Quickly estimate how many years it takes an investment to double.
Years to double ≈ 72 / annual interest rate. A well-known quick approximation, not exact for very high or low rates.
The Rule of 72 is a mental-math shortcut: divide 72 by an annual interest rate to estimate how many years it takes an investment to double, with no calculator required. It works because it approximates the true exact formula — years to double equals the natural log of 2 divided by the natural log of (1 + rate) — and 72 was chosen over the mathematically "purer" 69.3 specifically because 72 divides evenly by far more small numbers (2, 3, 4, 6, 8, 9, 12), making the mental arithmetic genuinely easier at everyday rates.
The approximation is most accurate in roughly the 6%-10% interest rate range, which happens to cover most typical long-term investment return assumptions — that's not a coincidence, it's why the rule became popular in the first place. Outside that range accuracy degrades: at very low single-digit rates or especially at high rates above 15-20%, the Rule of 72 increasingly underestimates the true doubling time. A useful refinement for rates far from 8%: adjust the divisor by roughly 1 for every 3 percentage points the rate differs from 8% (use 71 for a 5% rate, 73 for an 11% rate) to keep the estimate closer to the exact ln(2)/ln(1+rate) answer.
Very accurate in the roughly 6%-10% interest rate range, where it stays within a small fraction of the exact answer. Accuracy degrades outside that range — at very high rates (20%+) it noticeably underestimates the true doubling time, and it's also slightly less precise at very low single-digit rates.
The mathematically exact constant is approximately 69.3 (from the natural log of 2), but 72 is used instead because it divides evenly by many more small numbers — 2, 3, 4, 6, 8, 9 and 12 — making the mental division dramatically easier, while the small upward adjustment happens to also partially offset the approximation's error at typical rates.
Years to double = ln(2) ÷ ln(1 + rate), where 'ln' is the natural logarithm and rate is expressed as a decimal (e.g., 0.08 for 8%). This gives the mathematically exact doubling time for any compounding rate, unlike the Rule of 72's fixed approximation.
Yes — the same math applies to anything that grows (or shrinks) at a steady compounding rate, including inflation's erosion of purchasing power (72 ÷ inflation rate estimates years until prices double) or a country's GDP growth. The same 6%-10% accuracy sweet spot and high-rate limitations apply in every case.