The numerator of a fraction is decreased by `25%` and the denominator is increased by `250%`. If the resultant fraction is `6/5`, what is the original fraction?

Options

A. `22/5`
B. `24/5`
C. `27/6`
D. `28/5`

Answer

To solve the problem step by step, we will denote the original fraction as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Understand the changes to the fraction
The problem states that:
- The numerator \( x \) is decreased by \( 25\% \).
- The denominator \( y \) is increased by \( 250\% \).

### Step 2: Express the changes mathematically
- Decreasing the numerator \( x \) by \( 25\% \) means the new numerator becomes:
\[ x - 0.25x = 0.75x = \frac{3}{4}x \]

- Increasing the denominator \( y \) by \( 250\% \) means the new denominator becomes:
\[ y + 2.5y = 3.5y = \frac{7}{2}y \]

### Step 3: Set up the equation with the new fraction
According to the problem, the resultant fraction after these changes is \( \frac{6}{5} \). Therefore, we can set up the equation:
\[ \frac{\frac{3}{4}x}{\frac{7}{2}y} = \frac{6}{5} \]

### Step 4: Simplify the equation
To simplify, we can multiply both sides by \( \frac{7}{2}y \):
\[ \frac{3}{4}x = \frac{6}{5} \cdot \frac{7}{2}y \]
Calculating the right side:
\[ \frac{6 \cdot 7}{5 \cdot 2}y = \frac{42}{10}y = \frac{21}{5}y \]
Thus, we have:
\[ \frac{3}{4}x = \frac{21}{5}y \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 3 \cdot 5x = 21 \cdot 4y \]
This simplifies to:
\[ 15x = 84y \]

### Step 6: Solve for \( \frac{x}{y} \)
Rearranging gives:
\[ \frac{x}{y} = \frac{84}{15} \]
To simplify \( \frac{84}{15} \):
\[ \frac{84 \div 3}{15 \div 3} = \frac{28}{5} \]

### Conclusion
Thus, the original fraction is:
\[ \frac{28}{5} \]