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Inverting and Multiplying When Dividing Fractions

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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?
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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

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

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Associated Topics:
Middle School Fractions

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