Rule of Seventy Two

"Rule of 72"  is an approximate formula to find how much time it takes for an investment to double with compound interest.  It can also be used in other things that grow similar to interest.

Some scenarios are below. 

• If you invest 100$ with annual percentage of 6%, it takes approximately 72/6 or twelve years to double the original investment. 
• If you invest 100% with annual percentage of 9%, it takes approximately 72/9 or eight years to double the investment value. 
• If the inflation year over year is 4%, the money will lose half of the current value in eighteen years.
• If bacteria grows 3% every hour, it will double in size in twenty four hours or a day.
• As the population of India is growing at 1.2% from 1B people, the population if India will become 2B people in 72/1.2 or sixty years from now.
• If the trend of the GDP growing at 6% annually continues, it will become twice the current GDP in 72/6 or twelve years.
• Suppose an exponential program on n inputs runs for 10 seconds and the increasing input by one takes 11.2 seconds or 12% over 10 seconds, then increasing the number of inputs by six more inputs requires 20 seconds to run the program.

Seventy two is chosen so that it has several divisors and can be convenient for computation. Choosing sixty nine or seventy improves accuracy.  The formula is correct between 6% and 10%, The accuracy for every 3% away from 8% can be improved by adding additional 1 to the value of 72. If you invest 100$ with annual percentage of 25%, it takes 75.66/25 or little more than three years to double the investment value.

Verification from the basic mathematics

S = Sum, P = principal, R = interest rate percentage, r = interest rate fraction = R/100.
The sum after one periodic interval: S = P  + PR/100 =  P(1 + R/100).
The sum after two periodic intervals:  S = P(1+r)(1+r).
The sum after n periodic intervals:  S  = P(1+r)^N

When S = two times the value of P,   2   =  (1+r)^N
Number of periods can be calculated by taking logarithm on both sides.

log(1 + r) = r for small values between 5 to 10%.
N = log(2)/log(1 + r)  = 0.6933/r = 69.33/R

References:
Wikipedia - Rule of 72
Investopedia - Rule of 72
Exponential double time  - Rule of 72
Better Explained - Rule of 72