In analysis of companies, the ratios have to be strictly in the right order, because so many derivative ratios are computed from them.
In any case, as a general principle the X/Y ratio has to be the inverse of Y/X ratio if these terms are to be understood.
If the student/staff ratio is 12:1, the staff/student ratio must be 1:12.
There might be specific industries in which the same ratio can be expressed in either direction and be the same value, but I don't know of any.