Addition and Subtraction of Fractions

1. Adding and Subtracting Fractions With a Common Denominator
   
   No matter how tempting it may be, we can only perform addition and subtraction of fractions when the denominators are the same.  
   
   When the denominators are the same we just add the numerators
   
   Examples
   
   
   1.   2         1          3
              +         =                                  
        5         5          5
      
   2.    4            6          ²10          2
                 +           =             =        Always reduce the answer.                         
        15          15          ³15         3
      
   3.   4          5           9              2
               +         =           or 1        The result may be improper.                         
        7          7           7              7
      
   4.   5         1          ¹4           1
              -         =             =        We subtract the numerators.                         
        8         8          ²8           2
      
      

   Exercises
   
   

   1.    6          2     
                -                                
        11        11     
      Hold mouse over the yellow rectangle for the solution 
      
   2.   3         1      
              +                  
        4         4         
      Hold mouse over the yellow rectangle for the solution 
      
      
   3.    7          8        
               +                     
        20        20        
      Hold mouse over the yellow rectangle for the solution 
      
      

    

2. Adding and Subtracting Fractions Without a Common Denominator
   
   Now that we know how to add and subtract fractions with a common denominator, what can we do when the denominators are different?  In this case we must first build the fractions so that the denominators are the same.  We first find the LCD and then do the building.
   
   Example
   
   Add
   
             1          3        
                    +                    
             6          8        
   
   Solution
   
   The common denominator of 6 and 8 is 24, since 
   
           6  =  2 x 3     and  8  =  2 x 2 x 2
   
   so that the LCD is
   
           2 x 2 x 2 x 3  =  24
   
   We write
   
   
             1 x 4          3 x 3            4          9     
                          +               =           +            We now have the same denominator
             6 x 4          8 x 3           24        24
   
                  13      
            =            
                  24    
   
   Example
   
   Subtract
   
             3          1        
                    -                     
             4         10        
   
   Solution
   
   The common denominator of 4 and 10 is 20, since 
   
           4  =  2 x 2     and  10  =  2  x 5
   
   so that the LCD is
   
           2 x 2 x 5  =  20
   
   We write
   
   
             3 x 5           1 x 2           15         2     
                          -                =           -            
             4 x 5          10 x 2          20        20
   
                  13      
            =            
                  20    
   

   
   Example
   
   Find the value of x in the equation 
   
                   3          2         
        x  +           =         
                   7          3
   
   Solution
   
   The LCD of 7 and 3 is 21, so that we can write 
   
                   3 x 3          2 x 7        
        x  +                 =               
                   7 x 3          3 x 7
   
   or
   
                   9            14         
        x  +             =            
                   21           21
   
   Since the denominators are the same, the numerators must add up.  Since 
   
           5 + 9  =  14
   
                      5      
           x  =            
                     21    
   

   
   Exercises
   

   Add or subtract.  Simplify the answer.

    

   1.   2         1     
              +                               
        5         9     
      Hold mouse over the yellow rectangle for the solution 
      
      
   2.    3          1     
                -                                 
         4         12     
      Hold mouse over the yellow rectangle for the solution 
      

    

3. Application
   
   Sierra-At-Tahoe ski resort claims that they just received 3/4 feet of new snow on top of a base of 2/3 feet of snow.  How many total feet of snow does the resort have?
   
   Solution

   We add the two fractions to arrive at the total.
   
             3The LCD of 4 and 3 is 12.          
             4          3            4 x 3           3 x 4
   
             9The resort has 1 5/12 feet of snow (one foot five inches of snow).

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