A can do three-seventh of a work in $15$ days and B can do two-fifth of the same work in $14$ days. If A and B work together, then how long (in days) will they take to complete the work?

Find the total time taken by A and B individually to complete the entire work ($1$ unit of work).A completes $\frac{3}{7}$ of the work in $15$ days.
$$\text{Total time taken by A}=15\times \frac{7}{3}=35\text{ days}$$
B completes $\frac{2}{5}$ of the work in $14$ days.
$$\text{Total time taken by B}=14\times \frac{5}{2}=35\text{ days}$$
Work done by A in $1$ day = $\frac{1}{35}$ Work done by B in $1$ day = $\frac{1}{35}$ Combined $1$ -day work of A and B working together:
$$\text{Combined 1-day work}=\frac{1}{35}+\frac{1}{35}=\frac{2}{35}$$
Total days required by both together to complete the work:
$$\text{Total Days}=\frac{35}{2}=17\frac{1}{2}\text{ days}$$