Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $e$...

share|improve this question

1 Answer 1

up vote 19 down vote accepted

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 $72$.

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.