The sum of a fraction and its reciprocal is 113/56. Find the fraction.

Options

A. a. `7//8`
B. b. `5//8`
C. c. `8//9`
D. d. `3//7`

Answer

To solve the problem of finding a fraction such that the sum of the fraction and its reciprocal equals \( \frac{113}{56} \), we can follow these steps:

### Step-by-Step Solution:

1. **Let the fraction be represented as \( x \)**.
\[ \text{Let } x = \text{the fraction} \]

2. **Write the reciprocal of the fraction**.
\[ \text{The reciprocal of } x \text{ is } \frac{1}{x} \]

3. **Set up the equation based on the problem statement**.
\[ x + \frac{1}{x} = \frac{113}{56} \]

4. **Multiply both sides of the equation by \( x \) to eliminate the fraction**.
\[ x^2 + 1 = \frac{113}{56}x \]

5. **Rearrange the equation to form a standard quadratic equation**.
\[ x^2 - \frac{113}{56}x + 1 = 0 \]

6. **Multiply through by 56 to eliminate the denominator**.
\[ 56x^2 - 113x + 56 = 0 \]

7. **Use the quadratic formula to solve for \( x \)**.
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 56 \), \( b = -113 \), and \( c = 56 \).

8. **Calculate the discriminant**.
\[ b^2 - 4ac = (-113)^2 - 4 \cdot 56 \cdot 56 \]
\[ = 12769 - 12544 = 225 \]

9. **Substitute back into the quadratic formula**.
\[ x = \frac{113 \pm \sqrt{225}}{2 \cdot 56} \]
\[ = \frac{113 \pm 15}{112} \]

10. **Calculate the two possible values of \( x \)**.
- First value:
\[ x = \frac{113 + 15}{112} = \frac{128}{112} = \frac{8}{7} \]
- Second value:
\[ x = \frac{113 - 15}{112} = \frac{98}{112} = \frac{7}{8} \]

11. **Determine the fraction**.
The possible fractions are \( \frac{8}{7} \) and \( \frac{7}{8} \).

### Final Answer:
The fraction can be either \( \frac{8}{7} \) or \( \frac{7}{8} \).