The average weight of $36$ students is $50 \,kg$. It was found later that the figure of $37 \,kg$ was misread as $73 \,kg$. What is the correct average? (in $kg$)

$A$ professional institute's total expenditure on students for a particular course is partly fixed and partly varies linearly with the number of students. The average expense per student is $Rs.\,615$ when there are $24$ students and $Rs.\,465$ when there are $40$ students. What is the average expense when there are $60$ students? (in $Rs.$)

The average age of four brothers is $12 \, \text{years}$. If the age of their mother is also included, the average increases by $5 \, \text{years}$. The age of the mother (in $\text{years}$) is:

The average expenditure of a man for the first five months is $Rs.\, 3600$ and for the next seven months it is $Rs.\, 3900$. If he saves $Rs.\, 8700$ during the year,his average income per month is (in $Rs.$)

The average age of a board of directors of a company,having $10$ directors,was $48$ years. Coincidentally,when a director aged $53$ resigned from the board,another director died on the same day. $A$ new director aged $34$ joined the board. Next year in the same month,the average age of all the $9$ directors was found to be $46$ years. The age of the late (i.e.,dead) director at the time of his death was (in years):

What is the average of the squares of the first $19$ natural numbers?