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

Introduction / Context:
This aptitude question on fractions and algebra tests your ability to translate a word statement into an equation involving a number and its reciprocal. Such problems are very common in bank exams, SSC, and other competitive tests, and they check whether you can set up and solve a quadratic equation correctly.

Given Data / Assumptions:

• Let the fraction be x, where x is a non-zero rational number.
• Four times the fraction is written as 4x.
• Six times the reciprocal of the fraction is written as 6 * (1/x).
• The sum of these two expressions is given to be 11.

Concept / Approach:
The key idea is to convert the English statement into an algebraic equation. Once we have the equation 4x + 6/x = 11, we clear the denominator and obtain a quadratic equation in x. Solving the quadratic gives two possible values, and then we choose the one that matches the idea of a fraction less than 1 (since typical exam phrasing suggests so) and is present in the options.

Step-by-Step Solution:
Step 1: Let the required fraction be x.Step 2: Four times the fraction is 4x.Step 3: Six times its reciprocal is 6 * (1/x).Step 4: According to the question, 4x + 6 * (1/x) = 11.Step 5: Multiply through by x (x ≠ 0) to clear the denominator: 4x^2 + 6 = 11x.Step 6: Rearrange to standard quadratic form: 4x^2 - 11x + 6 = 0.Step 7: Factorize: 4x^2 - 11x + 6 = (4x - 3)(x - 2) = 0.Step 8: Therefore, x = 3/4 or x = 2.Step 9: Out of these, 3/4 is a proper fraction and appears in the options, so the required fraction is 3/4.

Verification / Alternative check:
Check x = 3/4: Four times the fraction = 4 * 3/4 = 3.Reciprocal of 3/4 is 4/3, so six times reciprocal = 6 * 4/3 = 8.Sum = 3 + 8 = 11, which matches the given condition. Hence 3/4 is correct. The value 2 also satisfies the equation but is not offered as a valid option and does not fit the natural reading of a fraction in this context.

Why Other Options Are Wrong:
Option b (4/3) is greater than 1. Substituting 4/3 will not give sum 11. Option c (4/7) and option d (7/4) similarly fail to satisfy the equation 4x + 6/x = 11 when substituted, so they are incorrect distractors.

Common Pitfalls:
Candidates often make errors by forgetting to multiply every term by x when clearing the denominator, which leads to an incorrect quadratic equation. Another mistake is to assume only one root will arise and to ignore that quadratics usually have two solutions; however, you must then check which solution suits the context and appears in the options. Mis-simplifying fractions such as 6 * 4/3 is another frequent source of small calculation errors.

Final Answer:
The required fraction is 3/4.