A fraction is greater than twice its reciprocal by 7/15. What is the value of this fraction?

Introduction / Context:
This problem again involves a relationship between a fraction and its reciprocal, but this time the fraction is greater than twice its reciprocal by a fixed amount. Such equations are a good test of algebraic translation skills and manipulation of quadratic equations. Mastering this pattern helps in many similar competitive exam problems.

Given Data / Assumptions:

• Let the fraction be x (x ≠ 0).
• Its reciprocal is 1/x.
• We are told that x is greater than 2 * (1/x) by 7/15.
• So the equation is x - 2/x = 7/15.

Concept / Approach:
The given verbal condition is transformed into an algebraic equation involving x and 1/x. To solve it, we remove the denominator by multiplying through by x, giving a quadratic equation. Quadratic equations are then solved using the quadratic formula or factorization. Finally, we inspect the obtained roots and select the one that appears as an option and fits the context.

Step-by-Step Solution:
Step 1: Start from the equation: x - 2/x = 7/15. Step 2: Multiply both sides by x to eliminate the denominator: x * x - 2 = (7/15)x. Step 3: This gives x^2 - 2 = (7/15)x. Step 4: Multiply through by 15 to clear the fraction: 15x^2 - 30 = 7x. Step 5: Rearrange to standard form: 15x^2 - 7x - 30 = 0. Step 6: Apply the quadratic formula: x = [7 ± sqrt(7^2 - 4 * 15 * (-30))] / (2 * 15). Step 7: Compute the discriminant: 7^2 = 49; 4 * 15 * 30 = 1800; so discriminant = 49 + 1800 = 1849. Step 8: The square root of 1849 is 43, so x = (7 ± 43) / 30. Step 9: The roots are x = (7 + 43) / 30 = 50/30 = 5/3 and x = (7 - 43) / 30 = -36/30 = -6/5.

Verification / Alternative Check:
Test x = 5/3 in the original condition: x - 2/x = 5/3 - 2 * (3/5). Compute 2 * (3/5) = 6/5. Now 5/3 - 6/5 = (25/15) - (18/15) = 7/15, which matches the given difference. The other root, -6/5, also solves the equation but is not among the answer options and is negative, whereas the provided options show positive fractions. Therefore, 5/3 is the valid choice.

Why Other Options Are Wrong:
If x = 3/5, then x - 2/x will be negative, not 7/15.
If x = 3/4 or x = 4/3, substituting into x - 2/x will not yield 7/15. Their differences from twice their reciprocals produce other values, so they do not satisfy the condition.

Common Pitfalls:
Students often misread the phrase “twice its reciprocal” and write x - 1/(2x) instead of x - 2/x. Another common mistake is sign error when moving terms across the equals sign. Careful algebra and clear writing of each step helps avoid these issues.

Final Answer:
The required fraction is 5/3.