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Division:

As subtraction is the opposite of addition, division is the opposite of multiplication. It may be expressed as:

----
   
3)12
 =
    

or alternatively:
12 : 3 =
   

and in both cases we are dividing 12 by 3.

You can also express this as how many times does 3 go into 12? In other words, if we divide the number 12 into blocks that are each equivalent to 3, how many such blocks are we going to find in 12? Look at the number line below:

--------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
0
1
2
3
4
5
6
7
8
9
10
11
12
 13 14

You can see that the first block of 3 ends at 3, the second ends at 6, the third ends at 9 and the fourth ends at 12. There are 4 block of 3 that go into 12 and therefore:

12 : 3 = 4
   

In the above example, we have 4 as our result and 4 is an integer (whole number), because 12 is divisible by 3. Very often, though, we have to divide a number that is not divisible and what happens then? Let us consider the division of:

13 : 3 =
   

We have two options. We have just established that 12 is divisible by 4, since 3 x 4 is 12. What is the difference between 13 and 12?
13 - 12 = 1
   

We can therefore say that:
13 : 3 = 4, R1
   

Where the R stands for remainder and what we are saying is that 13 divided by 3 equals 4, remainder 1 or that 1 is left over. That is the simple way of solving this division problem.

By using decimal numbers, we can be more precise.

All division should really be shown as follows:
13 : 3 = 4
   

We know that 4 x 3 equals 12, so we will write:
13 : 3 = 4
   
  12    
  ---    
    1    

We have subtracted 12 from 13 and we are showing the remainder of 1. Now, we will add a zero, which will allow us to go beyond the decimal point:
13 : 3 = 4.
    
  12    
  ---    
    10    

Notice that we have added the zero (in red) and the decimal point (also in red) and now we ask how many times does 3 go into 10? Well, we of course know that 3 x 3 = 9, so the answer has to be 3:
13 : 3 = 4.3
    
  12    
  ---    
    10    
      9    
     ---    
      1    

After we show the result of our multiplication of 3 by 3 and subtract the result of 9 from 10, we are again left with the remainder of 1 (shown in red). If you proceed, you will se that you can go on indefinitely beyond the decimal point and keep getting an answer of 3, with remainder 1 and you will have to round the number off.

Now try the following examples:

Examples:

1)  
 81 : 9 =  
11)  
  95 : 7 =
2)  
 82 : 9 =  
12)  
  113 : 21 =
3)  
 47 : 5 =  
13)  
  149 : 13 =
4)  
 65 : 8 =  
14)  
  229 : 33 =
5)  
 112 : 2 =  
15)  
  159 : 26 =
6)  
 95 : 5 =  
16)  
  79 : 12 =
7)  
 121 : 5 =  
17)  
  96 : 12 =
8)  
 39 : 3 =  
18)  
  226 : 15 =
9)  
 225 : 6 =  
19)  
  449 : 17 =
10)  
  114 : 8 =  
20)  
  229 : 74 =

Answers:

1)  
  9  
11)  
  13.571
2)  
  9.11  
12)  
  5.381
3)  
  9.4  
13)  
  11.462
4)  
  8.125  
14)  
  6.939
5)  
  56  
15)  
  6.115
6)  
  19  
16)  
  6.583
7)  
  24.2  
17)  
  8
8)  
  13  
18)  
  15.067
9)  
  37.5  
19)  
  26.412
10)  
  14.25  
20)  
  3.095
 
 
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