Let's break down the problem step by step:

Step 1: Initial setup

• The vessel contains 40 liters of pure milk.
• 10 liters of water is added to this vessel.
• Therefore, the total volume of the mixture becomes: $\text{Total mixture} = 40 \, \text{liters of milk} + 10 \, \text{liters of water} = 50 \, \text{liters of mixture}$
• The ratio of milk to water is: $\text{Milk : Water} = \frac{40}{10} = 4:1$

Step 2: Selling 25 liters of the mixture

• 25 liters of this mixture is sold.
• Since the ratio of milk to water is 4:1, out of the 25 liters, the quantities of milk and water sold are:
  • Milk sold = $\frac{4}{5} \times 25 = 20 \, \text{liters}$
  • Water sold = $\frac{1}{5} \times 25 = 5 \, \text{liters}$

Step 3: Remaining mixture in the vessel

• After selling 25 liters, the remaining quantities of milk and water are:
  • Remaining milk = $40 - 20 = 20 \, \text{liters}$
  • Remaining water = $10 - 5 = 5 \, \text{liters}$

Thus, after selling the mixture, the remaining liquid in the vessel consists of 20 liters of milk and 5 liters of water.

Step 4: Adding more milk and water in the ratio 2:1

• Let the amount of pure milk added be $2x$ liters and the amount of water added be $x$ liters.
• So, the total quantity of milk becomes $20 + 2x$ liters, and the total quantity of water becomes $5 + x$ liters.

Step 5: Final ratio of milk to water is 3:1

• According to the problem, after adding the milk and water, the ratio of milk to water in the vessel becomes 3:1. Therefore, we can set up the equation: $\frac{20 + 2x}{5 + x} = 3$

Step 6: Solve for $x$

• Cross-multiply to solve for $x$: $20 + 2x = 3(5 + x)$ $20 + 2x = 15 + 3x$ $20 - 15 = 3x - 2x$ $5 = x$

Step 7: Calculate the quantity of pure milk added

• Since $x = 5$, the amount of pure milk added is: $2x = 2 \times 5 = 10 \, \text{liters}$

Thus, the quantity of pure milk added to the vessel is 10 liters.

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Would you like any further details or clarification?

Here are 5 related questions you might find interesting:

1. How do you calculate ratios when quantities change?
2. How does the ratio of components in a mixture change when some part is removed?
3. What are other methods to solve mixture and ratio problems?
4. How do changes in one component of a mixture affect the whole system?
5. What are other real-life applications of mixture problems in different fields?

Tip: In mixture problems, it's often helpful to break down each step clearly and use equations to maintain the relationships between the different components (like milk and water here).