# The Rule of 69...er, I mean...72

## A nifty trick to work out how long it takes for savings to double

26 January 2022

The finance industry has a clever mathematical trick for working out how long it will take for the value of an investment to double. It's known as The Rule of 72.

To work out the doubling time, simply divide 72 by the compound interest rate. For example, if your savings are growing at 4% per year (lucky you!), then it will take 72/4 = 18 years for your savings to double. With a growth rate of 6% it'll take only 12 years. And if inflation is running at 5%, then it will take 72/5 or roughly 14 years for prices to double (or the purchasing power of £1 to halve).

The Rule of 72 works for the spread of viruses, too. At the peak of the covid pandemic, it was common for the number of infections to be growing at 10% per day, which meant the doubling time was 72/10 which is roughly every seven days.

The Rule of 72 is only an approximation, but it is a remarkably accurate one.

The maths behind it is at the bottom of this blog, and it reveals that the Rule of 72 should really be the ** Rule of 69. ** However, as I like to explain to school audiences, there's a very good reason why teenagers shouldn't use the Rule of 69.

The reason is that....

(pause for audience sniggering)....

...the number 69 doesn't have many factors, whereas 72 is exactly divisible by 2, 3, 4, 6, 8 and so on making it a much more convenient number. What other reason could there possibly be?

(Actually there is another reason - in the italicised footnote at the bottom of the blog.)

**THE MATHS BEHIND THE RULE OF 72 (or 69)**

Suppose you have an investment M which is growing at compound rate of R% per year. We want to know how many time periods (N) it will take for M to get to 2M.

In other words, for what value of N does:

M x (1+R)^N = 2M

The Ms cancel, and taking natural logs of both sides:

N ln(1+R) = ln (2) = 0.69 (approx)*

When R is small, ln (1+R) = R (one of the laws of small numbers) and so:

NxR = 0.69, and hence N = 0.69 / R or, in layman's terms N = 69% / R

**The approximation that changes ln(1+R) to R gives a value that is slightly too small, and this error increases as R increases. Changing the ratio from 0.69 to 0.72 makes the approximation more accurate for a wider range of values of R, which is another reason why Rule of 72 beats Rule of 69.*

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