Question

You are given two containers, the first containing one litre of wine and the second one litre of water. You also have a cup which has a capacity of r litres, where 0 < r < 1. You fill the cup from the first container and transfer the content to the second container, stirring thoroughly afterwards. Next, dip the cup in the second container and transfer r litres of liquid back to the first container. This operation is repeated again and again. Prove that as the number of iterations n of the operation tends to infinity, the concentrations of wine in both containers tend to equal each other.

Answer 1

Conservation of the respective liquids would impliy that the volume of wine in the barrel holding mostly water (let's call it Wa) has to be equal to the volume of water in the barrel holding mostly wine (let's call it Wi).

Now, let's see what the composition is after a few mixings:

Before first mixing: Wi holds 1 litre of wine and Wa holds 1 litre of water. (We will omit the unit "litre" henceforth.)

After first mixing: Wi holds 1-r of wine. Wa holds 1 of water and r of wine.

After second mixing: Wi holds 1-r+v of wine and r-v of water. (Here, we have used v to denote the amount of wine in the r litres of mixture taken from the second container.) Wa 1-(r-v) of water and r-v of wine.

Purity of wine in the first container:

Purity of water in the second container:

So, purity of wine in the second container:

Now, it can be seen that if you have water in Wa before a transfer, after a transfer in each direction you have , so is driven to zero in further transfers.