A boy on being asked what `6/7` of a certain fraction was, made the mistake of dividing the fraction by `6/7` and so got an answer which exceeded the correct answer by `13/70` Find the fraction :

Options

A. `2/5`
B. `3/5`
C. `4/5`
D. `7/9`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variable
Let the fraction be \( x \).

### Step 2: Write the correct calculation
The correct calculation for \( \frac{6}{7} \) of the fraction \( x \) is:
\[ \frac{6}{7} \times x \]

### Step 3: Write the incorrect calculation
The boy mistakenly divided the fraction \( x \) by \( \frac{6}{7} \). This can be expressed as:
\[ x \div \frac{6}{7} = x \times \frac{7}{6} = \frac{7x}{6} \]

### Step 4: Set up the equation
According to the problem, the incorrect answer exceeds the correct answer by \( \frac{13}{70} \). Therefore, we can set up the equation:
\[ \frac{7x}{6} = \frac{6x}{7} + \frac{13}{70} \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators, which is 42:
\[ 42 \left(\frac{7x}{6}\right) = 42 \left(\frac{6x}{7}\right) + 42 \left(\frac{13}{70}\right) \]

This simplifies to:
\[ 7 \times 7x = 6 \times 6x + \frac{42 \times 13}{70} \]
\[ 49x = 36x + \frac{546}{70} \]

### Step 6: Simplify the fraction
Now, simplify \( \frac{546}{70} \):
\[ \frac{546}{70} = \frac{273}{35} \]

### Step 7: Rearrange the equation
Now, we can rearrange the equation:
\[ 49x - 36x = \frac{273}{35} \]
\[ 13x = \frac{273}{35} \]

### Step 8: Solve for \( x \)
Now, divide both sides by 13:
\[ x = \frac{273}{35 \times 13} \]
Calculating the denominator:
\[ 35 \times 13 = 455 \]
Thus,
\[ x = \frac{273}{455} \]

### Step 9: Simplify the fraction
Now simplify \( \frac{273}{455} \):
The GCD of 273 and 455 is 91. Therefore:
\[ x = \frac{3}{5} \]

### Final Answer
The fraction is:
\[ \boxed{\frac{3}{5}} \]