If the average score of the x students is 81, their total score is 81x. After adding the additional student's score of 99, their total score is 81x + 99.
The average score, after the additional student takes the exam, is 83(x + 1) -- the new average is 83, and the number of students is one greater. These two expressions are equivalent; one contains the average of all but one student and adds the final test score; the other contains the average of all students. Set them equal to each other:
81x + 99 = 83(x+1)
81x + 99 = 83x + 83
16 = 2x
x = 8, choice (B).