$A$ work can be completed by $P$ and $Q$ in $12$ days,$Q$ and $R$ in $15$ days,and $R$ and $P$ in $20$ days. In how many days can $P$ alone finish the work?

$A$ and $B$ can complete a piece of work in $45$ and $40$ days respectively. Both started to work together,but after some days $A$ left and $B$ alone completed the remaining work in $23$ days. For how many days did $A$ work?

$6$ 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 men can do a piece of work in $x$ days. But $y$ women can do it in $3$ days. Then the ratio of the work done by $1$ man and $1$ woman is

$5$ men and $5$ women earn $Rs. 660$ in $3$ days. $10$ men and $20$ women earn $Rs. 3500$ in $5$ days. In how many days can $6$ men and $4$ women earn $Rs. 1060$?

$A$ and $B$ together can complete a work in $8$ days. $B$ alone can complete that work in $12$ days. $B$ alone worked for $4$ days. After that,how long will $A$ alone take to complete the remaining work?