## The Rule of 72

Did you know that you can instantly calculate in your head the number of years that a quantity will double in, given its annual rate of growth? Picture yourself in a conference or discussion where, for example, someone says that salaries in India are growing at an annual rate of 12% and you can say in a heartbeat “but that means salaries in India will double in about 6 years!”

As people wonder at your brilliance, you can pat yourself on the back for having learned “The Rule of 72.” No one need know that all you did was to divide 72 by 12, the annual growth rate, to arrive at 6, the approximate number of years in which the salaries would double.

I learned this trick, oddly enough, through a footnote in a high school economics textbook (Economics, by Lipsey and Steiner, now in its 13th edition). I was under the impression that everyone was in on this trick. However, I’ve come to realize that it is not as widely known as I expected. I decided to share it with my readership as a reward for your interest in my blog so that you, too, can amaze friends and colleagues.

In summary, here’s a table showing the number of years in which something will double, for annual growth rates ranging from 1% to 20%.  All you need to do is divide 72 by the annual growth rate. You will notice that most of the numbers in the “years to double” column are whole numbers, not fractions. This is why the Rule of 72 is so useful. In most cases, the division is very simple, since 72 has so many factors (2, 3, 4, 6, 8, 9, etc.).

Many users of the Rule of 72 don’t know that it should really be The Rule of 69. We use 72 because of this neat feature of easy divisibility.  You wouldn’t appear so sharp if you had to calculate 69 divided by 12 in your head, for example.

Why does The Rule of 72 work? It has to do with certain properties of natural logarithms. For those who are interested, here is the story.

FV = PV (1+r)^n

Where FV is the future value, PV is the present value, r is the annual interest rate (or annual rate of growth for our purposes) and n is the number of periods. In our example a period is a year. Knowing that the future value is twice the present value, the equation reduces to:

2 . PV = PV (1+r)^n

This in turn reduces to

2 = (1+r)^n

Taking natural logarithms on both sides of the equation, we get

ln  = ln [(1+r)^n]

Since we know from the properties of logarithms that ln a^b = b . ln a, and that the natural logarithm of 2 is approximately equal to 0.69, the equation reduces to

0.69 = n . ln (1+r)

Isolating n on the left hand side of the equation gives us

n = 0.69/ ln (1+r)

Now we take advantage of another property of natural logarithms, i.e., ln (1+r) is approximately equal to r when r is relatively small, to get

n = 0.69/r

Since .69 is not so nice to divide into (as discussed above), we replace it with .72 and then multiply numerator and denominator by 100 so we are dealing with whole numbers and the interest rate r can b expressed as a percentage rather than a decimal value.

n=72/r

Remember that this is an approximation. If you do the actual math using the compound interest formula you will get the exact answer. However, for most purposes, the Rule of 72 should work just fine. Enjoy! 1. Priyank says: