Mixing problem of two jars containing milk and water

Question

So i came across this question - there are $x$ gallons of milk in a container and $y$ gallons of water in another. Now $z$ gallons of milk are transferred from the first to the second container and the $z$ gallons of water are transferred to the first one . This is process is repeated (transferring $z$ gallons) another time and it is said that the quantity of milk in both containers is same as that after first transfer.

My approach- We equate the fractions of milk in both containers and hence obtain a relation between $x$, $y$ and $z$.

Now what i want to know is that after n transfers(quantity of milk is same as that after first transfer) how can we generalize this as a relation between $x$ , $y$ and $z$. PS: I am just being curious and i don't know even if it is possible.Any help would be awesome!

Answer 1

The statement to prove should be that there is the same amount of water in the milk container as milk in the water container. It doesn't matter how you do the transfers as long as you wind up with the same volume in the jars at the end as you started with. Let $w$ be the amount of water in the left jar at the end, so there is $x-w$ of milk in the left jar. In the right jar there is $w$ water missing, so you have $y-w$ of water and $w$ of milk. The only time you care about how the transfers are done is if you want to compute $w$ as a function of $z$.

Answer 2

As far as I undrerstand tge question, assume the discrete problem: $12$ black balls in box $1$ and $18$ white balls in box $2$. First $6$ balls are interchanged. Then again $6$ balls are interchanged. In the result the number of black (or white) balls in both boxes ($3+3$) is equal to the number of black (or white) balls transfered first time ($6$).