To find the value of P, we first need to determine the initial amounts of milk and water, and then set up an equation based on the changes described.

1. Calculate the Initial Quantities:

   The vessel contains 95 litres of a mixture with a Milk to Water ratio of 15 : 4.

   • Total ratio parts = 15 + 4 = 19.
   • Value of one part = 95 litres / 19 parts = 5 litres/part.
   • Initial Milk = 15 parts * 5 = 75 litres.
   • Initial Water = 4 parts * 5 = 20 litres.
2. Set up the Equation after Changes:

   When P litres of the mixture are removed, the milk and water are removed in the same 15:4 ratio. Afterwards, 18 litres of pure water are added.

   • Milk removed = (15 / 19) * P. The final amount of milk is 75 - (15/19)P.
   • Water removed = (4 / 19) * P. The final amount of water is (20 - (4/19)P) + 18.

   The new ratio of Milk to Water is 3 : 2. This gives us the equation:

   (75 - 15P/19) / (38 - 4P/19) = 3 / 2
3. Solve for P:

   Now, we solve the equation by cross-multiplying:

   2 * (75 - 15P/19) = 3 * (38 - 4P/19) 150 - 30P/19 = 114 - 12P/19

   Group the terms with P and the constant terms:

   150 - 114 = 30P/19 - 12P/19 36 = 18P/19

   Isolate P:

   P = (36 * 19) / 18 P = 2 * 19 = 38

Final Answer: The value of P is 38 litres.