Fun With Numbers: Pupil-teacher Ratio in English State Schools

There are about 7.5 million pupils in state schools in England.
"There are 922 thousand full-time equivalent people working in state-funded schools, this includes 451 thousand full-time equivalent teachers."
Further, "over 1.3 million people work in state-funded schools", which means, for example, that about 40% of the 1.3 million work full-time and the other 60% do half a working week (1,300 x 70% = 910).
So…
a) Slightly fewer than half the people on the schools payrolls are actually teachers. And there is an unknown number of people working in "education" in the wider sense (i.e. all the quangos) who don't actually "work in a state-funded school".
b) The pupil-teacher ratio of 16.7 doesn't seem too terrible to me. In other words, if a full-time teacher spend two-thirds of the day teaching, class sizes would be about 25, which is no different to most private schools.
c) For a given number of employees/expenditure, if they could could get the proportion of teachers up from 50% to 75% and reduce the number of pen-pushers down to 25% (which is normal for private schools), then we'd have 692 thousand full-time equivalent teachers and a teacher-pupil ratio of 10.8, which I think is pretty good (i.e. low) by any sort of standards. If a full-time teacher then only spends half the day teaching, that gives us a class size of just under 22 and reasonably well-rested teachers.