When dividing fractions, why does the method of multipying by the reciprocal of the second fraction work

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embizze | High School Teacher | (Level 1) Educator Emeritus

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When dividing fractions, why does the method of multipying by the reciprocal of the second fraction work.

(1) This rule is true for all real numbers except 0 -- division is simply multiplication by the multiplicative inverse (called the reciprocal). Thus dividing by 2 is the same as multiplying by 1/2.

(2) In the case of dividing by fractions: consider `(a/b)/(c/d)` . This is called a compound fraction, and like many other constructions is frowned upon-- usually for no other reason than historical precedent. Anyway, you should not have a fraction in the numerator or denominator.

We can multiply this expression by one without changing it -- the trick is to find a suitable form of one. One idea is to realize that (c/d)(d/c)=1. Thus if we multiply the original compound fraction `(a/b)/(c/d)` by `(d/c)/(d/c)` we have formed an equivalent expression -- better, the denominator is now 1, thus `(a/b*d/c)/1` . Dividing by one leaves the expression unchanged so `(a/b)/(c/d)=a/b*d/c` which is the rule you asked about as d/c is the reciprocal of c/d (as long as neither c nor d is 0)

(3) Another way to see this is to again start with `(a/b)/(c/d)` . This time multiply numerator and denominator by b/1. (Note that (b/1)/(b/1) is just 1, so the result is equivalent). Then you get `(a/1)/(bc/d)=a/(bc/d)` . Now multiply by (d/1)/(d/1) to get `(ad/1)/(bc/1)=(ad)/(bc)=a/b*d/c` .

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