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:

```
Years to double = 72 / Interest Rate
```

This formula is useful for **financial estimates** and understanding the nature of compound interest. Examples:

- At 6% interest, your money takes 72/6 or 12 years to double.
- To double your money in 10 years, get an interest rate of 72/10 or 7.2%.
- If your country’s GDP grows at 3% a year, the economy doubles in 72/3 or 24 years.
- If your growth slips to 2%, it will double in 36 years. If growth increases to 4%, the economy doubles in 18 years. Given the speed at which technology develops, shaving years off your growth time could be very important.

You can also use the rule of 72 for **expenses like inflation or interest**:

- If inflation rates go from 2% to 3%, your money will lose half its value in 24 years instead of 36.
- If college tuition increases at 5% per year (which is faster than inflation), tuition costs will double in 72/5 or about 14.4 years. If you pay 15% interest on your credit cards, the amount you owe will
**double**in only 72/15 or 4.8 years!

The rule of 72 shows why a “small” 1% difference in inflation or GDP expansion has a huge effect in forecasting models.

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.

## Deriving the Formula

Half the fun in using this magic formula is seeing how it’s made. Our goal is to figure out how long it takes for some money (or something else) to double at a certain interest rate.

Let’s start with $1 since it’s easy to work with (the exact value doesn’t matter). So, suppose we have $1 and a yearly interest rate R. After one year we have:

`1 * (1+R)`

For example, at 10% interest, we’d have $1 * (1 + 0.1) = $1.10 at the end of the year. After 2 years, we’d have

```
1 * (1+R) * (1+R)
= 1 * (1+R)^2
```

And at 10% interest, we have $1 * (1.1)^{2} = $1.21 at the end of year 2. Notice how the dime we earned the first year starts earning money on its own (a penny). Next year we create another dime that starts making pennies for us, along with the small amount the first penny contributes. As Ben Franklin said: “The money that money earns, earns money”, or “The dime the dollar earned, earns a penny.” Cool, huh?

This deceptively small, cumulative growth makes compound interest extremely powerful – Einstein called it one of the most powerful forces in the universe.

Extending this year after year, after N years we have

`1 * (1+R)^N`

Now, we need to find how long it takes to double — that is, get to 2 dollars. The equation becomes:

`1 * (1+R)^N = 2`

Basically: How many years at R% interest does it take to get to 2? Not too hard, right? Let’s get to work on this sucka and find N:

```
1: 1 * (1+R)^N = 2
2: (1+R)^N = 2
3: ln( (1+R)^N ) = ln(2) [natural log of both sides]
4: N * ln(1+R) = .693
5: N * R = .693 [For small R, ln(1+R) ~ R]
6: N = .693 / R
```

There’s a little trickery on line 5. We use an approximation to say that ln(1+R) = R. It’s pretty close – even at R = .25 the approximation is 10% accurate (check accuracy here). As you use bigger rates, the accuracy will get worse.

Now let’s clean up the formula a bit. We want to use R as an integer (3) rather than a decimal (.03), so we multiply the right hand side by 100:

`N = 69.3 / R`

There’s one last step: 69.3 is nice and all, but not easily divisible. 72 is closeby, and has many more factors (2, 3, 4, 6, 12…). So the rule of 72 it is. Sorry 69.3, we hardly knew ye. (We could use 70, but again, 72 is nearby and even more divisible; for a mental shortcut, go with the number easiest to divide.)

## Extra Credit

Derive a similar rule for tripling your money – just start with

`1 * (1+R)^N = 3`

Give it a go – if you get stuck, see the rule of 72 for any factor. Happy math.

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?

Your post rockks!!!!

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.

@Kar: Glad you liked it.

@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).

i really love this site . it gives me all the information that i needed.

Thanks, glad you enjoyed it.

This did really help me understand the meaning better than all the other web sites that I went to for the Definition.

@No One: Thanks.

I loved it i found out alot

@linda: Glad it was helpful!

i liked to read the description

It’s considered as the 8th wonder of the world! Great invention of mankind!

Rule of 72 rules

http://thegetwealthy.com

thnx

rockingggggggggggg!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1

Hi Kalid,

I love your website – your explanations are always excellent.

I thought you might appreciate my own article on this subject.

Regards,

David

@David: Thank you! Your article looks interesting too, I like the accuracy table comparison.

This type of math are always useful for your life so it is better to learn them as soon as possible.

I never realized math was so interesting!! Thanks a million!

@Anonymous: Glad you enjoyed it.

Thanks Kalid, this is very useful….

@Jai: You’re welcome!

The explanation is so good!

You are a very good math explaner

@Clarafy: Thanks!

So, if one earns 3% a year but the inflation rate is 2%, then the actual is 72/1 = 72 years? Okay, assuming that is correct, what if the difference is 0? Inflation and earned interest are the same? Then 72/0 = ? So, after 20 years, you have the same investment value, but we know the value of money will decrease over time. How is that factored? thanks

Thats really cool.

You can also do the same thing with smallish percentage chance getting to 50%

E.g. each trial gives you 7% of “winning” how long to 50/50 chance?

Answer is also about 72/7

This works as ln(1/2) = -ln2

and here we start with (1-R)^N = 1/2

So the to negative will divide returning to you rule of 72

@Ben: Thanks, that’s a really neat application!

Can you source your comment about Einstein calling it the most powerful force…?

@MJ: Hrm, I can’t! I looked around and it seems it’s undetermined (according to Snopes, anyway) if Einstein really said it:

http://www.snopes.com/quotes/einstein/interest.asp

Do you have a simple method of understanding the rule of 78 (front loaded interest used for most auto loans to penalize early payment)

i need answer for one question if possible please send the solution-

Q. A finance company offers to give Rs.8000 after 12 years in return for Rs.1000 deposited today. Using the rule 72 find out the approximate interest rate offered.

I read your site just to see what SOME of the different sites had to say about R of 72. I learned that about 25 years ago when working with A. L. Williams which is now Primerica. I have returned to Primerica after a long hietus, and math being my hight(est) point in school, love to be there, as I love to help people. Enough said. Thanks for the info that you have posted. May many many more read it!

@Enginerd (28 Apr 2008), 72 does work better than 69, it’s not just less dirty

You can approximate log R by just R, but a better approximation is R – R^2/2, or R(1 – R/2). So, the magic number that we’re looking for will be 100*(log 2)/(1 – R/2). If we use R=0, then the formula will work best for an interest rate of zero, not very handy. If the usual range of interest rates is between 0 and 15%, then using a value in the middle would make the most sense. So, using 7.5% would mean that the magic number would be close for that whole range, and the formula would be nearly perfect near 7.5%.

100*(log 2) / (1 – 7.5%/2) = 72.01529…

That’s why we use 72 instead of 69.

“You can approximate ln(1+R) by just R”, sorry for the typo!

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Thanks for this post. The rule 72 should be my futur friend. However, I need to go back in my high school lessons for refresh the log concept.

Hurrah! Finally I got a weblog from where

I know how to actually take useful information regarding my study and

knowledge.

“Fractional Reserve Banking” SUCKS!!!! So does the “Federal Reserve” & ALL other central banks.

I know only part of Jesus’ story. I have heard the only time he got really “pissed-off” was when he busted the “Money Changers” at some temple.

Thank YOU —- the Joe —-

That is a really nifty trick to know. Thanks for sharing Kalid.

—–

Ashwin

http://classof1.com

Reallly it is a magic…….thanks for shaire inf,,,,,,,,,,,,,,,,,,,,,,,,,,

8:):)