I agree with playfelt- if other families are being charged 40 then you should charge 40 for this family.

If it's really bothering you that they may think that they won...they haven't as your looking to replace them in a few months. Your bitter with them, and rightfully so! That day when you give them their notice, you will get the peace of mind knowing that you no longer have to work with these people who have been taking advantage.