Here's Why The 'Rule Of 72' Is Such A Great Math Hack Wikimedia CommonsLeonhard Euler

We recently posted a list of handy math tricks, and among them is a quick way to estimate how long it will take to double an investment with a given rate of return.

That trick is the rule of 72: Take your interest rate, and divide it into 72.

For example: If you expect a 6% average annual return, the doubling time will be 72/6 = 12 years.

That's much easier than trying to reverse engineer some complicated compound interest formula.

Why It Works

The reason this works comes from the basic formula for the future value of an investment receiving compound interest continuously.

If you have an initial investment principal of P, and an annual interest rate r, then after t years your investment will have a value of Pert, where e is Euler's number, an irrational number that shows up all over the place in mathematics.

We're interested in figuring out how long it will take for an investment to double with a given interest rate, so we want to know how long it will take for our investment to grow from P to 2P. In other words, we want to solve the equation 2P = Pert for the time t.

First, assuming our principal P is not zero, we can cancel out the P's on both sides of the equation, giving us 2 = ert.

Now, we have to get rid of the exponential function on the right hand side. Fortunately, the natural logarithm function, usually written as "ln", can do this for us by definition: Logarithms are the inverses of exponential functions, and the natural logarithm "undoes" an exponential function with a base of e. This gives us ln(2) = ln(ert) = rt.