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### 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
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/
```
Associated Topics:
Elementary Fractions

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