Multiplication of Fractions   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          2           4                2 x 4               8                 x            =                      =                                 3           5                3 x 5              15          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          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                  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          5           3                 x                       7           11             Hold mouse over the yellow rectangle for the solution           55           18                   x                       27           22             Hold mouse over the yellow rectangle for the solution          25           24                  x                       6            10             Hold mouse over the yellow rectangle for the solution    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                3           3          x    2                             8                5 Hold mouse over the yellow rectangle for the solution                 1                3           5          x    3                             3                4 Hold mouse over the yellow rectangle for the solution  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   Back to the Fractions page Back to the Math 187A page Back to the Math Department page e-mail Questions and Suggestions