



Most
people like multiplying more than they like dividing. 


It
would be great if we could change division into multiplication. 





WE
CAN! 





Say
we have 21 and we want to divide it by 7: 





21
÷ 7






Instead
of dividing by 7 


we
need to find something to multiply by instead. 





What? 





Here's
the deal, instead of dividing by 7 we multiply by ^{1}/7. 


We
multiply by the thing that times 7 equals 1. 


It
is called the Multiplicative Inverse
of 7. 





Step
1: 


Give
the number a denominator of 1 ... 











Now
we multiply this by ^{1}/7. 


Multiplying
fractions together is pretty easy. 


To
multiply
them, 


you
just multiply the numerators (numbers on top) 


and
the denominators (numbers on the bottom). 











Now
simplify if possible.



21
is equal to 7 x 3 so: 











Example: 





Divide
28 by 4. 





28
÷ 4 = ?






The
take the 4 and flip it over to get ^{1}/4
and multiply: 











Now
simplify if possible. 


28
= 4 x 7 so: 











We
could have broken those 4's into 2 x 2 


to
get all the way to prime factors before simplifying if we wanted to. 





OK,
so far this is pretty easy. 


In
fact, we have seen it all on earlier pages in a slightly different
way. 





But
here's where we really need this stuff: 





Divide
3 by ^{2}/3
. 





Icky!
Icky! Nasty! Nasty! 


What
do we do? 





Don't
sweat it. 


Just
flip over the thing you are dividing by and multiply: 











We
can write the 3 as ^{3}/1
to make things look nice if we want. 











Always
look to see if you can simplify these. 


This
time, the number on top breaks into prime factors as 9 = 3 x 3 


(hmm,
I think we saw those 3's a second ago). 


There
are no 3's that are in that denominator of 2. 


That
means we can't simplify this one. 





Here's
another one ... 





Example: 





Divide
^{3}/5
by ^{2}/3
: 











and 











The
prime factors in the top part are 3 x 3. 


The
prime factors in the bottom part are 5 x 2. 


No
numbers show up in both places so there's nothing to cancel. 





OK,
One More ... 





Example: 





Divide
 3 by ^{3}/4
: 





OH
NO! NOT A NEGATIVE NUMBER! 





Yeah
a negative number. No big deal. 


The
negative number doesn't change a thing. 


Do
the problem the same way. 


Put
a 1 under the 3 as a denominator and we're off: 











Now
multiply: 











Try
to simplify by breaking both parts into prime factors. 





The
prime factors of  12 are 1 x 2 x 2 x 3 





The
prime factors of 3 are, hmm, 3. 


So 











copyright 2005 Bruce Kirkpatrick 
