Explanation:

- The amount obtained by investing a sum for 3 years at a rate \( r\% \) per annum with simple interest is given by:

$$

\text{Total Amount} = \text{Principal} + \text{Simple Interest} = \text{Principal} \times (1 + \frac{r \times 3}{100})

$$

- The simple interest for the same principal and rate over 13 years is:

$$

\text{Simple Interest} = \text{Principal} \times \frac{r \times 13}{100}

$$

- We're told these two amounts are equal:

$$

\text{Principal} \times (1 + \frac{r \times 3}{100}) = \text{Principal} \times \frac{r \times 13}{100}

$$

- Simplifying, we find:

$$

1 + \frac{r \times 3}{100} = \frac{r \times 13}{100}

$$

$$

100 + 3r = 13r

$$

$$

100 = 10r

$$

$$

r = 10

$$

- The correct answer is Option 2: 10.