The average of $n$ observations is $40$. If two observations $50$ and $60$ are added, then the average of all the observations will be $41$. What is the value of $n$?

The sum of the initial $n$ observations is:
$${\text{Sum}}_1=40n$$
When two observations ($50$ and $60$) are added, the new total number of observations becomes $n+2$, and the new sum becomes:
$${\text{Sum}}_2=40n+50+60=40n+110$$
The new average is given as $41$:
$$\frac{40n+110}{n+2}=41$$
$$40n+110=41(n+2)$$
$$40n+110=41n+82$$
$$110-82=41n-40n$$
$$n=28$$
Thus, the value of $n$ is $28$.