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                     \ 10-Nov-2025  Bharat Arora   Proof-8                                    Reader’s Sign _______________________ Date __________

                                     ÷ 3        ÷ 3        ÷ 3       ÷ 3                     If  the  numerator  and the
                    Similarly:   81   =    27    =    9    =     3    =     1                denominator do not have
                                243        81         27         9          3                any common factor other
                                     ÷ 3        ÷ 3        ÷ 3       ÷ 3                     than 1, the fraction is said
                                                                                             to be in its simplest form.
                                                                                       2              1
                    example 1. Write the next two fractions equivalent to  .                 Here,      is the simplest
                                 2    2   2    4          2    2    3    6             4             3  81
                    Solution.   =× =               and      = ×=                             form of       .
                                 4    4   2    8          4    4    3    12                            243
                                                                      2       4       6
                    So, the next two equivalent fractions of            are  and         .
                                                                      4      8        12
                    example 2. Fill in the boxes to make equivalent fractions.
                                     1                                             4     12
                                 a.     =                                     b.       =
                                     2     12                                       7
                    Solution.  a.   We multiply 2 by 6 in order to make the denominator 12. So, we also

                                                                                1    1   6     6
                                     multiply the numerator by 6. Thus,  =× =
                                                                                2    2   6    12
                                 b.   We multiply 4 by 3 in order to make the numerator 12. So, we also multiply

                                                                        4    4   3    12
                                     the denominator by 3. Thus,  =× =
                                                                        7    7   3    21

                    To Check whether Two Fractions are equivalent or not

                    To check whether two fractions are equivalent or not, we cross multiply them, that is,

                    we multiply the numerator of one fraction with the denominator of the other. If the

                    products are the  same,  then  both  the  fractions  are  equivalent.  If the  products are
                    different, then the fractions are not equivalent.

                    example 3. Are the following fractions equivalent?

                                     2        4                    3        12                        5       15
                                 a.    and                     b.   and                           c.   and
                                     3        6                    4        16                        6        17
                                     2         4
                    Solution.  a.                . Here, 2 × 6 = 12 and 3 × 4 = 12. Both products are equal.
                                     3        6
                                               2       4
                                     Hence,   and          are equivalent fractions.
                                               3       6
                                     3         12
                                 b.               . Here, 3 × 16 = 48 and 4 × 12 = 48. Both products are equal.
                                     4        16
                                              3        12
                                     Hence,  and           are equivalent fractions.
                                              4        16

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                                                                                                    Fractions

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