What it works out
The rule of 72 is a piece of mental arithmetic. Divide 72 by a yearly rate of growth, as a percentage, and the answer is roughly the number of years a sum takes to double at that rate. At 6% it gives 12 years; the exact figure is 11.90.
It works because the exact answer, ln 2 ÷ ln(1 + r), is close to 69.3 divided by the percentage when the rate is small, and a little more than that as the rate rises. 72 is near the right number for everyday rates and divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes it easy to use in the head. The rule is closest at about 8%. Below that it gives a slightly long time and above it a slightly short one.
The rule is old: it appears in Luca Pacioli’s Summa de arithmetica of 1494, stated without a proof.
It describes anything that compounds. A debt on which interest is left unpaid, and prices under steady inflation, double on the same timetable.
What the sum leaves out
- The same rate is assumed every year, compounded once a year. No investment grows like that.
- At high rates the rule drifts. At 36% it says 2 years and the exact figure is 2.25.
- A doubling time describes a rate. It is not a promise that anything will double.
Questions people ask
- Why 72 and not 69 or 70?
- For continuous compounding the exact number is about 69.3, and 70 is sometimes used. With compounding once a year the best number rises with the rate, and 72 fits rates near 8%. It also has many divisors, which is the practical reason it is the one remembered.
- How accurate is the rule of 72?
- Within about a tenth of a year for rates between 6% and 11%. Outside that it drifts: at 2% it says 36 years against an exact 35.00, and at 20% it says 3.6 against 3.80.
- Does the rule of 72 work for inflation?
- Yes, for a steady rise in prices. At 3% a year prices double, and a fixed sum loses half of what it buys, in roughly 72 ÷ 3 = 24 years. The exact figure is 23.45.
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