Can you give me the formula to prorate a lessee rent?

Divide the number of days in the month into the amount they pay. This will tell you what you charge them to live there each day.

But as noted, normally rent is not prorated. If they give notice, it is from basic first of the month to first of the month, not date of notice to that day a month latter.

Also on a lease, since they can't normally break the lease, there is seldom a reason to prorate on a lease. Even on month to month rental, they are to give notice from payment due date to payment due date.