Milk/Water puzzle

After the first transfer you have
W + TM and M - TM
The second transfer is TW(W/(W+TM)) + TM(TM/(W+TM))
resulting in
W-TW(W/(W+TM)) + TM-TM(TM/(W+TM)) and M-TM+TM(TM/(W+TM))  + TW(W/(W+TM))

TM-TM(TM/(W+TM)) =
TM(W+TM)/(W+TM) -TM(TM/(W+TM))
= (TM(W)+TM(TM) -TM(TM))/(W+TM)
= (TM)(W+TM-TM)/(W+TM)
= TM(W)/(W+TM)

TM(W) = TW(W) so there is as much water in the milk/water mixture as milk in the water/milk mixture.

it suffices to know that one glass starts and ends with
M = M1+M2 = M1+W1
and the other starts and end with
W = W1+W2 = W2+M2
You can even start with M != W and transfer any amout back and forth as many times as you want,
as long as each glass starts and ends with its original volume , conservation of liquids tells you that
there is as much water in the milk/water mixture as milk in the water/milk mixture
(so this doesn't work with water and ethanol)

It is a good puzzle, but I really love the way it mixes the English and Metric Units.

Can anybody really cook in metric???

a tablespoon is a Meric unit = 0.015L or 0.020L in Australia

Hi InteractiveMind,

Why is everyone resorting to all of the maths? It may be correct but it's not necessary. You started and ended with the same volume in each glass, after transferring a tablespoonful from one to the other and then back again. In the process there was an exchange of milk/water between the glasses. Since the volume hasn't changed in either, whatever milk was transferred to the water glass was replaced by an equal amount of water from the water glass.

Regards
  Kelvin

"And the amount of water in the milk/water solution: TW"
This last statement of yours is wrong. You put the teaspoon from the water glass into the milk
after you had put the milk in the water glass. Therefore that teaspoon included at least a little milk.

>whatever milk was transferred to the water glass was replaced by an equal amount of water from the water glass.

Um, not exactly.  Once you put some milk in the water glass, it's not all water anymore, so when you scoop out a spoonful, it has some milk in it, which means it's not all water.

 

grg99,

What do you mean "not exactly"? You started with a litre of water and a litre of milk. You ended with a litre of milky water and a litre of watery milk. Since the volumes of the individual containers have not changed, whatever milk was transferred to the water container must have been replaced by an equal amount of water being transferred to the milk container. Granted, a small amount of milk may have made a two-way trip, but that's irrelevant to the final outcome.

>What do you mean "not exactly"? You started with a litre of water and a litre of milk. You ended with a litre of milky water and a litre of watery milk. Since the volumes of the individual containers have not changed, whatever milk was transferred to the water container must have been replaced by an equal amount of water being transferred to the milk container. Granted, a small amount of milk may have made a two-way trip, but that's irrelevant to the final outcome.


One of us is confused.  LEt me go thru it again:

(1)  You scoop out a tablespoon of milk.

(2)  You drop it into the cup of water.

(3)  You now have a cup that's mostly water, but also has a small percentage of milk.

(3)  You scoop up a tablespoon of this water + a little milk mixture.

(4)  You drop this mixture into the cup of 100% milk.

(5)  So in (2) you dropped a tablespoon of 100% milk, but in (4) you're dropping 9X% water.  The two spoonfuls are different.

*That's* what seems different.  I don't see how you can draw the conclusion you do from these facts.

Now my take on it was:  since the amount moved is in no way special, you can go to an extreme and move ALL of one cup to the other and the answer should be the same.  Therefore since you're pouring ALL of one cup into another, the stuff gets completely mixed and is a 50/50 mixture.

 

grg99,

You're getting confused by the fact that the spoonfuls are different in your (5). It makes no difference to the final outcome. In the end you have transferred some milk to the water glass and some water to the milk glass. Since the volume in the water glass is the same at the end as it was in the beginning any milk that has been added to it must be offset by an equal amount of water being removed, and it is that water which has gone to the milk glass so that it also ends with the same volume.

If you want to look at it another way you dropped a tablespoon of 100% milk into a 100% of litre of water. Then you drop a tablespoon of 9X% water into 9X% of a litre of milk, because you removed some of the milk earlier. The change in volume compensates for the change in concentration, which is what all the maths in the posts above is showing.

>Since the volume in the water glass is the same at the end as it was in the beginning any milk that has been added to it must be offset by an equal amount of water being removed,

Sorry, but I don't see the logic here.  Just because the volumes are equal, how does that guarantee the ratios are equal?      Maybe I'm dense, but I don't see the logical connection.   Can you explain it so even I can understand it?  :)

Okay, I get it now!

BTW does anybody see any holes in my argument making use of continuity instead of arithmetic?  

>> x=99%water (1% milk) after milk was poured
wrong : 100 parts water and 1 part milk gives a mixture of slightly over 99% water and slightly under 1% milk.

The rest of your calculations are thus incorrect.

KelvinY --

I read your reply to grg99's post and what you said is very sensible, but it seems that bringing back the same volume but different concentration of liquid does not compensate enough to cancel out the effect. after phase 2 (please refer to post above) while the milk cup is 99.01% milk, the water remains at 99% water even if we remove a tbsp of it, or any amount for that matter (assuming homogeneity)


Centeris

Infinity, I used 0.01 L = 1 tbsp for demo purposes to show observable effect in solution

Centeris

>> Infinity, I used 0.01 L = 1 tbsp for demo purposes to show observable effect in solution
So, after adding one teaspoon of milk in the water, you have :

    1/1.01 % water  ~=  0.9901  =  99.01 %  !=  99%
    0.01/1.01 % milk  ~=  0.0099  =  0.99 %  !=  1%

So, I repeat : this :

>> x=99%water (1% milk) after milk was poured

is incorrect ...

Oh i get it now. i stand corrected. Thanks Infinity08! :-)

Centeris

"BTW does anybody see any holes in my argument making use of continuity instead of arithmetic?  "
Sounds good to me

Since the ending volumes are the same, all we need to do is compare:
 the tablespoons of liquid we add to the milk
the tablespoons of liquid we add to the water

The tablespoons we add to the water is 100% milk.

The tablespoons we add to the milk is 100% milk.

OOPs:


 Since the ending volumes are the same, all we need to do is compare:
 - the tablespoons of liquid we add to the milk
 - the tablespoons of liquid we add to the water

The tablespoons of liquid we add to the water is 100% milk.
The resulting ratio of milk in the water is: 1TBS/CUP

The tablespoons of liquid we add to the milk is less than 100% water AND IS PART MILK.
The resulting ratio of water in the milk is: LESS THAN 1TBS/CUP (since part of the TBS is milk)


<<<Is there more water in the milk/water mixture or more milk in the water/milk mixture?>>>

There's more milk in the water mixture because we added pure milk.

Bob, you were doing so well, then you fell down the well.  As many others.

Try it this way, assume your tablespoon is really large, like the same size as the cup.
Then you're pouring 100% over, then the same amount back.   Much simpler to analyze.

>>Try it this way, assume your tablespoon is really large, like the same size as the cup.
Then you're pouring 100% over, then the same amount back.   Much simpler to analyze.<<

I agree that it's much simpler to analyze.
Anybody looking at what I said?

Here's an easy way of picturing the problem:

You have a gross of white eggs and a gross of brown eggs.  A gross is twelve cartons where each carton is twelve eggs.


Take a white carton (12 eggs) and "mix it" into the brown eggs.

The brown eggs now have 156 eggs (12 white).

Now take a proportionate mix of the brown eggs and make a carton and move it back to the white eggs.  In this case 1/13 eggs in the brown mix is white.  So if put made even a single white egg in the carton, you'd have too much white in the carton.  The twelfth egg would have to be mostly white but part brown.  So in the brown mix, there would be less than 11 white eggs.  In the white, there would be more than 11 brown eggs.

So, yes [oops], if the white eggs are milk and the brown eggs are water, there'd be more milk (white eggs) in the water (brown eggs) than the reverse.

d-glitch, I love it. Now try this -
Box 1 has 1 black marble and Box 2 has 1 white marble,
You tabke the 1 marble from Box and add it to Box 2, then take 1 marble from Box 2 and put it back into Box 1.

No matter which one goes back into Box 1 the number of black marbles in box 2 is going to be the same as the number of white marbles in box 1 - either none or 1!

Box 1         Box 2
================
144.000 W       144.000 B                   Move 1/12th of the eggs from Box 1 to Box 2           144/12 = 12

108.000 W       144.000 B                   Move 1/13th of the eggs from Box 2 to Box 1            144/13 = 11.077
                        12.000 W                                                                                               12/13 = 0.923

108.923 W       108.923 B                   144 - 11.077 = 108.923 Brown eggs in Box 2
  11.077 B          11.077 W

Good job d-glitch.

I picked eggs because they were simple to visualize.  Unfortunately 108.923 eggs aren't going to sell well at the grocery.

There is, I'm sure, a convenient set of numbers to use.  If the cartons each had 12*13 eggs in them (i.e. the big box would have 1872 eggs in it) all the numbers would be integers, but it's still an ugly example.

I think the moral of this story has to be to let water be water and milk be milk.

We'd have never gotten into this mess if no one pulled out a tablespoon to being with.

Let's drink to that, Bob!!!

In terms of continuity, whatever milk has left the milk container has to be made up of something, in this case water, since the volume is the same. Also true for the water, whatever water is gone, has to be made up from the milk. If there was more of one item, milk or water, in either mixture, the volumes would no longer be equal.

The Intuitive REALLY BIG SPOON solution:

Let's use really a BIG spoon, in fact it holds 0.5 litre. Now that's a BIG spoon!

After the first transfer:
 >> 0.5 litres of milk in the first container
 >> 0.5 litres of milk + 1.0 litres of water in the second container;
       ie,  1/3 milk, 2/3 water.

After the second transfer:
 >> 0.5 +0.5*(1/3)  = 2/3 litres of milk in the first container
 >> 1 - 0.5*(2/3) = 2/3 litres of water in the second container.

TA-DA: same amount of milk and water.

The smaller you make the spoon the more milk you'll have in the first container, and the more water you'll have in the second, but, inituitively, they will always be equal.

Still not convinced...Try it again with THE REALLY REALLY BIG (1.0 litre) spoon.

The EGGHEAD solution:

Let Mi be the amount of milk in container i, in litres.
Let Wi be the volume of water in container i, in litres.
Let i =1, 2.
Let T be the volume of one tablespoon, in litres.

Initial conditions:
   M1o = 1, W1o = 0, V1o = 1
   M2o = 0, W2o = 1, V2o = 1

After first transfer:
   M11 = 1-T,  W11 = 0,
   M21 = T,    W21 = 1,

After second transfer:

   M12 = M11   + T*( T/(1+T) )
       = (1-T) + T*( T/(1+T) )
       = (1–T) + T*T/(1+T)

          (1-T)*(1+T) + T*T
       = -----------------
                (1+T)

           (1-T*T) + T*T
       = -----------------  = 1/(1+T)
                (1+T)


   W22 = W21 – T*( 1/(1+T) )
       = 1   - T/(1+T)

         1*(1+T)       T
       = -------  -  -------
          (1+T)       (1+T)

         (1+T) - T
       = ----------  =  1/(1+T)
           (1+T)


Therefore M12 = W22. In other words, the amount of milk in container 1 equals the amount of water in container 2.

GETTING BACK TO THE QUESTION

Hi All,
It seems we've all been so eager to solve the puzzle that we've forgotten InteractiveMind's question, which is:
<..I justify that there would be more water in the milk/water solution than otherwise... BUT WHY AM I WRONG?>

Hi InteractiveMind,
You are correct when you say you are wrong, so at least you're ahead of the people who think you're wrong because you think your wrong!

And here's the reason you're wrong:

You say:
  <The composition within the water/milk solution will be: W + TM - T(W+TM)>

This is incorrect because T(W+TM) is not what was transferred back to the milk container. T(W+TM) has a volume T *(1+T) litres, whereas it should have a volume of T litres.

So, the correct composition within the water/milk solution is:
     W + TM - T(W+TM)/(1+T)    (Expression 1)

Similarly, the correct composition within the milk/water solution is:
      M - TM + T(W+TM)/(1+T)    (Expression 2)

Therefore, from Expression 1, the amount of milk in the water/milk solution is:

   TM   -   T*(TM)/(1+T)

   TM * (1+T)         T*T*M
= --------------  -   ---------
           (1+T)         (1+T)

=   (TM + T*T*M - T*T*M) / (1+T)

=  T/(1+T) litres

and, from Expression 2, the amount of water in the milk/water mixture is:

      T*(W)/(1+T)

=    T/(1+T) litres

How about I indulge you all with a little simpler equation?
1 liter Milk (1L M)  and 1 liter Water (1L W)
1 L M - 1 Tbls. M    and  1 L W + 1 Tbls. M
So, we still have 1 Liter of Milk just divided into 2 glasses.  But Milk = Milk + Water.

Well, the next question is:  What type of Milk is it?  This makes a huge difference.
http://hypertextbook.com/facts/2002/AliciaNoelleJones.shtml

If the density is less than water, you are taking some milk back.  If the density of milk is more, then you are taking pure water.

Because most milk has a mixture of water leads me to think that milk is more dense than water.  Hence, the milk will sink to the bottom causing the teaspoon at the top only to take water.

Test this out though.  Fifth graders did it.
http://sciconn.mcb.arizona.edu/Sink_Float_lesson.html

Mathematically, you will be missing a critical formula of density.  Which no one actually mentioned.  What is the density of this milk to water.

This is a cooking problem.
Oh yeah, he did not say if we spilled any.   Ooops!  Well, also there are remains in the glasses.  Oh, can I get one of those one liter glasses.  I want it as a souvenir of something that I have not seen for a long time.  Let's have a Lactose in tolerant person test this out. Just kidding.  Of course, this person will not like the pure milk cup.  ha ha ha ...