Professional artist beginning to tutor amateur in proportion (3,4)
I believe the answer is:
pro rata
'in proportion' is the definition.
'professional artist beginning to tutor amateur' is the wordplay.
'professional' becomes 'pro' ('pro' can be a synonym of 'professional').
'artist' becomes 'ra' (Royal Academician, member of Royal Academy of Arts).
'beginning to' suggests taking the first letters.
The first letters of 'tutor amateur' is 'ta'.
'pro'+'ra'+'ta'='PRO RATA'
(Other definitions for pro rata that I've seen before include "Proportional, according to the rate" , "Proportionally" , "maintaining correspondence" , "fairly divided" , "In proportion (Latin)" .)