Volume of Water in an Urn

Date: 11/12/96 at 12:09:14
From: Andrew McElroy
Subject: Volume of water in an urn

Dear Dr. Math,

An urn contains 1 liter of water while a second urn nearby is empty.  
After 1/2 of the water in the first urn is emptied into the second 
urn, 1/3 of the water in the second urn is emptied into the first urn.  
Then, after 1/4 of the contents of the first urn is poured into the 
second urn, 1/5 of the contents of the second urn is poured into the 
first urn.  At each succesive pouring from alternate urns, the 
denominator of the fractional part poured increases by 1. How many 
liters of water remain in both urns after 1,200,000 pourings?

I have not even begun to solve this problem because I have no idea 
how to do it.

Sincerely,
Andrew McElroy

Date: 12/11/96 at 01:20:42
From: Doctor Rob
Subject: Re: Volume of water in an urn.

The trick is to see how much water the urns contain after odd numbers 
of pourings.

After one pouring, each urn contains 1/2 liter.
After two pourings, the first urn contains 2/3 liter.
After three pourings, the first urn contains 1/2 liter again.
After four pourings, the first urn contains 3/5 liter.
After five pourings, the first urn contains 1/2 liter again.

Make a table with the number of pourings versus the contents of one of 
the urns.  See if you can find a pattern.  If you can, try to prove 
that the pattern is correct using the principle of mathematical
induction.

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