Growing 1% a month, how long to double, and what annual rate?

Question

Something grows at a steady 1\% per month. Solve in your head. How many months until it doubles, and what effective annual rate is that? Aim: under 40 seconds.

Accepted Answer

Apply the rule of 72 in months: \frac{72}{1} = 72 	ext{ months} = 6 	ext{ years.} For the annual rate, compound twelve months: 1.01^{12} - 1. Approximate the exponent by the doubling logic or directly: 1.01^{12} \approx 1 + 12(0.01) + \binom{12}{2}(0.01)^2 \approx 1 + 0.12 + 0.0066 \approx 1.127. So the effective annual rate is about 12.7\%, and the doubling takes 72 months (6 years). Takeaway. The rule of 72 works in any time unit, feed it a monthly rate and you get a doubling time in months. And note 1\% monthly is not 12\% a year but nearly 12.7\%, because the monthly gains compound on e…

Growing 1% a month, how long to double, and what annual rate?

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