If the difference between the reciprocal of a positive proper frac tion and the fraction itself be `(9)/(20)` then the fraction is

Options

A. `(3)/(5)`
B. `(3)/(10)`
C. `(4)/(5)`
D. `(5)/(4)`

Answer

To solve the problem, we need to find a positive proper fraction \( \frac{x}{y} \) such that the difference between its reciprocal \( \frac{y}{x} \) and the fraction itself \( \frac{x}{y} \) is \( \frac{9}{20} \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We start by expressing the condition given in the problem mathematically:
\[ \frac{y}{x} - \frac{x}{y} = \frac{9}{20} \]

2. **Find a Common Denominator**:
The common denominator for the left-hand side is \( xy \):
\[ \frac{y^2 - x^2}{xy} = \frac{9}{20} \]

3. **Cross Multiply**:
To eliminate the fractions, we cross multiply:
\[ 20(y^2 - x^2) = 9xy \]

4. **Rearrange the Equation**:
Rearranging gives us:
\[ 20y^2 - 9xy - 20x^2 = 0 \]

5. **Recognize as a Quadratic Equation**:
This is a quadratic equation in terms of \( y \):
\[ 20y^2 - 9xy - 20x^2 = 0 \]

6. **Use the Quadratic Formula**:
We can apply the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 20 \), \( b = -9x \), and \( c = -20x^2 \):
\[ y = \frac{9x \pm \sqrt{(-9x)^2 - 4 \cdot 20 \cdot (-20x^2)}}{2 \cdot 20} \]

7. **Calculate the Discriminant**:
Calculate the discriminant:
\[ (-9x)^2 - 4 \cdot 20 \cdot (-20x^2) = 81x^2 + 1600x^2 = 1681x^2 \]

8. **Substitute Back into the Formula**:
Now substitute back into the quadratic formula:
\[ y = \frac{9x \pm 41x}{40} \]

9. **Find Possible Values for \( y \)**:
This gives us two possible solutions for \( y \):
\[ y = \frac{50x}{40} = \frac{5x}{4} \quad \text{or} \quad y = \frac{-32x}{40} = -\frac{8x}{10} \]
Since \( y \) must be positive, we take \( y = \frac{5x}{4} \).

10. **Determine the Fraction**:
Since \( \frac{x}{y} \) is a proper fraction, we substitute \( y \) back:
\[ \frac{x}{y} = \frac{x}{\frac{5x}{4}} = \frac{4}{5} \]

### Final Answer:
The fraction is \( \frac{4}{5} \).