# This without a calculator, you may ask? By

This lesson provides the definition and illustrates examples of compatible numbers, as well as giving explanations of what makes them compatible. The examples are taken from real-life situations using money and covering all four mathematical operations: addition, subtraction, multiplication, and division.

## Definition of a Compatible Number

When we think of the word, ‘compatible,’ we often think of two people that go quite well together, like the historical Shakespearian couple, Romeo and Juliet, or American outlaws, Bonnie and Clyde. We may also think of foods that taste great together, like peanut butter and jelly or hot dogs and ketchup. Similarly, when we are speaking of compatible numbers, we’re referring to numbers that fit, look, and work well together, but for mathematical computation purposes.

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Simply speaking, compatible numbers are numbers that are easy to add, subtract, multiply, or divide in your head. They are visually easier to compute mentally than other numbers for various reasons.

## Why Do We Use Compatible Numbers?

When we are asked to find the sum (in addition), difference (in subtraction), product (in multiplication), or quotient (in division), our heads can sometimes spin. We use compatible numbers to make the problem easier to solve in our head by rounding each number to the nearest ten, twenty, fifty or hundred.For example, if given the problem 297 + 348, our heads may start to spin with confusion. But if we make the numbers compatible and round up to the nearest hundred or ten spot, 300 and 350 are much easier to compute in our heads. So, we know that the answer is about 650! How easy was that?You can see why use of compatible numbers makes mental mathematical computation so much easier!

## Examples of Compatible Number Usage

Let’s take a look at some examples of compatible number usage from each of the four common mathematical operations: addition, subtraction, multiplication, and division.

To make things more fun, let’s use examples from real life using money. Who doesn’t love money?

You are at the grocery store and only have a \$20 bill to spend on ingredients for homemade pasta with meat sauce (yum!). You already have the pasta and meat at home, but you just need some fresh fruits and vegetables to fulfill the ingredient list. As you rummage through the produce department, you keep track of the cost of the items you pick up by adding them in your head.How is this even possible without a calculator, you may ask? By using compatible numbers, of course! You may want to round up to the nearest dollar, though, to make sure you don’t overspend because you have to account for sales taxes!

• 3 pounds of tomatoes: It is \$1.

89/pound. Use the compatible number \$2.00 instead of \$1.

89, which helps you easily compute about \$6 for tomatoes.

• 2 onions: It is \$0.40 for an onion, so \$0.80 for two.

But you’re adding up a bunch of things in your head in the store, so it’s easier to use the compatible number of \$1.00 for two onions.

So far, we have spent about \$7 on tomatoes and onions and have \$13 more to spend on the other ingredients.

### Subtraction

You have \$798 in the bank and spend \$196 buying new school clothes.

In order to compute about how much money you have left in your account in your head, you have to use subtraction! And you’ll probably want to use compatible numbers to make it easier.\$796 (compatible number = \$800) – \$196 (compatible number = \$200)So, \$800 – \$200 = \$600.So, you have about \$600 left in your bank account! Note that the exact difference is \$602.

### Multiplication

You’ve agreed to tutor 8-year-old Moira Stanley for \$18 an hour, 22 hours a week.

What a great job! You want to quickly compute in your head about how much you can expect on your weekly paycheck from Mr. Stanley.If you turn \$18 into a compatible number, it would be \$20. If you turn 22 hours to a compatible number, it would be 20, as well. Therefore, it’s much easier to multiply 20 * 20, which would be 400 (2 * 2 = 4, add two zeros, which makes it 400!).So, you know that you’ll receive roughly \$400 a week from tutoring little Moira.

Note that the exact product is \$396.

### Division

You’re at a restaurant with 8 friends. The waiter forgot to separate the checks, so you’re left with the big bill. The bill total is \$245. You all want to leave a 20% tip, too.

You can turn \$245 into a compatible number by making it \$250. If you divide this by 5 (because 20 is 1/5 of 100), it’s clear to see that 20% of \$250 is \$50. You tack on the \$50 and get \$300 that needs to be divided amongst all of your friends.If you want to get a rough estimate of how much you owe as an individual, you can turn 8 people into 10, which is more compatible than 8, and easily compute that each person will have to pay about \$30 each.

## Lesson Summary

Remember that compatible numbers are numbers that work well together for mental computation. When taking the quiz for this lesson, remember that computing math is easier when you make numbers more compatible by either:

• A: Rounding to the nearest ten, twenty, fifty or hundred place
• B: Rounding to a number that is more compatible with the other number in the mathematical equation

Hint: Look for common divisors and multiples.

## Quick Study Points

Terms & Examples Explanations
Why? to make the problem easier to solve in our heads by rounding each number to the nearest ten, twenty, fifty or hundred
Examples Addition: limited funds at grocery store for a meal; Subtraction: calculating what’s in your bank account after a purchase; Multiplication: determining the amount due when earning hourly wages; Division: paying a restaurant bill with friends

## Learning Outcomes

After finishing this lesson, you should be able to:

• Note the characteristics of compatible numbers
• Explain why they are helpful
• Give examples of the various uses of compatible numbers
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