Fraction Puzzle Using Digits from 0 to 9

Date: 02/04/2008 at 19:50:14
From: Gabi
Subject: fractions less then 1/2 

Using all ten numbers 0 through 9, make 5 fractions less then 1/2.  I
only could come up with 1/4, 2/6, 3/9.  I couldn't find fractions for
the 0,5,7 and 8 even trying to change the 1/4, 2/6 and the 3/9.

Date: 02/04/2008 at 21:09:11
From: Doctor Ian
Subject: Re: fractions less then 1/2

Hi Gabi,

This is the kind of problem where you can stumble around forever
unless you set out with some kind of plan. 

I'd probably make a table, like this:

           1    2    3    4    5    6    7    8    9


  1
  
  2
   
  3

  4

  5
  
  6

  7

  8

  9
  
Anything divided by itself is 1, so those can't be possibilities:

           1    2    3    4    5    6    7    8    9

  1        -    
  
  2             -
      
  3                  -

  4                       -

  5                            -
  
  6                                 -

  7                                      -

  8                                           -

  9                                                -

  
1 divided by anything greater than 2 is less than 1/2, so let's mark
those as good, with an x.  (And we'll mark 1/2 as '-', so we know
we've already considered it.)

           1    2    3    4    5    6    7    8    9  

  1        -    -    x    x    x    x    x    x    x
  
  2             -
      
  3                  -

  4                       -

  5                            -
  
  6                                 -

  7                                      -

  8                                           -

  9                                                -


2 divided by anything greater than 4 is less than 1/2:

           1    2    3    4    5    6    7    8    9

  1        -    -    x    x    x    x    x    x    x
  
  2             -    -    -    x    x    x    x    x
      
  3                  -

  4                       -

  5                            -
  
  6                                 -

  7                                      -

  8                                           -

  9                                                -

We can do the same kind of reasoning for 3, 4, and so on:

           1    2    3    4    5    6    7    8    9

  1        -    -    x    x    x    x    x    x    x
  
  2             -    -    -    x    x    x    x    x
      
  3                  -    -    -    -    x    x    x

  4                       -    -    -    -    -    x

  5                            -    -    -    -    -
  
  6         all these               -    -    -    -
             would be
  7         greater than 1               -    -    -

  8                                           -    -

  9                                                -

How does this help?  Well, it gives us a complete list of all the
possible fractions that could be less than 1/2:

  1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9

            2/5, 2/6, 2/7, 2/8, 2/9

                      3/7, 3/8, 3/9

                                4/9

So now we have to find a set of 5 of these that use up all the digits.
That's not necessarily easy, but we can be systematic about it, and
that's half the battle. 

The first thing I notice is the diagonal:

  1/3, 
       2/5, 
            3/7, 
                 4/9

That's almost a solution right there:

        *
  1  2  3  4
  -, -, -, -       We've used everything but 0, 6, and 8.
  3  5  7  9       We've used 3 twice.
  *

Can you see how to get from here to a complete solution?  Keep in mind
that 0 over anything is 0, which is less than 1/2.  

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