Two persons $A$ and $B$ can complete a task in $10\text{ days}$ and $15\text{ days}$, respectively. They work together and $A$ leaves $5\text{ days}$ before the task is completed. The number of days taken to complete the whole task is:

Let the total work be $30\text{ units}$ (LCM of $10$ and $15$).Efficiency of $A=\frac{30}{10}=3\text{ units/day}$ Efficiency of $B=\frac{30}{15}=2\text{ units/day}$ Let the total time taken to complete the task be $t\text{ days}$.Since $A$ leaves $5\text{ days}$ before completion, $A$ works for $(t-5)\text{ days}$, while $B$ works for the entire $t\text{ days}$.The total work done equation is:
$$3(t-5)+2t=30$$
$$3t-15+2t=30$$
$$5t=45⟹t=9\text{ days}$$
Thus, the total number of days taken is $9$.