Q&A #12670

Dividing fractions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Dec 09, 2003 at 03:36:29
Subject: Re: Dividing fractions

>If I divide the fraction 6/10 by 2/5 and simply divide across the top and
>divide across the bottom like I do when I multiply, it gives me the same
>answer as multiplying by the reciprocal (after simplifying). Why?

Because division is the inverse operation of multiplication, so they should,
and do, work according to the same principals. It is easy to show this
always works by applying the usual multiply by the reciprocal of the divisor
approach and then reorder the numerator and denominator factors.

As an example if we look at 6/10 divided by 2/5 we would traditionally
proceed by doing this...

    6         5
   ---   *   ---    but that is
   10         2    

   (6 * 5) / (10 * 2)  

If we apply the commutative property of multiplication to the denominator we
have (6*5) / (2*10) and now breaking this back into products we get 
(6/2) * (5/10)  and reversing the "multiply by the reciprocal rule" we can
change this to a division problem that is (6/2)/(10/5)

The problem with this method is that often you just exchange one problem
for another...

For example if we have (3/5) divided by (4/7)  then using the top divided
by top, bottom divided by bottom rule we would get (3/4) divided by (5/7)
which is equal to the same value, but brings us no closer to a simplified
rational answer.

 -Pat Ballew, for the T2T service

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