The rule of 72 estimates how many years money takes to double: divide 72 by the yearly interest rate in percent, so 8% gives 9 years. We tested it against the exact doubling time at every hundredth of a percentage point from 0.5% to 30%. When the rate is a nominal yearly rate with interest added monthly, as with an APR, 72 was never within 1% of the exact answer, giving 2.6% to 3.85% too many years throughout. Dividing 70 by the rate instead stayed within 1% across the range. With interest added once a year, 72 is exact at about 7.85%, a crossover Michael James published in 2010, and within 1% only from about 5.7% to 10%.
The divisor 72 hits the exact doubling time at a single yearly rate
The doubling time is the years a balance needs to reach twice its size. Interest added to the balance, which then earns interest itself, is compounding, and each point at which it is added is a rest. With one rest a year at rate r, the exact doubling time is ln 2 divided by ln(1 + r), where ln 2 is about 0.693.
That makes 69.3 the right divisor for interest added continuously; yearly rests grow more slowly, and the extra in 72 offsets that near one rate only. Wikipedia derives the rate as about 7.79%; Michael James put it at "about 7.85%". Below that rate the rule gives too many years, 36 against an exact 35.0 at 2%, and above it too few, 5.3% short at 20%. Wikipedia's table already marks 70 as closer at low rates; our sweep puts the switch at 4.9%.
Estimates from the rule shoot high at every rate when an APR is compounded monthly
An APR is a nominal rate split into twelfths, one added each month, so the balance outgrows the headline rate and 72 overshoots. An effective rate (EIR, APY or AER) already includes that effect, so the yearly formula is exact for it however often interest is added.
| Rate | Exact years, added yearly | 72, added yearly | Exact years, APR added monthly | 72, APR monthly | 70, APR monthly |
|---|---|---|---|---|---|
| 4% | 17.67 | +1.85% | 17.36 | +3.70% | +0.82% |
| 8% | 9.01 | −0.07% | 8.69 | +3.53% | +0.65% |
| 15% | 4.96 | −3.22% | 4.65 | +3.23% | +0.36% |
| 30% | 2.64 | −9.16% | 2.34 | +2.60% | −0.25% |
Plus means the rule gives too many years and minus too few, by our arithmetic. Robinhood and Wikipedia already favour 69 or 70 for frequent compounding; the measured gap is our addition.
Until 16 October 2026 our rule of 72 calculator assumed interest was added once a year without saying so, even in its own 22% credit card example: US$5,000 over 10 years came to about 36,500, where the same APR added monthly gives about 44,200. Zero and negative rates blanked the result silently, the error had no sign, and a fact card placed the rule's undershoot "above about 10%", where it begins above 7.85%. The revised version asks how interest is added. With 8 entered and "Monthly (rate is an APR)" chosen, it returns 9.00 years by the rule against 8.69 exact, and says the rule gives 0.31 years too many; a negative rate now returns an exact halving time.
An adjusted divisor zeroes in on exact doubling for yearly interest
W. E. Smith, in the UNSW school magazine Parabola in 2000, derived a divisor that rises with the rate, about 69.3 plus 0.347 times the rate. A rounded form adds a third to 72 for every point above 8%, so 20% gets 76. Across our yearly sweep it stayed within 0.10% of the exact time, a little closer than Smith's own slope.
It suits yearly rests alone: on an APR added monthly it gives 6.6% too many years at 15% and 13% too many at 30%, where plain 70 does better, as the table shows.
Pacioli aims the rule at a yearly rate without explaining the number
The rule is printed in Luca Pacioli's Summa de arithmetica (Venice, 1494), folio 181 recto, problem 44, read here in the Wellcome Collection's scan. Wikipedia and most pages quote Giovanni Vacca's 1915 transcription, reproduced by MacTutor with a rough translation: "you hold the rule 72 in mind, which always you will divide by the interest".
Vacca shortened and modernised it. The print adds "a far capo al ãno" after the rule and "a capo d'anno" after the 6% example, which we translate as "closing the account at each year's end": interest added yearly. Problem 40 on the same page opens "a non far capo d'anno", for simple interest, which supports that reading. Vacca also dropped a second example: at 8%, divide 72 by 8 to get 9 (our translation). That reading of the abbreviated type is our own, unchecked by a specialist.
Pacioli gives no reason for 72. Vacca wrote that it was "impenetrable" to later arithmeticians such as Tartaglia.
CPF floor rates and the SSB average make 70 the closer shot
The CPF Board pays 2.5% on the Ordinary Account for October to December 2026, its legislated minimum, and 4% on the Special, MediSave and Retirement Accounts, a floor extended to 31 December 2027. Its interest page says interest is "computed monthly, compounded and credited yearly", so the yearly formula applies; the extra 1% on the first S$60,000 is left out.
At 2.5% the exact doubling time is 28.07 years; 72 gives 2.6% too many and 70 gives 0.25% too few. At 4% the exact figure is 17.67. For that rate Singapore's MoneySense gives "72/4 = 18 years", 1.85% too many; 70 gives 17.50, 0.98% too few and only just inside 1%.
A Singapore Savings Bond is a 10-year government bond whose coupons (interest payments) step up. MAS's notice for the October 2026 issue, GX26100Z, gives a 10-year "average annual return" of 2.32%. We could reproduce it only as a half-yearly yield on the April and October coupons, our reconstruction of the method. The exact doubling time is then 30.05 years; 72 gives 3.3% too many and 70 only 0.4% too many. A holder's balance does not double: the bond matures after 10 years and pays its coupons out, so the arithmetic assumes reinvestment at the same rate. These rates are arithmetic inputs, implying no view on any account or bond.
Halving estimates fall wide of the mark at negative rates
A balance that loses a fixed share r each year halves in ln 2 divided by minus ln(1 − r). Dividing 72 by the size of the loss always gives too many years, more so than for an equal gain. At a 5% yearly loss the rule says 14.4 years against an exact 13.51, 6.6% too many; for a 5% gain the error is 1.4%. At a 20% loss it is 15.9% too long.
Inflation, often described as halving, follows the doubling formula. Prices rising 3% a year double in 23.45 years, and cash loses half its purchasing power over the same span, so the rule's 24 is 2.3% too many, the same as for savings at 3%. Which price measure such a rate comes from is the subject of the guide to MAS core inflation.
