Not quite free to give the result of the division (8)
I believe the answer is:
quotient
'result of the division' is the definition.
'quotient' can be an answer for 'result' (I have seen 'Result of division ' mean 'quotient' so perhaps 'result' could also mean 'quotient'). I am not certain of the 'of the division' bit.
'not quite free to give' is the wordplay.
'not' becomes 'o' (I've seen this in other clues).
'free' indicates anagramming the letters.
'to give' becomes 'nt' (I am not sure about this - if you are sure you should believe this answer much more).
'o'+'quite'='oquite'
'oquite' is an anagram of 'quotie'.
'quotie'+'nt'='QUOTIENT'
'the' acts as a link.
Can you help me to learn more?
(Other definitions for quotient that I've seen before include "Three, when seven divided by two" , "In maths, it's obtained by dividing one quantity by another" , "Not quite (anag) - ratio" , "Result of division (maths)" , "Result obtained by division" .)