# The so-called rule of 72 (or rather, 69)

This BBC article discusses the 'rule of 72' - essentially along the lines that questions to do with economic growth and inflation and so forth can be approximated by a simple formula using the number 72. At the end of the article, it says that a more accurate number to use is '70 or even 69', which leads me to suspect that the 'real' number is $69 + \epsilon$, for $\epsilon \in (0,0.5)$. The reason that 72 is used instead is that it has a large number of small divisors.

My question is this: where does this number come from?

I suspect it will be derived from $e$...

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Please see the following article It is thorough, well-written, and will tell you everything that you need, and more. However, if you prefer a less well-written summary, please read on.

Suppose that your investment accrues interest at the rate $r$ per period, with interest compounded every period. The period might be a year, a half-year, a day, or interest might even be compounded continuously. Or else, to be pessimistic, you have a debt, and interest accrues on it at rate $r$ per period.

Please note that we are using the mathematician's notion of interest rate. For example an interest rate of $8\%$ gives $r=0.08$. Actually, this is the same as the ordinary notion, since $8\%$ is an abbreviation for $8$ per centum, that is, $8$ per $100$, or $8/100$. But non-mathematicians are ordinarily more comfortable with $8$ than with $0.08$.

The Rule of $72$, and variants, have to do with the approximate doubling time, in periods, of your investment or debt.

By the formula for compounded growth at interest rate $r$, $A$ dollars grow to $$A(1+r)^t$$ in $t$ periods. Note that $t$ need not be an integer.

We want the doubling time, so we want to solve the equation $$A(1+r)^t=2A.$$ Divide both sides by $A$, then take the natural logarithm of both sides. We obtain $$t\ln(1+r)=\ln 2$$ or equivalently $$t=\frac{\ln 2}{\ln(1+r)}.$$

We want a quick and easy approximation for $t$, given $r$. More precisely, we wanted a quick approximation. Calculators are cheap and widely available, so we can quickly get a practically exact answer. Or ask Google. But back to the past.

By using the Taylor series for $\ln(1+x)$, or otherwise, we have $\ln(1+x)\approx x$ if $x$ is not too far from $0$.

So the exact formula above suggests the approximation $$t \approx\frac{\ln 2}{r}.$$

But $r$ is the mathematician's version of interest rate, where $8\%$ gives $r=0.08$. Let $R$ be the "layman's" number, which would be $8$.

Since $R=100r$, the approximate doubling time above, in terms of layman's interest rate, is given by

$$t \approx \frac{100 \ln 2}{R}.$$

Note that $100\ln 2$ is approximately $69.3$. And $72$ is reasonably close to $69.3$.

Let's take our numerical example with $r=0.08$. Then $72/8=9$, while $69.3/8 \approx 8.66$. Not a great deal of difference.

But please note that neither estimate is exact, since we used the approximation $\ln(1+r)\approx r$ in deriving the rule.

In fact, $\ln(1.08) \approx 0.07696$, so a very good approximation to the true doubling time is, in this case, $9.006$. Interestingly, the Rule of $72$, is, in this case, substantially more accurate than the Rule of $69$, or $69.3$. The use of $72$ makes up, to a large degree, for the inaccuracy involved in approximating $\ln(1.08)$ by $0.08$.

Details about the accuracy of the approximation are strongly dependent on $r$. For example, you might want to solve the following problem.

Exercise: You have borrowed some money from Break-Your-Legs Loans. The interest rate is $50\%$ per month, compounded monthly. Find the doubling time of your debt, and the approximate doubling time given by the Rule of $72$.

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Whoops! Forgot the first rule: check the internet... Thanks. – David Roberts Jul 21 '11 at 2:27
(+1) for this "less well-written" exposition! – The Chaz 2.0 Jul 21 '11 at 4:01
Also, 72 is easy to divide by lots of things in your head, while 69.3 is rather harder. – Ross Millikan Jul 21 '11 at 4:07