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