Andy's Averaging Machine

Date: 6/23/96 at 21:22:34
From: Anonymous
Subject: Andy's Averaging Machine

Andy has a special averaging machine which works in the following 
ways:

Into the machine you can place 2 or more cards, each card with one 
number on it, and turn a handle. The machine will then produce a new 
card whose number is the average of all the cards you had put in it.
The machine then returns this card to you, together with the cards you 
place in it. You can use the machine as many times as you wish, but 
the machine will not produce a second card with the same number on it.

a) How can Andy get a card with 3/32 on it?

b) How can he get a card with 1/5 on it?

c) How can he get a card with a fraction whose denominator is 29?

d) How can he get a card with 1/29  ? 

Date: 6/24/96 at 7:30:42
From: Doctor Anthony
Subject: Re: Andy's Averaging Machine

(a) If a and b are the numbers you put in, then (a+b)/2 = 3/32

      so a+b = 3/16  and we could make a = 2/16 = 1/8
                                              b = 1/16

(b) In this case (a+b)/2 = 1/5  so  a+b = 2/5

Assuming a and b must be different we have a+b = 4/10 and we make 

        a = 3/10   b = 1/10
 
(c) To get a denominator of 29, we have (a+b)/2 = c/29 

So a+b = 2c/29   then let a = (c-1)/29   b = (c+1)/29

This works provided c is not equal to 1.  If c = 1 then we put

a+b = 4/58 then let a = 3/58  and b = 1/58

(d) To get an average of 1/29 we enter 3/58 and 1/58

-Doctor Anthony,  The Math Forum
 Check out or web site!  http://mathforum.org/dr.math/