The difference between a positive fraction and its reciprocal is `6(39)/(160)`. What is the fraction under consideration ?

Options

A. `(15)/(8)`
B. `(16)/(5)`
C. `(32)/(5)`
D. `(13)/(8)`

Answer

To solve the problem, we need to find a positive fraction \( \frac{x}{y} \) such that the difference between the fraction and its reciprocal equals \( 6 \frac{39}{160} \).

### Step-by-step Solution:

1. **Understand the problem**: We are given that the difference between a positive fraction \( \frac{x}{y} \) and its reciprocal \( \frac{y}{x} \) is \( 6 \frac{39}{160} \).

2. **Convert the mixed fraction to an improper fraction**:
\[ 6 \frac{39}{160} = 6 + \frac{39}{160} = \frac{6 \times 160 + 39}{160} = \frac{960 + 39}{160} = \frac{999}{160} \]

3. **Set up the equation**: The difference between the fraction and its reciprocal can be expressed as:
\[ \frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy} \]
Therefore, we have:
\[ \frac{x^2 - y^2}{xy} = \frac{999}{160} \]

4. **Cross-multiply to eliminate the fractions**:
\[ 160(x^2 - y^2) = 999xy \]

5. **Rearrange the equation**:
\[ 160x^2 - 160y^2 = 999xy \]

6. **Factor the left-hand side**:
\[ 160(x^2 - y^2) = 999xy \implies 160(x - y)(x + y) = 999xy \]

7. **Substituting possible values**: We need to check the given options to find a fraction \( \frac{x}{y} \) that satisfies the equation. The options are:
- Option 1: \( \frac{15}{8} \)
- Option 2: \( \frac{16}{5} \)
- Option 3: \( \frac{32}{5} \)
- Option 4: \( \frac{13}{8} \)

8. **Testing Option 3: \( \frac{32}{5} \)**:
- Here, \( x = 32 \) and \( y = 5 \).
- Calculate \( x^2 - y^2 \):
\[ x^2 = 32^2 = 1024, \quad y^2 = 5^2 = 25 \]
\[ x^2 - y^2 = 1024 - 25 = 999 \]
- Calculate \( xy \):
\[ xy = 32 \times 5 = 160 \]
- Substitute back into the equation:
\[ 160(x^2 - y^2) = 160 \times 999 = 999 \times 160 \]
- This confirms that the equation holds true.

9. **Conclusion**: The fraction under consideration is \( \frac{32}{5} \).