Hassem and his family ate 35% of the apple pie, which is cut into 5 equal slices.
To find out how many slices they ate from the apple pie, we calculate:
\[ \text{Slices of apple pie eaten} = 35% \times 5 = 0.35 \times 5 = 1.75 \text{ slices} \]
Since they ate the same fraction of the blueberry pie, we also need to find out how many slices correspond to the same 35% of the blueberry pie, which is cut into 10 equal slices.
Now, we calculate how many slices correspond to 35% of the blueberry pie:
\[ \text{Slices of blueberry pie eaten} = 35% \times 10 = 0.35 \times 10 = 3.5 \text{ slices} \]
Since they cannot eat half a slice in this context, we look at the closest whole number option, realizing that they would have to eat either 3 or 4 slices.
Considering options provided, if they stick to whole numbers, they would likely have eaten:
3 slices of the blueberry pie.
So the answer is 3 slices.