How to Divide Fractions?

Date: Sun, 6 Nov 94 22:23:41 EST
From: Paul Robert Oliver
Subject: I'm stuck!

Dear Dr. Math,

When dividing fractions, for example: 3/11 divided by 5 ... I
know that you reciprocate the "5" and multiply, but I don't
understand why ... except that it works!  Can you explain it?
Thanks.
                                        Sincerely,
                                        I'm Stuck               

Date: Sun, 6 Nov 1994 23:10:57 -0500
From: Gabe Farmboy Cavallari

Paul, good question.  I'm glad you're not approaching this problem by
doing it by rote.  This is Gabe, one o' the math doctors.  Let's look at the
problem like this:

(3/11)/5.  This way we can see that the fraction 3/11 is in the numerator.
Now, if you multiply this expression by 1, then the expression stays the 
same, right?  Go with me on this one.  Let's multiply (3/11)/5 by a form 
of 1 which would be helpful, say 11/11.  I'll try to type this in the best 
way I can think of...

__3/11__     *  __11__
   5              11

And then the expression becomes [(3*11)/11]/(5*11).  In the fraction of the
numerator, the 11's cancel out, while in the denominator the product of 5 
and 11 is 55.  So our fraction becomes 3/55.

I hope this helped.  If I didn't explain it well enough, please write back and
I'll try again.  Thanx for the question.

-gabe, math doc
__________

Date: Thu, 10 Nov 1994 18:59:28 -0500
From: Anonymous
Subject: Re: I'm stuck!

Hi Paul, just checking to see if you received Gabe's explanation and
understood. ?

Not only does what he did look like we've inverted and multiplied 
but in fact, it can be done that way.  Why?

Let's look at a similar situation in "everyday life":

1) Turning something over is the "inverse" of setting it down without 
turning.

2) Upside down is the inverse of right side up as far as turning things
over is concerned.

3) Turning over something that's upside down is the same as setting 
down something that is right side up.

When we do the inverse to the inverse and we get the same thing.

Back to numbers and math words:

1) Multiplication and division are inverse operations.

2) 5 is the inverse of 1/5 for multiplication or division.

3) Dividing by 5 is the same as multiplying by 1/5.

I have a choice:  I can divide 3/11 by 5 or multiply 3/11 by 1/5.  The
answer will be the same.  So it's a question of how I want to write the
problem. Which is easier for me to do?

-- steve