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 
      

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

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 
      

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

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.        6         1         11  
                    -        ÷                                      
             7         3         21   
      Hold mouse over the yellow rectangle for the solution 
      

   2.     
      Hold mouse over the yellow rectangle for the solution 
      
      

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