Comparing Ratios and Fractions

Date: 01/06/2002 at 16:55:28
From: Sarah
Subject: Fractions

What are the differences between ratios and fractions?

Date: 01/07/2002 at 09:54:12
From: Doctor Ian
Subject: Re: Fractions

Hi Sarah,

That's a good question! The short answer is that a fraction is one way 
to express some kinds of ratios.

For example, suppose that there are 10 boys and 8 girls in a class.  
There are 18 students in total, so the ratio of boys to students is 
10:18, and in this case it's natural to say that the boys make up 
10/18 of the class. 

On the other hand, the ratio of boys to girls is 10:8, and this 
doesn't naturally suggest any particular fraction.  

You could summarize this by saying that fractions are useful for 
expressing ratios like

  size of subset : size of set

where a subset of a set contains only elements drawn from a particular 
set. For example, given the set 

  {fred, wilma, barney, betty}

the following are all subsets:

  {}                             

  {fred}

  {wilma, betty}

  {wilma, barney}

  {fred, barney, betty}

  {fred, wilma, barney, betty}   

Note that the empty set is a subset of any set; and a set is always a 
subset of itself.  (These would correspond to fractions like 0/4 and 
4/4.)  

But a subset can't contain any elements that aren't in the set.  So 

  {fred, wilma, pebbles}

is NOT a subset of 

  {fred, wilma, barney, betty}

This is why the ratio of boys to girls in the original example isn't 
naturally expressible as a fraction - the boys are not a subset of the 
girls, and vice versa.  However, you might write something like

  number of boys    number of kids with y chromosomes
  --------------- = ---------------------------------
  number of girls   number of kids with x chromosomes

So what's going on here?  Well, note that it makes perfect sense to 
write the fractions

  number of boys
  --------------------------------- = 1
  number of kids with y chromosomes

  number of girls
  --------------------------------- = 1
  number of kids with x chromosomes

because these are subset:set relations.  

Now, since both of these fractions are equal to the same thing (1), 
they are equal to each other. So

 number of boys                      number of girls
 --------------------------------- = --------------------------------
 number of kids with y chromosomes   number of kids with x chromosomes

and with a little algebraic manipulation, we end up with

  number of boys    number of kids with y chromosomes
  --------------- = ---------------------------------
  number of girls   number of kids with x chromosomes

So as you can see, the distinction between ratios and fractions is 
blurry at best... blurry enough, really, that it's not clear that it's 
really something worth worrying about. 

And of course, fractions are more than just a notation for expressing 
ratios.  Here are a couple of ways to think about fractions:

   Numbers in a Fraction
   http://mathforum.org/library/drmath/view/57191.html   

   Real and Other Numbers
   http://mathforum.org/library/drmath/view/57052.html   

Does this help?

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