To find the number of members in the middle school band, we can use the information that the middle school band has \( \frac{2}{9} \) as many members as the high school band.
Given that the high school band has 502 members, we can calculate the number of members in the middle school band as follows:
\[ \text{Middle school band members} = \frac{2}{9} \times 502 \]
Now we perform the multiplication:
\[ \frac{2 \times 502}{9} = \frac{1004}{9} \approx 111.56 \]
Rounding to the nearest whole number, we get approximately 112.
So, the answer is 112.