How many years does it take to double your money?
Divide 72 by your return, says the famous rule — and at 8% it is astonishingly right: the rule says 9.00 years, the exact answer is 9.006, a two-day error on a nine-year wait. Slide away from 8%, though, and the 500-year-old shortcut starts lying: at 2% it overshoots by a full year (36 vs 35.0), and at a 24% return it promises 3 years when the truth is 3.22 — nearly three months of phantom speed. The widget below shows exactly where the rule holds and where it bends.
| Return | Rule of 72 | Exact |
|---|---|---|
| 2% | 36.0 yrs | 35.00 yrs |
| 4% | 18.0 yrs | 17.67 yrs |
| 6% | 12.0 yrs | 11.90 yrs |
| 8% | 9.0 yrs | 9.01 yrs |
| 10% | 7.2 yrs | 7.27 yrs |
| 15% | 4.8 yrs | 4.96 yrs |
| 20% | 3.6 yrs | 3.80 yrs |
Assumes a constant annual compound return, which no real investment delivers — treat every figure as a planning estimate, not a promise. General information, not financial advice.
A rule older than logarithms
The Rule of 72 first appears in print in Luca Pacioli's Summa de arithmetica, geometria, proportioni et proportionalita, published in Venice in 1494 — the same volume that codified double-entry bookkeeping. Pacioli states the rule as common merchant knowledge and offers no proof, which is the remarkable part: John Napier would not publish the logarithms that actually explain the rule until 1614, a hundred and twenty years later. Renaissance traders were using a correct consequence of a mathematics that did not exist yet.
Where the shortcut bends
The exact doubling time is ln(2) ÷ ln(1 + r). For small rates, ln(1 + r) ≈ r, which would make the ideal constant 100 × ln(2) ≈ 69.3 — the "Rule of 69.3" that textbooks mention and nobody uses, because 72 divides beautifully and 69.3 doesn't. The choice of 72 over 69.3 builds in a small overestimate that, by luck and design, almost perfectly offsets the approximation error from annual compounding at rates near 8%. Below that, the rule overshoots (it claims 36 years at 2%; reality is 35.0); far above it, it undershoots badly enough to matter for anyone modelling debt at credit card rates. A useful patch quants use: add 1 to the constant for every 3 points above 8% — at 24%, "Rule of 77" lands within days of the truth.
The dark-side uses are the most persuasive ones
Doubling time is most vivid where compounding works against you. A credit card balance parked at 20% doubles in 3.8 years — untouched, a $5,000 debt becomes $10,000 before a federal election cycle ends. Inflation at 3% halves the buying power of cash in about 23.4 years, which is a polite way of saying a retirement fund "safely" in a zero-interest account loses half its groceries in a generation. And a 2% annual investment fee, compounding against a 7% gross return, is the difference between doubling in 10.2 years and doubling in 14.2 — the fee eats four years of your life per doubling. Same rule, three villains.
Questions people actually ask
Why 72 and not 69.3?
The mathematically "natural" constant is 100 × ln(2) ≈ 69.3, which is exact for continuous compounding. But 69.3 is miserable to divide in your head, while 72 splits cleanly by 2, 3, 4, 6, 8, 9 and 12 — and the small upward bias of 72 happens to correct for annual (rather than continuous) compounding at everyday rates. It is a rule optimised for mental arithmetic, not for precision, and around 8% those two goals coincide almost perfectly.
Does the rule work for debt and inflation too?
Yes — anything that compounds. At a 20% credit card rate, an untouched balance doubles in about 3.8 years (the rule says 3.6). At 3% inflation, money under the mattress halves in buying power in roughly 23.4 years. The rule is symmetric: it tells you how fast compounding works for you or against you.
How old is the Rule of 72?
At least as old as printed arithmetic. It appears in Luca Pacioli’s Summa de arithmetica, published in Venice in 1494 — the same book that gave the world double-entry bookkeeping — where Pacioli presents the rule as already-known practice among merchants, without deriving it. Traders were mentally doubling ducats with it before anyone had invented the logarithms that explain why it works.
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Sources
- Luca Pacioli, Summa de arithmetica, geometria, proportioni et proportionalita (Venice, 1494) — earliest known printed statement of the rule.
- John Napier, Mirifici Logarithmorum Canonis Descriptio (1614) — the logarithms that later explained it.
- Exact values on this page computed as ln(2)/ln(1+r); the table's rule-vs-exact pairs are reproducible from those two formulas alone.
Related
Five centuries on, the rule's real lesson isn't the 72 — it's that the merchants who memorised it understood compounding, and most of the people it compounds against still don't.