Fractions and Lowest Terms

Date: 07/19/2001 at 20:34:42
From: Bethany Boruta
Subject: Algebra lowest terms

Problem:
               2
       6a   14b
       __ . __ 
              2
       7b   2a 

                 2
        3    14b
       ___ .  ___  
       
        7b     a

How do you finish cancelling and then what do you multiply?
What is the lowest term?

Date: 07/20/2001 at 16:40:59
From: Doctor Ian
Subject: Re: Algebra lowest terms

Hi Bethany,

To see what's going on, sometimes it helps to expand the exponential 
terms:

  6a   14b^2   6 * a * 14 * b * b
  -- * ----- = ------------------
  7b    2a^2   7 * b *  2 * a * a

Now you can rearrange the factors to get

               6 * 14 * a * b * b
             = ------------------
               2 *  7 * a * a * b 

Now, every time you see something in both the numerator and 
denominator, you can get rid of it, e.g., 

               6 * 14 * a * b    b
             = --------------- * -
               2 *  7 * a * a    b 

               6 * 14 * a * b    
             = --------------- * 1
               2 *  7 * a * a     

because multiplying by something and then dividing by the same thing 
has no effect. (More precisely, it has the same effect as multiplying 
by 1.)

An easy way to do this is to line similar terms up and cross them off 
in pairs:

                        x       x
               6 * 14 * a *     b * b
             = ----------------------
               2 *  7 * a * a * b 
                        x       x

                        
               6 * 14 * b
             = ----------
               2 *  7 * a

What do you do about the numbers? Well, here's where all that stuff 
you learned about prime factors becomes useful. If you break down the 
numbers into their prime factors, you can cross off common factors in 
the same way that we just crossed off common variables:

                x             x
               (2 * 3) * (2 * 7) * b
             = ---------------------
                2      *      7 * a
                x             x

               3 * 2 * b
             = ---------
                   a

               6b
             = --
                a

Now, this is kind of a pain, although when you're just starting to 
learn a new technique, it's really helpful to go through all the 
steps. But here's how I'd do it now:

                      6a   14b^2    <--- drop this exponent by 1
                      -- * -----         
  remove this b --->  7b    2a^2 

                  
  remove this a --->  6a   14b^1    <--- drop this 1
                      -- * -----
                       7    2a^2    <--- drop this exponent by 1

                       6   14b      <--- divide 14 by 7
                       - * ---
  divide 7 by 7 --->   7    2a


                       6   2b       <---
                       - * --             Cancel the 2's 
                       1   2a       <---

                        
                       6   b  
                       - * - = 6b/a  
                       1   a  

Can you follow the same steps to simplify the second expression?  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/