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There are two cups on a table. One is filled with tea, the other with milk. If we take a spoon of tea from the first cup and place that into the cup with milk and stir and then do the same but vice versa. Which was there more of in the end? Tea in the cup of milk or milk in the cup of tea?

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    $\begingroup$ There is no difference (and I did not downvote) $\endgroup$ – imranfat Oct 11 '16 at 13:54
  • $\begingroup$ As the answers show, it is a simple thing to solve. I personally had this experience once: A group of mathematicians had dinner at a restaurant. They had a few drinks with dinner. Then someone asked this question. It is remarkable how long it took us to find the answer. $\endgroup$ – GEdgar Oct 11 '16 at 14:45
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Suppose a spoon volume is $q$ of a cup volume (certainly, $0 < q < 1$). Then, after the first action we have $(1-q)$ of tea in the first cup and $(1+q)$ of liquid in the second cup, $1$ of milk and $q$ of tea.

In the second move we take a $q$ of the mixture, that is $\frac q{1+q}$ of milk and $\frac{q^2}{1+q}$ of tea. After adding it into the first cup we have $(1-q)+\frac{q^2}{1+q} = \frac{(1-q^2)+q^2}{1+q} = 1/(1+q)$ of tea in the first cup and $1-\frac q{1+q} = \frac {(1+q)-q}{1+q} = 1/(1+q)$ of milk in the second one.

So the amount of tea in the first cup is equal the amount of milk the second cup.

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  • $\begingroup$ "In the second move we take a $q$ of the mixture, that is $\dfrac{q}{1+q}$ of milk and $\dfrac{q^2}{1+q}$ of tea". How did you get this? @CiaPan $\endgroup$ – Wolgwang Jan 26 at 7:18
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    $\begingroup$ @YouKnowMe The volume of a mixture is $(1+q)$, of which $1$ is milk, $q$ is tea. So the concentration of milk in the mixture is $\frac 1{1+q}$ and the concentration of tea is $\frac q{1+q}$. When you take a spoon of it, i.e. a volume of $q$, then that volume times a concentration equals amount of a specific component, hence you have $q\cdot\frac 1{1+q}$ of milk and $q\cdot\frac q{1+q}$ of tea in the spoon. $\endgroup$ – CiaPan Jan 26 at 19:50
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If as a result of all the operation some amount of milk get into the cup of tee, the same amount of tea must have been taken out because the total volume did not change. And this volume got into the other cup. So the volume of tea in milk must be the same as volume of milk in tea, no matter how many operations we have done. It's only required that the total volume in each cup did not change.

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I think also the two cups may have different sizes and still, the amount of tea in the milk cup is the same as the amount of milk in the cup of tea. Consider the following:

The volume of the spoon is $V$, the amount of the tea transferred from the cup of tea into the cup of milk is $T_1$ (first spoon). So $$V=T_1$$Then you transfer a mixture of tea and milk from the cup of the mixture to the cup of tea (second spoon) where: $$V=T_2+M$$ Where $T_2$ and $M$ are the amount of tea and milk in the second spoon respectively. In the end, you will have $$(T_2+M)=T_1=V$$ So:

  • The amount of milk in the cup of tea is $M$
  • The amount of tea in the second cup is $T_1-T2$ (the amount of tea removed from the cup of tea) which equals to $M$

Please correct me if I am wrong.

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If the spoon has capacity $C$, you take $C$ units of tea to the milk cup in the first step. In the second step, there are $A$ units of tea and $B$ units of milk in the spoon, which are taken to the tea cup. The final amount of milk taken to the tea cup is $B$. The final amount of tea taken to the milk cup, considering the two steps, is $C-A$, and this equals $B$. So there is no difference.

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  • $\begingroup$ The final amount of milk taken to the teacup is not B. This is because the B amount of milk transferred to the teacup gets mixed. You cannot just extract all the B amount of milk. $\endgroup$ – Chuan Jul 15 '20 at 0:44
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No matter how you mix the same amount of two substances into two equally sized cups, the ratios of the substances will be inverses of each other. Why? Let's say we have cups 1 and 2, and we have Milk and Tea, the same amount of both.

This means that after any mixing, $M_1+M_2=T_1+T_2$ (we have the same amounts of milk and tea) and $M_1+T_1=M_2+T_2$ (both cups are equally filled). We easily derive that $M_1/T_1 = T_2/M_2$. To do this, you just do the sum and the difference of the two equalities above, getting $M_2-T_1=T_1-M_2$ and $2M_1=2T_2$, meaning that $M_2=T_1$ and $M_1=T_2$, whence the above formula.

(This is one of the magics, you can put any amount of one into the other one, and then the same amount of the (now mixture) back, and the ratios will be equal.)

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  • $\begingroup$ I think this is the best answer because it formalizes the intuition that > because the size of cup 1 is invariant, the amount of tea missing in cup 1 is the same amount as the milk added to cup 1 (missing from cup 2). $\endgroup$ – Chuan Jul 15 '20 at 1:23

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