Division of Whole Numbers

 

  1. Definition of Division

    Example:

    Suppose that we have twelve students in the class and we want to divide the class into three equal groups.  How many should be in each group?

    Solution:

    We can ask the alternative question,  "Three times what number equals twelve?"
    The answer to this question is four.

    Division is defined as this reverse of multiplication.  

    We write

                   4
            3 | 12        
    or        12 3  =  4

    We call the number 12 her the dividend, the number 3 the divisor, and the number 4 the quotient.               


                           quotient
            divisor | dividend           
    or        dividend divisor  =  quotient
  2. Properties of Division

    1. Division by Oneself

      Example

      Suppose that you had $100 and had to distribute all the money to 100 people so that each person received the same amount of money.  How much would each person get?  

      Solution

      If you gave each person $1 you would achieve your goal.  This comes directly from the identity property of one.  Since the the questions asks what number times 100 equals 100.  

      In general we conclude, 


      Any number divided by itself equals 1



      Examples

      100 100  =  1        38 38  =  1        15 15  =  1

       

       

    2. Division by 1

      Example

      Now lets suppose that you have twelve pieces of paper and need to give them to exactly one person.  How many pieces of paper does that person receive?

      Solution

      Since the only person to collect the paper is the receiver, that person gets all twelve pieces.  This also comes directly from the identity property of one, since one times twelve equals twelve. 

      In general we conclude, 

      Any number divided by 1 equals itself



      Examples

      12 1  =  12        42 1  =  42        33 1  =  33

       

       

    3. When Zero is the Dividend

      Example

      Now lets suppose that you have zero pieces of pizza and need to distribute your pizza to four friends so that each person receives the same number of pieces.  How many pieces of pizza does that person receive?

      Solution

      Since you have no pizza to give, you give zero slices of pizza to each person.  This comes directly from the multiplicative property of zero, since zero times four equals zero.

      In general we conclude, 

      Zero divided by any nonzero number equals zero



      Examples

      0 4  =  0        0 1  =  0        0 24  =  0

       

       

    4. The Problem With Dividing by Zero

      Example

      Finally lets suppose that you have five bags of garbage and you have to get rid of all the garbage, but have no places to put the garbage.  How can you distribute your garbage to no places and still get rid of it all?

      Solution

      You can't!  This is an impossible problem.  There is no way to divide by zero.

      In general we conclude, 

      Dividing by zero is impossible



      Examples

      5 0  =  undefined        0 0  =  undefined        1 0  =  undefined

       

       

  3. Division With Remainder

    Often when we work out a division problem, the answer is not a whole number.  We can then write the answer as a whole number plus a remainder that is less than the divisor.

    Example

            34 5 

    Solution

    Since there is no whole number when multiplied by five produces 34, we find the nearest number without going over.  Notice that 

            5 x 6  =  30         and         5 x 7  =  35

    hence 6 is the nearest number without going over.  Now notice that 30 is 4 short of 34.  We write 

            34 5  =  6 R 4    "6 with a remainder of 4"

    Example

            4321 6 

    Solution

                   720
            6 | 4321  
                 42       
    6 x 7  = 42
                   12      
    43 - 42  =  1 and drop down the 2
                   12      
    6 x 2  =  12 
                     01    
    12 - 12  =  0 and drop down the 1
                       0     
    6 x 0  =  0
                       1     
    1 - 0  =  1


    We can conclude that 

            4321 6  =  720 R1

     

    In general we write

            (divisor x quotient) + remainder  =  dividend

     

    Example

                       511
            37 | 18932  
                   185         
    37 x 5  = 185
                       43       
    189 - 185  =  4 and drop down the 3
                       37       
    37 x 1  =  37 
                         62    
      43 - 37  =  6 and drop down the 2
                         37     
    37 x 1  =  37
                         25     
    62 - 37  =  25

    We can conclude that 

            18932 37  =  511 R25

    Exercises   (To check your answer hold mouse over the yellow rectangle)

    Divide

    A.  6275 8              784 R3

B.  3828 7              546 R6

C.  324337 43        7542 R31

D.  6749 103          65 R54

 

  1. Applications


    Example

    You are the manager of a ski resort and noticed that during the month of January you sold a total of 111,359 day ski tickets.  What was the average number of tickets that were sold that month?

    Solution

    Since there are 31 days in January, we need to divide the total number of tickets by 31

                       3589
            31 | 111259  
                     93         
    31 x 3  = 93
                     182       
    111 - 93  =  18 and drop down the 2
                     155       
    31 x 5  =  155 
                       275    
      182 - 155  =  27 and drop down the 5
                       248     
    31 x 8  =  248
                         279   
    275 - 248  =  27
                        
    279    31 x 9  =  279
                            
    0

    The ski resort averaged 3,589 ticket sales per day in the month of January.

     

    Exercise

    You are buying a custom refrigerator with a rectangular front.  If you only have enough space for the width to be 48 inches and you need the face to have an area of 2,976 square inches, how high must the refrigerator be?

    Solution 

    The frig must be 62 inches tall.

 



Back to the Arithmetic of Whole Numbers page

Back to the Math 187A page

Back to the Math Department page

e-mail Questions and Suggestions