Improper Fractions and Mixed Numbers

 

  1. Definitions of Improper Fractions and Mixed Numbers

    In your pocket, you have five quarters.  Their are two ways of mentioning your cash.  The first way is five quarters or 5/4 dollars.  The second way is one dollar and a quarter or 1 dollars.  Both of these represent the same amount of money, however they are written in very different ways.  We define

    A proper fraction is a fraction smaller than one (the denominator is larger than the numerator)

            2        5         7         3        5   
                ,           ,          ,         ,          
    are all proper fractions                       
            9        10      32        4        7


    An improper fraction is a fraction greater than or equal to one (the numerator is larger or the same as than the denominator

           12       8        16       21       1   
                ,           ,          ,         ,          
    are all improper fractions                       
            5        8        13        2        1

    A mixed fraction is the sum of a whole number greater than zero and a proper fraction

              1            4             2             7           11   
         6       ,    4           ,  1          ,  7       ,    2          
    are all mixed fractions                       
              4            7            13            8           12


  2. Changing an Improper Fraction to a Mixed Fraction

    We have already learned how to do all of the work in changing an improper fraction to a mixed fraction when we learned how to perform long division with a remainder.  Now we just need to learn how to write down the answer.  

    Example

    Convert 

            22  
                                   
             7     

    into  a mixed number

    Solution

    We divide

                   3
            7 | 22
                 21
                   1

    The result is 3R1.  Now to write this as a mixed number, the whole number part is 3 the numerator is 1 and the denominator is the original denominator 7


            22               1
                      =  3                      
             7                7


    Exercises

    Convert the following improper fractions to mixed numbers

    1.         53  
                                     
               6     
      Hold mouse over the yellow rectangle for the solution  8 5/6

    2.         92  
                                     
              63     
      Hold mouse over the yellow rectangle for the solution  1 29/63

    3.         9321  
                                       
               32     

      Hold mouse over the yellow rectangle for the solution  291 9/32


  3. Changing a Mixed Fraction to an Improper Fraction

    Warm up problem

    Write the number 4 as an improper fraction with denominator 3.  

    Solution

    We write 

                     4 x 3               12
         4  =                    =              
                        3                   3

    To change a mixed fraction to an improper fraction we write the whole number as a fraction with the given denominator and then add the numerators.  

    Example

    Write the mixed fraction

            3

    as an improper fraction

    Solution

    Write 

                  3           3 x 4           3
            3           =                +                    
                  4              4              4

                    12 + 3           15
              =                   =                      
                         4                4


    Exercises

    Convert the following mixed fractions to improper fractions

    1.           5  
          7                         
                6     
      Hold mouse over the yellow rectangle for the solution  47/6
       

    2.            1  
          2                          
                16     
      Hold mouse over the yellow rectangle for the solution  33/16

    3.           12  
          5                          
                13     
      Hold mouse over the yellow rectangle for the solution  77/13


  4. Reducing Improper Fractions and Mixed Numbers

    To reduce a mixed fraction, we need only reduce the fractional part of the mixed fraction by pulling out the common factor.

    Example

    Reduce

                 9  
        18                          
                30     

    Solution

    We have 

             9                3 x 3                3
                    =                          =                            
            30            2 x 3 x 5            10

    So

                 9                    3
        18             =   18                  
                30                  10



    To reduce an improper fraction, we can cancel common factors just as we did with proper fractions.

    Example

    Reduce 

           
    102  
                                     
             68

    Solution

            102               2 x 51            2 x 3 x 17
                        =                      =                                                 
             68                2 x 34            2 x 2 x 17

                    3
           =              
                    2

    For an improper fraction with a large numerator, it is easier to first convert the fraction to a mixed fraction then reduce.


    Example

    Reduce

            35682  
                                        
               15

    Solution

    Divide first

                      2378
              15| 35682
                 - 30
                      56
                    - 45
                      118    
                    - 105
                        132
                     -  120  
                          12

    Hence

           
    35682                       12
                           =    2378                                   
               15                          15

    Now reduce the fractional part

            
    12             2 x 2 x 3          4
                     =                        =                  
             15                3 x 5             5

    Putting the two results together gives

           
    35682                       12                      4
                           =    2378             =  2378                                    
               15                          15                      5

    Exercises

    Reduce the following.  Write your answer as a mixed fraction.

    1.           24  
          4                          
                64     
      Hold mouse over the yellow rectangle for the solution  4 3/8
       

    2.         42  
                                     
              36     
      Hold mouse over the yellow rectangle for the solution  1 1/6

    3.         26234  
                                          
                 18     
      Hold mouse over the yellow rectangle for the solution  1457 4/9

 



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