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### Simplifying Fractions

```Date: 12/08/2002 at 21:01:13
From: Lillian
Subject: Simplifying Fractions

How do I simplify a fraction?
```

```
Date: 12/08/2002 at 22:05:45
From: Doctor Ian
Subject: Re: Simplifying Fractions

Hi Lillian,

It depends on the fraction, but the general idea is this. Suppose you
have a fraction,

bibbity
-------
bobbity

and it turns out that both the numerator and denominator share a
common factor, e.g.,

bibbity   bibble * itty
------- = -------------
bobbity   bobble * itty

then you can eliminate the common factor:

bibbity   bibble * itty
------- = -------------
bobbity   bobble * itty

bibble   itty
= ------ * ----
bobble   itty

bibble
= ------ * 1
bobble

bibble
= ------
bobble

In practice, it might look like this:

12   6 * 2   6
-- = ----- = -
18   9 * 2   9

Of course, there can be more than one common factor!

12   6 * 2   6   2 * 3   2
-- = ----- = - = ----- = -
18   9 * 2   9   3 * 3   3

So the easiest way to deal with numbers is to find all the prime
factors right away, and kill them off in pairs:

/   /
12   2 * 2 * 3       2
-- = ------------- = -
18       2 * 3 * 3   3
/   /

It might be too soon for you to be thinking about 'variables', but
when you get to algebra you'll want to use the same rule:

/   /           /   /
21 x^3 y^2   3 * 7 * x * x * x * y * y
---------- = -------------------------------------
35 x   y^5   5 * 7 * x         * y * y * y * y * y
/   /           /   /

3 x^2
= -----
y^3

I'm just telling you this now because it's not uncommon for students
who have no trouble reducing fractions with numbers to freeze up when
they have to do the same thing to a fraction with variables, because
they don't realize that it _is_ the same thing.

The important point is not to get confused or intimidated when the
numbers get big, or you have lots of variables, or whatever. The main
idea is _always_ the same: If you have

a bunch of stuff multiplied together
------------------------------------------
another bunch of stuff multiplied together

then if you can identify common factors, you can kill them off (cancel
them) in pairs. What is tricky is that sometimes it's hard to remember
that something like 12 actually _is_ a bunch of stuff multiplied
together.

12   a bunch of stuff multiplied together
-- = ------------------------------------------
18   another bunch of stuff multiplied together

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

And one of the secrets to happiness in math is to get into the habit
of looking at every integer you see this way. For example, as soon as
you _see_ a number like 72, you should already be thinking

72 = 8 * 9

= (2 * 2 * 2) * (3 * 3)

It's kind of like what Superman does when he uses his x-ray vision to
look through people's clothes so he can see if they're hiding things.

It takes a little work to get into this habit (learning the most
common divisibility rules,

Divisibility Rules - Dr. Math FAQ
http://mathforum.org/dr.math/faq/faq.divisibility.html

can help a lot - especially the rules for 2, 3, 5, 9, and 10), but you
wouldn't believe how much work it can save you down the road.

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Factoring Numbers
Middle School Fractions

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