I'm an Alchemist and I want to prepare Universal Panacea. That's a precious thing and I will be able to sell it for good money.
The complete recipe for the Panacea is of course very complex (and secret), but the final step is actually quite simple: I need to boil the component X in a silver cauldron with the right quantity of saltpeter dust and then seal it in vials.
I have a good amount of spare vials. They can contain most of the liquids that exist and have a very precise graduation line to know when they are full. They don't have intermediary lines so you can't directly measure half a vial of liquid.
I have only one silver cauldron. Its shape makes it difficult to judge precisely the amount of liquid it contains but by experience I know eleven vials of liquid can fit inside. There is a tap in the side that make it possible to slowly recover liquid from it but you can't access the bottom half of the cauldron this way (as the tap is in the middle of the side).
Component X is a liquid. I have an opaque barrel of it but as it tends to evaporate I don't know how much there is left. I am quite sure there is enough to fill twice the cauldron though. Component X boils at a temperature around 60°C and I manage to boil it without loosing more that 10% of it with evaporation (sometimes less than 10% evaporates). It also is lighter than water and not miscible with any mundane liquid.
Saltpeter is a powder that reacts with boiling component X to make the Panacea. I need exactly (no more and no less) one pinch and a half of saltpeter per vial of the final product. Yes, I am sure all my pinches will be of the same quantity, I'm an expert alchemist after all. I can't take precise half-pinches: my pinches are regular but their size is not on-demand.
The Panacea is produced by the reaction between saltpeter and boiling component X. Panacea is liquid but it never evaporates, but it can mix with component X, with which it shares all the other physical properties. I need to fill the bigger number possible of vials with it, while being sure there is no remaining saltpeter of component X in the vials.
The only other equipment for chemistry I have is a mundane bathtub. That's where I wash my cauldron and vials so I can be sure there is no unwanted materials in my potions.
How can I manage to get the biggest amount possible of Panacea vials out of my single barrel of component X ?