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 
      
      
   2.          55           18
                        x             
               27           22            
      Hold mouse over the yellow rectangle for the solution 
      
      
   3.         25           24
                       x             
               6            10            
      Hold mouse over the yellow rectangle for the solution 
      

    

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