If a fraction is multiple by itself and then divided by the reciprocal of the same fraction, the result is `18 ( 26)/( 27)` . Find the fraction.

Options

A. ` ( 8 )/( 27)`
B. `1(1)/( 3)`
C. `2( 2)/( 3)`
D. `3( 2)/( 3)`

Answer

To solve the problem, we need to find a fraction such that when it is multiplied by itself and then divided by its reciprocal, the result is \( 18 \frac{26}{27} \).

### Step-by-step Solution:

1. **Let the fraction be represented as \( \frac{x}{y} \)**.

2. **Multiply the fraction by itself**:
\[ \left(\frac{x}{y}\right) \times \left(\frac{x}{y}\right) = \frac{x^2}{y^2} \]

3. **Find the reciprocal of the fraction**:
\[ \text{Reciprocal of } \frac{x}{y} = \frac{y}{x} \]

4. **Divide \( \frac{x^2}{y^2} \) by its reciprocal \( \frac{y}{x} \)**:
\[ \frac{x^2}{y^2} \div \frac{y}{x} = \frac{x^2}{y^2} \times \frac{x}{y} = \frac{x^3}{y^3} \]

5. **Set the equation equal to the given result**:
\[ \frac{x^3}{y^3} = 18 \frac{26}{27} \]

6. **Convert the mixed number to an improper fraction**:
\[ 18 \frac{26}{27} = \frac{18 \times 27 + 26}{27} = \frac{486 + 26}{27} = \frac{512}{27} \]

7. **Now we have the equation**:
\[ \frac{x^3}{y^3} = \frac{512}{27} \]

8. **Cross-multiply to eliminate the fractions**:
\[ x^3 \cdot 27 = y^3 \cdot 512 \]

9. **Rearranging gives us**:
\[ \frac{x^3}{y^3} = \frac{512}{27} \]

10. **Taking the cube root of both sides**:
\[ \frac{x}{y} = \frac{8}{3} \]

11. **Thus, the fraction is**:
\[ \frac{x}{y} = \frac{8}{3} \]

12. **Expressing it as a mixed number**:
\[ 8 \div 3 = 2 \quad \text{remainder } 2 \quad \Rightarrow \quad 2 \frac{2}{3} \]

### Final Answer:
The required fraction is \( 2 \frac{2}{3} \).