A table will help explain why the rule of 72 works. You will also see where the 72 came from.
Consider the compound interest formula.
A = P( 1 + r/100 )n
A is the resulting amount of money.
P is the principal invested for n interest periods at r% annually.
We need to investigate what happens when the money is doubled or when A = 2P.
When A = 2P, the equation above becomes 2P = P( 1 + r/100 )n
After dividing both sides of the equation by P, we get 2 = ( 1 + r/100 )n
Now, solve for n.
Take the log of both sides of the equation.
log 2 = log ( 1 + r/100 )n
log 2 = n log ( 1 + r/100 )
n = log2 / log (1 + r/100)
Now, here is a table showing different values for r, n, and nr.
The average of the nr values gives 71.81 which is close to 72 and this is where our rules of 72 came from.
Notice that n in the table represents the number of years it will take the money to double. For example, when nr = 72.389, and the interest rate or r is 9, it will take about 8.043 years for the money to double.
Thus, the formula nr / r or 72 / r makes sense!
Check the exponential and logarithmic functions unit if you do not know how to solve these logarithmic equations.