Ordering Fractions

Date: 01/17/2003 at 15:07:18
From: Jasmine
Subject: Ordering Fractions

How do I order the fractions 2/3, 3/5, and 2/7? 

Date: 01/17/2003 at 15:32:03
From: Doctor Ian
Subject: Re: Ordering Fractions

Hi Jasmine,

One way to think about a fraction like 2/3 is this:  If I cut
something into 3 pieces, 

  +-------------------+
  |                   |
  +-------------------+ 
  |                   |
  +-------------------+ 
  |                   |
  +-------------------+
  
and keep 2 of them, 

  . . . . . . . . . . .
  .                   .
  +-------------------+ 
  |                   |
  +-------------------+ 
  |                   |
  +-------------------+

that represents 2/3 of the thing.  Similarly, if I cut the thing 
into 5 pieces, and keep 3 of them, I have 3/5 of the thing:

  +---+---+---+ . . . .
  |   |   |   |   .   .
  |   |   |   |   .   .
  |   |   |   |   .   .
  |   |   |   |   .   .
  |   |   |   |   .   .
  +---+---+---+ . . . .

Now the question is:  Which way do we end up with more of the 
thing? Well, suppose we cut each thing into 3 pieces, and cut 
each of those pieces into 5 smaller pieces.  Then we end up 
with 15 pieces:

  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+

Since 15 is divisible by 3, it's easy to pick out 2/3 of 
the pieces:

  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  | x | x | x | x | x |      2/3 = 10/15
  +---+---+---+---+---+
  | x | x | x | x | x |
  +---+---+---+---+---+

And since 15 is also divisible by 5, it's easy to pick out 
3/5 of the pieces:

  +---+---+---+---+---+
  | x | x | x |   |   |
  +---+---+---+---+---+
  | x | x | x |   |   |      3/5 = 9/15
  +---+---+---+---+---+
  | x | x | x |   |   |
  +---+---+---+---+---+
  
So now the comparison is easy:  10/15 is greater than 9/15, 
so 2/3 is greater than 3/5. 

Is there a way to do this without drawing pictures?  How did 
I know to divide the thing up into 15 pieces?  

Remember that what made this easy is that the number of 
smaller pieces was divisible by both of the numbers of 
larger pieces.  The easiest way to find a number that is 
divisible by both 3 and 5 is to multiply 3 by 5 to get 15!

So to compare 2/3 and 3/5, I want to use 3 * 5 = 15 as 
their common denominator:

  2   5   10                3   3    9
  - * - = --                - * - = --
  3   5   15                5   3   15
  
Now, to compare 2/3 to 2/7, I could go through this whole 
thing again... or I could notice that the same number of bigger 
pieces must be a bigger fraction!  Since 1/3 of a thing is more 
than 1/7 of a thing, 2/3 must be bigger than 2/7.  

So now I know that 

     3/5 < 2/3       
     
     2/7 < 2/3

But I still have to figure out whether 2/7 is bigger or 
smaller than 3/5.   To find a common denominator, I do what 
I did before:  7 * 5 = 35, and

  2   5   10                3   7   21
  - * - = --                - * - = --
  7   5   35                5   7   35

So now it's pretty clear that 2/7 is smaller than 3/5, so we 
have the order:

  2   3   2
  - < - < -
  7   5   3

Now, if we have a lot of fractions, we'll have to do a lot of
comparisons, which means finding a lot of common denominators.  
We can make life easier if we just find _one_ common denominator 
for all the fractions.  How can we do that?  By multiplying all 
our denominators together:  3 * 5 * 7 = 105, and 

  2   3*5    30
  - * --- = ---
  7   3*5   105
  
  3   3*7    63
  - * --- = ---
  5   3*7   105
  
  2   5*7    70
  - * --- = ---
  3   5*7   105
  
That's kind of an ugly denominator... but now we can just order 
the numerators to order the fractions, so it's a good trade-off,
don't you think? 

Does this help?

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