If the numberator of a fraction is increased by 250% and the denominator is increased by 300%, the new fraction thus formed is `(15)/(28)`. What will be the resultant fraction if the numerator of the original fraction is increased by 400% and denominator is increased by 450%?

Options

A. `(300)/(539)`
B. `(400)/(539)`
C. `(300)/(359)`
D. `(300)/(953)`

Answer

To solve the problem step by step, let's denote the original fraction as \( \frac{x}{y} \).

### Step 1: Understand the percentage increases
- Increasing the numerator \( x \) by 250% means the new numerator becomes:
\[ x + 2.5x = 3.5x \]
- Increasing the denominator \( y \) by 300% means the new denominator becomes:
\[ y + 3y = 4y \]

### Step 2: Set up the equation with the new fraction
According to the problem, when these increases are applied, the new fraction becomes:
\[ \frac{3.5x}{4y} = \frac{15}{28} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3.5x \cdot 28 = 15 \cdot 4y \]
This simplifies to:
\[ 98x = 60y \]

### Step 4: Simplify the equation
Dividing both sides by 2, we get:
\[ 49x = 30y \]
Thus, we can express \( y \) in terms of \( x \):
\[ y = \frac{49}{30}x \]

### Step 5: Determine the new increases for the numerator and denominator
Now, we need to find the new fraction when:
- The numerator \( x \) is increased by 400%, which means:
\[ x + 4x = 5x \]
- The denominator \( y \) is increased by 450%, which means:
\[ y + 4.5y = 5.5y \]

### Step 6: Form the new fraction
The new fraction after these increases will be:
\[ \frac{5x}{5.5y} \]

### Step 7: Substitute \( y \) in terms of \( x \)
Substituting \( y = \frac{49}{30}x \) into the new fraction:
\[ \frac{5x}{5.5 \cdot \frac{49}{30}x} \]
This simplifies to:
\[ \frac{5x}{\frac{269.5}{30}x} \]
Cancelling \( x \) from the numerator and denominator gives:
\[ \frac{5}{\frac{269.5}{30}} = \frac{5 \cdot 30}{269.5} = \frac{150}{269.5} \]

### Step 8: Simplify the fraction
To simplify \( \frac{150}{269.5} \), we can multiply both the numerator and denominator by 2 to eliminate the decimal:
\[ \frac{300}{539} \]

### Step 9: Final Result
Thus, the resultant fraction after the increases is:
\[ \frac{300}{539} \]