The sum of four times a fraction and six times its reciprocal is 11. What is the value of this fraction?

Introduction / Context:
Yet again, we are given a relationship between a fraction and its reciprocal, this time involving four times the fraction and six times its reciprocal. This type of problem is standard in algebra for competitive exams and reinforces the technique of forming and solving quadratic equations.

Given Data / Assumptions:

• Let the fraction be x, with x ≠ 0.
• Its reciprocal is 1/x.
• The equation is 4x + 6 * (1/x) = 11.
• We must find x among the given options.

Concept / Approach:
We translate the verbal condition into algebra, multiply to clear the denominator, and solve the resulting quadratic equation. After finding the roots, we choose the root that appears in the given choices and fits the idea of a reasonable fraction in this context.

Step-by-Step Solution:
Step 1: Begin with the equation 4x + 6/x = 11. Step 2: Multiply both sides by x to eliminate the denominator: x * 4x + x * (6/x) = 11x. Step 3: Simplify: 4x^2 + 6 = 11x. Step 4: Rearrange to standard form: 4x^2 - 11x + 6 = 0. Step 5: Try to factor the quadratic: we look for two numbers whose product is 4 * 6 = 24 and whose sum is -11. Step 6: The pair (-3) and (-8) multiplies to 24 and adds to -11. So factor as (4x - 3)(x - 2) = 0. Step 7: Set each factor equal to zero: 4x - 3 = 0 or x - 2 = 0. Step 8: Solve: from 4x - 3 = 0, we get x = 3/4; from x - 2 = 0, we get x = 2.

Verification / Alternative Check:
Check x = 3/4 in the original equation. The reciprocal is 4/3. Compute 4x + 6/x = 4 * (3/4) + 6 * (4/3). First, 4 * (3/4) = 3. Next, 6 * (4/3) = 24/3 = 8. Sum: 3 + 8 = 11, which matches the condition. The other root x = 2 also satisfies the equation but is not among the options, and the problem clearly focuses on fractional forms. Therefore, 3/4 is the correct answer from the given choices.

Why Other Options Are Wrong:
4/3, 4/7 and 7/4 produce different values when substituted into 4x + 6/x. None of them give exactly 11, so they do not satisfy the equation and must be discarded.

Common Pitfalls:
Some students attempt to cross-multiply incorrectly or make errors in factorization. Another mistake is to forget that a quadratic can yield two roots and then failing to verify which root matches the answer set. For accuracy, always test each candidate solution in the original equation.

Final Answer:
The required fraction is 3/4.