A vessel initially contains $1000$ liters of acid A and $500$ liters of acid B. First, $x$ liters of the mixture is removed from the vessel, and then $y$ liters of acid B is added to the remaining mixture. After this, $z$ liters of the new mixture is again taken out. Determine the possible values of $x$, $y$, and $z$ such that the final quantities of acid A and acid B in the vessel are equal.

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**Solution:**

1. **Initial quantities:**
   - Acid A = $1000$ liters
   - Acid B = $500$ liters
   - Total mixture = $1500$ liters

2. **After removing $x$ liters of mixture:**
   - The ratio of acid A to acid B is $1000:500 = 2:1$.
   - Amount of acid A removed = $\frac{2}{3}x$ liters
   - Amount of acid B removed = $\frac{1}{3}x$ liters

So, remaining acid A = $1000 - \frac{2}{3}x$

Remaining acid B = $500 - \frac{1}{3}x$

3. **Adding $y$ liters of acid B:**
   - New acid B quantity = $500 - \frac{1}{3}x + y$

4. **After adding $y$ liters, total mixture becomes:**
   $$	ext{Total} = 1500 - x + y$$

5. **Removing $z$ liters of the new mixture:**
   - The ratio of acid A to acid B now is:
     $$\frac{1000 - \frac{2}{3}x}{500 - \frac{1}{3}x + y}$$

- Amount of acid A removed in this step = $\frac{1000 - \frac{2}{3}x}{1500 - x + y} 	imes z$

- Amount of acid B removed in this step = $\frac{500 - \frac{1}{3}x + y}{1500 - x + y} 	imes z$

6. **Final quantities after second removal:**
   - Acid A = $\left(1000 - \frac{2}{3}xight) - \frac{1000 - \frac{2}{3}x}{1500 - x + y} 	imes z$

- Acid B = $\left(500 - \frac{1}{3}x + yight) - \frac{500 - \frac{1}{3}x + y}{1500 - x + y} 	imes z$

7. **Condition for equal quantities:**
   $$	ext{Final Acid A} = 	ext{Final Acid B}$$

Substitute the expressions:

$$\left(1000 - \frac{2}{3}xight) - \frac{1000 - \frac{2}{3}x}{1500 - x + y} z = \left(500 - \frac{1}{3}x + yight) - \frac{500 - \frac{1}{3}x + y}{1500 - x + y} z$$

8. **Simplify the equation:**

Multiply both sides by $1500 - x + y$ to eliminate denominators:

$$\left(1000 - \frac{2}{3}xight)(1500 - x + y) - \left(1000 - \frac{2}{3}xight) z = \left(500 - \frac{1}{3}x + yight)(1500 - x + y) - \left(500 - \frac{1}{3}x + yight) z$$

Rearranging terms and simplifying leads to a relationship between $x$, $y$, and $z$.

9. **Testing the given options:**

- For $(x, y, z) = (300, 400, 200)$:

- Calculate final acid A and acid B quantities using the above formulas.

- Both come out equal, confirming this option satisfies the condition.

10. **Conclusion:**

The only set of values from the options that equalizes the final quantities of acid A and acid B is $(300, 400, 200)$.

**Answer:** $(x, y, z) = (300, 400, 200)$

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