The sum of a fraction and three times its reciprocal is 31/6. What is the value of the fraction?

Introduction / Context:
This problem involves a fraction and its reciprocal in an algebraic relationship. Such questions are popular in aptitude tests as they require forming and solving quadratic equations based on reciprocal relationships.

Given Data / Assumptions:

• Let the fraction be x, with x not equal to 0.
• Its reciprocal is 1/x.
• We are told that x + 3 * (1/x) = 31/6.
• We must find x from the given options.

Concept / Approach:
We convert the word statement into an equation, clear denominators, and solve the resulting quadratic equation. Then we match the solutions with the given options and select the appropriate fraction.

Step-by-Step Solution:

Verification / Alternative check:
Check x = 9/2: 9/2 + 3 * (2/9) = 9/2 + 6/9 = 9/2 + 2/3 = 27/6 + 4/6 = 31/6. This satisfies the equation. The other root x = 2/3 also satisfies the relation, but 2/3 is not among the options, whereas 9/2 is. Therefore 9/2 is the correct answer for this multiple choice question.

Why Other Options Are Wrong:
Substituting 2/9, 9/4 or 4/9 into x + 3/x does not yield 31/6. They produce different sums, so those options cannot be correct.

Common Pitfalls:
Common mistakes include incorrect multiplication when clearing denominators, miscomputing the discriminant of the quadratic, or assuming there will be only one algebraic solution. Always check which allowed root appears in the answer choices.

Final Answer:
The fraction that satisfies the given condition and matches the options is 9/2.