Actually, the formulation is not accurate.
It does not allow to distinct if these 23 are among 85 and 51, or they are 23 others students.


So, there are two alternative formulations:


1. "At a high school, 85 students are enrolled in fine arts, 51 are enrolled in history. Among them, 23 are enrolled in both subjects. 
    How many students are taking fine arts only or history only?"


2. "At a high school, 85 students are enrolled in fine arts, 51 are enrolled in history. Beside it, 23 other students are enrolled in both subjects. 
    How many students are taking fine arts or history?"


Any of these formulations make sense and is unambiguous. 

Regarding your original formulation, the answer is just contained in the condition. Do you agree?


For the formulation #1 the solution is:


85 - 23 = 62 students are enrolled in fine arts only.
51 - 23 = 28 students are enrolled in history only.


For the formulation #2 the solution is:


85 + 23 = 108 students are enrolled in fine arts.
51 + 23 = 74 students are enrolled in history only.


In Math, including the school Math problems, every word counts and every word is significant. 
Those who read old textbooks know it very well. 
People of younger generation are more careless about correct wording. 
Read and use right sources and improve your skills. It will help you in your life.