Learn how to use the compound interest formula and how it’s related to the rule of 72.
Does the Rule of 72 Always Work?
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 firstname.lastname@example.org.
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