Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
Elementary Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/