The mean weight of $34$ students of a school is $42\, kg$. If the weight of the teacher is included, the mean weight increases by $400\, g$. Find the weight (in $kg$) of the teacher.

The average weight of $21$ boys was recorded as $64 \, kg$. If the weight of the teacher was added,the average increased by $1 \, kg$. What was the teacher's weight (in $, kg$)?

$A$ family consists of grandparents,parents,and three grandchildren. The average age of the grandparents is $67 \text{ years}$,that of the parents is $35 \text{ years}$,and that of the grandchildren is $6 \text{ years}$. What is the average age of the family? (in $\text{years}$)

What is the average of all numbers between $100$ and $200$ which are divisible by $13$ (in $.5$)?

The average of a collection of $20$ measurements was calculated to be $56 \, cm$. But later,it was found that a mistake had occurred in one of the measurements which was recorded as $64 \, cm$,but should have been $61 \, cm$. The correct average must be? (in $cm$)

The mean of $11$ numbers is $35$. If the mean of the first $6$ numbers is $32$ and that of the last $6$ numbers is $37$, then find the sixth number.