Explanation:

To solve this problem, we use the concept of weighted averages. Here's a breakdown:

- We need to find the ratio \( x : y \) where \( x \) and \( y \) are the amounts of rice costing Rs65 and Rs95 respectively.

- Use the alligation rule, which helps mix two ingredients at different prices to find the desired price.

- Formula:

$$

\text{Ratio} = \frac{\text{Price of costlier rice} - \text{Price of mixture}}{\text{Price of mixture} - \text{Price of cheaper rice}}

$$

- Applying the values:

$$

\frac{95 - 83}{83 - 65} = \frac{12}{18} = \frac{2}{3}

$$

- Therefore, the correct mixing ratio is 2 : 3.

Given this analysis:

- Option 1 (19 : 13): Incorrect, as it doesn’t match the derived ratio.

- Option 2 (13 : 19): Incorrect, as it doesn’t match the derived ratio.

- Option 3 (2 : 3): Correct, aligns with the calculation.

- Option 4 (3 : 2): Incorrect, as the numbers are in the opposite order.

Using the above logic:

Correct Answer: Option 3 (2 : 3)

.