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We also arrange with each Head of School to give us time blocked out in their diaries in advance each month. We are then at liberty to fill that with appointments for them to interview children. If they do not get used they get the time back (which they love!). It does help to then be able to say to families who are making applications that we have some set times for interviewing available on XX dates & times and you can book them in – for instance you could say to families we interview students for Year 7 on every 2nd Wednesday, and Junior School on Tuesdays/Thursdays and so on.
That may help you with still being reactive to processing a new application quickly, but not having to try and arrange interviews in already very busy diaries.