# Why the rule of 72 works

A table will help explain why the rule of 72 works. You will also see where the 72 came from and learn how to use the rule of 72 to solve investment problems. For example, how long it takes to double an investment of \$2500 if the interest rate is 5%?

When entering the value of r in the formula, do not convert the rate into a decimal. If r = 2%, just replace r with 2.

Time it takes to double an investment of \$2500 is 72 / 5 = 14.4 years.

Notice that the amount of money you invest is irrelevant when using the rule of 72.

For example, how long it take to double an investment of \$500,000 if the interest rate is 5%? Again, it will take about 14.4 years to double any amount as long the following two conditions are met.

• The interest is compounded annually.

• Money is not added to the account.

## Now, why the rule of 72 works

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 the rule 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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