If a fraction is divided by its reciprocal and then multiplied 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, we will follow the reasoning provided in the video transcript.

### Step 1: Define the Fraction
Let the fraction be represented as \( x \).

### Step 2: Find the Reciprocal
The reciprocal of the fraction \( x \) is \( \frac{1}{x} \).

### Step 3: Divide the Fraction by its Reciprocal
When we divide the fraction \( x \) by its reciprocal \( \frac{1}{x} \), we have:
\[ \frac{x}{\frac{1}{x}} = x \cdot x = x^2 \]

### Step 4: Multiply by the Fraction Itself
According to the problem, we then multiply the result by the fraction itself:
\[ x^2 \cdot x = x^3 \]

### Step 5: Set the Equation
According to the problem, this result equals \( \frac{512}{125} \):
\[ x^3 = \frac{512}{125} \]

### Step 6: Simplify the Right Side
Notice that \( \frac{512}{125} \) can be expressed as:
\[ \frac{512}{125} = \left( \frac{8}{5} \right)^3 \]

### Step 7: Equate the Bases
Since the powers are the same, we can equate the bases:
\[ x = \frac{8}{5} \]

### Step 8: Find the Reciprocal
Now, we find the reciprocal of \( x \):
\[ \text{Reciprocal} = \frac{1}{x} = \frac{1}{\frac{8}{5}} = \frac{5}{8} \]

### Step 9: Calculate the Difference
Now we need to find the difference between the actual fraction \( x \) and its reciprocal:
\[ \text{Difference} = x - \frac{1}{x} = \frac{8}{5} - \frac{5}{8} \]

### Step 10: Find a Common Denominator
To subtract these fractions, we find 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 11: Perform the Subtraction
Now we can subtract the two fractions:
\[ \text{Difference} = \frac{64}{40} - \frac{25}{40} = \frac{64 - 25}{40} = \frac{39}{40} \]

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

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