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.

 Okay. 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 ...
Approximately!
>... 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 =