$A$ and $B$ can do a piece of work in $8$ days,$B$ and $C$ can do the same work in $12$ days,and $A, B,$ and $C$ together complete it in $6$ days. The number of days required to finish the work by $A$ and $C$ together is:

Ramesh takes twice as much time as Mahesh and thrice as much time as Suresh to complete a job. If working together,they can complete the job in $4 \, \text{days}$,then the time taken by each of them separately to complete the work is (in $\text{days}$)

$A$ piece of work can be completed by $10$ men and $6$ women in $18$ days. Men work $9$ hours per day while women work $7.5$ hours per day. The per hour efficiency of a woman is $\frac{2}{3}$ of a man's efficiency. In how many days will $10$ men and $9$ women complete the work?

$10$ women can complete a work in $7$ days and $10$ children take $14$ days to complete the work. How many days will $5$ women and $10$ children take to complete the work?

$10$ men and $15$ women together can complete a work in $6$ days. It takes $100$ days for one man alone to complete the same work. How many days will be required for one woman alone to complete the same work?

$A$ and $B$ together can do a piece of work in $30$ days. $B$ and $C$ together can do it in $20$ days. $A$ starts the work and works on it for $5$ days,then $B$ takes over and works for $15$ days. Finally,$C$ finishes the work in $18$ days. The number of days in which $C$ alone can do the work is (in days):