In a vessel , there are 10 litres of alcohol . An operation is defined as…

Concept: When a volume is removed from a mixture and a pure substance is then added, each component (alcohol, water) leaves the vessel in proportion to its current share of the mixture, and the added substance simply joins its own component afterward. Because here the volume removed (5 L) differs from the volume added (10 L), the total volume changes at every step, so each component must be tracked stage by stage rather than through the equal-volume replacement shortcut.

Step-by-step:

1. Start: the vessel holds 10 L of alcohol and 0 L of water (total 10 L).

2. Operation 1 - remove 5 L: the vessel is pure alcohol at this point, so all 5 L removed is alcohol. Remaining: 5 L alcohol, 0 L water (total 5 L).

3. Operation 1 - add 10 L water: alcohol stays at 5 L, water becomes 10 L (total 15 L).

4. Operation 2 - remove 5 L: the mixture is now 5 L alcohol and 10 L water out of 15 L total, so the 5 L removed carries alcohol and water in that same 1:2 proportion - (5 x 5/15) = 5/3 L alcohol and (5 x 10/15) = 10/3 L water leave with it.

5. Remaining after this removal: alcohol = 5 - 5/3 = 10/3 L; water = 10 - 10/3 = 20/3 L (total 10 L).

6. Operation 2 - add 10 L water: alcohol stays at 10/3 L, water becomes 20/3 + 10 = 50/3 L (total 20 L).

Cross-check: the alcohol volume can also be found by multiplying the fraction retained at each removal: 10 x (10-5)/10 x (15-5)/15 = 10 x 1/2 x 2/3 = 10/3 L, matching the step-by-step value; the running totals (15 L after operation 1, 20 L after operation 2) also match the volumes added and removed at each stage.

Result: Alcohol : Water = 10/3 : 50/3 = 10 : 50 = 1 : 5.