I mean, if you tack on a hospital to a university, the correct denominator to compare against is "patients served," not "students educated," at least for the portion of the headcount you're sticking in the numerator.
Hospitals attached to universities aren't in general "tacked on" but are a part of the educational environment. They exist not only to serve patients but to educate students.
No, of course, but is the primary focus of the bulk of the staff educational or patient care? Seems disingenuous to pretend it's the former just to make a point.