$4$ men and $6$ women earn $Rs. \, 1600$ by doing a piece of work in $5$ days. $3$ men and $7$ women earn $Rs. \, 1740$ by doing the same work in $6$ days. In how many days can $7$ men and $6$ women complete the same work while earning $Rs. \, 3760$?

$A$ and $B$ can do a piece of work in $30$ days,while $B$ and $C$ can do the same work in $24$ days and $C$ and $A$ in $20$ days. They all work together for $10$ days when $B$ and $C$ leave. How many days more will $A$ take to finish the work?

Hemant decided to build a farmhouse in $40$ days. He employed $100$ men in the beginning and $100$ more after $35$ days and completed the construction in the stipulated time. If he had not employed the additional men,how many days behind schedule would it have been finished? (in days)

$P$ can complete a work in $12$ days working $8$ hours a day. $Q$ can complete the same work in $8$ days working $10$ hours a day. If both $P$ and $Q$ work together,working $8$ hours a day,in how many days can they complete the work?

$A$ and $B$ can do a work in $45 \text{ days}$ and $40 \text{ days}$,respectively. They began the work together,but $A$ left after some time and $B$ completed the remaining work in $23 \text{ days}$. After how many days of the start of the work did $A$ leave?

$A, B$ and $C$ can do a job in $6$ days,$12$ days and $15$ days,respectively. After $\frac{1}{8}$ of the work is completed,$C$ leaves the job. The rest of the work is done by $A$ and $B$ together. The total time taken to finish the work is (in days):