Learn how to use the “casting out nines” method to quickly and easily check your answers to arithmetic problems.
In the last few articles we’ve talked about two techniques that you can use to check your answers to arithmetic problem. The first trick is to use the signs of the numbers in a problem to quickly and easily check that your answer has the correct sign. And the second trick is to use the parities of the numbers in a problem (that’s whether the numbers are even or odd) to check that your answer has the correct parity.
While these are both great techniques, today we’re going to talk about something called “casting out nines” that in many ways is even better. Why am I so fond of this trick? Because it’s super easy to use and it’s very good at telling you when you made a mistake...which is a pretty tough combination to beat.
How to Calculate “Check Digits”
Before we get into the nitty-gritty details of using casting out nines, we first need to learn how to turn any number into what we’ll call a “check digit.” We’ll see why this is important in a minute, but for now all you need to know is that any number—no matter how many digits long it is—can be turned into a single digit number by adding together all the digits of the number, and repeatedly doing this each time the resulting number has more than a single digit.
Calculating Check Digit Examples
This might sound complicated in words, but it’s really not too hard to do in practice…as a few examples will show. Let’s start with an easy one: What check digit does the number 42 give? Well, all we have to do is add together the digits of this two-digit number—that’s a 4 and a 2—to get the check digit of 4+2=6. Okay, how about 48 instead? Once again, add the two digits together to get 4+8=12. Is this a single digit number? Nope, we’ve still got two digits here…so let’s repeat the process and add these two digits to get 1+2=3—that’s the check digit for the number 48.
Calculating Check Digit Practice
Now it’s your turn. What’s the check digit for the number 233? Did you get 8? I hope so since 2+3+3=5+3=8.
How about the number 555? What’s the check digit? Did you get 6? You should have since 5+5+5=15, and then 1+5=6.
Once you’ve got this idea down and you can calculate check digits easily, you’re ready to move on to learning how to apply this new skill to the technique of casting out nines.
How to Use Casting Out Nines to Check Addition
We won’t talk about why what I’m about to tell you is true right now (we’ll save that for a later article), but it turns out that whenever you add a group of numbers, there’s a remarkable relationship between the check digits of those numbers and the check digit of the answer to the problem. Namely, if you add up the individual check digits of each of the numbers in an addition problem and then find the check digit for that new sum, the number you get will always be the same as the check digit of the correct answer to the problem.
Demonstration of Casting Out Nines to Check Addition
Again, this might sound a little confusing, but this example should clear things up:
Let’s say you’re adding 253 + 827 + 131, and you (as we’ll soon see) mistakenly come up with an answer of 1201. The bad news is that you got the wrong answer. But the good news is that you’re about to find that out before it’s too late by using the casting out nines technique we just learned. Here’s how it works:
Start by finding the check digits of the three numbers in the addition problem: 253, 827, and 131. The check digit of 253 is 2+5+3=10, and then 1+0=1; the check digit of 827 is 8+2+7=17, and then 1+7=8; and, as you can check for yourself, the check digit of 131 is 5. Now you need to add together these three check digits—1, 8, and 5—and then find the check digit of that number. So the sum of the three numbers is 1+8+5=14, and the check digit is therefore 1+4=5.
Now all we have to do is compare this check digit to the check digit of the answer we calculated, 1201. If the two numbers disagree, then we know we have a problem. So, what’s the check digit of 1201? Well, it’s 1+2+0+1=4. Oops! This check digit of 4 is not the same as the check digit of 5 we got from the individual numbers in the addition problem. Which means that the answer of 1201 is wrong (it should really be 1211), and that you’d better go back and try again.