$A$ 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?

$3$ persons undertake to complete a piece of work for $Rs. 1,200$. The $first$ person can complete the work in $8$ days,the $second$ person in $12$ days,and the $third$ person in $16$ days. They complete the work with the help of a $fourth$ person in $3$ days. What does the $fourth$ person get? (in $Rs.$)

$P, Q$ and $R$ undertake to do a work for $Rs. 707$. $P$ and $Q$ together do $\frac{5}{7}$ of the work and the rest is done by $R$ alone. How much should $R$ get? (in $Rs.$)

$A$ 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)

$12$ men and $16$ boys can do a piece of work in $5$ days. $13$ men and $24$ boys can do it in $4$ days. Then,the ratio of the daily work done by a man to that of a boy is:

$250$ men can finish a work in $20 \, \text{days}$ working $5 \, \text{hours}$ a day. To finish the work within $10 \, \text{days}$ working $8 \, \text{hours}$ a day,the minimum number of men required is: