A grouped frequency table of student exam scores is structured as follows:

| Score Interval | Frequency | | :--- | :--- | | $[50, 60)$ | $10$ | | $[60, 70)$ | $15$ | | $[70, 80)$ | $k$ | | $[80, 90)$ | $20$ | | $[90, 100]$ | $10$ |

If the median of this dataset, calculated using linear interpolation, is exactly $76.25$, find the value of $k$.