lisa, they haven't said no 
I don't think there is a set ratio - it will depend on a lot, including how much they 'want' your dd to attend (not just on a SN front, but in terms of how gifted etc)
I went to a private school, on nearly 100% scholarship/bursary. the official line was a 2/3 scholarship, but there was always extra money to be found to enable me to stay there - extra essay prizes, or music tuition fees, or 3rd/4th/5th year 'exhibition' (extra bursary). this was secondary, but I had the same (although not as extreme - 'only' 60% of fees at prep too)
I would go and look around, without dd initially (I am sure school would understand your reasoning on that - and you can always go again with her if it works out), and have a good talk through with the school.
do be aware about uniforms/extras. there will probably be a reasonable second hand shop, but anything new is likely to be pricey. trips and extras - well, again, if the trip was fairly essential for a subject, money was found for me to go (we were really poor - think homeless/B&B/council temporary housing).
I did not suffer particularly from being 'the poor one' either. yes, there were a few snidey comments, but then those girls would have been bitchy about something else - it was just their character, unfortunately.