The present ages of Prajwal and Lavanya are in the ratio $5:6$ respectively. After $3$ years, the ratio of Prajwal's age to Lavanya's age will be $6:7$. What is Prajwal's present age?

Let the present ages of Prajwal and Lavanya be $5k$ and $6k$ respectively, where $k$ is a constant multiplier.
According to the given condition, after $3$ years, their ages will be $(5k+3)$ and $(6k+3)$, and their ratio becomes $6:7$:

$$\frac{5k+3}{6k+3}=\frac{6}{7}$$

Cross-multiply to solve for $k$:

$$7(5k+3)=6(6k+3)$$

$$35k+21=36k+18$$

Rearrange the terms:

$$36k-35k=21-18$$

$$k=3$$

Now, calculate Prajwal's present age:

$$\text{Prajwal’s present age}=5k=5\times 3=15\text{ years}$$

Therefore, Prajwal's present age is $15$ years.