If the numerator of a fraction is increased by 250% and the denominator is increased by 400%. The resultant fraction is `(7)/(19)`. What is the original fraction ?

Options

A. A) `(10)/(19)`
B. B) `(5)/(9)`
C. C) `(9)/(5)`
D. D) `(19)/(7)`

Answer

To find the original fraction, we will denote the original fraction as \( \frac{a}{b} \).

### Step 1: Understand the changes in the numerator and denominator
- The numerator \( a \) is increased by 250%.
- The denominator \( b \) is increased by 400%.

### Step 2: Calculate the new numerator and denominator
- Increasing \( a \) by 250% means:
\[ \text{New numerator} = a + 2.5a = 3.5a \]
- Increasing \( b \) by 400% means:
\[ \text{New denominator} = b + 4b = 5b \]

### Step 3: Set up the equation based on the resultant fraction
According to the problem, the new fraction is given as:
\[ \frac{3.5a}{5b} = \frac{7}{19} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3.5a \cdot 19 = 7 \cdot 5b \]
This simplifies to:
\[ 66.5a = 35b \]

### Step 5: Simplify the equation
To simplify, we can divide both sides by 0.5:
\[ 133a = 70b \]

### Step 6: Express \( a \) in terms of \( b \)
Rearranging gives:
\[ \frac{a}{b} = \frac{70}{133} \]

### Step 7: Find the original fraction
The original fraction \( \frac{a}{b} \) can be simplified:
\[ \frac{70}{133} \text{ cannot be simplified further.} \]

### Step 8: Check the options
The original fraction is \( \frac{10}{19} \) after dividing both the numerator and denominator by 7.

### Final Answer
The original fraction is:
\[ \frac{10}{19} \]

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