The average weight of $8$ persons increases by $2.5\text{kg}$ when a new person comes in place of one of them weighing $65\text{kg}$. What might be the weight of the new person?

Let the total initial weight of the $8$ persons be $W$.
The original average is $\frac{W}{8}$.

When a person weighing $65\text{kg}$ is replaced by a new person of weight $x\text{kg}$, the new total weight becomes $W-65+x$.
The new average increases by $2.5\text{kg}$:

$$\frac{W-65+x}{8}=\frac{W}{8}+2.5$$

Multiplying both sides by $8$:

$$W-65+x=W+8\times 2.5$$

$$W-65+x=W+20$$

$$x-65=20$$

$$x=20+65=85\text{kg}$$

Thus, the weight of the new person is $85\text{kg}$.