# Mixing Drinks

### Question:

You have two drinks of the same size as on the left of the picture, A is red and B is blue. You take a spoonful from A and put it into B. After mixing well, you take an equal spoonful from B and put it into A.

Which cup has the more contaminated mixture? In other words, does A have more blue in the mix, or does B have more red in the mix?

say there is $$100$$ units in each of A and B to start, and a spoonful is $$10$$ units, these numbers are arbitrary. After the first swap, A has $$90$$ of red liquid, B has $$110$$ of liquid, of which $$\frac{1}{11}$$ is red.
After the second swap, A has $90+\frac{1}{11}\times 10 = 90 + \frac{10}{11}$ of red, so $100-\left(90 + \frac{10}{11}\right)=9+\frac{1}{11}$ of blue. B has $100-\frac{10}{11}\times 10 = 90 + \frac{10}{11}$ of blue and $9+\frac{1}{11}$ of red!