Quick answer: the Rule of 72 says money doubles in about 72 ÷ rate years. At 9% it is nearly perfect (8.0 vs an exact 8.04 years). At 3% it overshoots by 2%, and at 24% it overshoots by 23% — the rule is a 6–12% specialty tool wearing a universal costume.
Exact doubling time solves (1 + r)ⁿ = 2, which gives n = ln(2) ÷ ln(1 + r). That is not mental math. The Rule of 72 replaces it with division — and division is mental math. The constant 72 is a compromise chosen because it divides cleanly by 3, 4, 6, 8, 9 and 12, and because it lands close to the exact answer in the rate range where most investing actually happens.
| Rate | Rule of 72 | Exact years | Rule's error |
|---|---|---|---|
| 3% | 24.0 | 23.45 | +2.3% too slow |
| 6% | 12.0 | 11.90 | +0.8% too slow |
| 9% | 8.0 | 8.04 | −0.5% (nearly perfect) |
| 12% | 6.0 | 6.12 | −2.0% too fast |
| 24% | 3.0 | 3.26 | −8% too fast… see below |
Two things worth noticing. First, the error changes sign around 8–9% — below that the rule overestimates doubling time (pessimistic), above it, the rule underestimates (optimistic). Second, at high rates the drift compounds: 24% is not 2% off but 7.9% off (3.0 claimed vs 3.26 actual years), and the gap keeps widening as rates climb. Credit-card debt at 42% APR does not double "every 1.7 years" — it doubles every 1.96 years, which is worse.
The mathematically honest shortcut for continuous compounding is the Rule of 69.3 — because ln(2) = 0.693. For annual compounding, 69.3 ÷ rate is closer than 72 ÷ rate at low rates: at 3%, it gives 23.1 vs an exact 23.45 (1.5% error instead of 2.3%). The practical hierarchy: 69.3 when you're modeling, 72 when you're talking, and the compound interest calculator when the answer actually moves money.
The same rule runs in reverse, and this is the version people skip. At 6% inflation, purchasing power halves in about 72 ÷ 6 = 12 years. Cash sitting at 3% while inflation runs 6% is not "safe" — it loses a third of its value in roughly 13.5 years (the exact half-life at −3% real is ln(2) ÷ ln(1.03) ≈ 23.4 years, but at −3% real rate the rule's error direction flips too — compute it). Your real return is approximately nominal minus inflation, and the doubling rule applies to that real number, not the headline rate.
For rates between 6% and 10%, yes — errors stay under about 2%. Outside that band, or for anything you'll act on, run the exact formula. The rule's job is to make compounding feel real in conversation, not to price an investment.
Tradition plus divisibility: 72 is divisible by 2, 3, 4, 6, 8, 9 and 12, which makes mental answers clean, and it happens to minimize error near the 8–10% rates where the rule was most used. 69.3 is the mathematically pure constant; 70 is a common middle-ground.
Yes, with the same caveats. A −6% real return halves purchasing power in about 12 years. The rule is symmetric — it just describes doubling or halving time for any constant growth rate.
Divide ln(2) by ln(1 + rate): at 12%, ln(2) ÷ ln(1.12) = 6.12 years. Or skip the logarithms entirely — the compound interest calculator prints exact growth timelines for any rate and contribution schedule.
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