$A$ can do a piece of work in $30$ days while $B$ can do it in $40$ days. In how many days can $A$ and $B$ working together do it?

$A$ contractor undertakes to dig a canal $12 \, km$ long in $350 \, days$ and employs $45$ men. After $200 \, days$ he finds that only $4.5 \, km$ of the canal has been completed. Find the number of extra men he must employ to finish the work in time (in $men$).

$A$ farmer can plough a field working $6$ hours per day in $18$ days. How many hours per day must the farmer work to finish the same work in $12$ days?

$P$ can do a piece of work in $9$ days. $Q$ is $50\%$ more efficient than $P$. The number of days it takes for $Q$ to do the same piece of work is

$4$ men and $6$ women together can complete a work in $8$ days,while $3$ men and $7$ women together can complete it in $10$ days. How many days will $20$ women working together take to complete the same work?

$A$ can do a work in $15\, days$ and $B$ in $20\, days.$ If they work on it together for $4\, days,$ then the fraction of the work that is left is