Look at the u-shaped tube at the top of the transparency. why did the number of water molecules on each side of the membrane change, whereas the number of sugar molecules stayed the same?

Because of the U shaped tube having a selective permeable membrane, the number of the water molecules on the sides changes but the number of the sugar molecules remain same. This particular cell membrane is such that it lets only specific kind of molecules and ions pass through.

The answer to that question is a