Shortcut for Comparing Fractions

Date: 01/14/2003 at 07:50:56
From: Joe
Subject: Comparing fractions

We are going over comparing fractions in class and my teacher showed a 
little shortcut. He used the example of 2/3 and 3/5. He said you can 
cross multiply and get 9/10. The numerator (the product of the 
interior terms, 3 and 3) always corresponds to the second of the two 
fractions, and the denominator (the product of the exterior terms, 2 
and 5) corresponds to the first fraction. 
      
He showed us many examples and I have tried many myself and this 
always works. 9 is less than 10, so 3/5 is less than 2/3. My question 
is WHY?

Date: 01/14/2003 at 11:20:56
From: Doctor Ian
Subject: Re: Comparing fractions

Hi Joe,

Suppose I have two quantities, and I want to know which is larger. 
One way to do that is to divide one by the other, and see if I end up
with something less than, greater to, or equal to 1.  

This is trivial when you do it with integers:

  7 / 5 > 1  so 7 > 5

  3 / 4 < 1  so 3 < 4

But it also works with fractions.  Suppose I use '?' to represent '=',
'>', or '<'.  That is, I'm going to use it as a 'variable' for these
relational operators.  Then if I want to compare two fractions, a/b
and c/d, I can do this:

   a/b
  ----- ? 1
   c/d

Of course, to divide by a fraction, I invert and multiply, so this is
the same as 

  a   d
  - * - ? 1
  b   c

If I get something greater than 1, a/b must be larger. If I get
something less than 1, a/b must be smaller. If I get 1, the two 
fractions must be equal.  

Let's try it with your example:

   2/3
  ----- ? 1
   3/5

  2   5
  - * - ? 1
  3   3

     10
     -- > 1
      9

So 'cross-multiplying' is just what you have to do to divide the first
fraction by the second. Comparing the result to 1 tells you whether 
the numerator (first fraction) or denominator (second fraction) is
larger.  

Does that make sense?  

You can get a better feel for what is going on if you use letters
instead of numbers. For example, suppose my two fractions are a/b and
(a+1)/b.  I know that the second one has to be larger, right?  Let's
see what happens when we divide:

       a/b
  --------- ? 1
   (a+1)/b

  a     b
  - * ----- ? 1 
  b   (a+1)

         a
       ----- < 1         
       (a+1)

Let's try comparing two equivalent fractions. If we have a fraction
a/b, we can make an equivalent fraction by multiplying by k/k, where k
is anything other than zero:

     a/b
  --------- ? 1
  (ka)/(kb)

   a    kb
   - * ---- ? 1 
   b    ka
  
        kab
        --- = 1
        kab

For fun, you might consider trying to see which is larger: a/b, or
(a+k)/(b+k). That is, when you increment both the numerator and
denominator of a fraction by the same amount, is the resulting 
fraction larger, or smaller? 

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

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