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How to Check Your Arithmetic, Part 2

Learn how to quickly check your answers to arithmetic problems.

By
Jason Marshall, PhD,
Episode #081

Nobody enjoys that nagging feeling that they just messed something up. But, unfortunately, that’s exactly how many people feel after doing arithmetic. If that sounds like you, I’ve got great news!

As we began to learn in How to Check Your Arithmetic, Part 1, there are lots of simple techniques that you can use to quickly and easily check your arithmetic and rid yourself of that feeling of doubt and uncertainty forever. And the best part is that you don’t even need to resort to pulling out a calculator to do it.

How to Check Your Arithmetic

In last week’s article, we learned how to quickly check the sign of the answers of multiplication and division problems. While that was a great start, it would be nice if there was some easy way to see if you got the actual numbers right too and not just the overall sign. Yes, that certainly would be very nice. But, unfortunately, the only way to completely ensure that you got exactly the right number for your answer is simply to repeat the problem carefully and check.

While that might seem like a bit of a bummer, it really isn’t because I’ve still got a couple of tricks for you to keep tucked up your sleeves that will usually (although not always) tell you when your answer can’t be right. Why is that helpful? Well, if you use one of these tricks and find out that your answer can’t be right, then you know that you’d better go back and try the problem again!

How to Use Parity to Check Addition and Subtraction of Two Numbers

One way to test if your answers to addition and subtraction problems are viable (meaning if they can be correct) is by checking what’s called their parity. That sounds complicated, I know—but it’s actually pretty simple…once you know that parity is just a fancy word that means whether the number is even or odd. With that cleared up, the quick and dirty tip for testing whether or not an answer to an addition or subtraction problem can be correct is to know the following things:

  • The sum or difference of two odd numbers is always an even number.

  • The sum or difference of two even numbers is always an even number.

  • The sum or difference of one even and one odd number is always an odd number.

Let’s check this and make sure it’s true: 3 + 5 = 8 (two odds makes an even), 4 + 4 = 8 (two evens makes an even), and 4 + 5 = 9 (an even and an odd make an odd).

How to Use Parity to Check Addition and Subtraction of a List of Numbers

But what if we’re not talking about only two numbers? What if we’re talking about adding and subtracting a bunch of numbers? In that case, all you have to know is:

  • Adding or subtracting a list containing an odd number of odd numbers always results in an odd number.

  • Adding or subtracting a list containing an even number of odd numbers always results in an even number.

With that in mind, here’s an example for you to try.

Imagine we get an answer of 74 for the problem 22 + 51 – 3 – 11 + 14.

Can that answer be right? Well, we can quickly check the viability of the answer without recomputing the entire problem by looking to see if the result of 22 + 51 – 3 – 11 + 14 has to be an even or an odd number? In this case, there is an odd number of odd numbers (three to be precise—51, 3, and 11), which means that the result must be an odd number. So the answer we got, 74, can’t be right. This method doesn’t tell us what the right answer is, but it does tell us that our original answer is wrong and that we’d better go back and try again.

How to Use Parity to Check Multiplication

Is there a similar parity check for multiplication problems? There sure is! The quick and dirty tip is to know the following:

  • The product of two odd numbers is an odd number.

  • The product of two even numbers is an even number.

  • The product of one odd and one even number is an even number.

And if we’re not just talking about two numbers but a whole list of numbers, then the rule is:

  • If there are no even numbers in the list, the product of those numbers must be an odd number.

  • If there are one or more even numbers in the list, the product of those numbers must be an even number.

Why does this work? Because any even number you multiply by must be divisible by two (that’s why it’s even), which means that the result of the multiplication must be a number that itself is divisible by two…and thus an even number!

Can Parity Be Used to Check Division?

So that’s how to use parity to check whether your answers to addition, subtraction, and multiplication problems can be correct. But what about division? Well, division is funny because dividing two integers doesn’t always result in another integer. In other words, while the answer to the problem 8 / 2 = 4 is an integer, the answer to 8 / 3 is not. Why does that matter? Because only integers can be even or odd—there’s no such thing as an even or odd fraction! Frequently the result of a division problem will not be an integer, and in those situations you can’t use the parity of the answer to check your work. Sometimes the result will be an integer, and there are some parity rules you can use. But honestly, those situations are few and far between, so my suggestion is to not bother with trying to use parity to check your answers to division problems…at least not at this point.

Number of the Week

Before we finish up, it’s time for this week’s featured number from my post on QDT’s blog The Quick and Dirty. This week’s number is pi billion seconds…aka, about 3.14 billion seconds. Why is that an interesting number? Well, if we use the fact we talked about last week that one year is about pi x 10 million seconds, we can see that if a person is lucky enough to live for a century, their lifetime will span about 100 x 3.14 x 10 million = 3.14 billion seconds.

In other words, pi billion seconds is about how many seconds there are in a good long healthy and fortunate human life. Thankfully, that's quite a few seconds. Well, sort of. I mean, it's comforting to see that the number of seconds in a human life is in the billions (and not millions), but it's also not in the trillions or quadrillions. Which is a good reminder to make as many of those however many billions of seconds count!

Wrap Up

Okay, that’s all the math we have time for today. Be sure to check out next week’s article for the exciting conclusion to this series in which we’ll talk about one last method—a very powerful one—for checking your arithmetic called “casting out nines.”

In the meantime, remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.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!

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