$A$ and $B$ can do a piece of work in $20$ days and $12$ days,respectively. $A$ started the work alone and then after $4$ days $B$ joined him until the completion of the work. How long did the work last? (in days)

$A$ worker can complete a piece of work in $50$ days. On the first day,he works alone. On the second day,one more person joins him,on the third day,one more person joins them,and this process continues until the work is completed. How many days are needed to complete the work?

$A$ can do a piece of work in $30$ days,$B$ in $50$ days,and $C$ in $40$ days. If $A$ is assisted by $B$ on one day and by $C$ on the next day alternately,the work will be completed in (in days):

$S, T$ and $U$ can complete a work in $40, 48$ and $60$ days respectively. They received $Rs. 10800$ to complete the work. They begin the work together,but $T$ left $2$ days before the completion of the work and $U$ left $5$ days before the completion of the work. $S$ completed the remaining work alone. What is the share of $S$ (in $Rs.$) from the total money?

$A$ 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 $\text{days}$)

$5$ men can prepare $10$ toys in $6$ days working $6$ hours a day. Then,in how many days can $12$ men prepare $16$ toys working $8$ hours a day? (in days)