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)/(3)`
B. `(3)/(5)`
C. `(4)/(5)`
D. `(7)/(9)`

Answer

To solve the problem, we need to find the certain fraction (let's denote it as \( x \)) based on the information given.

### Step 1: Understand the correct calculation
The correct calculation of \( \frac{6}{7} \) of the fraction \( x \) is:
\[ \text{Correct Answer} = \frac{6}{7} \times x \]

### Step 2: Understand the boy's mistake
The boy mistakenly divided the fraction \( x \) by \( \frac{6}{7} \). The result of this operation is:
\[ \text{Mistaken Answer} = x \div \frac{6}{7} = x \times \frac{7}{6} = \frac{7x}{6} \]

### Step 3: Set up the equation based on the information provided
According to the problem, the mistaken 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 4: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by \( 42 \) (the least common multiple of the denominators 6, 7, and 70):
\[ 42 \times \frac{7x}{6} = 42 \times \frac{6x}{7} + 42 \times \frac{13}{70} \]
This simplifies to:
\[ 49x = 36x + 13 \times 6 \]
Calculating \( 13 \times 6 \):
\[ 49x = 36x + 78 \]

### Step 5: Solve for \( x \)
Now, we can isolate \( x \):
\[ 49x - 36x = 78 \]
\[ 13x = 78 \]
\[ x = \frac{78}{13} = 6 \]

### Step 6: Conclusion
The certain fraction is:
\[ \boxed{6} \]