Coffee and Tea 


Problem:
  You have a cup of coffee and a cup of tea.  You take one teaspoon of tea
and pour it into the coffee.  You take one teaspoon from the new mixture,
and pour it back into the tea.  Ignore losses and determine whether there
is more tea in the coffee or more coffee in the tea after this operation is
performed.  (I don't know where this problem originated. It was just now
proposed to me over undernet, but i've heard it before.  Oddly it must have
taken me 10 minutes to solve...I'm getting rusty.)

Answer:
There is just as much coffee in the tea as there is tea in the coffee.

Solution:

Let C = Coffee (the base unit, whatever that is).
Let T = Tea (the base unit, whatever that is).
Let a = an amount of volume equal to one teaspoon.
Let b = an amount of volume equal to one cup minus one teaspoon.

Follow he steps below....on the left is the Cup that starts out all coffee
and on the right is the one that starts out all tea.

We stgart with step one -- the same amount in each cup (but 'pure'):

Step 1  (b+a)C			(b+a)T


We now put one teaspoon of T into the first cup, giving....

Step 2  (b+a)C + aT		bT

We now have a new substance on the left. Call it X, which contains a fraction
of C and a fraction of T.  X has the following 'chemical formula':

          b+a           a
    X  =  ---- C   +   ---- T
          b+2a         b+2a

We can now look at the left side as being composed of an amount (b+2a) of
the substance X. So we can rewrite 2) as 

Step 2	(b+2a)X			bT

So now we can take one teaspoon of the mixture and put it back into the cup of
'pure' tea, yielding....

Step 3	(b+a)X			bT + aX

We can rewrite the left side as

      (b+a         a     )    (b+a)^2         a(b+a)
(b+a) (---- C  +  ---- T )  = -------  C   +  ------ T
      (b+2a       b+2a   )      b+2a           b+2a

So the ratio of T to C is:

a(b+a)   (b+2a)     a
------ * ------- = ---
b+2a     (b+a)^2   a+b

Remember this ratio: a/(a+b)

Rewriting the right side, we get

      (b+a         a     )    (      a^2 )      a(b+a)
bT + a(---- C  +  ---- T )  = ( b +  ----) T  + ------ C
      (b+2a       b+2a   )    (      b+2a)       b+2a


We can rewrite the factor of T in the following manner

(      a^2 )     b^2 + 2ab +a^2   (a+b)^2
( b +  ----)  =  -------------- = -------
(      b+2a)         b+2a           b+2a


So the right side of step 3 becomes finally,

(a+b)^2     a(b+a)
------- T + ------ C
  b+2a       b+2a


We now take the ratio of the amount of C to the amount of T:

a(b+a)    b+2a      a
-----  * ------- = ---
b+2a     (a+b)^2   a+b

Which is the same ratio we had in the corresponding case with
the other cup.

Numbers.  Gottaluvum.
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