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How Does Division Work?

So you want to divide things up. You want to break them down into equal groups. You want to take a big pile of quarters and make them into small stacks. You ordered a pizza and you want to divide it up evenly. It's all division. It's easy to slice up a pizza or make stacks of coins, but how does it work on paper? How does division really work?

We'll keep things really simple for this explanation. In division, you start with a number (dividend) and you determine how many times another number (divisor) can be divided into it. For example, you start with the number three (3) and figure out how many times the number one (1) can be divided into it. You will discover that if you break a pile of three down into groups of one, you will get three groups.

That's actually one of the neat tricks in division. Any number that is divided by one (1) gives you itself for the answer.

Examples:
5 ÷ 1 = 5
10 ÷ 1 = 10
58 ÷ 1 = 1
5,800,657,268 ÷ 1 = 5,800,657,268

Start With Some Easy Numbers

We're not going to start your division work with numbers in the billions. Let's start small with numbers less than ten (10). We'll just walk through all of the numbers less than ten that can be divided by other numbers less than ten. You already know how the ones work.

1 ÷ 1 = 1
2 ÷ 1 = 2
3 ÷ 1 = 3
4 ÷ 1 = 4
5 ÷ 1 = 5
6 ÷ 1 = 6
7 ÷ 1 = 7
8 ÷ 1 = 8
9 ÷ 1 = 9

But what about twos? Remember that you are always looking to divide things into equal groups. We'll ask you the first problem in a simple way. How many groups of two (2) nuts can be made from a pile with two (2) nuts? If you only have two nuts, you can only make one pile of two nuts. When you write it out with numbers, it goes like this...
2 ÷ 2 = 1

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We're going to talk about numbers called remainders in the next section. Remainders happen when one number is not evenly divided by another. They aren't difficult, but we're starting with the basics on this page. Let's look at the way other numbers can be divided by 2.
4 ÷ 2 = 2 (You can make two groups of two from the number four.)
6 ÷ 2 = 3 (You can make three groups of two from the number six.)
8 ÷ 2 = 4 (You can make four groups of two from the number eight.)

And now we move to the threes. It's getting a little harder to do in your head as the numbers get bigger. When you're done, you should be able to do this simple division without thinking.
3 ÷ 3 = 1 (You can make one group of three from the number three.)
6 ÷ 3 = 2 (You can make two groups of three from the number six.)
9 ÷ 3 = 3 (You can make three groups of three from the number nine.)

How many groups of four (4) can you make from the number four? The answer is one group. What about eight? If you have eight flowers and create groups with four flowers, how many groups can you make? Use your fingers if you want. It's good practice. Here are the answers for fours...
4 ÷ 4 = 1 (You can make one group of four from the number four.)
8 ÷ 4 = 2 (You can make two groups of four from the number eight.)

There's only one division problem for five (5) when you're working with numbers below ten (10), but we're going to toss an extra equation on the list because fives are really great numbers in division.
5 ÷ 5 = 1 (You can make one group of five from the number five.)
10 ÷ 5 = 2 (You can make two groups of five from the number ten.)

Things to Notice

We already told you about numbers that are divided by one. The answer is always the number you started with. Have you noticed any other patterns? What did you notice about numbers that are divided by themselves?
4 ÷ 4 = 1
8 ÷ 8 = 1
100,258,159 ÷ 100,258,159 = 1

Do you see that? Any number divided by itself is equal to one. That's a handy rule.

One other rule for you to know is that you can never divide by zero. That's it, no fancy stuff. In the math we are doing, you can have a zero as your dividend and the quotient will be zero. You can never have a zero as your divisor, because there is no true answer. Think about it for a second. Can you divide something by nothing? No.
0 ÷ 4 = 0
4 ÷ 0 = Nononononononononono! (Mathematicians call that undefined.)

Before we go, think about a division problem for a second. Look at the way it is written out. We want you to notice a pattern with these numbers. Look at the way these division problems relate to your multiplication problems. It's almost as if they are backwards versions of the problems.

Examples:
4 ÷ 2 = 2
4 = 2 x 2

9 ÷ 3 = 3
9 = 3 x 3

That's the connection between multiplication and division. It's so much more than just moving some symbols around. Multiplication puts groups together and division breaks groups apart.

Other Formats

Because this is a web page, we will be writing out division problems from left to right on one line. We also use the division sign called an obelus (÷) much of the time. When you do your work in math class, you will use a different format that looks like an "L" on its side. It's technically called the "close parenthesis and vinculum", but that's a mouthful, so it's okay to refer to it as a "division bracket". It helps to separate all of your numbers and organize your long division problems.

2 r 2 6 ) 14       - 12         2

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