What this calculator does
The rule of 72 is a mental shortcut: divide 72 by the annual percentage rate and you get the approximate number of years for money to double. At 8%, that is nine years. It requires no calculator, which is exactly why it has survived for centuries.
This page reports the shortcut alongside the exact answer so you can see how good the approximation is. It is most accurate between about 6% and 10%, and drifts at the extremes: but the error is small enough that the rule remains a genuinely useful piece of mental arithmetic.
The formula
The exact doubling time comes from solving (1+r)ⁿ = 2, which gives n = ln2 / ln(1+r). Because ln2 ≈ 0.693 and ln(1+r) ≈ r for small r, the expression collapses to roughly 69.3 divided by the percentage rate. The number 72 is used instead because it divides cleanly by many integers and happens to correct for the approximation error in the range people care about most.
| Term | Meaning |
|---|---|
| n | The number of years to double. |
| r | The annual growth rate. |
| ln | The natural logarithm, to base e. |
The inputs explained
| Field | What to enter |
|---|---|
| Annual rate (%) | The annual rate of growth: an investment return, an inflation rate, or any compounding growth rate. |
When to use it
Quick investment sanity checks
If a fund claims to double money in three years, that implies about 24% a year sustained. The rule turns extraordinary claims into a rate you can judge.
Understanding inflation
The same arithmetic applies to prices. At 3% inflation, prices double in a little over 23 years: a useful frame for any long-term plan.
Explaining compounding to someone else
The rule needs no equipment and produces a memorable number, which makes it the most effective way to convey what compound growth actually does.
Estimating debt growth
An unpaid 18% credit card balance doubles in about four years. That single number communicates the danger more effectively than the rate itself.
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.
Doubling time at different rates
The shortcut alongside the exact calculation, so the size of the approximation error is visible.
| Annual rate | Doubling time (rule of 72) | Exact doubling time | Tripling time |
|---|---|---|---|
| 1% | 72.0 years | 69.7 years | 110.4 years |
| 2% | 36.0 years | 35.0 years | 55.5 years |
| 3% | 24.0 years | 23.4 years | 37.2 years |
| 5% | 14.4 years | 14.2 years | 22.5 years |
| 7% | 10.3 years | 10.2 years | 16.2 years |
| 8% | 9.0 years | 9.0 years | 14.3 years |
| 10% | 7.2 years | 7.3 years | 11.5 years |
| 15% | 4.8 years | 5.0 years | 7.9 years |
| 20% | 3.6 years | 3.8 years | 6.0 years |
Questions
Why 72 and not 69.3?
The mathematically exact numerator for continuous compounding is ln2, about 0.693, giving 69.3. But 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes the mental arithmetic much easier, and it also happens to compensate for the approximation error at the rates people most often use.
How accurate is the rule?
Within a few percent for rates between about 5% and 12%. Outside that band the error grows, the rule overstates at low rates and understates at high ones, though it stays small in absolute terms. Both figures are shown here so you can see the difference directly.
Is there a rule for tripling?
Dividing 114 by the rate approximates tripling time, and 144 approximates quadrupling. The exact tripling time is shown in the results panel.
Does it work for losses?
A related rule does. Dividing 72 by a rate of decline approximates the years for a value to halve: useful for thinking about inflation eroding cash.
For the full compounding picture with contributions, use the compound interest calculator. To see prices rise instead, try the inflation calculator.