To find the number of members in the middle school band, we need to calculate \( \frac{2}{9} \) of the number of members in the high school band.
The high school band has 502 members.
Now, we calculate \( \frac{2}{9} \) of 502:
\[ \frac{2}{9} \times 502 = \frac{1004}{9} \approx 111.56 \]
Rounding this to the nearest whole number, we find that there are approximately 112 members in the middle school band.
Thus, the answer is 112.