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The question asks to find the volume of vessel C given the following:

- Vessel A contains 10 liters of alcohol.
- Vessel B contains 10 liters of water.
- Vessel C is filled with alcohol from vessel A and then poured into vessel B.
- Vessel C is then filled with the mixture from vessel B and poured back into vessel A.
- The final ratio of alcohol to water in vessel A is \(3:1\).

Let the volume of vessel C be denoted by \(x\).

### Step-by-step solution:

1. Initially, vessel A contains 10 liters of pure alcohol and vessel B contains 10 liters of pure water.

2. Vessel C, which has volume \(x\), is first filled with alcohol from vessel A. So, after this, \(x\) liters of alcohol is removed from vessel A. The remaining alcohol in vessel A becomes \(10 - x\) liters.

3. This alcohol from vessel C is poured into vessel B, which initially had 10 liters of water. So now, vessel B has a mixture of \(x\) liters of alcohol and 10 liters of water.

4. Next, vessel C is filled with this mixture from vessel B. Since vessel B now has a total of \(x + 10\) liters of liquid, the fraction of alcohol in the mixture is:
   \[
   \text{Fraction of alcohol in B} = \frac{x}{x + 10}
   \]
   Therefore, when vessel C is filled with this mixture, the amount of alcohol in vessel C is:
   \[
   \text{Alcohol in C} = \frac{x}{x + 10} \times x = \frac{x^2}{x + 10}
   \]
   The remaining volume in vessel C will be water.

5. Vessel C is then poured back into vessel A. After this process, the alcohol in vessel A increases by the amount of alcohol in C, and the amount of water in A increases by the amount of water in C. So, the final volume of alcohol in vessel A becomes:
   \[
   10 - x + \frac{x^2}{x + 10}
   \]
   The volume of water in vessel A becomes:
   \[
   \text{Water in A} = \frac{10x}{x + 10}
   \]

6. We are given that the final ratio of alcohol to water in vessel A is \(3:1\). Therefore, we have:
   \[
   \frac{10 - x + \frac{x^2}{x + 10}}{\frac{10x}{x + 10}} = 3
   \]

7. Solving this equation for \(x\) will give us the volume of vessel C. Let's solve this algebraically.The volume of vessel C is \( \frac{10}{3} \) liters.

Would you like more details or have any questions? Here are five related questions for practice:

1. How would the solution change if the ratio of alcohol to water in vessel A was 4:1?
2. What would happen if vessel C's volume was equal to vessel A's volume?
3. How does the amount of water in vessel A change after the entire process?
4. Can you derive a general formula for any initial volumes and final ratios?
5. What if vessel B had a different initial amount of water?

**Tip**: When solving mixture problems, it's helpful to track both the substance removed and returned at each step, especially when dealing with ratios.

Vessel A has 10 liters alcohol and vessel B has 10 liters water. Vessel C is filled with alcohol of vessel A and then the entire quantity of C is poured into B. Then vessel C is again filled with mixture of vessel B and poured into A. Finally, the ratio of alcohol to water in vessel A becomes 3:1. Find the volume (in liters) of vessel C.

Solve a mixture problem involving vessels with alcohol and water, where the final ratio of alcohol to water in vessel A is 3:1. This solution involves algebraic equations and mixture concepts, suitable for students in Grades 10-12.

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