If the numerator of a fraction is increased by 25% and the denominater is doubled, the fraction thus obtained is `5//9`, What is the original fraction?

Options

A. `2//3`
B. `4//9`
C. `8//9`
D. None of these

Answer

To solve the problem, let's denote the original fraction as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Express the changes to the numerator and denominator
According to the problem, if the numerator is increased by 25%, the new numerator becomes:
\[ x + 0.25x = 1.25x \]
The denominator is doubled, so the new denominator becomes:
\[ 2y \]

### Step 2: Set up the equation
The new fraction after these changes is given as \( \frac{5}{9} \). Therefore, we can set up the equation:
\[ \frac{1.25x}{2y} = \frac{5}{9} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 9 \cdot 1.25x = 5 \cdot 2y \]
This simplifies to:
\[ 11.25x = 10y \]

### Step 4: Rearrange the equation
Now, we can rearrange this equation to express \( y \) in terms of \( x \):
\[ y = \frac{11.25}{10}x = 1.125x \]

### Step 5: Substitute back to find the original fraction
Now we can express the original fraction \( \frac{x}{y} \) using the relationship we found:
\[ \frac{x}{y} = \frac{x}{1.125x} \]
This simplifies to:
\[ \frac{x}{y} = \frac{1}{1.125} = \frac{1}{\frac{9}{8}} = \frac{8}{9} \]

### Conclusion
Thus, the original fraction is:
\[ \frac{8}{9} \]

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