I have a pint of lager in one glass and a pint of real ale in another glass. Neither glass is full to the rim. I take a teaspoon of lager from the first glass and add it to the real ale mixing thoroughly. I then take a teaspoon of this mixture and put it back in the lager glass.
Is there more lager in the beer, or beer in the lager?
Same amount of lager in beer as there is beer in lager (as long as mixed perfectly).
Start with 100 units of beer in glass 1 and 100 of lager in glass 2, and say your spoon is 5 units
Decant step gives 100 of beer and 5 of lager in glass 1 = 105 units total volume, and 95 units of lager in glass 2
Mix up and decant back – you’ll be moving 5 units back and that will be (100/105) beer and (5/105) lager back into glass 2
At end glass 1 contains 100 units which is 100*(100/105) units beer and 100*(5/105) units lager so 95.238 units beer and 4.762 units lager.
Glass 2 contains 100 units which is 95 + (5*5/105) units of lager and (5*100/105) units of beer so 95.238 lager and 4.762 beer.