The rule of 69

The rule of 69 is used for estimating an investment's doubling time. The rule number (69) is divided by the interest percentage per period to obtain the approximate number of periods (usually years) required for doubling. The rule is useful for quick, mental calculations i.e. To estimate how long it would take for an investment growing at 3% annually, you would do 69/3 = 23 years.

If a is the amount growing at x % and n the number of years needed to double the amount then, the following relation must be met:

(1+x/100)^n . a = 2 . a <=> (1+x/100)^n = 2
n . ln (1+x/100) = ln(2) <=> n = ln(2) / ln(1+x/100)

For small values of x/100 we can approximate ln(1+x/100) with x/100 and the equation above becomes n ~= ln(2) / (x/100) = 100.ln(2)/x
100.ln(2) = 69.315
n ~= 69.3/x