The reciprocal of the sum of the reciprocals of `3//8 and 8//11` is

Options

A. `(97)/(24)`
B. `(24)/(97)`
C. `(88)/(97)`
D. `(97)/(88)`

Answer

To solve the problem, we need to find the reciprocal of the sum of the reciprocals of \( \frac{3}{8} \) and \( \frac{8}{11} \).

### Step-by-step Solution:

1. **Find the reciprocal of \( \frac{3}{8} \)**:
\[ \text{Reciprocal of } \frac{3}{8} = \frac{8}{3} \]

2. **Find the reciprocal of \( \frac{8}{11} \)**:
\[ \text{Reciprocal of } \frac{8}{11} = \frac{11}{8} \]

3. **Sum the two reciprocals**:
\[ \text{Sum} = \frac{8}{3} + \frac{11}{8} \]

4. **Find a common denominator**:
The least common multiple (LCM) of 3 and 8 is 24.
- Convert \( \frac{8}{3} \) to have a denominator of 24:
\[ \frac{8}{3} = \frac{8 \times 8}{3 \times 8} = \frac{64}{24} \]
- Convert \( \frac{11}{8} \) to have a denominator of 24:
\[ \frac{11}{8} = \frac{11 \times 3}{8 \times 3} = \frac{33}{24} \]

5. **Add the two fractions**:
\[ \text{Sum} = \frac{64}{24} + \frac{33}{24} = \frac{64 + 33}{24} = \frac{97}{24} \]

6. **Find the reciprocal of the sum**:
\[ \text{Reciprocal of } \frac{97}{24} = \frac{24}{97} \]

### Final Answer:
The reciprocal of the sum of the reciprocals of \( \frac{3}{8} \) and \( \frac{8}{11} \) is \( \frac{24}{97} \).

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