Lesson 4

Whole Number Division

Introduction

Here you'll learn how to divide whole numbers.

Vocabulary

Here are the vocabulary words used in this Concept.

Dividend  the number being divided

Divisor  the number doing the dividing

Quotient  the answer to a division problem

Remainder  the value left over if the divisor does not divide evenly into the dividend

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Guidance

You have now learned how to add, subtract and multiply. The last whole number operation that we will learn is long-division.

What does “division” actually mean. The best way to understand it is to think of splitting into groups.

We saw that  multiplication means to add groups of things together to get a product or total. Therefore division which is splitting a total into groups is the opposite of multiplication.

7 2 ÷ 9 =  

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In this problem we are trying to discover how many groups there will be if we split 72 into groups of 9.

72 is the number being divided, it is the dividend. 9 is the number doing the dividing, it is the divisor.

We can complete this problem by thinking of our multiplication facts and working backwards. Ask yourself “What number multiplied by 9 equals 72?”

Think back to your 9 times tables. If you said “8”, you're right!

9 × 8 = 7 2  

So 72 can be split into 8 groups of 9. Therefore:

7 2 ÷ 9 = 8  

The answer to a division problem is called the quotient. Quotient is a really old word that means "how many times".

Remainders

Sometimes, a number won’t divide evenly. When this happens, we have a remainder.

1 5 ÷ 2 = ?

In this example we aredividing by two. Because fifteen is not an even number there is going to be a number left over. We call that the remainder. If we look at the the nearest possible multiplication solution:

7 × 2 = 1 4 + 1

 We get 14 and their is an extra 1 required to get to fifteen. So when we divide and have a remainder we write it like this:

1 5 ÷ 2 = 7 r 1  

We can use an “r” to show that there is a remainder.

Here is another example with a larger remainder value.

3 5 ÷ 9 = 3 r 8  

How did we get that? We use our 9 times table to see to the closest value to 35.

9 × 1 =   9
9 × 2 = 1 8
9 × 3 = 2 7
9 × 4 = 3 6

Our dividend number 35 is smaller than 36 so we have to use 3 × 9 = 27.

Then the remainder is the dividend 35 minus 27. The result is 8. So in this problem our remainder is 8.

In this example we can see the remainder must always be a smaller number than the divider.

Guided Tutorial 1. Simple Long-Division

Larger numbers can be divided by using a division box. This is a tool we can use to organize the problem so we can get an answer.

Lets have a look at dividing a longer number and how to present the problem so we can work it out.We are going to solve:

7 9 8 ÷ 7 = ?

Use this tutorial to learn how to do this step by step.

 Step 1. Setting up

           
7   7 9 8  
           
           

We are going to divide 798 by 7. We set the sum up with a division box.

Division looks a little different from the other sums, but we still have to be aware of place values as we solve the problem.

The small number on the left is the divisor.

The big number on the right is the dividend.

 Step 2. Hundreds

    1      
7   7 9 8  
           
           

With division we work from the biggest place value to the smallest.

The division steps start from the left and move to the right.

So we divide the first digit 7 (Hundreds) by the divider 7 and we get 1.

The 1 is put into the quotient line above the 7.

 Step 3. Check the number

    1        
7   7 9 8    
  - 7        
    0        
             

When we are doing long division it is good to check each step with a multiplication.

1 × 7 = 7

We put this in the first work-line in the calculation area under the division box.

Subtract the values. 7 - 7 = 0 so there is no remainder. We can see that clearly.

 Step 4. Tens

    1 1      
7   7 9 8    
  - 7        
    0 9      
    - 7      
      2      

Next we divide the second digit 9 by the divider 7.

The answer is 1 so we put that on the quotient line above the 9.

Now we are going to do our multiplication check. 

Bring down the 9 to the work-line. Multiply 7 x 1 and put that value in the next work-line. 

Subtract. This gives us a tens place-value left-over of 2 Tens.

 Step 5. Find the Unit value

    1 1 4    
7   7 9 8    
  - 7        
    0 9      
    - 7      
      2 8    

We have 2 left over in the tens place value. This is 20 Units. We bring-down the 8 next to the 2 and have a total of 28 Units.

We can now divide 28 by the divider 7.

Using our times tables we can easily work out the answer is 4. So we put 4 above the 8 in the quotient line.

 Step 6. Final checks

    1 1 4  
7   7 9 8  
  - 7      
    0 9    
      7    
      2 8  
    - 2 8  
      0 0  

Now we are going to do our last multiplication check.

Multiply 4 X 7 and place it in the last work-line of the calculation area.

Do the subtraction. The answer is 0. So there is no remainder.

 Step 7. Answer & Review

    1 1 4  
7   7 9 8  
  - 7      
    0 9    
      7    
      2 8  
    - 2 8  
      0 0  

We now know 798 ÷ 7 = 114.

We can check with multiplication.

114 × 7 = 798

Even though the problem was simple we did the multiplication checks at each stage. 

This keeps the division tidy and we can easily review our work and spot any mistakes.

= END OF TUTORIAL =

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Guided Tutorial 2. Long Division with Remainders

Now that we understand the basic long-division method we can look at another example but this time with a remainder because the divider does not go equally into the dividend.

We are also going to make the problem a little longer.

 Step 1. Setting up

                 
6   4 5 8 2      

We are going to divide 4582 by 6. We set the sum up with a division box.

Division looks a little different from the other sums, but we still have to be aware of place values as we solve the problem.

The small number 6 on the left is the divisor.

The big number 4,582 on the right is the dividend.

 Step 2. Thousands

    0            
6   4 5 8 2      
                 

The division steps start from the left and move to the right.

So we divide the first digit 4 by the divider 6. This doesn't go so we put 0 into the quotient line above the 4.

 Step 3. Hundreds

    0 7          
6   4 5 8 2      
  - 4 2          
      3          

Now we look at the first two digits in the dividend. They are 45 Hundreds. We use our multiplication test to find how many times 6 goes into 45.

6 × 6 = 36
6 × 7 = 42
6 × 8 = 48

So the answer is 42. We put the 42 in the work-line under the division box.

Subtract the values. 45 - 42 = 3 Hundreds is left over.

 Step 4. Tens

    0 7 6        
6   4 5 8 2      
  - 4 2          
      3 8        
    - 3 6        
        2        

Now we have to bring down the 8 Tens next to the 3 Hundreds. That makes a value of 380 Tens.

Again we have to work out how many time 6 can go into 38. From our previous multiplication tests we can see this will be 6.

We put 6 in the quotient line in the Tens place value position above the 8.

We do our multiplication check and subtract. The left-over value for the Tens place value is 2.

 Step 5. Units

    0 7 6 3 r 4  
6   4 5 8 2      
  - 4 2          
      3 8        
    - 3 6        
        2 2      
      - 1 8      
          4      

We repeat the bring down process for the units.

The dividend Unit digit 2 is brought down next to the 2 tens left-over value. This makes 22 Units.

The multiple of 6 that is closest to 22 is

3 × 6 = 18

We put 3 in the quotient line.

 

 Step 6. Calculating the Remainder

    0 7 6 3 r 4  
6   4 5 8 2      
  - 4 2          
      3 8        
    - 3 6        
        2 2      
      - 1 8      
          4      

Now we are going to do our last multiplication check.

Multiply 3 × 6. We get 18. 

Place 18 in the last work-line of the calculation area.

Do the subtraction. The answer is 4. The remainder for this problem is 4.

We write the remainder into the quotient line using r4.

 Step 7. Answer & Review

    0 7 6 3 r 4  
6   4 5 8 2      
    4 2          
      3 8        
      3 6        
        2 2      
        1 8      
          4      

We now know 4,582 ÷ 6 = 764 with a remainder of 4.

We can check with multiplication.

764 × 6 = 4578 + 4 = 4582

Now we understand how to do long division where the divider does not go equally into the dividend.

Remember to keep your division tidy so you can easily review your work and spot any mistakes.

= END OF TUTORIAL =

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Guided Tutorial 3. Long-Division 2-Digit Divisor

Now we are going to look at a problem with a two-digit divisor. We are also going to increase the size of the dividend to show how you can easily use your new long-division skills to solve a wide range of dividing problems.

1 7   7 8 6 8 4

 That looks pretty challenging. But getting the answer is just a matter of going step by step.

 Step 1. Setting up

                 
17   7 8 6 8 4    

We are going to divide 78,684 by 17. We set the sum up with a division box. Look at the dividend place values.

7 Ten-thousands (TTh)

8 Thousands (T)

6 Hundreds (H)

8 Tens (T)

4 Units (U)

We are going to use place value terms in this tutorial explanation.

 Step 2. Ten-thousands

    0            
17   7 8 6 8 4    
  - 0            
    7            

7 Ten-thousands cannot be be divided into 17 equal parts. Therefore the quotient is 0.

Write 0 on top of the 7.

Now do the multiplication check.

17 × 0 = 0

Place this into the first work-line then subtract 0 from 7.

The answer is 7 Ten-thousands remainder.

 

 Step 3. Thousands

    0 4          
17   7 8 6 8 4    
  - 0            
    7 8          
                 

Now bring down the 8 Thousands.

7 Ten-thousands and 8 Thousands together make 78 Thousands.

Let's use the 17 times tables.

17 × 1 = 17
17 × 2 = 34
17 × 3 = 51
17 × 4 = 68
17 × 5 = 85

85 is greater than 78, and 68 is less than 78. Therefore the quotient will be 4.

 Step 4. Thousands cont.

    0 4          
17   7 8 6 8 4    
  - 0            
    7 8          
  - 6 8          
    1 0          

We put the 4 into the quotient line above the 8 in the Thousands position.

4 × 17 = 68

This is placed into the next work-line.

When 68 is subtracted from 78 the remainder is 10.

Remember this is 10 Thousands.

 Step 5. Hundreds

    0 4 6        
17   7 8 6 8 4    
  - 0            
    7 8          
  - 6 8          
    1 0 6        
  - 1 0 2        
        4        

We repeat the bring down process for the 6 Hundreds.

10 Thousands makes 100 Hundreds. 100 Hundreds and 6 Hundreds  together are 106 Hundreds. Let's divide 106 by 17.

17 × 6 = 102
17 × 7 = 119

119 is greater than 106, 102 is less than 106. Therefore the quotient is 6. We do the subtraction and the remainder is 4.

 Step 6. Tens

    0 4 6 2      
17   7 8 6 8 4    
  - 0            
    7 8          
  - 6 8          
    1 0 6        
  - 1 0 2        
        4 8      
      - 3 4      
        1 4      

These 4 Hundreds make 40 Tens. We bring down the 8 Tens. We now have 48 Tens.

Again looking at our 17 times table
17 × 2 = 34
17 × 3 = 51

51 is greater than 48 and 34 is less than 48. The quotient is 2 and is placed on the quotient line in the Tens position.

We put the 34 into the work-line and subtract. The remainder is 14 Tens.
 

 Step 7. Units

    0 4 6 2 8 r 8
17   7 8 6 8 4    
  - 0            
    7 8          
  - 6 8          
    1 0 6        
  - 1 0 2        
        4 8      
      - 3 4      
        1 4 4    
      - 1 3 6    
        0 0 8    

These 14 Tens make 140 Units. We bring down the 4 Units from the dividend.

We now have a total 144 Units.

17 × 8 = 136

We can see immediately that 8 is the quotient, and after subtracting 144 - 136 the remainder is also 8.

We write 8 in the Units position on the quotients line. We also write r8 at the end of the answer.

 Step 8. Summary

    0 4 6 2 8 r 8
17   7 8 6 8 4    
  - 0            
    7 8          
  - 6 8          
    1 0 6        
  - 1 0 2        
        4 8      
      - 3 4      
        1 4 4    
      - 1 3 6    
        0 0 8    

So 78,684 divided by 17 is 4,628 with a remainder of 8.

This example shows how using multiplication helps long division.

By writing down the times table for the divider at different values you can quickly work-out the correct value to go into the quotient for all place values.

Now practice with the Exercises.

= END OF TUTORIAL =

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Video Review

If you have an Internet connection here are some videos for you to look at that will help you learn long division.

      Take the time to complete all the Exercises and use the Practice and Test Interactive tools to make sure division becomes your friend. Learn to enjoy the numbers. They are fun and these exercises let you do it all at your own speed anytime you want.

 

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