Rule of 72
There is a curious and helpful trick that allows us to mentally estimate annual compound interest amounts, where we are interested in doubling our money.
The Rule of 72 works as follows.
If we want to know how long it will take for our money to double, just divide 72 by the interest rate.
So for example, if the interest rate is 10%,
72 ÷ 10 = 7.2 years
So it will take just over 7 years to double our money.
If the interest rate is 8%, to double our money it will take
72 ÷ 8 = 9 years
Finding Interest
We can use the Rule of 72 the other way around too. Say we have a 15 year time span and we want to double our money in that time. What interest rate do we need so that the money will double?
Answer: 72 ÷ 15 = 4.8%
How Does Rule of 72 Work?
From the last section (Interesting Interest), the amount of money we have after investing P dollars for t years at r% interest (as a decimal) is given by:
A = P(1 + r)t
We want to know how long it takes to double our $A to $2A.
2A = A(1 + r)t
Cancelling gives:
2 = (1 + r)t
Using logarithms to solve this equation, we have:
ln 2 = t ln(1 + r)
We can find the value of the right hand side for different values of r. When we multiply these values by r, an interesting thing occurs − the values are very near 72.
Example
If r = 3% = 0.03, then:
This means it would take more than 23 years to double our money at an interest rate of 3%.
Now multiplying 23.45 by 3 and we find
23.45 × 3 = 70.35.
We see that this value of years × interest rate is quite close to 72.
Let's now do the same for a range of typical interest rates from r = 2 through to r = 14.
We get:
(Rounded to 2 decimal places)
Rate Years Rate × Years 2% 35.00 70.01 3% 23.45 70.35 4% 17.67 70.69 5% 14.21 71.03 6% 11.90 71.37 7% 10.24 71.71 8% 9.01 72.05 9% 8.04 72.39 10% 7.27 72.72 11% 6.64 73.06 12% 6.12 73.40 13% 5.67 73.73 14% 5.29 74.06
We observe that the values in the last column are near 72. So we can approximate t (the time it takes to double our money for a given interest rate, r) as:
The Rule of 72 gives us an easy "back of the envelope" calculation for the time it will take to double our money. It is good for a range of typical interest rates, from 5% to about 12%. Even for high interest rates like 20%, the value is 76.04.
If you are interested, go back to 
Interesting Interest.
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