Finding Fractions between Fractions

Date: 01/08/2002 at 20:43:46
From: Kristen
Subject: Fractions

What are four fractions that are between 1/3 and 1/2?

Date: 01/08/2002 at 21:42:46
From: Doctor Ian
Subject: Re: Fractions

Hi Kristen,

If I have two numbers like 4 and 10, I can find a number right in the 
middle of them by adding the numbers and dividing the sum by 2:

  4 + 10     14
  ------  =  --  =  7
    2         2

It works with fractions, too!  (In fact, it works with _any_ numbers.)

So to find a fraction halfway between 1/3 and 1/2, 

  1/3 + 1/2   2/6 + 3/6   5/6
  --------- = --------- = --- = 5/12
      2           2        2

So now I have three fractions, in order:

  1/3,  5/12,  1/2

I can do the same trick to find another fraction between 1/3 and 5/12,

  1/3 + 5/12   4/12 + 5/12   9/12
  ---------- = ----------- = ---- = 9/24
      2             2          2

which gives me

  1/3,  9/24, 5/12,  1/2

and another fraction between 5/12 and 1/2, or between 1/3 and 9/24, 
and so on, and so on. 

Now, this seems like a lot of work. Is there an easier way? Actually, 
there is. Note that I could have found 5/12 in this way:

    1/3 <  ?  < 1/2

   4/12 <  ?  < 6/12

It's pretty clear that 5/12 goes in the middle there. So what if I 
used an even bigger denominator?  

    1/3 <  ?  < 1/2

   8/24 <  ?  < 12/24

Now you can get three fractions in there: 9/24, 10/24, 11/24.  What 
would happen if you used 48 as a denominator? 
     
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/