The question can be solved in under a minute if you understand the concept of concentration and volume.

Removal and addition happen 19 times so:

C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})
All terms get canceled (4 in num with 4 in den, 5 in num with 5 in den etc) and you are left with C_f = \frac{1}{77}
Since Volume now is 22 lt, Volume of wine = 22*(\frac{1}{77}) = \frac{2}{7}

Theory:
1. When a fraction of a solution is removed, the percentage of either part does not change. If milk:water = 1:1 in initial solution, it remains 1:1 in final solution.

2. When you add one component to a solution, the amount of other component does not change. In milk and water solution, if you add water, amount of milk is the same (not percentage but amount)

3.
Amount of A = Concentration of A * Volume of mixture
Amount = C*V
( e.g. In a 10 lt mixture of milk and water, if milk is 50%, Amount of milk = 50%*10 = 5 lt)
When you add water to this solution, the amount of milk does not change.
So Initial Conc * Initial Volume = Final Conc * Final Volume
C_i * V_i = C_f * V_f
C_f = C_i * (V_i/V_f)
In the question above, we find the final concentration of wine. Initial concentration C_i = 1 (because it is pure wine)
When you remove 1 lt out of 3 lt, the volume becomes 2 lt which is your initial volume for the addition step. When you add 2 lts, final volume becomes 4 lt.
So C_f = 1 * 2/4
Since it is done 19 times, C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})

The concentration of wine is 1/77 and since the final volume is 22 lt (the last term has V_f as 22, you get amount of wine = 1/77 * 22 = 2/7 lt

If the operation is only done 19 times then where and why does "22" Lt pop up in the final volume of mixture I was following how the demoninators increased but dont understand the "22".

Also if 1 L of wine is removed every operation how is the concentration of the wine mixture go up since part of it is being removed...only thing that is increasing the total volume of the solution..

Thanks a lot.

Kudos +1 Karishma

Is there an fast way to compute the result of the multiplacation series like we have for Cf? I actually did the long way .