The difference between a fraction and its reciprocal is `9/11` If the cubes of both the fraction and its reciprocal are considered, what will be the difference between them?

Options

A. `3996/1331`
B. `1331/2538`
C. `729/1331`
D. `2538/1331`

Answer

To solve the problem step by step, we will start with the information given and apply algebraic identities to find the required difference between the cubes of the fraction and its reciprocal.

### Step-by-Step Solution:

1. **Define the Fraction**:
Let the fraction be \( x \). Then, its reciprocal is \( \frac{1}{x} \).

2. **Set Up the Equation**:
According to the problem, the difference between the fraction and its reciprocal is given by:
\[ x - \frac{1}{x} = \frac{9}{11} \]

3. **Cube Both Sides**:
We need to find the difference between the cubes of the fraction and its reciprocal, which is \( x^3 - \frac{1}{x^3} \). We can use the identity:
\[ (a - b)^3 = a^3 - b^3 - 3ab(a - b) \]
Here, let \( a = x \) and \( b = \frac{1}{x} \). So, we have:
\[ x - \frac{1}{x} = \frac{9}{11} \]
Therefore,
\[ (x - \frac{1}{x})^3 = x^3 - \frac{1}{x^3} - 3 \cdot x \cdot \frac{1}{x} \cdot (x - \frac{1}{x}) \]
Simplifying gives:
\[ (x - \frac{1}{x})^3 = x^3 - \frac{1}{x^3} - 3(x - \frac{1}{x}) \]

4. **Substitute the Known Value**:
Substitute \( x - \frac{1}{x} = \frac{9}{11} \) into the equation:
\[ \left(\frac{9}{11}\right)^3 = x^3 - \frac{1}{x^3} - 3 \cdot \frac{9}{11} \]

5. **Calculate \( \left(\frac{9}{11}\right)^3 \)**:
\[ \left(\frac{9}{11}\right)^3 = \frac{729}{1331} \]

6. **Substitute and Rearrange**:
Now we have:
\[ \frac{729}{1331} = x^3 - \frac{1}{x^3} - \frac{27}{11} \]
Convert \( \frac{27}{11} \) to have a denominator of \( 1331 \):
\[ \frac{27}{11} = \frac{27 \times 121}{11 \times 121} = \frac{3267}{1331} \]

7. **Combine the Fractions**:
Rearranging gives:
\[ x^3 - \frac{1}{x^3} = \frac{729}{1331} + \frac{3267}{1331} \]
This simplifies to:
\[ x^3 - \frac{1}{x^3} = \frac{729 + 3267}{1331} = \frac{3996}{1331} \]

8. **Final Answer**:
Thus, the difference between the cubes of the fraction and its reciprocal is:
\[ \frac{3996}{1331} \]