Ok I am no math whiz but if you are saying it takes 48 days to cover the entire lake with the patch then wouldn't it take half that many days to cover half of the lake. I am going to say it would take 24 days to cover half the lake.
Waldorff is correct. The growth of the patch of lily pads is exponential. If every day the patch doubles in size, that means that every day the patch is half the size it will be next the day. If on day 48 it takes up the whole lake, on day 47 it will take up half, and on day 46 it will take up a quarter, etc.