Combining Mixed Numbers and Order of Operations

  1. Adding Mixed Numbers

    In the last section we learned how to add and subtract two fractions.  

    If we have two mixed numbers to add, we can just add the whole number parts and then add the fraction parts.

    Example

    Add

                2           1      
            3        + 2                                   
                7           5   

    Solution

    We first add the fractional parts:

              2         1             2 x 5         1 x 7
                    +          =                 +               
    the LCM of 7 and 5 is 35                           
              7         5               35             35

                  17     
            =                  
    2 x 5 + 1 x 7  =  10 + 7  =  17                       
                  35

    Now add the whole number parts


            3 + 2  =  5

    The final answer is

            
                2           1             17
            3        + 2        =  5                                    
                7           5             35

    Example

    Add

                3           5      
            4        + 5                                   
                4           6   

    We first add the fractional parts:

              3         5             3 x 3         5 x 2   
                    +          =                 +                                     
              4         6               12             12    
    The LCM of 4 and  6 is 12

                  19     
            =                  
    3 x 3 + 5 x 2  =  9 + 10  =  19                       
                  12

                     7     
            =  1                
    3 x 3 + 5 x 2  =  9 + 10  =  19                       
                    12

    The whole number parts are the numbers from the original mixed fractions and the one from the
    1 7/12.  This extra "1" is analogous to the idea of carrying in addition of whole numbers.  We have

            4 + 5 + 1  =  10  

    The final answer is

            
                3           5                7
            4        + 5         =  10                                   
                4           6               12



    Exercises

    Add

    1.              3             2      
              5          + 8                                     
                  10           15   
      Hold mouse over the yellow rectangle for the solution  13 13/30

    2.             5            11      
              3        +  2                                     
                  9            12   
      Hold mouse over the yellow rectangle for the solution  6 17/36



  2. Subtracting Mixed Numbers

    Subtraction of mixed numbers is similar to addition, in that we treat the whole numbers separately from the fractional part.

    Example

    Subtract

                3          1      
            8        - 3                                   
                5          6   

    Solution

    We first subtract the fractional parts:

              3         1             3 x 6         1 x 5
                    -          =                 -                
    the LCM of 5 and 6 is 30                     
              5         6               30             30

                  13     
            =                  
    3 x 6 - 1 x 5  =  18 - 5  =  13                       
                  30

    Now subtract the whole number parts


            8 + 3  =  5

    The final answer is

            
                3           1             13
            8        - 3        =  5                                    
                5           6             30

    Sometimes, we need to borrow a 1 from the whole number in order to subtract the fractions.

    Example

    Subtract

                3           7      
            6        -  3                                   
                4           8   

    Solution

    If we try to subtract the fractional parts, we soon see that we cannot do this.

              3         7              6         7
                    -           =            -                                                   
              4         8              8         8

    Since 

              6          7
                    <                                                    
              8          8

    we must borrow from the whole number.

                3                   3              1 x 4 + 3               7
            6        =  5 + 1       =  5 +                      =  5               
                4                   4                   4                      4


    Now we can subtract the fractional parts:

              7         7             7 x 2         7
                    -          =                 -          
    the LCM of 4 and 8 is 8                           
              4         8               35           8

                  7     
            =                
    7 x 2 - 7  =  14 - 7  =  7                       
                  8

    Now subtract the whole number parts


            5 - 3  =  2

    The final answer is

            
                3           7             7
            6        - 3        =  2                                   
                4           8             8


    Exercises

    Subtract

    1.              5            4      
              9         - 4                                     
                   9           21   
      Hold mouse over the yellow rectangle for the solution  5 23/63

    2.              3             3      
              7          -  3                                     
                  10            4   
      Hold mouse over the yellow rectangle for the solution  3 11/20

  3. Fractions and Order of Operations

    The same rules of order of operations apply to expressions with fractions.  In particular, the order is:

    1. Parentheses

    2. Exponents

    3. Multiplication and Division

    4. Addition and Subtraction

    With left to right when there is a tie.

    The pneumonic,  "Please Excuse My Dear Aunt Sally"  could help us remember this order. 
          

    Example

              1           4         7
                    +          x                                       
              3           7        30       
    Multiplication comes before addition

                  1           2            24           17
            =          +                 
             x                              
                  3           15         
    17          1530

                  1 x 5           2                7        
            =                +             =                                                       
                    15            15              15       

    Example

           

              
         2           16     
          =                               
    Exponents come first                     
                3           25        

                   12          25              25                 1
          =          x                =                or   1                    
                     
                3          
    816              24                24


    Exercises

    Evaluate using the correct order of operations

    1.                1         11  
                    -                                              
             7         3         21   
      Hold mouse over the yellow rectangle for the solution  17/77

    2.     
      Hold mouse over the yellow rectangle for the solution  17/24

 



Back to the Fractions page

Back to the Math 187A page

Back to the Math Department page

e-mail Questions and Suggestions