If a work can be completed by $\mathrm{A}$ in $30$ days and by $\mathrm{B}$ in $60$ days, then the number of days taken by them to complete the work together is:

X and Y together can finish a work in $8$ hours. If X alone can finish in $12$ hours, then Y alone can finish the same in:

A can finish a work in $18$ days and B can do it in half the time taken by A. Then, working together, what part of the same work they can finish in a day?

A can do $\frac{1}{3}$ of the work in $5$ days and B can do $\frac{2}{5}$ of the work in $10$ days. In how many days both A and B together can do the work?