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### Comparing Ratios and Fractions

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

What are the differences between ratios and fractions?
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```
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

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/
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Associated Topics:
Middle School Fractions
Middle School Ratio and Proportion

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