Coffee or Tea?

Date: 07/09/2001 at 14:39:55
From: Traylor
Subject: Mixtures

Here is the problem:

Person A has a cup of coffee. Person B has a cup of tea. Person A 
takes a teaspoon of coffee and puts it in the tea. Person B mixes the 
teaspoon of coffee and tea, then takes a teaspoon of the mixture and 
returns it to the coffee. Is there more coffee in the tea, or more tea 
in the coffee, or are they the same?

My thought is there is more coffee in the tea. I have been told they 
are the same. Could you please explain this, because I don't get it!

Date: 07/09/2001 at 18:03:43
From: Doctor Rob
Subject: Re: Mixtures

Thanks for writing to Ask Dr. Math, Traylor.

They are the same.

The volume of each cup after the swap is the same.  That means that
whatever coffee is missing from the first cup must have been replaced
with exactly the same amount of tea.

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

Date: 07/16/2001 at 17:39:18
From: Doctor Ian
Subject: Re: Mixtures

Hi Traylor,

Suppose I have a school full of boys, and a separate school full of 
the same number of girls. I fill a bus with boys, drive them over to 
the girls' school, and let them mingle. Then I grab enough boys and 
girls to fill the bus, and drive them back over to the boys' school.

Now, are there more girls at the boys' school, or more boys at the 
girls' school? 

Do you see why I could have accomplished the same thing by filling the 
bus with boys, driving over to the girls, telling _some_ of the boys 
to get off, and picking up enough girls to take their places?

When you think about it this way, it becomes pretty obvious that every 
boy who goes over to the girls' school is replaced by one girl who 
goes back to the boys' school.  

In your problem, the teaspoon leaves the coffee cup full of some 
amount of coffee. When it comes back, some of the coffee has been 
replaced with tea. How much tea comes back? One unit of tea comes back 
for each unit of coffee left behind. 

This is a problem that has driven a lot of people crazy over the 
years. You can a very extensive description of it here:

  Brain teaser: the wine in the water problem - Donald Sauter
  http://www.geocities.com/CapitolHill/lobby/7049/wine.htm   

Focusing on mixtures in this context can be a red herring. Mixtures 
are something that most people don't have a lot of experience with, so 
they get caught up in trying to reason about what's happening when two 
things get mixed, which leads them away from the solution, rather than 
toward it.

I hope this helps. Write back if you have more questions, about this 
or anything else. 

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