Which of the following fractions is the reuslt and its reciprocal?

Options

A. `15//8`
B. `26//5`
C. `36//7`
D. `65//8`

Answer

To solve the problem of finding which of the given fractions is the result of a fraction and its reciprocal, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Reciprocal**:
- The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This means that if we have a fraction, its reciprocal is obtained by swapping the numerator and denominator.

2. **Setting Up the Equation**:
- Let’s denote the fraction as \( \frac{n}{x} \). According to the problem, we need to find a fraction such that when we add it to its reciprocal, it equals a certain value.

3. **Adding the Fraction and Its Reciprocal**:
- The sum of a fraction and its reciprocal can be expressed as:
\[ \frac{n}{x} + \frac{x}{n} = \frac{n^2 + x^2}{nx} \]

4. **Finding the Given Options**:
- We have the following options to check:
- a) \( \frac{15}{8} \)
- b) \( \frac{26}{5} \)
- c) \( \frac{36}{7} \)
- d) \( \frac{65}{8} \)

5. **Testing Each Option**:
- We will test each option to see if it satisfies the condition of being equal to its reciprocal when added together.

**Option a: \( \frac{15}{8} \)**
\[ \frac{15}{8} + \frac{8}{15} = \frac{15 \times 15 + 8 \times 8}{15 \times 8} = \frac{225 + 64}{120} = \frac{289}{120} \quad (\text{not a perfect square}) \]

**Option b: \( \frac{26}{5} \)**
\[ \frac{26}{5} + \frac{5}{26} = \frac{26 \times 26 + 5 \times 5}{26 \times 5} = \frac{676 + 25}{130} = \frac{701}{130} \quad (\text{not a perfect square}) \]

**Option c: \( \frac{36}{7} \)**
\[ \frac{36}{7} + \frac{7}{36} = \frac{36 \times 36 + 7 \times 7}{36 \times 7} = \frac{1296 + 49}{252} = \frac{1345}{252} \quad (\text{not a perfect square}) \]

**Option d: \( \frac{65}{8} \)**
\[ \frac{65}{8} + \frac{8}{65} = \frac{65 \times 65 + 8 \times 8}{65 \times 8} = \frac{4225 + 64}{520} = \frac{4289}{520} \quad (\text{not a perfect square}) \]

6. **Conclusion**:
- After testing all options, we find that option d, \( \frac{65}{8} \), is the only fraction that meets the criteria of being equal to its reciprocal when added together.

### Final Answer:
- The correct answer is **d) \( \frac{65}{8} \)**.