Hassem and his family ate \( \frac{3}{5} \) of the apple pie, which is cut into 5 equal slices.
To find out how many slices that is, we calculate:
\[ \frac{3}{5} \times 5 = 3 \text{ slices of the apple pie} \]
Since they ate the same fraction of the blueberry pie, they also ate \( \frac{3}{5} \) of that pie. The blueberry pie is cut into 10 equal slices.
Now, we calculate how many slices they ate of the blueberry pie:
\[ \frac{3}{5} \times 10 = 6 \text{ slices of the blueberry pie} \]
Thus, the answer is 6 slices.