A run was to find `6//7` of a fraction. Instead of multiplying, he divided the fraction by `6//7` and the result obtained was `13//70` greater than original value. Find the fraction given to Arun ?

Answer

To solve the problem step-by-step, we will denote the unknown fraction as \( \frac{x}{y} \).

### Step 1: Define the original fraction and the intended operation
Arun was supposed to find \( \frac{6}{7} \) of the fraction \( \frac{x}{y} \). This means he should have calculated:
\[ \frac{6}{7} \cdot \frac{x}{y} = \frac{6x}{7y} \]

### Step 2: Define the operation Arun actually performed
Instead of multiplying, Arun divided the fraction by \( \frac{6}{7} \). Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[ \frac{x}{y} \div \frac{6}{7} = \frac{x}{y} \cdot \frac{7}{6} = \frac{7x}{6y} \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the result of Arun's division was \( \frac{13}{70} \) greater than the original value. Therefore, we can write the equation:
\[ \frac{7x}{6y} = \frac{6x}{7y} + \frac{13}{70} \]

### Step 4: Clear the fractions by finding a common denominator
To eliminate the fractions, we can multiply through by \( 42y \) (the least common multiple of the denominators):
\[ 42y \cdot \frac{7x}{6y} = 42y \cdot \frac{6x}{7y} + 42y \cdot \frac{13}{70} \]

This simplifies to:
\[ 49x = 36x + \frac{546y}{70} \]

### Step 5: Simplify the equation
Now, we can simplify the equation:
\[ 49x - 36x = \frac{546y}{70} \]
\[ 13x = \frac{546y}{70} \]

### Step 6: Solve for \( \frac{x}{y} \)
Now, we can express \( \frac{x}{y} \):
\[ \frac{x}{y} = \frac{546}{70 \cdot 13} \]
Calculating \( 70 \cdot 13 = 910 \):
\[ \frac{x}{y} = \frac{546}{910} \]

### Step 7: Simplify the fraction
Now we simplify \( \frac{546}{910} \):
- The GCD of 546 and 910 is 182.
- Dividing both the numerator and the denominator by 182:
\[ \frac{546 \div 182}{910 \div 182} = \frac{3}{5} \]

### Conclusion
Thus, the fraction given to Arun is:
\[ \frac{3}{5} \]