In an examination, the number of those who passed and the number of those who failed were in the ratio 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.

In first case,
Let the number of students passed = 3x and failed = x.
Total candidates appeared = 3x + x = 4x.

In second case,
Total candidates appeared = 4x + 8
Number of passed students = 3x - 6
Number of failed students = (4x + 8) - (3x - 6)
= 4x + 8 - 3x + 6
= x + 14.

According to question, ratio of number of passed to failed student in this case = 2 : 1.

$\therefore \dfrac{3x - 6}{x + 14} = \dfrac{2}{1} \\[0.5em] \Rightarrow (3x - 6) = 2(x + 14) \\[0.5em] \Rightarrow 3x - 6 = 2x + 28 \\[0.5em] \Rightarrow 3x - 2x = 28 + 6 \\[0.5em] \Rightarrow x = 34.$

∴ x = 34, 4x = 136.

Hence, the number of students who appeared were 136.