$P$ and $Q$ can do a piece of work in $30$ and $40$ days respectively. They began the work together,but $P$ left after some days and $Q$ finished the remaining work in $12$ days. After how many days did $P$ leave?

$6$ men can complete a piece of work in $12$ days. $8$ women can complete the same piece of work in $18$ days,whereas $15$ children can complete the piece of work in $10$ days. $4$ men,$12$ women,and $20$ children work together for $2$ days. If only men were to complete the remaining work in $1$ day,how many men would be required?

Two women,Sita and Gita,working separately can mow a field in $8$ and $12 \, hrs$ respectively. If they work in stretches of one hour alternatively,with Sita beginning at $9:00 \, a.m.$,when will the mowing be finished? (in $p.m.$)

$A$ and $B$ working separately can do a piece of work in $10 \text{ days}$ and $15 \text{ days}$ respectively. If they work on alternate days beginning with $A$,in how many days will the work be completed?

$15$ 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?

$A$ and $B$ can do a work in $12$ days,$B$ and $C$ in $15$ days,and $C$ and $A$ in $20$ days. If $A, B,$ and $C$ work together,they will complete the work in (in days):