If the numerator of a fraction is increased by 250% and the denominator of the fraction is increased by 450% the resultant fraction is `(15)/(11)`, what is the original fraction ?

Options

A. `(13)/(15)`
B. `(15)/(7)`
C. `(7)/(15)`
D. `(15)/(13)`

Answer

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

### Step 1: Understand the percentage increases
- If the numerator \( x \) is increased by 250%, it becomes:
\[ x + \frac{250}{100} \times x = x + 2.5x = 3.5x \]

- If the denominator \( y \) is increased by 450%, it becomes:
\[ y + \frac{450}{100} \times y = y + 4.5y = 5.5y \]

### Step 2: Set up the equation for the resultant fraction
The resultant fraction after these increases is given as \( \frac{15}{11} \). Therefore, we can set up the equation:
\[ \frac{3.5x}{5.5y} = \frac{15}{11} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3.5x \cdot 11 = 15 \cdot 5.5y \]
This simplifies to:
\[ 38.5x = 82.5y \]

### Step 4: Simplify the equation
To simplify, we can divide both sides by 38.5:
\[ x = \frac{82.5}{38.5}y \]

### Step 5: Calculate the ratio of \( x \) to \( y \)
Calculating \( \frac{82.5}{38.5} \):
\[ \frac{82.5}{38.5} = \frac{825}{385} = \frac{165}{77} = \frac{15}{7} \]
Thus, we have:
\[ x = \frac{15}{7}y \]

### Step 6: Express the original fraction
The original fraction \( \frac{x}{y} \) can now be expressed as:
\[ \frac{x}{y} = \frac{\frac{15}{7}y}{y} = \frac{15}{7} \]

### Conclusion
The original fraction is:
\[ \frac{15}{7} \]