by Jason Marshall
In the last several articles we’ve talked about the compound interest formula and the rule of 72. Now that we know what these things are, it’s time to talk a bit more about how, why, and when you can use them. While we’re at it, we’ll also finally solve the mystery of why the rule of 72 works in the first place. And we’ll even see that sometimes it doesn’t actually work at all!
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Recap: the Compound Interest Formula and the Rule of 72?
As we’ve found, you can use the compound interest formula to calculate how long it will take money to grow. Which means that you can use it to figure out how long it might take for your money to double. But hang on a minute! You might be thinking: “Isn’t that exactly what the rule of 72 is for? Doesn’t the rule of 72 just say that you can find the doubling time of an investment by dividing the number 72 by the interest rate?” Why yes, yes it does. And the rule of 72 certainly works wonderfully…so long as you’re interested in the doubling time.
But what if you’re interested in knowing how long it will take to grow your money by 20%? Or 50%? Or how about the time it takes to triple your money? Well, for that you need something more than the rule of 72. And that something is the compound interest formula that we discovered in the last article.
How to Use the Compound Interest Formula
Let’s say you have $1000 invested in an account earning 5% annual interest and you want to figure out how long it’s going to take to triple your money. In other words, how long will it take for your account to be worth $3000? Well, the compound interest formula says that:
FV = PV x (1 + rate)^years
In this case we can plug our present value PV = $1000, the future value that we’re interested in FV = $3000, and the interest rate = 0.05 (the decimal form of 5%) into this equation to figure out how many years it’s going to take. After plugging in and simplifying, we’re left with the equation:
1.05^years = 3
You can solve this equation using algebra and something called logarithms, or you can solve it using trial and error by trying different numbers of years until you get close enough to the answer. Today, we’ll stick to the latter method and save logarithms for a future date. In this case, 22 years gives 1.05^22 = 2.93 (which being a bit less than 3 is too small) and 23 years gives 1.05^23 = 3.07 (which being a bit more than 3 is too big). So the real answer must be somewhere in between. As it turns out, and as you can check by plugging numbers in, it takes about 22.5 years for an account earning 5% interest to triple.
How is the Compound Interest Formula Related to the Rule of 72?
Instead of using the compound interest formula to figure out how long it takes your money to triple, we could have used it to figure out how long it takes your money to double—exactly what we do using the rule of 72. Does that mean that the two things are related? Yes! Let’s return to the compound interest formula to see why. The formula says that
FV = PV x (1 + rate)^years
Now, if we’re talking about the doubling time as we are with the rule of 72, then FV / PV = 2…since that’s what it means to double! So for the rule of 72, the compound interest formula looks like:
(1 + rate)^years = 2
Now, we’re not going to go into all the details here, but it turns out that you can use logarithms and an approximation having to do with logarithms to turn this into the equation that you know as the rule of 72:
years = 72 / rate
If you’re interested in seeing more of the details, check out the bonus section at the very end of the article. But remember, the details aren’t all that important right now. The important thing is to understand that the rule of 72 isn’t some bit of magic—it’s an approximation to its bigger and more powerful sibling: the compound interest formula.
Does the Rule of 72 Always Work?
You can use the compound interest formula to figure out how long it will take to triple your money.
At the beginning of the article, I said that the rule of 72 works great when you’re interested in calculating doubling times. But is that always true? Does the rule of 72 ever not work? Well, think about this: How long does the rule of 72 say it will take to double your money in an account earning 72% interest? Hmm, the rule of 72 says it’ll take 72 / 72 = 1 year. But the account only earns 72% interest in a year…which means that it doesn’t actually double! What’s wrong?
Well, it turns out that the approximation we had to use earlier to come up with the rule of 72 from the compound interest formula is only accurate when the interest rate is small. As you move to larger and larger interest rates—like 72%—the approximation gets worse and worse. Go ahead and try plugging in some numbers starting from small interest rates and working up to large ones to see for yourself. In truth, that really isn’t much of a real world problem when it comes to calculating actual doubling times since interest rates on investment accounts are almost always fairly small. But when in doubt you can always skip the rule of 72 and just use the compound interest formula instead since it’s always accurate.
Number of the Week
Before we finish up, it’s time for this week’s featured number selected from the various numbers of the day that I posted to the Math Dude’s Facebook page over the past week:
This week’s number is 106.5 billion. Why? Because that’s the number of humans who have ever lived! How do we know this? Well, in truth, we don’t—it’s not like there are good records going back to the beginning of humanity. But, using some very reasonable assumptions, the population reference bureau concluded in 2002 that this number is a pretty good estimate. And, since there are currently almost 7 billion humans on the planet, it means that about 6.5% of all humans who have ever lived are currently alive today.
So if you ever hear anybody try to claim that something like 75% of all humans ever born are alive today—which is a fairly common myth—you now have the facts to straighten them out.
If you think that number was interesting, be sure to become a fan of the Math Dude on Facebook where you’ll learn a new number that’s just as interesting as this one every single weekday. If that’s not enough to convince you to check it out, I’ve also started posting daily math puzzles that have proven to be quite popular. Head over to the Math Dude’s Facebook page and see for yourself. Of course, if you’re on Twitter, please follow me there too and keep up to date on the podcast, the numbers of the day, the daily math puzzles, and all the latest math and science news. Finally, remember to email any math questions that you may have to email@example.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!
Web Bonus: Where Does the Rule of 72 Come From?
We can solve the equation
(1 + rate)^years = 2
for “years” by taking the natural logarithm of both sides (and doing a bit of rearranging) to get:
years = ln(2) / ln(1 + rate)
Now, here’s the real key to deriving the rule of 72: It turns out that as long as the interest rate in ln(1+rate) is much smaller than 1, then ln(1+rate) is approximately just equal to the rate. Which means that we get the approximate equation
years = ln(2) / rate
If we plug in the value of the natural logarithm of 2 and multiply the right side by 100 so that we can write the interest rate as a whole number and not as a decimal, we get
years = 69.3 / rate
But that’s not quite the rule of 72! What’s going on? Well, it turns out that 69.3 isn’t nearly as easy to work with as its nearby neighboring number 72. After all, 72 can be evenly divided by 2, 3, 4, 6, 8, 12, 18, and so on. Since it’s so much easier to work with, we have the rule of 72:
years = 72 / rate