Inverting and Multiplying When Dividing Fractions

Date: 9/28/95 at 0:30:35
From: Anonymous
Subject: dividing fractions

I have always been taught that to divide fractions I must 
invert and multiply.  Why does this algorithm work?  Is there 
any other way to divide fractions?

Date: 9/28/95 at 2:30:58
From: Doctor Andrew
Subject: Re: dividing fractions

Division can be defined as x = a / b if and only if b * x = 
a.  This says that if b times x equals a, then x equals a 
divided by b.  It also says that if x equals a divided by b, 
then b times x equals a.  "If and only if" is an expression 
that mathematicians like to use a lot so they can say one 
sentence instead of two. 

Note: when I write a/b I mean a fraction and when I write 
a / b I mean division.

So if we have a fraction p/q and we want to divide it into a 
number n the result is the number x such that

p/q * x = n

Let's solve this for x.  Start by multiplying both sides by q 
(make sure q isn't zero)

px = n * q.

now divide both sides by p (p had better not be zero):

x = n * q / p  = n * q/p

So, n / (p/q) = n times the reciprocol of p/q (which is q/p.)

You can divide fractions any way you want so long as the 
statement x = a / b if only if b * x = a is always true.  For 
instance, instead of multiplying the reciprocols you could, 
if you were dividing two fractions, divide the numerators and 
the denominators separately like this:

a   c   a / c
- / - = -----
b   d   c / d

But if (a/c) is not an integer (say 0.12623456 and not a 
number like 10) or (b/d) is not an integer, then you're going 
to have a fraction with decimal numbers, which is just not 
allowed.  The reciprocol method allows you to keep
integers in the numerator and denominator of the quotient 
(that's what the result is called) when you divide.

I hope this helps.  Please send us any more questions or ask 
me to clarify anything here that might be confusing.

- Doctor Andrew,  The Geometry Forum