I just had someone inquire about half days, and I turned them down as I am not willing to do them right now. It was a one year old, and they only needed afternoons. That seemed like it may be a pretty difficult transition period, only a few afternoons a week.
Having said that, I think $20 per day is pretty reasonable, based on your full day rate, and the fact that you are not providing lunch. It has to be worthwhile!!