Explanation:

Let’s break this down:

- First vessel: Ratio 3:5 (milk : water).

Milk = 3/8, Water = 5/8.

- Second vessel: Ratio 5:3 (milk : water).

Milk = 5/8, Water = 3/8.

- Want: Final mixture with ratio 3:4 (milk : water), i.e., Milk = 3/7, Water = 4/7.

Here's how you set it up:

- Let’s say x liters from vessel 1, y liters from vessel 2.

- Milk in final: (3/8)x + (5/8)y.

- Water in final: (5/8)x + (3/8)y.

- Set up the ratio:

$$

\frac{\frac{3}{8}x + \frac{5}{8}y}{\frac{5}{8}x + \frac{3}{8}y} = \frac{3}{4}

$$

Cross-multiplied:

$$

4(3x + 5y) = 3(5x + 3y) \\

12x + 20y = 15x + 9y \\

11y = 3x \\

x/y = 11/3

$$

- That’s option 1: 11 : 3.

- Option 2: 7 : 3 — Not correct, does not match calculation.

- Option 3: 8 : 5 — No.

- Option 4: 10 : 9 — Also not matching.

- Option 5: 11 : 6 — Close but not correct.

Option 1 (11 : 3) is the answer.