The average weight of $6$ persons is increased by $5 \, kg$ when one of them who weighs $40 \, kg$ is replaced by a new man. The weight of the new man is (in $kg$):

The average weight of $4$ men $A, B, C,$ and $D$ is $67 \, kg.$ When a $5^{th}$ man $E$ is included,the average weight decreases by $2 \, kg.$ $A$ is replaced by $F.$ The weight of $F$ is $4 \, kg$ more than $E.$ The average weight decreases because of the replacement of $A,$ and now the average weight is $64 \, kg.$ Find the weight of $A.$

The average weight of $15$ oarsmen in a boat is increased by $1.6\, kg$ when one of the crew,who weighs $42\, kg$,is replaced by a new man. Find the weight of the new man (in $kg$).

$10$ years ago,the average age of all $25$ teachers of a girls' college was $45$ years. $4$ years ago,the principal retired at the age of $60$ years. One year later,a new principal aged $54$ years was recruited. Considering the principal as a teacher,what is the present average age of all $25$ teachers?

The average weight of $3$ men $A, B,$ and $C$ is $84 \, kg$. Another man $D$ joins the group and the average weight now becomes $80 \, kg$. If another man $E$,whose weight is $3 \, kg$ more than that of $D$,replaces $A$,then the average weight of $B, C, D,$ and $E$ becomes $79 \, kg$. The weight of $A$ in $kg$ is:

$A$ man spends $Rs. 1800$ monthly on an average for the first four months and $Rs. 2000$ monthly for the next eight months and saves $Rs. 5600$ a year. His average monthly income is (in $Rs.$)