Reducing Fractions

Date: 02/25/97 at 10:24:45
From: Tori Johnson
Subject: Reduce 18/35,35/48 & 6/20

I'm having trouble reducing fractions. I need to reduce 18/35, 35/48 
and 6/20.  Can you help?

Date: 02/25/97 at 11:56:00
From: Doctor Mike
Subject: Re: Reduce 18/35,35/48 & 6/20

Dear Tori,
  
This is an interesting bunch of problems.  Let's start with the last 
one.  You will soon see why I want to do it that way.
  
There are some tricks sometimes used for reducing fractions, but once 
you understand what is going on, you will not need them. Ten years old 
is a perfect age to gain this understanding. 
  
The numerator of 6/20 is six, which factors into 2*3.  That is, 2 
times 3 equals 6.  The denominator of 6/20 is 20, which factors into 
2*10.  That is, 2 times 10 equals 20.  Let's remember this.
  
I hope you have already seen the easy and very natural rule for 
multiplying fractions.  You multiply the two numerators together to 
get the new numerator, and you multiply the two denominators together 
to get the new denominator.  In symbols, it looks like this:

          A     X       A*X 
         --- * ---  =  -----
          B     Y       B*Y 
    
This shows how to multiply 2 fractions together.  BUT ALSO, it shows 
how to "UN-multiply" if you want to do that.  What I mean is that if 
you have a fraction like (A*X)/(B*Y), then this rule tells how you can 
change it to A/B times X/Y.   
  
Okay, back to 6/20 and how we are going to reduce it.  Remember:
  
            6       2*3  
          ----  =  ------
           20       2*10 
    
I am going to use the rule for multiplying fractions to continue 
working with 6/20 like this: 
  
            6       2*3        2     3           3      3 
          ----  =  ------  =  --- * ---  =  1 * ---  = ---
           20       2*10       2     10          10     10
   
What I did was to use the fraction multiplication rule, then change 
2/2 to 1, then do the multiplication by 1.  This shows EXACTLY WHY you 
can reduce 6/20 to 3/10.  
  
It is important for you to know why, because if you really understand 
something you will never forget it.  Some people say that about riding 
a bicycle, that if you learn how but then don't do it for 30 years you 
will still know how.  Same thing about really understanding fractions 
and factoring numbers.  If you really understand the ideas, you will 
still know it when your grandchildren are ten years old.
  
Let's take a look at another one of those problems of yours, say 
18/35.  We want to factor the numerator and denominator to see if we 
can find the same number as a factor in both places. That is how we 
did the 6/20 problem; we found a 2 in both places:
  
           18       2*3*3 
          ----  =  -------
           35        5*7  
  
I have factored the numerator and the denominator both as much as I 
possibly can.  A number like 2 or 3 or 5 or 7 that cannot be factored 
any more is called a prime number.  As you can clearly see, there is 
no prime number factor that is both on the top and on the bottom.  
That means that the fraction 18/35 is already in lowest terms.  You 
should try the other one 35/48 and see if it can be reduced.   
  
Here's another practice problem that is a little harder.  See if you 
can work with 130/143 to reduce it.  Good luck.  
  
I hope this helps.
  
P.S.  I have tried to explain why this works.  Maybe your 
      teacher will talk about this too.  Maybe he or she
      will teach other methods, and maybe some tricks to
      do this kind of problem faster.  That's OK.  You can
      learn those other ways too.
  
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/