$P$ 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?

$A$ can do a certain work in the same time in which $B$ and $C$ together can do it. If $A$ and $B$ together could do it in $10$ days and $C$ alone in $50$ days,then $B$ alone could do the work in (in days)

$A$ 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?

$A$ alone would take $27 \, \text{days}$ more to complete the job than if both $A$ and $B$ worked together. If $B$ worked alone, he would take $3 \, \text{days}$ more to complete the job than $A$ and $B$ working together. What time would they take if both $A$ and $B$ worked together? (in $\text{days}$)

$A$ and $B$ together can complete a task in $20$ days. $B$ and $C$ together can complete the same task in $30$ days. $A$ and $C$ together can complete the same task in $40$ days. What is the ratio of the number of days taken by $A$ when completing the same task alone to the number of days taken by $C$ when completing the same task alone?

$A$ does $\frac{4}{5}$ of a work in $20 \, \text{days}$. He then calls in $B$ and they together finish the remaining work in $3 \, \text{days}$. How long would $B$ alone take to do the whole work? (in $\text{days}$)