When dividing by a fraction, why can you not take the reciprocal of term involving addition/subtraction?Given something like: a a b You would multiply the numerator a by the reciprocal of the denominator, b a to get: a ⋅ b a = a b a = bGiven 1 1 a + 1 b By taking the LCM and adding the denominators you get: 1 ( a + b a b ) Given the reciprocal division rule in example one: a b a + b Why can you not take the reciprocal of 1 a + 1 b to begin with? I did this and ended up with: 1 ⋅ ( a 1 + b 1 ) = a + bHowever 1 1 / a + 1 / b is not the same as a + b so this is incorrect. I was trying to find a similar example online but I could not, why is this incorrect? Does the rule only work with one fraction as the denominator and not terms linked by addition and/or subtraction?

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When dividing by a fraction, why can you not take the reciprocal of term involving addition/subtraction?Given something like:      a          a      b      You would multiply the numerator a by the reciprocal of the denominator,             b      a       to get:  a  ⋅      b    a    =            a      b        a    =  bGiven      1                  1        a            +              1        b            By taking the LCM and adding the denominators you get:      1          (                        a          +          b                          a          b                    )      Given the reciprocal division rule in example one:            a      b              a      +      b      Why can you not take the reciprocal of       1    a    +      1    b   to begin with? I did this and ended up with:  1  ⋅      (          a      1        +          b      1        )    =  a  +  bHowever       1          1              /            a      +      1              /            b       is not the same as   a  +  b so this is incorrect. I was trying to find a similar example online but I could not, why is this incorrect? Does the rule only work with one fraction as the denominator and not terms linked by addition and/or subtraction?

gasskadeu7 · Accepted Answer

When you took the reciprocal of       1    a    +      1    b   to get   a  +  b, you implicitly assumed that the reciprocal of a sum is the sum of the reciprocals. This is not true.Note that       1    a   is       a          −      1      . Just as the square of a sum is not the sum of the squares (in general), that is,   (  x  +  y      )    2    ≠      x    2    +      y    2  , so also the reciprocal of a sum is not in general the sum of the reciprocals:   (  x  +  y      )          −      1        ≠      x          −      1        +      y          −      1

Kendra Hudson · Accepted Answer

The reciprocal of       1    a    +      1    b  , is by definition      1                  1        a            +              1        b              =      1                  a        +        b                    a        b              =            a      b              a      +      b        .So your reasoning fails because       a    1    +      b    1   is not the reciprocal of       1    a    +      1    b  .

When dividing by a fraction, why can you not take the reciprocal of term involving addition/subtraction? Given something like: a/(a/b) You would multiply the numerator a by the reciprocal of the denominator, b/a to get: a*b/a= (ab)/a=b Given 1/( 1/a + 1/b)

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