The difference between a positive proper fraction and its reciprocal is 9 / 20. What is the value of the fraction?

Introduction / Context:
This problem involves fractions and their reciprocals. We are told that the difference between a positive proper fraction and its reciprocal is a specific rational number. The goal is to set up and solve an equation involving the fraction and its reciprocal. Such questions test algebraic manipulation skills and understanding of proper fractions.

Given Data / Assumptions:
Let the fraction be x.The fraction is positive and proper, so 0 < x < 1.The difference between the reciprocal and the fraction is 9 / 20, that is, 1 / x − x = 9 / 20.

Concept / Approach:
We translate the verbal condition into an algebraic equation: 1 / x − x = 9 / 20. Multiplying through by a common denominator removes fractions and gives a quadratic equation in x. We then solve the quadratic, pick the root that fits the description of a positive proper fraction and reject any invalid root.

Step-by-Step Solution:
Step 1: Start with the equation 1 / x − x = 9 / 20.Step 2: Multiply both sides by 20x to clear denominators: 20x * (1 / x) − 20x * x = 20x * 9 / 20.Step 3: Simplify each term: 20 − 20x^2 = 9x.Step 4: Rearrange to standard quadratic form: −20x^2 − 9x + 20 = 0.Step 5: Multiply through by −1 to make the leading coefficient positive: 20x^2 + 9x − 20 = 0.Step 6: Solve the quadratic equation 20x^2 + 9x − 20 = 0 using the quadratic formula.Step 7: The discriminant is 9^2 − 4 * 20 * (−20) = 81 + 1600 = 1681, whose square root is 41.Step 8: So x = [−9 ± 41] / (40). This gives two roots: x1 = (−9 + 41) / 40 = 32 / 40 = 4 / 5 and x2 = (−9 − 41) / 40 = −50 / 40 = −5 / 4.Step 9: Since x must be a positive proper fraction (between 0 and 1), we discard −5 / 4 and keep x = 4 / 5.

Verification / Alternative check:
Check the solution directly. If x = 4 / 5, then 1 / x = 5 / 4. The difference 1 / x − x = 5 / 4 − 4 / 5. Compute this difference: 5 / 4 − 4 / 5 = (25 − 16) / 20 = 9 / 20, which matches the given condition. So 4 / 5 is correct.

Why Other Options Are Wrong:
For x = 3 / 5, we have 1 / x = 5 / 3 and the difference is 5 / 3 − 3 / 5 = (25 − 9) / 15 = 16 / 15, not 9 / 20. For x = 3 / 10, the reciprocal is 10 / 3 and the difference is 10 / 3 − 3 / 10 = (100 − 9) / 30 = 91 / 30, which is far from 9 / 20. The option 5 / 4 is greater than 1, so it is not a proper fraction and is rejected by the conditions of the problem.

Common Pitfalls:
Some learners reverse the order of subtraction and set up x − 1 / x = 9 / 20, which leads to a different equation. Others make algebraic errors when clearing denominators or solving the quadratic. Careful step-by-step manipulation and checking the nature of the roots help avoid these mistakes.

Final Answer:
The positive proper fraction that satisfies the condition is 4/5.