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Let's solve the problem step by step:

### Problem Breakdown:

1. The initial ratio of milk to water is given as 3:2.
   Let the original volume of the solution be \( V \) liters.
   Therefore, the amount of milk = \( \frac{3}{5}V \) liters, and the amount of water = \( \frac{2}{5}V \) liters.

2. The volume is increased by 50% by adding water.
   New volume of the solution = \( V + \frac{50}{100}V = 1.5V \) liters.
   Since only water is added, the amount of milk remains the same, but the amount of water increases.
   
   New amount of water = \( 1.5V - \frac{3}{5}V \).

3. From the resultant solution, 30 liters are withdrawn, and the same amount of water is added back. The ratio of milk to water in the final solution becomes 3:7.

### Let’s derive the equations:

#### After 30 liters are removed:
When 30 liters are withdrawn from the solution, it contains both milk and water. Therefore, the amount of milk withdrawn = \( \frac{3}{10} \times 30 = 9 \) liters, and the amount of water withdrawn = \( \frac{7}{10} \times 30 = 21 \) liters.

Amount of milk left = \( \frac{3}{5}V - 9 \).

Amount of water left = (new amount of water before withdrawal) - 21 + 30 (since 30 liters of water are added back).

#### Final ratio:
After this process, the milk to water ratio is 3:7.

Now, I'll calculate the original volume \( V \).The original volume of the solution is 60 liters.

Would you like further details on any step or clarification?

Here are five questions related to this topic for further exploration:

1. What is the concept behind the ratio of two quantities?
2. How does changing the volume affect the concentration in solutions?
3. Can you explain the method of solving equations involving ratios?
4. How do you handle problems involving partial replacement in mixtures?
5. What are the key steps to follow in word problems involving ratios and percentages?

**Tip:** In mixture problems, keeping track of each component separately (like milk and water) helps in forming correct equations!

A vessel contains milk and water in the ratio 3:2. The volume of the contents is increased by 50% by adding water to this. From this resultant solution, 30 L is withdrawn and then replaced with water. The resultant ratio of milk to water in the final solution is 3:7. Find the original volume of the solution.

Learn how to solve a mixture problem where the original solution's volume and ratios change. This problem involves key mathematical concepts such as ratios, proportions, and partial replacement. Suitable for Grades 8-10 students.

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