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

Introduction / Context:
This problem is another example of an equation involving a fraction and its reciprocal. The fraction plus three times its reciprocal gives a specific value. Such expressions naturally lead to quadratic equations when we remove the denominator. Solving these equations efficiently is a key algebra skill for competitive exams.

Given Data / Assumptions:

• Let the fraction be x (x ≠ 0).
• Its reciprocal is 1/x.
• We are given: x + 3 * (1/x) = 31/6.
• We must find x from the given answer choices.

Concept / Approach:
We express the verbal condition as an algebraic equation. Then we multiply through by x to clear the denominator, giving a quadratic equation in x. Using the quadratic formula or factorization, we obtain the roots. After that, we check which root appears among the options and suits the meaning of “fraction” in the context of the question.

Step-by-Step Solution:
Step 1: Start with the equation: x + 3/x = 31/6. Step 2: Multiply both sides by 6x to remove the denominator: 6x * x + 6x * (3/x) = 6x * 31/6. Step 3: Simplify: 6x^2 + 18 = 31x. Step 4: Rearrange into standard quadratic form: 6x^2 - 31x + 18 = 0. Step 5: Use the quadratic formula: x = [31 ± sqrt(31^2 - 4 * 6 * 18)] / (2 * 6). Step 6: Compute the discriminant: 31^2 = 961; 4 * 6 * 18 = 432; so discriminant = 961 - 432 = 529. Step 7: The square root of 529 is 23, so x = (31 ± 23) / 12. Step 8: The roots are x = (31 + 23) / 12 = 54/12 = 9/2 and x = (31 - 23) / 12 = 8/12 = 2/3.

Verification / Alternative Check:
Test x = 9/2 in the original expression. The reciprocal is 2/9. So compute x + 3/x = 9/2 + 3 * (2/9). First, 3 * (2/9) = 6/9 = 2/3. Now, 9/2 + 2/3 = (27/6) + (4/6) = 31/6, which matches the given value. The other root, 2/3, also satisfies the equation but is not listed among the options. Among the choices, 9/2 is the only correct match.

Why Other Options Are Wrong:
2/9, 9/4 and 4/9 do not satisfy the equation when substituted. For example, with x = 9/4, the sum x + 3/x leads to a different value, not 31/6. Therefore, those options must be rejected.

Common Pitfalls:
One common error is failing to multiply through by the correct factor (here, 6x) to clear all denominators, which leads to an incorrect quadratic. Another pitfall is forgetting that a quadratic can have two valid roots; you still need to check the answer choices rather than assuming only one solution exists.

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