$A$ man,a woman,and a boy can together complete a piece of work in $3$ days. If a man alone can do it in $6$ days and a boy alone in $18$ days,how long will a woman alone take to complete the work? (in days)

$A$ certain number of men can do a piece of work in $40 \, days$. If there were $45 \, men$ more,the work could have been finished in $25 \, days$. Find the original number of men employed in the work.

$A$ and $B$ together can do a piece of work in $30$ days. $A$ having worked for $16$ days,$B$ finishes the remaining work alone in $44$ days. In how many days shall $B$ finish the whole work alone?

$P$,$Q$ and $R$ contracted to do a work for $Rs. 4200$. $P$ can do the work in $6$ days,$Q$ in $10$ days and $R$ in $12$ days. If they work together to do the work,what is the share of $R$ (in $Rs.$)?

$15$ men take $20$ days to complete a job working $8$ hours a day. How many hours a day should $20$ men work to complete the same job in $12$ days?

$A$ can complete a work in $10$ days,$B$ can complete the same work in $20$ days,and $C$ in $40$ days. $A$ starts working on the first day,$B$ works on the second day,and $C$ works on the third day. Again,$A$ works on the fourth day,$B$ on the fifth day,and so on. If they continue working in the same way,in how many days will the work be completed?