Simplifying Expressions with Fractions

Date: 9/2/96 at 16:39:19
From: Anonymous
Subject: Simplifying Expressions

I have tried to do my homework and I don't understand how to work it 
out. It works with a calculator but I want to do it for myself on 
paper. One of the problems is:

9(-1/7)(1/3)(-28)    

I am supposed to simplify it.  Please help!!!

Thank you, Veronica

Date: 9/3/96 at 15:22:42
From: Doctor Mike
Subject: Re: Simplifying Expressions

Hello Veronica, 
  
This is a great problem.  It brings up several important points
but it is not computationally difficult.  Here we go.
  
First of all, since exactly two of the four numbers multiplied
together are negative, the result is positive.  So, you can safely
drop the minus signs as a simplification, getting 9(1/7)(1/3)(28).
  
Second, remember that 9 is the same as (9/1).  My next move may
not seem like a simplification, and it's really not, but it will
help anyway.  Change the expression to (9/1)(1/7)(1/3)(28/1). 
  
Third, remember the rule for multiplying fractions: (a/b) times
(x/y) is (ax)/(by).  That is, multiply the numerators to get the new
numerator, and multiply the denominators to get the new denominator.
In your case, it results in a simplification to (9*28)/(7*3). 
  
Fourth, you COULD start cancelling.  You know, 7 goes into 28 four 
times and 3 goes into 9 three times, and 4 times 3 is 12. But since I 
have you as a captive audience for a few minutes, I want to show you 
WHY cancellation works.  You will need to remember just two simple 
things: (a) multiplication is the same in any order, and (b) the rule 
for multiplying fractions works also for UN-multiplying them :
  
    9 * 28       28 * 9       28     9   
   --------  =  --------  =  ---- * ---  = 4 * 3  =  12
    7 * 3         7 * 3       7      3 
  
I'm not saying you should not cancel.  Far from it.  But you should 
always remember what is really happening when you take the quick 
cancellation shortcut.  Then you will never forget the important rules 
of when and where cancellation is permitted, and when and where it is 
not.
  
I hope this helps.  Write back if you have more questions.  

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