If a fraction is divided by its reciprocal and then mltiplied by itself then the fraction becomes `(512)/125`. The difference between the actual fraction and its reciprocal fraction is :

Options

A. `29/30`
B. `39/40`
C. data insufficient
D. none of these

Answer

To solve the problem step by step, let's denote the fraction as \( \frac{x}{y} \).

### Step 1: Set up the equation
When the fraction \( \frac{x}{y} \) is divided by its reciprocal \( \frac{y}{x} \), we can write:
\[ \frac{x}{y} \div \frac{y}{x} = \frac{x}{y} \times \frac{x}{y} = \frac{x^2}{y^2} \]

### Step 2: Multiply by the fraction itself
Now, we multiply this result by the fraction \( \frac{x}{y} \):
\[ \frac{x^2}{y^2} \times \frac{x}{y} = \frac{x^3}{y^3} \]

### Step 3: Set the equation equal to \( \frac{512}{125} \)
According to the problem, this result equals \( \frac{512}{125} \):
\[ \frac{x^3}{y^3} = \frac{512}{125} \]

### Step 4: Find the cube roots
Taking the cube root of both sides gives us:
\[ \frac{x}{y} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}} = \frac{8}{5} \]

### Step 5: Determine the actual fraction and its reciprocal
Now we have the fraction:
\[ \frac{x}{y} = \frac{8}{5} \]
The reciprocal of this fraction is:
\[ \frac{y}{x} = \frac{5}{8} \]

### Step 6: Calculate the difference between the fraction and its reciprocal
Now, we need to find the difference between the fraction and its reciprocal:
\[ \frac{8}{5} - \frac{5}{8} \]

### Step 7: Find a common denominator
To subtract these fractions, we need a common denominator, which is \( 40 \):
\[ \frac{8}{5} = \frac{8 \times 8}{5 \times 8} = \frac{64}{40} \]
\[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \]

### Step 8: Perform the subtraction
Now, we can subtract the two fractions:
\[ \frac{64}{40} - \frac{25}{40} = \frac{64 - 25}{40} = \frac{39}{40} \]

### Final Answer
The difference between the actual fraction and its reciprocal is:
\[ \frac{39}{40} \]