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.
r | n | nr |
1 | 69.660 | 69.660 |
2 | 35.002 | 70.004 |
5 | 14.206 | 71.03 |
9 | 8.043 | 72.389 |
12 | 6.116 | 73.392 |
15 | 4.959 | 74.392 |
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.
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