### Back of the envelope calculations with The Rule of 72

Monday, June 20th, 2011For example, consider Moore’s Law, which describes how "the number of transistors that can be placed inexpensively on an integrated circuit doubles approximately every two years." If something doubles every two years, at what rate does it increase per month, on average? If you know the rule of 72, you’ll instantly know that the monthly growth rate is about 3%. You get the answer by dividing 72 by 24 (the number of months).

Computer scientists are usually very familiar with powers of two. It’s often convenient to take advantage of the fact that 2^10 is about 1,000. That means that when something increases by a factor of 1,000, it has doubled about 10 times. By extension, and with a little more error, an increase of a million corresponds to 20 doublings, and a billion is 30 doublings (log base two of a billion is actually 29.897, so the error isn’t too wild). You can use this to ballpark the number of doublings in a process really easily, and go directly from that to a growth rate using the rule of 72.

For example, the bottom of this page tells us that there were about 16,000 internet domains on July 1st 1992, and 1.3M of them on July 1st 1997. Let’s think in thousands: that’s a jump from 16 to just over 1,000 in 5 years. To get from 1 to 16 is four doublings, so from 16 to 1,000 is six doublings (because 1,000 is ten doublings from 1). So the number of domains doubled 6 times in 5 years, or 6 times in 60 months, or once every 10 months (on average). If you want something to double in 10 months, the rule of 72 tells us we need a growth rate of 7.2% per month. To check: 16,000 * (1.072 ^ 60) = 1,037,067. That’s a damned good estimate (remember that we were shooting for 1M, not 1.3M) for five seconds of mental arithmetic! Note that the number of domains was growing much faster than Moore’s law (3% per month).

You can quickly get very good at doing these sorts of calculations. Here’s another easy example. This page shows the number of internet users growing from 16M in December 1995 to 2,072M in March of 2011. That’s just like the above example, but it’s 7 doublings in 15.25 years, or 183 months. That’s pretty close to a doubling every 24 months, which we know from above corresponds to 3% growth per month.

You can use facility with growth rates to have a good sense for interest rates in general. You can use it when building simple (exponential) models of product growth. E.g., suppose you’re launching a product and you reckon you’ll have 300K users in a year’s time. You want to map this out in a spreadsheet using a simple exponential model. What should the growth rate be? 300K is obviously not much more than 256 * 1,024, which is 18 doublings in 365 days, or a doubling roughly every 20 days. The rule of 72 gives 72/20 = ~3.5, so you need to grow 3.5% every day to hit your target. Is that reasonable? If it is, it means that when you hit 300K users, you’ll be signing up about 3.5% of that number, or 10,500 users per day. As you can see, familiarity with powers of two (i.e., estimating number of doublings) and with the rule of 72 can give you ballpark figures really easily. You can even use your new math powers to avoid looking stupid in front of VCs.

The math behind the rule of 72 is easy to extend to triplings (rule of 110), quadrupling (rule of 140), quintupling (rule of 160), etc.

Finally, you can use these rules of thumb to do super geeky party tricks. E.g., what’s the tenth root of two? Put another way, what interest rate do you need for something to double after ten periods? The rule of 72 tells you it’s 72/10 = 7.2%, so the tenth root of two will be about 1.072 (in fact 1.072 ^ 10 = 2.004). What’s the 20th root of 5? The rule of 160 tells you you need 160/20 = 8% growth each period, so 1.08 should be about right (the correct answer is ~1.0838).

As with all rules of thumb, it’s good to have a sense of when it’s most applicable. See the wikipedia page or this page for more detailed information. It’s also of course good to understand that it may not be suitable to model growth as an exponential at all.