Rule of 72 (Doubling Time)

Divide 72 by the percentage rate and you have the years to double: 6% doubles in about twelve years, 9% in eight, 12% in six. It is the most useful piece of mental arithmetic in personal finance, and it has been in print since Luca Pacioli's Summa de Arithmetica of 1494, stated without proof as something merchants already knew.

The exact constant is not 72. Doubling requires $n \ln(1+i) = \ln 2$, and for small rates that is close to $0.693/i$, so 69.3 would be more accurate, and exactly right for continuous compounding. Seventy-two is used because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because the small upward fudge happens to compensate for the approximation across the range of rates people actually meet. It is at its best between about 6% and 10% and drifts noticeably above 20%.

The rule cuts both ways, which is the part worth remembering. At 3% inflation, prices double in 24 years, so a fixed pension halves in purchasing power over an ordinary retirement. The same arithmetic that makes savings look encouraging makes inflation look alarming.