# 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

Here is a graph showing $1 doubling at different interest rates.

The growth of `A=$1` at 3% (the slowest growing), 5%, 8%, 10% and 12% (the fastest growth), showing (with a pink dot) when it doubles to value $2.

#### 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
$2*A*.

2

A=A(1 +r)^{t}

Cancelling gives:

2 = (1 +

r)^{t}

Using logarithms to solve this equation, we have (recall `ln` means `log_e`):

ln 2 =

tln(1 +r)`t=(ln\ 2)/(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:

`t=(ln\ 2)/(ln(1+r))=(ln\ 2)/(ln (1.03))=23.45`

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, we find

`23.45 xx 3 = 70.35`.

We see that this (value of years) `xx` (interest rate) is quite close to 72.

### Range of interest rates

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 `xx` 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:

`t~~72/r`

Equivalently, we can approximate *r* (the interest rate needed to double our money for a given interest time, *t*) as:

`r~~72/t`

Here is a graph of the curve `r=72/t` with the values given in the above table (the pink dots). We can see the curve very closely approximates the situation for a wide range of interest rates.

The graph of `r=72/t` (our approximation), with the actual times for doubling our money for various interest rates from the table above.

### Conclusion

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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