A proper fraction is greater than its reciprocal by 72/77. If the fraction is denoted by x, satisfying x − 1/x = 72/77, then what is the value of this fraction x?

Introduction / Context:
This question uses algebra to describe a relationship between a fraction and its reciprocal. The problem is typical in quantitative aptitude where you must translate a verbal condition into an equation, solve a quadratic, and then select the fraction that meets the condition.

Given Data / Assumptions:

• Let the fraction be x, with x not equal to zero.
• x − 1 / x = 72 / 77.
• We must find x from the options given.

Concept / Approach:
The equation x − 1 / x = 72 / 77 is nonlinear due to the reciprocal term. Multiplying through by x gives a quadratic equation. We then solve the quadratic using factorisation or the quadratic formula and check which solutions match the options. Because the condition mentions a fraction, we expect rational roots that can be simplified to one of the listed fractions.

Step-by-Step Solution:
Start with x − 1 / x = 72 / 77. Multiply both sides by x (x is non zero): x^2 − 1 = (72 / 77) x. Multiply through by 77 to clear denominators: 77x^2 − 77 = 72x. Rearrange to standard quadratic form: 77x^2 − 72x − 77 = 0. Solve 77x^2 − 72x − 77 = 0, which factorises to (7x − 11)(11x + 7) = 0. Hence, x = 11 / 7 or x = −7 / 11. From the list of options, 11 / 7 is present and is greater than 1, which is consistent with x being greater than its reciprocal.
Verification / Alternative check:
Check x = 11 / 7 directly: 1 / x = 7 / 11. Then x − 1 / x = 11 / 7 − 7 / 11. Using denominator 77, this is (121 − 49) / 77 = 72 / 77, which matches the given condition exactly.

Why Other Options Are Wrong:
Option a (7 / 11) leads to 7 / 11 − 11 / 7 which equals (49 − 121) / 77 = −72 / 77, the negative of the required value. Options c and d do not satisfy the equation when substituted and do not arise as roots of the quadratic. Option e (11 / 4) also fails the equation x − 1 / x = 72 / 77.

Common Pitfalls:
A common mistake is to perform the subtraction x − 1 / x incorrectly or to forget to multiply every term by x, leading to an incorrect quadratic. Also, some learners pick the first root they see without verifying it matches the wording that the fraction is greater than its reciprocal by a positive amount.

Final Answer:
The fraction that satisfies the given condition is 11 / 7.