the Rule of 72
We invest $1000 and, after a certain length of time, our investment has doubled in value. If our annual return was 6%, how long did it take to double?

>I'd say twelve years.

How'd you get that?
>Easy. The Rule of 72. I divide 6 into 72 and I get 12, because N x R= 72, eh?
I assume N is the number of years and R% is the return.

Okay, at 6% annual return, how long would it take to triple?
>Triple? I have no idea ... but I have a funny feeling you're going to tell me.

We start with $A and, with an annual return of R%, we'd have
  • A(1+R/100) after one year, and
  • A(1+R/100)2 after two years, and
  • A(1+R/100)3 after three years, and
  • ... and
  • A(1+R/100)N after N years.
In order for our $A to double, we'll need

A(1+R/100)N = 2A
or cancelling the A
(1+R/100)N = 2
or taking logarithms
N log(1+R/100) = log(2)
or dividing by log(1+R/100)
N = log(2)/log(1+R/100)
so multiplying by R
NxR = {log(2)R}/log(1+R/100)

So, is N x R = 72?

In Fig. 1 we've plotted this expression for NxR
... namely {log(2)R}/log(1+R/100) versus R
and find (surprise!) that it's roughly 72. That means that:

NxR = 72
approximately ... to multiply by 2

so Number-of-Years x Annual-Percentage-Return = 72 (approximately)

For example, using R = 7%, we'd get
(using logarithms to the base 10, tho' any base will do):
log(2)*7/log(1.07) = (0.301)(7)/0.0294 = 71.7
Neat, eh?

>I assume that, at the red dot, it's exact.
Yes, and ...

>And to triple ... to multiply our investment by 3?

If we wanted our $A to be multiplied by a factor F (instead of 2) we'd need

NxR = {log(F)R}/log(1+R/100)

and I've plotted these, too, for 3 and 5 so we get ...

>Wait! To multiply by 3 we have the Rule of 114, since NxR = 114 ...
>... and if we wanted our R = 6% per year investment to triple, it'd take N = 114/6 = 19 years.
Approximately, remember.

>... and to multiply by 5 we have the Rule of 166, since NxR = 166, and ...
Remember, it's approximately!

>... so it'd take N = 166/6 = ... uh, how many years is that?

Suppose you want your R = 6% investment to increase by a factor of 10. Since 10 = 2x5 we can use N = 72/6 = 12 years to get the multiply by 2 then another N = 166/6 = 27.7 years to get the additional multiple by 5 so it'd take 12 + 27.7 = 39.7 years. Neat, eh?

>You forgot to say approximately! By the way, what's the exact answer?

Roughly 39.51653063577150 years

You can play, here (to get the exact answer ... for comparison):

Annual Interest Rate: R = %       Increase by what factor?
Years to increase: N =