The rule of 72

Question

Bankers say that money growing at r per cent a year doubles in about 72 divided by r years: 9 years at 8 per cent, 12 years at 6 per cent, 24 years at 3 per cent. 1. Where does the number 72 come from? 2. How accurate is the rule at 1 per cent, 8 per cent and 30 per cent? 3. Why do people use 72 rather than the "correct" constant?

Accepted Answer

The exact doubling time is ln 2 divided by ln(1 + r/100). For small rates that is about 69.3 divided by r, and 72 is a convenient nearby number with many divisors that also corrects for the approximation at typical rates. The derivation. Doubling means (1 + r/100) to the n = 2, so n = ln 2 / ln(1 + r/100). Since ln 2 = 0.693 and, for small x, ln(1 + x) is approximately x, this is about 0.693 / (r/100) = 69.3 / r years. Why 72 and not 69. The approximation ln(1 + x) is about x slightly overstates the log, so 69.3/r slightly understates the doubling time. A better fit is 69.3/r + 0.33, and arou…

The rule of 72

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