The "Rule of 72": A Gentle Reminder

Did you know you can instantly calculate the number of years a quantity will double in, given its annual growth rate?

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 learning “The Rule of 72.” No one needs to 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 through a footnote in a high school economics textbook (Economics, by Lipsey and Steiner). I was under the impression that everyone was in on this trick. However, I’ve realized it is not as widely known as expected.

Everyone can be a math hero in a meeting!

Look at the table below showing the years in which something will double for annual growth rates ranging from 1% to 20%. To calculate the number of years, you need only 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 helpful. The division is usually straightforward since 72 has many factors (2, 3, 4, 6, 8, 9, etc.).

Many users of the Rule of 72 don’t know it should 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.

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

We start with the formula for compound interest:

FV = PV (1+r)^n

FV?is the future value,?PV?is the present value,?r?is the periodic interest rate (or annual growth rate 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 [2] = 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 0.69 is not so nice to divide into (as discussed above), we replace it with 0.72 and then multiply the numerator and denominator by 100. Thus, we are dealing with whole numbers, and the interest rate r can be expressed as a percentage rather than a decimal value.

n = 72/r

Remember that this is an approximation. You will get the exact answer if you do the math using the compound interest formula. However, the Rule of 72 should work fine for most purposes. Enjoy!

(I published the original version of this article in The Nelson Touch blog in the early days of people analytics):

https://nelsontouchconsulting.wordpress.com/2011/02/28/the-rule-of-72/.