, with the rule of 72 traced to Luca Pacioli's Summa de arithmetica of 1494. For further reading, see Doubling time.

1. The doubling time formula, ln(2) ÷ ln(1 + r), giving the time for a quantity to double under steady growth.
2. The rule of 70 and the rule of 72, mental approximations of doubling time; the rule of 72 appears in Luca Pacioli's Summa de arithmetica (1494).