Tips for Simplifying Fractions

Date: 06/03/98 at 21:03:24
From: Cambree
Subject: I have no idea how to reduce or simplify a fraction

I have no idea how to reduce or simplify a fraction. Could you 
please send me the solution to how to reduce or simplify a fraction 
(like 2 and 4/5 or 4/5)? It would help a lot. Please send me an 
example and how to do it.

Thanks,
Cambree W.

Date: 06/06/98 at 13:08:12
From: Doctor Peterson
Subject: Re: I have no idea how to reduce or simplify a fraction

Hi, Cambree, 

Being able to reduce fractions is an important skill, so you're right 
to want to get it clear before you move on too far. I'm glad you've 
asked for help. 

I'll try to give you a quick summary of how to do it. If these 
examples aren't enough, don't hesitate to ask your teacher for some 
extra help. Sometimes all it takes is a chance to ask your questions 
privately and give a teacher a chance to point out one little step you 
might be missing. Or you could send us a specific problem and show us 
what you've done and where you're stuck, and we can help you figure 
out what you need.

The two examples you gave actually don't need reducing. 2 4/5 is a 
mixed number, which is equal to the improper fraction 14/5, and 
there's nothing you can do to simplify it. And 4/5 is already as 
simple as it can get.

The basic idea of reducing or simplifying fractions is that two 
fractions are the same if the numerator and denominator are multiplied 
by the same thing, because that's just a way of multiplying the 
fraction by one. For example:

   4    8         
   - = -- 
   5   10         

because: 

   4    4   2    4 * 2     8
   - =  - * - =  ----- =  --
   5    5   2    5 * 2    10

Simplifying means find the simplest fraction that is equal to the one 
you're given. That's useful because it can save a lot of work if you 
have to do more with the fraction - you'll have smaller numbers to 
work with. You could say you're trying to find a small fraction (one 
with small numerator and denominator) hidden inside the given 
fraction. If you take my example above in reverse, you can see that if 
you are given 8/10, you have to recognize that 8 = 4*2 and 10 = 5*2, 
then divide both by 2.

Let's take a harder example. If I'm given 36/54, I have to do 
something like this:

   36   18*2   18   2   2
   -- = ---- = -- * - = -
   54   18*3   18   3   3

The hard part is how to find the Greatest Common Factor (or Divisor) 
of the two numbers, which is that 18 that appeared magically in what I 
just did. For some problems, you might just happen to see that 18, and 
you're almost done. I didn't. I'll show you what I actually did a 
little later. The important thing to realize is that you don't have to 
be a whiz at this to get it done.

Some people will very carefully factor each number completely, turning 
each one into a product of prime numbers, and then match up any 
factors that appear in both numbers and cancel them:

                  /   / /
   36   2*2*3*3   2*2*3*3      2
   -- = ------- = --------- = ---
   54   2*3*3*3   2  *3*3*3    3
                  /   / /

(In case that confuses you, canceling factors really means something 
like this:

   2*2*3*3   2*2*3*3     2   2   3   3   1                   1   2
   ------- = --------- = - * - * - * - * - = 1 * 2 * 1 * 1 * - = -
   2*3*3*3   2*  3*3*3   2   1   3   3   3                   3   3

so that any factor on both the top and bottom turn into a 1.)

This way of doing it makes the answer very neat, but it can be a 
little intimidating, because it takes some practice to completely 
factor a number. The fact is that you don't really have to do it that 
way. Often some factors jump out at you, but others hide better. You 
can just cancel out whatever common factors you do see, then go back 
to looking for more factors. In this example, I saw immediately that 
both numbers are even. I looked again and recognized that both are 
multiples of 6. So I divided each number by six, making the problem a 
lot simpler:

   36   6*6   6
   -- = --- = -
   54   6*9   9

But I'm not finished yet. I look again, and I see another common 
factor, 3:

   6   3*2   2
   - = --- = -
   9   3*3   3

So each time I found a number that divides both numbers evenly, I 
divide it out and keep looking. I never actually found the GCF, but 
eventually removed all the factors.

Now, even what I said was "obvious" may not be obvious to you. It 
takes a lot of practice to recognize factors. One thing you can do to 
make it easier is just to "play" with fractions. If you get used to 
them, they'll become friends, and friends like to help you out when 
they can! What I mean is, if you take simple fractions and try to make 
them complicated by multiplying the numerator and denominator by 
something, you'll get used to what fractions that can be simplified 
look like.

It may be very helpful that you're working with ratios, because a lot 
of ratio problems are really just fraction problems in disguise. If 
you see that the ratios 6/5 and 30/25 are the same, look at them and 
think, what does that tell me about simplifying 30/25?

I hope this helps. Let me know how you do.

-Doctor Peterson,  The Math Forum
Check out our web site! http://mathforum.org