a solution has two letters of vinegar and 6 l of water in order to make the vinegar proportion 2/5 how much vinegar show be added

To make the proportion of vinegar in the solution \( \frac{2}{5} \), we need to determine how much vinegar should be added.

### Given:
- Initial amount of vinegar = 2 liters
- Initial amount of water = 6 liters
- Total solution = 2 liters vinegar + 6 liters water = 8 liters
- Desired vinegar-to-solution ratio = \( \frac{2}{5} \)

Let \( x \) be the amount of vinegar to be added. After adding \( x \) liters of vinegar, the total amount of vinegar will be \( 2 + x \) liters, and the total volume of the solution will be \( 8 + x \) liters.

### Setup the equation for the proportion:
\[
\frac{2 + x}{8 + x} = \frac{2}{5}
\]

### Solve for \( x \):
1. Cross multiply to eliminate the fraction:
\[
5(2 + x) = 2(8 + x)
\]

2. Expand both sides:
\[
10 + 5x = 16 + 2x
\]

3. Move all terms involving \( x \) to one side:
\[
5x - 2x = 16 - 10
\]

4. Simplify:
\[
3x = 6
\]

5. Solve for \( x \):
\[
x = \frac{6}{3} = 2
\]

### Conclusion:
You need to add **2 liters of vinegar** to achieve the desired vinegar proportion of \( \frac{2}{5} \).

Would you like more details on the steps, or do you have any questions?

### Related Questions:
1. How would the solution change if there were more water initially?
2. What if the target proportion was \( \frac{3}{5} \) instead?
3. How can you generalize this method for any given target proportion?
4. What would happen if you increased both vinegar and water but kept the ratio constant?
5. What if the initial vinegar amount was unknown, and only the target ratio and total volume were provided?

### Tip:
In mixture problems, setting up an equation with the desired ratio is key to solving for the unknown.

A solution has two liters of vinegar and 6 liters of water. In order to make the vinegar proportion 2/5, how much vinegar should be added?

Learn how to adjust the amount of vinegar in a solution to achieve a 2/5 vinegar-to-solution ratio. This step-by-step guide covers key algebraic concepts, ratio setup, and cross-multiplication, suitable for students in Grades 6-8.

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