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

Introduction / Context:
This question is essentially the same algebraic relationship as a previous one, again using a fraction and three times its reciprocal. It checks whether you can consistently set up and solve this type of equation.

Given Data / Assumptions:

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

Concept / Approach:
The approach is to form a quadratic equation by clearing the denominator. Solving the quadratic yields two algebraic solutions, and then we check which solution appears among the options.

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, so 9/2 satisfies the equation. The other root x = 2/3 also satisfies it, but only 9/2 appears as an answer choice.

Why Other Options Are Wrong:
Options 2/9, 5/4 and 4/5 do not satisfy the equation when substituted for x. Their sums x + 3/x differ from 31/6, so they are incorrect.

Common Pitfalls:
Common errors include arithmetic mistakes when computing the discriminant or mishandling the quadratic formula. Some students forget to check which root is actually present among the options before deciding the final answer.

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