# The Rule of 72

The Rule of 72 is a great mental math shortcut to estimate the effect of any growth rate, from quick financial calculations to population estimates. Here’s the formula:

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/the-rule-of-72/

[…] But even after training, we get caught again. At 5% interest we’ll double our money in 14 years, rather than the “expected” 20. Did you naturally infer the Rule of 72 when learning about interest rates? Probably not. Understanding compound exponential growth with our linear brains is hard. […]

[…] Several minor, 2% improvements add up over time (see the rule of 72). Everyday efficiencies are great — you don’t always need a breakthrough to make a difference. Browse the photos below and read my comments at flickr to see what the fuss was about. In upcoming posts I’ll expand on why these examples of innovation made me shriek with delight and what we can learn from them. You need to upgrade or install Adobe Flash Player […]

[…] The Rule of 72 is a mental math shortcut to estimate the time needed to double your money. We’re going to derive it (yay!) and even better, we’re going to understand it intuitively. […]

[…] Rule of 72: Years To Double = 72/Interest Rate (derivation) […]

[…] The Rule of 72 | BetterExplained  Annotated By the way, the Rule of 72 applies to anything that grows, including population. Can you see why a population growth rate of 3% vs 2% could be a huge problem for planning? Instead of needing to double your capacity in 36 years, you only have 24. Twelve years were shaved off your schedule with one percentage point. […]

[…] The Rule of 72 | BetterExplained  Annotated By the way, the Rule of 72 applies to anything that grows, including population. Can you see why a population growth rate of 3% vs 2% could be a huge problem for planning? Instead of needing to double your capacity in 36 years, you only have 24. Twelve years were shaved off your schedule with one percentage point. […]

[…] Rule of 72: Years To Double = 72/Interest Rate (derivation) […]

Thanks for the great explanation. It’s a neat tip for those of us who can’t calculate ln[2^(1/r)] mentally…

Thanks, glad you liked it :).

why do we say that if an object travels the speed of light it will go into the future/past? Does this then mean that theoretical an object can never be created to travel faster than light?

[…] Treating interest in this funky way (trajectories and factories) will help us understand some of e’s cooler properties, which come in handy for calculus. Also, try the Rule of 72 for a quick way to compute the effect of interest rates mentally (that investment with 6% APY will double in 12 years). Happy math. […]

Hey, I really love this site. I especially love the decryption of e on another page. But anyway, I did the extra credit, I think it would be 110/R.

N = ln(3)/ln(1+R) and as you said, for small R’s, the ln(1+R) approximates R. After multiplying by 100, it turns out to be about 109.86/R, which I rounded up for simplicity to 110/R. That was fun! hahaha.

@Devin: Thanks, happy you’re enjoying the site! Yep, you got it: it’s the rule of 110 for tripling your money (if you need to remember it, think about “always giving 110 percent”).

Also, for quadrupling your money, you can use the rule of 72 twice to get the “Rule of 144”. Though at that point the rounding errors start to add up – the rule of 140 would be better :).

I’m willing to bet that it was changed from 69 to 72 so people wouldn’t feel dirty talking about the “Rule of 69”.

On the other hand, the ln(1+R)~R approximation tends to underestimate the doubling time. Changing from 69 to 72 corrects for that. Doing a few test cases, 72 seems a bit more accurate than 69.

Ah, good points. Also, 69 only has the factors 3 x 23 so it doesn’t divide that easily (for mental math).

[…] The Rule of 72 looks like this: Years to Double = 72 / Interest Rate and sounds like this: Money doubles every 8 years if it grows at at rate of around 9% per year. […]

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