Can you help me to find a formula to this experiment?

Question

So you have the following experiment:

Could you find a function or sum for me to be able to calculate it for each step?

I have thought about something like this (b and a are the pipettes, x and y are the containers):

Im not a math guy so this is propably wrong but it would be nice if you helped me out.

Thanks in advance for any responses!

Answer 1

Let's go together: For $x_0$ and $y_0$ which are initial values of $A$ and $B$, respectively. in first exchange we get \begin{cases} x_1=x_0+\dfrac{1}{10}y_0-\dfrac{1}{10}x_0=\dfrac{9}{10}x_0+\dfrac{1}{10}y_0,\\ y_1=y_0+\dfrac{1}{10}x_0-\dfrac{1}{10}y_0=\dfrac{9}{10}y_0+\dfrac{1}{10}x_0. \end{cases} next \begin{cases} x_2=\dfrac{9}{10}x_1+\dfrac{1}{10}y_1,\\ y_2=\dfrac{9}{10}y_1+\dfrac{1}{10}x_1. \end{cases} so we have following two sequences simultaneously: \begin{cases} x_{n+1}=\dfrac{9}{10}x_n+\dfrac{1}{10}y_n,\\ y_{n+1}=\dfrac{9}{10}y_n+\dfrac{1}{10}x_n. \end{cases}

Answer 2

We start with $1$ unit of water. After one drop, we have $100-10=90\%$ water left:

$$t_1=0.9$$

And again:

$$t_2=0.9t_1=0.9^2$$

And again...

$$t_n=0.9^n$$