Comparing Ratios to Form a Proportion

Date: 01/08/2004 at 21:45:06
From: china
Subject: I don't quite understand how to do proportions

Which two of the following four ratios form a proportion?

3 to 4     4 to 6     8 to 12    10 to 12

Date: 01/09/2004 at 10:31:48
From: Doctor Ian
Subject: Re: I don't quite understand how to do proportions

Hi China,

A proportion is a statement that two ratios are equal.  We use ratios
when we compute percentages, for example:

   15    ?
   -- = ---         15 is what percent of 40? 
   40   100

So you need to find two of those ratios that are equal.  How can you
check that?  Well, let's try two of them:

  3   4
  - = -
  4   6

Is this true?  We can find out by cross-multiplying:

  3*6 = 4*4

   18 = 16       Oops!

If the cross products aren't equal, then neither are the ratios. 
Here's what it would look like if you started with two ratios that are
the same:

     3    5
     - = --
     9   15

  3*15 = 9*5

    45 = 45      They're the same.

Now, you might look at this cross-multiplication stuff and wonder what
it has to do with anything.  Well, here's another way to think about
it.  Suppose we have two ratios, like 3/9 and 5/15, and we want to
know if they're the same.  If two numbers are the same, then dividing
either by the other has to give us a result of 1, right?

So let's try that:

    3/9
  ------- = ...
   5/15

To divide by a fraction, we multiply by its reciprocal:

    3/9     3   15   3*15   45
  ------- = - * -- = ---- = -- = 1
   5/15     9    5   9*5    45

And only if the numerator and denominator are equal will the fraction
be equal to 1.  But notice that the numerator and denominator are just
the same things we get by cross-multiplying.  Does that make sense?

One more thing you can do is this.  Note that if two ratios are the
same, they can be _reduced_ to the same ratio.  For example, 

  3   1 * 3   1
  - = ----- = -
  9   3 * 3   3
          ^
          |______________ These cancel

and 

   5   1 * 5   1
  -- = ----- = -
  15   3 * 5   3

So you don't have to do comparisons at all if you just reduce all your
ratios to their lowest terms.  For example, if I have the ratios

  3 to 15, 10 to 40, 5 to 60, and 20 to 80

I can reduce them to 

  1/5, 1/4, 1/12, and 1/4

which tells me right away that 10 to 40 and 20 to 80 form a
proportion.  To check, 

       10   20
       -- = --
       40   80

  10 * 80 = 40 * 20

      800 = 800

Does this help? 

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