How to Divide Numbers? Long Division explanied and Alternative Methods

Division is splitting into equal parts or groups. Learn more >>

Before we begin:

Before proceeding with this lesson, it would be helpful for you ;

• to review the concept of division. 
• Knowing the multiplication table, so make sure you know it completely without any gaps. If you have any challenges with the multiplication table, you need to address this as soon as possible.

We will learn what's actually happening in the division operations that your teachers teach in schools, understand the logic behind it, and discover alternative methods of division.

Dividing a two-digit number by a one-digit number

This is how your teachers teaches you division. Lets take a look;

Let's divide 34 by 2. How many times does 2 go into 34?

Step1:

How many times does 2 go into 3? only once ! Write the result above the 3, to the tens place.

Step 2:

Multiple 

Write the result of the multiplication under the 3 and and subtract.

The first digit is done.Let's bring down the next number and move on to the second digit.

How many times does 2 go into 14? it is 7! 

Let's write 7 above the 4, that is, in the ones place.

Just like in the previous step, let's multiply 7 by 2 and subtract.

In this division, we didn't have any remainder, and as a result, we found that when we divide 34 by 2, we get 17.

When we divide, we are essentially distributing the dividend (in this case, 34) into equal parts of the divisor (in this case, 2).

In our example, we are asking "how many groups of 2 can we make from 34?" And we found that we can make 17 groups. That's why the answer, or quotient, is 17.

It's like having 34 apples and you want to distribute them into baskets, with each basket holding 2 apples. You would end up with 17 baskets.

So, division is really a process of distribution or grouping.

What's actually happening?

Why did we ask how many times 2 fits into 3, i.e., why did we start from the left (from 3) instead of starting from the right (from 4)?

Our goal is to divide 34 into 2 groups, click here for the meaning of division >>>  If we break down 34 into its digits, we can take it as 3 tens and 4 ones, or we could say directly that there are 34 ones

Division means equal sharing, starting from the largest digit and distributing, and trying to distribute what can't be distributed by breaking it down into smaller pieces.

I distributed 2 tens into 2 buckets, and I can break down and distribute the remaining tens (10 ones). That's why I need to understand first whether the tens can be distributed or not. If these tens couldn't be distributed, then I would need to break them down into smaller pieces, that is, into ones. Therefore, I start from the left and ask, 'How many 2s are in 3?

Let me give you an example, if the division was as follows;

Let's start from the largest digit, 1. 

There isn't a 2 in 1, so we were trying to find out how many times 2 fits into 18, weren't we?

One block of ten cannot be distributed into 2 groups, so the block of ten needs to be broken down into smaller parts. 

The block of ten can be divided into 10 ones, and we already had 8 ones in our hand, making a total of 18 ones. Now, I can distribute these ones into 2 groups. 

Why didn't we write 1 over 4, but wrote it over 3 instead? Why didn't we write it randomly somewhere? What did writing it over 3 provide us? 

3 is in the tens place, and by writing 1 over 3, we're actually writing 10. This means that there are 10 sets of 2 in 34.

Why are we multiplying 2 and 1? ( Actually 10 )

We need to find out how much more of our number needs to be divided into groups of 2.

We found that there are 10 sets of 2 in 34, so we have divided our count up to 20 into groups of 2, and now we need to find out how much of our number is left.

So 14 is left..When we also devide 14 to 2, it will be done.

Why do we drop down the 4?

Actually, the above explanation also explains why we drop down the 4. Our goal is to divide the number incrementally. I need to subtract 20 from 34 (because I have divided the number up to 20), and I also need to divide the remaining 14. The number 34 is composed of three 10s and four 1s. When I subtract 2 from 3, due to the place values, I am subtracting 20 from 30, leaving 10. I still have 4 left. Therefore, by dropping down the 4, I'm making the number 14.

Another Example ;

How many times does 4 go into 69?

4 goes into 6 once, so we wrote 1 above the 6, in other words, in the tens place.

We multiplied 1 by 4, wrote the result of the multiplication under the 6, and subtracted it.

Step 2:

Drop down the next number and continue dividing.

Now we are going to divide 29 by 4. How many 4s are there in 29? There are 7! I am writing 7 in the ones place.

 I will multiply 7 by 4 and subtract the product from 29 so that I need to check whether the remaining number can also be divided by 4.

We have no more numbers left to drop down next to 1, and we cannot divide 1 by 4 either, so our division operation has ended. We will write 1 as a remainder. The result is: if I divide 69 by 4, the quotient is 17 and the remainder is 1.

Another Example

As we know, we were checking if there is a 6 in 4, but there's a problem here, there is no 6 in 4!

Since there is no 6 in 4, in other words, since there are 0 of them, you can write 0 above 4, or you may not (continue reading to learn why).

I am including the number next to 4, so I'm looking for how many 6s are there in 45. 7 times !

I'm writing 7 in the ones place above the 5, multiplying 7 by 6, and subtracting it from 45.

Our division operation is done. There are 7 sixes in 45, and our remainder is 3.

An alternative way for division

This alternative way isn't too different from what we call long division, but we can say it's a more logical approach.

Just as we sometimes take a big bite and sometimes a small bite to finish our food, in division too, we can completely finish the number to be divided by sometimes taking a big bite and sometimes a small bite.

Let's divide 45 by 6.

How many 6s are there in 45? You need to estimate this approximately, it doesn't have to be exact.. For example, I start with 10, let's think as if there were 10, 10 times 6 equals 60 and that's too much, so let's take 5. So, let's take a bite of 5 units...

We will try to divide the remaining 15 by 6 as much as we can

How many times does 6 fit into 15? It fits 2 times!

To find the remainder from 2 times 6, I multiply 6 by 2 and subtract it from 15

I divided 45 by 6, there are a total of 7 sixes and a remainder of 3. 

Dividing 3 Digit Number by 1 Digit Number

The principle of dividing a number of any digit by another number of any digit is essentially the same. However, let's see a few examples.

Let's divide 848 by 6.

How many 6s are there in 8? There is one... I write that one above the 8, in the hundreds place.

I multiply 1 by 6, then subtract the result from 8.

I am continuing the division by bringing down the 4 next to the 2.

How many times does 6 go into 24? 4 times! I write 4 in the quotient

I multiply 4 by 6 and subtract the result from 24."I still haven't brought down the 8, I need to bring down the 8.

I am bringing down the 8.Now I need to divide the 8.

How many 6s are there in 8? There's 1! We write 1 in the ones place of the quotient, next to the other numbers.

I need to find how much is left, so I multiply 1 by 6 and subtract it from 8.

There are no more numbers left to bring down, so my division is complete.

What actually happened? Let's learn what's behind these operations.

We start the division from the largest place value, here the largest place value is hundreds. If we break down 848 into its place values, it consists of 8 hundreds, 4 tens, and 8 ones.

The aim in division is to group, so we will divide into 6 groups. If I distribute the hundreds into 6 groups, there will be 1 hundred in each group, and 2 hundreds will be left over.

Here you are seeing the operation of the grouping shown above.

The 1 above the 8 in the quotient actually represents 100 because it's in the hundreds place.

Now we have 2 hundreds, 4 tens, and 8 ones. Since I can't distribute the 2 hundreds into 6 groups, I convert these 2 hundreds into tens. 2 hundreds mean 20 tens. When I bring down the 4, Bingo! Both the hundreds have been converted into tens and they have joined with the 4 tens. You see 24, which actually means 24 tens.

If the 24 tens-blocks are distributed into the 6 buckets below, each bucket receives an equal amount of 4 tens-blocks.

That's why I wrote 4 in the tens place of the quotient.

Now I only have ones left. If I group these ones as well, my division will be complete.

Alternative Method

We can divide using the 'bite method' I mentioned earlier, we can look at 'How many 6s are there in 800?' You don't have to find a very close result, since multiplying by 10, 100, 1000 is easy, let me take a bite of 6 hundreds at first, that is, we divide the number up to 600.

I can take 60 again and again... until I reach 240.

And finally, we can take a bite of 6 from 8.

Example:

Let's divide 143 by 7.

Let's start with the division in the traditional way.

7 is contained in 14 twice. I write 2 in the tens place of the quotient.

I bring down the next digit and here's where the problem starts.

So far, I've grouped 140, only 3 units are left, I can't distribute 3 units into 7 groups, so 3 should stay as it is. So, my remainder is 3, but there seems to be a problem in the quotient.

Result cant be 2 ! Yes, it's not 2 anyway.

Since 2 is in the tens place, its actual value is 20... So, we can add a zero next to the 2.