Multiplication of Fractions

 

  1. Multiplying Two Fractions

    Suppose that you are making lemonade and have found a recipe that requires three-quarters of a cup of sugar to make one pitcher of lemonade.  You want to make a half pitcher of lemonade.  How much sugar do you need?

    Solution

    The operation of arithmetic needed here is multiplication, that is, what is 

             3           1
                    x            
             4           2            

    The pictures below show how to proceed.  We draw three shaded regions of four possible and than cut the shaded regions in half.

           

           

    The result is that there are still three shaded regions, but their are 4 x 2 total pieces.

    We conclude that 

             3           1                3 x 1               3
                    x            =                      =                       
             4           2                4 x 2               8


    In general, to multiply two fractions, multiply the numerators and multiply the denominators.

    Example

    1.          2           4                2 x 4               8
                      x            =                      =                       
               3           5                3 x 5              15

    2.          4          15               4 x 15       
                      x            =                                
               9           28              9 x 28         

                         2 x 2 x 3 x 5               5
              =                                   =                
      We have canceled common factors
                      3 x 3 x 2 x 2 x 7           21

    3.          8           3               8 x 3       
                      x            =                        
        Improper fractions multiply similarly       
               3          16              3 x 16         

                         2 x 2 x 2 x 3              1
              =                                   =          
      If all factors factors, we are left with a one
                      3 x 2 x 2 x 2 x 2           2
    4.                  2              5       2  
            5   x            =                    
      A whole number can always be put over 1.         
                       3              1       3    

                        
      10
                  =                
                          3

    Notice that we always cancel out common factors before multiplying

    Exercises

    Multiply the following fractions

    1.          5           3
                      x             
               7           11            
      Hold mouse over the yellow rectangle for the solution  15/77

    2.          55           18
                        x             
               27           22            
      Hold mouse over the yellow rectangle for the solution  5/3

    3.         25           24
                       x             
               6            10            
      Hold mouse over the yellow rectangle for the solution  10

     

  2. Multiplying Mixed Numbers

    If we need to multiply mixed numbers, we must first write them as improper fractions.  

    Example

    Multiply

                   3                1
              2          x    1             
                   4                3

    Solution

    First we write the two fractions as improper fractions

                   3                1                2 x 4 + 3              1 x 3 + 1
              2          x    1             =                          x                                 
                   4                3                      4                          3

                 11               4               11 x 2 x 2          
                           x                 =                             
                  4                3                 2 x 2 x 3        

                11          
                               
                 3       

    Now we can convert back to a mixed fraction by doing long division

                   3
            3 | 11
              -   9
                   2   

    We can conclude that 

                   3                1                  2  
              2          x    1             =   3            
                   4                3                  3

    Exercises

    Multiply

    1.                1                3
                3          x    2             
                     8                5
      Hold mouse over the yellow rectangle for the solution  9 3/8

    2.                1                3
                5          x    3             
                     3                4
      Hold mouse over the yellow rectangle for the solution  12


  3. Applications

    You just bought a 2 1/3 acre lot overlooking Lake Tahoe.  The Tahoe Regional Planning Agency has determined that because the land contains environmentally sensitive soil, you are only allowed to build on 1/5 of your land.  How many acres of land can your new house cover?    

    Solution

    This is a multiplication problem, so

                   1              1               2 x 3 + 1          1
              2          x                 =                       x               
                   3              5                    3                 5

                     7          1                7
              =           x             =                
                     3          5               15

    You can build on exactly 7/15 of an acre

 



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