I got this problem from Rustan Leino, who read it in a puzzle book he had.
I solved it and wrote up my solution.
You have two jars. One contains vinegar, the other oil. The two jars contain the same amount of their respective fluid.
Take a spoonful of the vinegar and transfer it to the jar of oil. Then, take a spoonful of liquid from the oil jar and transfer it to the vinegar jar. Stir. Now, how does the dilution of vinegar in the vinegar jar compare to the dilution of oil in the oil jar?
It's the same, by the following reasoning.
Let $Q$ be the volume of fluid in each jar at the start. Let $V_v$ be the volume of vinegar in the vinegar jar at the end, and let $V_o$ be the volume of vinegar in the oil jar at the end. Similarly, let $O_v$ and $O_o$ be the volumes of oil in the jars at the end.
Because the procedure moves the same volume into each jar as out of it (one spoonful), at the end the volume of fluid in each jar reverts to its initial value of $Q$. Thus, $V_v + O_v = V_o + O_o = Q.$
We start with $Q$ of vinegar, and that vinegar is now distributed among the jars, so $V_v + V_o = Q.$ Similiarly for the oil, so $O_v + O_o = Q.$
By simple algebra, we see that $O_v = V_o$ and $O_o = V_v.$ Thus, the dilution of vinegar in the vinegar jar, $V_v / Q,$ is the same as the dilution of oil in the oil jar, $O_o / Q.$