The sum of a fraction and seven times its reciprocal is 11/2. What is the value of the fraction?

Introduction / Context:
This question involves an unknown fraction and its reciprocal in a linear relation that leads to a quadratic equation. It tests algebraic manipulation and solving quadratic equations that arise from reciprocal relationships.

Given Data / Assumptions:
- Let the fraction be x, where x is non zero.
- The condition is x + 7 * (1 / x) = 11 / 2.
- We must find the value of x that satisfies this equation and matches the style of an ordinary fraction.

Concept / Approach:
We convert the verbal statement into an equation and clear the denominator by multiplying both sides by x. This creates a quadratic equation in x. Solving the quadratic gives possible values for x, and then we interpret which of those values matches the intended answer in the context of the question and the options.

Step-by-Step Solution:
Start with x + 7 * (1 / x) = 11 / 2. Multiply both sides by x: x^2 + 7 = (11 / 2) * x. Multiply both sides by 2 to eliminate the fraction: 2x^2 + 14 = 11x. Rearrange into standard quadratic form: 2x^2 − 11x + 14 = 0. Factorise: 2x^2 − 11x + 14 = (2x − 7)(x − 2) = 0, so x = 7 / 2 or x = 2.

Verification / Alternative Check:
Check x = 7 / 2: the reciprocal is 2 / 7. Then x + 7 * (1 / x) becomes 7 / 2 + 7 * (2 / 7) = 7 / 2 + 2 = 7 / 2 + 4 / 2 = 11 / 2, which matches the given value. Check x = 2: the reciprocal is 1 / 2. Then 2 + 7 * (1 / 2) = 2 + 7 / 2 = 4 / 2 + 7 / 2 = 11 / 2, so it also satisfies the equation. Both values are mathematically correct, but among the answer options, only 7 / 2 is offered and it is explicitly written as a proper fraction form used in the statement, so this is chosen as the correct option.

Why Other Options Are Wrong:
2 / 7 would give 2 / 7 + 7 * (7 / 2), which is much larger than 11 / 2.
3 / 4 leads to a different sum when combined with seven times its reciprocal and does not produce 11 / 2.
4 / 3 and 5 / 2 similarly do not satisfy the original equation when substituted.

Common Pitfalls:
Students may forget to multiply across correctly when clearing denominators and thus derive an incorrect quadratic equation. Another issue is assuming there will be only one solution and ignoring other roots. In multiple choice settings it is critical to solve accurately and then see which of the valid roots is present among the options.

Final Answer:
The fraction whose sum with seven times its reciprocal equals 11 / 2 is 7/2.