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How to Check Your Arithmetic, Part 1

Learn how to quickly and easily check your answers to arithmetic problems without using a calculator.

By
Jason Marshall, PhD,
Episode #080

Do you ever solve a simple arithmetic problem and then wonder if you got the answer wrong? Perhaps this happens to you during tests or maybe it’s while you’re performing the highly-endangered act of balancing a checkbook. No matter the circumstances, rest assured that there are ways for you to rid yourself of those gnawing pangs of dread and ensure that your work is error free. And, as we’ll see in the next several articles, you don’t have to use a bulky calculator to do it.

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How to Check the Sign of Simple Multiplication Problems

The first anxiety relieving tip is designed to help you check that your answers to multiplication problems have the correct sign. As you might guess, this tip is based upon the by-now-familiar sayings about multiplying positive and negative numbers that we talked about in the last article:

  • A positive times a positive makes a positive.

  • A negative times a negative makes a positive.

  • A positive times a negative makes a negative.

When multiplying pairs of numbers, your results had best not disagree with these trusty slogans. So 3 x 3 must be a positive number, –3 x 3 must be a negative number, and –3 x –3 must be a positive number.

How to Check the Sign of More Complex Multiplication Problems

But that’s not all these sayings are saying. Not only do they tell us what the sign of the answer must be when two numbers are multiplied together, they also tell us what the sign of the answer must be when any number of numbers are multiplied together! Here’s the quick and dirty tip:

  • If the list of numbers you’re multiplying contains an odd number of negative numbers, the result must be a negative number.

  • If the list of numbers you’re multiplying contains an even number of negative numbers, the result must be a positive number.

So 2 x –4 x 5 x 3 must result in a negative number since there is an odd number of negative numbers in the problem. And 2 x –4 x 5 x –3 must result in a positive number since there is an even number of negative numbers in the problem.

Try this example: –1 x –4 x 10 x –37 x 9 = 13,320…is that the right answer?

Well, there are three negative numbers here: –1, –4, and –37. And since this is an odd number of negative values, we can immediately tell that the result has to be a negative number. So the answer 13,320 can’t be right. Although, as you can check, it’s very close—the real answer is –13,320.

Why does this method work? Well, it goes back to those old sayings that a negative times a negative (which, you’ll notice, is an even number of negative numbers) makes a positive and a positive times a negative (which, you’ll notice, is an odd number of negative numbers) makes a negative. All we’re doing with these longer problems is applying the basic rules for pairs of numbers several times.

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