Doubling Time Calculator - Growth Rate & Rule of 72

About the Doubling Time Calculator

Anything that grows by a steady percentage — money earning interest, prices rising with inflation, a town's population, a colony of bacteria, a website's audience — grows exponentially. The most useful single number for understanding exponential growth is the doubling time: how long the quantity takes to double. It stays the same however large the amount becomes, which is why steady growth that looks gentle at first soon produces enormous numbers.

This doubling time calculator finds the doubling time from a growth rate, or from two values measured some time apart. It gives the exact result, compares it with the popular rule of 72 and rule of 70, and shows how long the quantity takes to grow fourfold and tenfold. Enter a negative rate and it gives the halving time of a decline instead.

How to Use the Doubling Time Calculator

Choose From a growth rate or From two values.

For a rate, enter the growth rate per period as a percentage, such as 7 for 7 percent a year. Use a negative number for a decline.

For two values, enter the starting value, the ending value and the time between them.

Choose whether growth happens once per period, like interest paid annually, or continuously, like a population growing all the time.

Choose the period — years, months, days or hours — so the answer is labelled correctly.

How Doubling Time Is Worked Out

  growth once per period:   doubling time = ln 2 ÷ ln(1 + r)
  continuous growth:        doubling time = ln 2 ÷ r
  rule of 72:               doubling time ≈ 72 ÷ rate %
  rate from two values:     r = (end ÷ start)^(1 ÷ time) − 1
  halving time (r < 0):     ln 0.5 ÷ ln(1 + r)

Here r is the rate as a decimal, 0.07 for 7 percent, and ln is the natural logarithm.

Step-by-Step Example

Growth of 7 percent a year.

  Exact:        ln 2 ÷ ln 1.07 = 0.6931 ÷ 0.06766 = 10.24 years
  Rule of 72:   72 ÷ 7 = 10.3 years
  Rule of 70:   70 ÷ 7 = 10 years
  Continuous:   0.6931 ÷ 0.07 = 9.9 years

At 7 percent a year, anything doubles in about 10.2 years. It grows fourfold in about 20.5 years and tenfold in about 34 years.

From two values: 100 grows to 150 in 5 years.

  Rate:       (150 ÷ 100)^(1/5) − 1 = 8.45% a year
  Doubling:   ln 2 ÷ ln 1.0845 = 8.55 years

A decline of 5 percent a year halves in ln 0.5 ÷ ln 0.95, about 13.5 years, falls to a quarter in about 27 years and to a tenth in about 44.9 years.

The Rule of 72

The rule of 72 is a mental shortcut: divide 72 by the percentage rate. It works because ln 2 is about 0.693, and for small rates ln(1 + r) is close to r, so the doubling time is roughly 69.3 ÷ rate. Using 72 instead of 69.3 corrects for the slight curve at typical interest rates, and 72 divides neatly by 2, 3, 4, 6, 8, 9 and 12.

  Rate    Exact    Rule of 72
   2%     35.0       36.0
   5%     14.2       14.4
   8%      9.0        9.0
  12%      6.1        6.0
  20%      3.8        3.6

The rule is most accurate between about 6 and 10 percent. For continuous growth, or for low rates, the rule of 70 is closer, and population scientists often prefer it.

Doubling in Everyday Life

Money. Savings earning 5 percent a year double in about 14.2 years; at 8 percent, in about 9 years.

Inflation. Prices rising at 3 percent a year double in about 23.4 years, so money kept as cash loses half its buying power in that time.

Population. A population growing at 2 percent a year doubles in about 35 years.

Debt. A credit card balance at 24 percent a year, left unpaid, doubles in just over 3 years.

Why Exponential Growth Surprises People

People tend to picture growth as a straight line, adding the same amount each year. Exponential growth adds the same percentage, so the amount added keeps increasing. Every doubling time, the quantity adds as much as all the growth that came before. A pond weed doubling every day that covers the whole pond on day 30 covered only half of it on day 29 — and just one thousandth on day 20.

That is also why real growth rarely stays steady for long. Resources run out, markets saturate and rates change, so long-range doubling calculations are best read as what would happen if nothing changed, not as forecasts.

Compounding Once or Continuously

Growth that happens in steps, such as interest credited once a year, uses ln(1 + r). Growth that happens all the time, such as a population or radioactive decay, uses the continuous formula ln 2 ÷ r, which gives a slightly shorter doubling time for the same rate: 9.9 years rather than 10.24 at 7 percent. Interest credited monthly or daily falls between the two and sits very close to the continuous figure.

Understanding Your Result

The headline is the doubling time, or the halving time for a decline, in the period you chose.

The growth rate line shows the rate used, including the rate found from two values.

The rule of 72 line gives the quick estimate and the rule of 70 for comparison.

The longer horizons line shows the time to grow fourfold and tenfold, or to fall to a quarter and a tenth.

When Should You Use This Calculator?

Use it to see how long savings or investments take to double at a given return.

Use it to understand how quickly inflation erodes money.

Use it to measure the growth rate of sales, users or a population from two figures.

Use it in science and economics lessons on exponential growth and decay.

Common Mistakes

Using the rule of 72 at extreme rates. It drifts at very low or very high rates; use the exact figure.

Mixing periods. A monthly rate gives a doubling time in months, not years.

Adding rates instead of compounding them. Growth of 50 percent over 5 years is 8.45 percent a year, not 10 percent.

Assuming growth continues forever. Treat long horizons as illustrations.

Forgetting declines halve. A negative rate gives a halving time, not a doubling time.