The arithmetic mean of $100$ observations is $24$. If $6$ is added to each of the observations and then each of them is multiplied by $2.5$,find the new arithmetic mean.

The average age of $6$ members of a family is $20$ years. If the age of a servant is included,the average age increases by $25\%$. What is the age (in years) of the servant?

The average age of a cricket team reduces by $2$ when two senior players aged $32$ and $30$ are replaced by $2$ youngsters of equal age. What is the age of these youngsters?

There are $35$ students in a hostel. If the number of students increased by $7$,the expenses of the mess were increased by $Rs. 42$ per day while the average expenditure per head decreased by $Rs. 1$. Find out the actual expenditure of the mess (in $Rs.$).

$A, B, C$ and $D$ are four consecutive even numbers,respectively,and their average is $65$. What is the product of $A$ and $D$?

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):