         i.            A solution of vinegar and water contains 2 liters  of vinegar and 6 liters of  water. How much pure vinegar must be added to this solution so that the resulting solution will be   vinegar.

Let's solve this problem step-by-step.

### Problem Summary:
- Initial solution contains:
  - Vinegar: 2 liters
  - Water: 6 liters
- We want to add a certain amount of pure vinegar, \(x\) liters, to make the resulting solution 50% vinegar.

### Step-by-Step Solution:

1. **Define the Variables:**
   - Let \(x\) be the amount of pure vinegar to be added (in liters).

2. **Calculate the Total Amounts After Adding \(x\):**
   - Total vinegar after adding \(x\) liters: \(2 + x\) liters.
   - Total water remains the same: 6 liters.
   - Total solution after adding \(x\) liters: \(2 + 6 + x = 8 + x\) liters.

3. **Set Up the Equation for 50% Vinegar Concentration:**
   - We want the final solution to be 50% vinegar. Therefore, the amount of vinegar should be half of the total solution:
   
   \[
   \frac{2 + x}{8 + x} = 0.5
   \]

4. **Solve the Equation:**

   \[
   2 + x = 0.5(8 + x)
   \]

   Simplify:

   \[
   2 + x = 4 + 0.5x
   \]

   Rearrange to isolate \(x\):

   \[
   x - 0.5x = 4 - 2
   \]

   \[
   0.5x = 2
   \]

   \[
   x = \frac{2}{0.5}
   \]

   \[
   x = 4
   \]

### Conclusion:
To make the resulting solution 50% vinegar, you must add **4 liters of pure vinegar** to the original solution.

Would you like more details on this solution, or do you have any questions?

### Related Questions:
1. How would the solution change if the original mixture had different amounts of vinegar and water?
2. What if the target concentration was 60% instead of 50%?
3. How can this problem be modeled using a system of equations?
4. Can this problem be visualized using a graph? If so, how?
5. What are other real-life applications of mixing solutions with desired concentrations?

**Tip:** When dealing with percentages in mixture problems, always convert them to their decimal forms to simplify the calculations.

Learn how to calculate the concentration of vinegar in a mixture using a step-by-step solution. This problem involves understanding mixtures, percentages, and solving equations to achieve a desired vinegar concentration. Suitable for students in Grades 7-9.

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