The denominator of a fraction is 1 more than double the numerator. On adding 2 to the numerator and subtracting 3 from the denominator, we obtain 1. Find the original fraction.

Options

A. `2/5`
B. `1/9`
C. `4/9`
D. `1/3`

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \). According to the problem, the denominator is one more than double the numerator. Therefore, we can express the denominator as:
\[ \text{Denominator} = 2x + 1 \]

### Step 2: Write the Fraction
The fraction can now be written as:
\[ \text{Fraction} = \frac{x}{2x + 1} \]

### Step 3: Set Up the Equation
The problem states that when we add 2 to the numerator and subtract 3 from the denominator, the result is 1. We can express this mathematically as:
\[ \frac{x + 2}{(2x + 1) - 3} = 1 \]
This simplifies to:
\[ \frac{x + 2}{2x - 2} = 1 \]

### Step 4: Cross Multiply
To eliminate the fraction, we can cross-multiply:
\[ x + 2 = 1 \cdot (2x - 2) \]
This simplifies to:
\[ x + 2 = 2x - 2 \]

### Step 5: Solve for \( x \)
Now, we can rearrange the equation to solve for \( x \):
\[ x + 2 + 2 = 2x \]
\[ 4 = 2x - x \]
\[ 4 = x \]

### Step 6: Find the Denominator
Now that we have \( x \), we can find the denominator:
\[ \text{Denominator} = 2x + 1 = 2(4) + 1 = 8 + 1 = 9 \]

### Step 7: Write the Original Fraction
Now we can write the original fraction:
\[ \text{Original Fraction} = \frac{x}{\text{Denominator}} = \frac{4}{9} \]

### Conclusion
Thus, the original fraction is:
\[ \frac{4}{9} \]