Improper Fractions and Mixed Numbers   Definitions of Improper Fractions and Mixed Numbers In your pocket, you have five quarters.  Their are two ways of mentioning your cash.  The first way is five quarters or 5/4 dollars.  The second way is one dollar and a quarter or 1¼ dollars.  Both of these represent the same amount of money, however they are written in very different ways.  We define A proper fraction is a fraction smaller than one (the denominator is larger than the numerator)         2        5         7         3        5                ,           ,          ,         ,           are all proper fractions                                9        10      32        4        7 An improper fraction is a fraction greater than or equal to one (the numerator is larger or the same as than the denominator        12       8        16       21       1                ,           ,          ,         ,           are all improper fractions                                5        8        13        2        1 A mixed fraction is the sum of a whole number greater than zero and a proper fraction           1            4             2             7           11         6       ,    4           ,  1          ,  7       ,    2           are all mixed fractions                                  4            7            13            8           12 Changing an Improper Fraction to a Mixed Fraction We have already learned how to do all of the work in changing an improper fraction to a mixed fraction when we learned how to perform long division with a remainder.  Now we just need to learn how to write down the answer.   Example Convert          22                                            7      into  a mixed number Solution We divide                3         7 | 22              21                1 The result is 3R1.  Now to write this as a mixed number, the whole number part is 3 the numerator is 1 and the denominator is the original denominator 7         22               1                   =  3                                7                7 Exercises Convert the following improper fractions to mixed numbers         53                                            6      Hold mouse over the yellow rectangle for the solution          92                                           63      Hold mouse over the yellow rectangle for the solution          9321                                              32      Hold mouse over the yellow rectangle for the solution  Changing a Mixed Fraction to an Improper Fraction Warm up problem Write the number 4 as an improper fraction with denominator 3.   Solution We write                   4 x 3               12      4  =                    =                                   3                   3 To change a mixed fraction to an improper fraction we write the whole number as a fraction with the given denominator and then add the numerators.   Example Write the mixed fraction         3¾ as an improper fraction Solution Write                3           3 x 4           3         3           =                +                                   4              4              4                 12 + 3           15           =                   =                                            4                4 Exercises Convert the following mixed fractions to improper fractions           5       7                                    6      Hold mouse over the yellow rectangle for the solution               1       2                                     16      Hold mouse over the yellow rectangle for the solution            12       5                                     13      Hold mouse over the yellow rectangle for the solution  Reducing Improper Fractions and Mixed Numbers To reduce a mixed fraction, we need only reduce the fractional part of the mixed fraction by pulling out the common factor. Example Reduce              9       18                                       30      Solution We have           9                3 x 3                3                 =                          =                                     30            2 x 3 x 5            10 So              9                    3     18             =   18                               30                  10 To reduce an improper fraction, we can cancel common factors just as we did with proper fractions. Example Reduce          102                                              68 Solution         102               2 x 51            2 x 3 x 17                     =                      =                                                           68                2 x 34            2 x 2 x 17                 3        =                               2 For an improper fraction with a large numerator, it is easier to first convert the fraction to a mixed fraction then reduce. Example Reduce         35682                                                   15 Solution Divide first                   2378           15| 35682              - 30                   56                 - 45                   118                     - 105                     132                  -  120                         12 Hence         35682                       12                        =    2378                                               15                          15 Now reduce the fractional part          12             2 x 2 x 3          4                  =                        =                            15                3 x 5             5 Putting the two results together gives         35682                       12                      4                        =    2378             =  2378                                                15                          15                      5 Exercises Reduce the following.  Write your answer as a mixed fraction.           24       4                                     64      Hold mouse over the yellow rectangle for the solution            42                                           36      Hold mouse over the yellow rectangle for the solution          26234                                                   18      Hold mouse over the yellow rectangle for the solution    Back to the Fractions page Back to the Math 187A page e-mail Questions and Suggestions