Doubling Time
With growth rate per period 7 %, starting amount 10000 USD, doubling time comes to 10.24 — periods to double. It is reached in 6 steps, the last of which is log(2) / log(1 + 7 / 100), and each one is printed on the page with its numbers filled in. The formula is the one published by Luca Pacioli, Summa de Arithmetica (1494), not an approximation fitted to it.
How long a quantity growing at a fixed rate takes to double, with the Rule of 72 shortcut beside the exact figure and the error between them.
Formula and sources checked · How we check
Growth rate per period 7, Starting amount 10000
10.24
Periods to double for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Periods to double
log(2) / log(1 + 7 / 100)10.245- Rule of 72 estimate
72 / 710.286- How far the shortcut is off
(10.285714 - 10.244768) / 10.244768 * 1000.4 %- Periods to triple
log(3) / log(1 + 7 / 100)16.238- Periods to reach ten times
log(10) / log(1 + 7 / 100)34.032- What the starting amount becomes after ten periods
10000 * pow(1 + 7 / 100, 10)19,671.514 USD
Worked example
At 7% a year money doubles in 10.24 years. The Rule of 72 says 10.29 — close enough for mental arithmetic, and that is why the rule uses 72 rather than the mathematically cleaner 69.3, which is exact only at rates approaching zero.
How to work it out yourself
- 1.Enter the rate per period, whatever the period is. At 7% a year the answer is in years; at 0.6% a month it is in months.
- 2.The Rule of 72 column divides 72 by the rate as a whole number. It is a mental shortcut, accurate to about 1% between 4% and 12%.
- 3.Use it on anything that compounds, not just money — population, bacteria, an epidemic, inflation eroding the value of a currency.
- 4.For inflation, doubling time is how long prices take to double. At 3% that is 23.4 years, which is why a 1990 price looks so strange today.
Doubling time against growth rate
| Growth rate per period (%) | Periods to double | Rule of 72 estimate | How far the shortcut is off |
|---|---|---|---|
| 1 | 69.66 | 72 | 3.36 % |
| 2 | 35.00 | 36 | 2.85 % |
| 3 | 23.45 | 24 | 2.35 % |
| 4 | 17.67 | 18 | 1.85 % |
| 5 | 14.21 | 14.4 | 1.36 % |
| 6 | 11.90 | 12 | 0.88 % |
| 7 | 10.24 | 10.29 | 0.4 % |
| 8 | 9.01 | 9 | -0.07 % |
| 9 | 8.04 | 8 | -0.54 % |
| 10 | 7.27 | 7.2 | -1 % |
| 12 | 6.12 | 6 | -1.9 % |
| 15 | 4.96 | 4.8 | -3.22 % |
| 20 | 3.80 | 3.6 | -5.31 % |
The Rule of 72 is closest around 8%, where it is almost exact. It drifts low below 3% and high above 15%, which is the range worth knowing: at 1% it under-reads by 3.4% and at 20% it over-reads by 5.2%.
The formula
- Periods to double
log(2) / log(1 + 7 / 100) - Rule of 72 estimate
72 / 7 - How far the shortcut is off
(10.285714 - 10.244768) / 10.244768 * 100 - Periods to triple
log(3) / log(1 + 7 / 100) - Periods to reach ten times
log(10) / log(1 + 7 / 100) - What the starting amount becomes after ten periods
10000 * pow(1 + 7 / 100, 10)
Source: Luca Pacioli, Summa de Arithmetica (1494) — the earliest known statement of the rule, SEC Investor.gov — compound interest, US Bureau of Labor Statistics — CPI inflation, for applying doubling time to prices
Questions people actually ask
- What is the Rule of 72?
- Divide 72 by the annual growth rate to get the years to double. At 8% that is 9 years, against an exact 9.01. It works because ln(2) is 0.693, and 72 is close to 69.3 while dividing far more easily.
- What is the exact doubling time formula?
- ln(2) ÷ ln(1 + r), where r is the rate as a decimal. At 7% that is 0.6931 ÷ 0.0677 = 10.24 years.
- Why 72 and not 69?
- Because 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, and because the extra makes up for the discrete compounding that the continuous formula ignores. 69.3 is exact for continuous compounding but useless in your head.
- How long to double my money at 5%?
- 14.21 years exactly, or 14.4 by the Rule of 72. At 10% it is 7.27 years, and at 2% it is 35 — which is the argument for equities over savings accounts stated in one number.
- Does the rule work for tripling?
- A rule of 114 approximates tripling and a rule of 144 approximates quadrupling, on the same reasoning. The exact figures are in the table above; the shortcuts are less accurate than the doubling one because the errors compound with the multiple.
- Does it work for shrinking things?
- Yes — as halving time, with the rate negative. At 3% inflation money halves in purchasing power in 22.8 years, and the Rule of 72 gives 24. The gap is wider than for growth, so use the exact form for anything that matters.
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