If the numerator of a fraction is increased by 200% and the denominator is increased by 160%, the resultant fraction is `7/13`. What is the original fraction?

Options

A. `7/15`
B. `2/15`
C. `8/15`
D. `5/7`

Answer

To find the original fraction, let's denote the original fraction as \( \frac{x}{y} \).

### Step 1: Understand the changes to the numerator and denominator
According to the problem:
- The numerator \( x \) is increased by 200%. This means the new numerator becomes:
\[ x + 200\% \text{ of } x = x + 2x = 3x \]

- The denominator \( y \) is increased by 160%. This means the new denominator becomes:
\[ y + 160\% \text{ of } y = y + 1.6y = 2.6y \]

### Step 2: Set up the equation for the new fraction
The new fraction after these changes is given as:
\[ \frac{3x}{2.6y} = \frac{7}{13} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3x \cdot 13 = 7 \cdot 2.6y \]
This simplifies to:
\[ 39x = 18.2y \]

### Step 4: Rearrange the equation to find the ratio of \( x \) to \( y \)
Rearranging the equation gives:
\[ \frac{x}{y} = \frac{18.2}{39} \]

### Step 5: Simplify the ratio
To simplify \( \frac{18.2}{39} \):
1. Convert \( 18.2 \) to a fraction:
\[ 18.2 = \frac{182}{10} = \frac{91}{5} \]
2. Now substitute back into the ratio:
\[ \frac{x}{y} = \frac{91/5}{39} = \frac{91}{5 \cdot 39} = \frac{91}{195} \]

### Step 6: Final simplification
Now we can simplify \( \frac{91}{195} \):
- The fraction \( \frac{91}{195} \) cannot be simplified further since 91 and 195 have no common factors other than 1.

### Conclusion
Thus, the original fraction is:
\[ \frac{91}{195} \]