“At this rate, how long until my money doubles?” There’s an answer that needs no calculator and no spreadsheet: divide 72 by the annual rate. At 6%, doubling takes 12 years. At 4%, 18. At 9%, 8. That’s the rule of 72, the longest-lived piece of mental math in finance — and the calculator above also tells you exactly how wrong it is.
How it works
A sum growing at a steady rate has doubled when it reaches twice what you started with. The rule approximates that moment with one division:
years to double ≈ 72 / annual rate in %
The exact answer, for the formula-curious, is t = ln(2) / ln(1 + r): you ask after how many years (1 + r)ᵗ equals 2, and logarithms do the rest. The calculator shows both, side by side, with the gap in months.
A worked example
Say you have $10,000 growing at a hypothetical 7% a year, gains reinvested. The rule says 72/7 ≈ 10.3 years to reach $20,000. The exact answer is 10.2 years — the rule is off by about two weeks over a decade. For arithmetic you can do at a red light, that’s a forgivable error.
The part that surprises people comes next: the second doubling costs no extra time. Another 10.2 years and the $20,000 becomes $40,000. Same rate, same wait, twice the money — that’s compounding, and the rule of 72 is the fastest way to feel it. For the year-by-year picture, there’s the compound interest calculator.
Why 72, exactly
The “true” constant would be about 69.3 — that’s ln(2) × 100, which falls out of the exact formula when rates are small. But 69 is miserable to divide in your head. 72 isn’t: it splits cleanly by 1, 2, 3, 4, 6, 8, 9 and 12 — nearly every rate you’ll meet in practice. And the slight rounding up cancels the approximation error right where rates usually live: at 8% the rule is essentially exact. Below that it overestimates a little (at 1% it says 72 years against a true 69.7); above, it underestimates (at 20%, 3.6 against 3.8). Try the extremes above and watch the error line.
This isn’t a modern advisor’s trick, either: the rule’s first appearance in print is in Luca Pacioli’s Summa de arithmetica — Venice, 1494, digitized in the sources below. Pacioli presents it as already well known, with no explanation: people were estimating doublings with it long before logarithms existed to prove it.
Read it backwards
The same division measures what grows against you:
- Inflation. At 3% a year, prices double in about 23 and a half years — which means cash sitting still loses half its purchasing power in the same time. The calculator shows that line for whatever rate you enter.
- Debt. A balance at 12% you never pay down doubles in just over 6 years. The 72 is neutral: it measures compound growth, whichever direction it cuts.
One honest caveat: the rate you type in is a steady assumption, not a promise. Real returns wobble year to year, and higher returns always travel with higher risk. The rule is for comparing orders of magnitude in two seconds — for real decisions, run the full numbers. And if doubling times got your attention, the natural next step is what waiting to start actually costs: that’s where doubling years turn into real dollars.