Driving at a constant speed, Emily usually takes 240 minutes to drive from her home to her grandmother's house. One day Emily begins the drive at her usual speed, but after driving $\frac{1}{2}$ of the way, she encounters a detour and reduces her speed by 10 miles per hour. This time the trip takes her a total of 300 minutes. How many miles is the drive from Emily's home to her grandmother's house?