$X$ and $Y$ together finish a task in $12\text{ days}$. If $Y$ alone can finish the same task in $15\text{ days}$, then how many days will $X$ alone take to finish the task?

Let the total work be the LCM of $12$ and $15$, which is $60\text{ units}$.Efficiency of $X+Y=\frac{60}{12}=5\text{ units/day}$ Efficiency of $Y=\frac{60}{15}=4\text{ units/day}$ Therefore, the efficiency of $X$ is:
$$\text{Efficiency of }X=(\text{Efficiency of }X+Y)-\text{Efficiency of }Y=5-4=1\text{ unit/day}$$
The time taken by $X$ alone to complete the task is:
$$\text{Time}=\frac{\text{Total Work}}{\text{Efficiency of }X}=\frac{60}{1}=60\text{ days}$$