When the numerator of a positive fraction is incresed by 2 and the denominator of the same fraction is multiplied by 2, the new fraction can be reduced to `1/2` to its lowest term. The sum of the numerator and denominator of the original fraction can be :

Options

A. 13
B. 45
C. 16
D. any even integer greater than 3

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: Set up the equation based on the problem statement.
According to the problem, when the numerator \( x \) is increased by 2, and the denominator \( y \) is multiplied by 2, the new fraction becomes:
\[ \frac{x + 2}{2y} \]
This new fraction is said to be equal to \( \frac{1}{2} \).

### Step 2: Write the equation from the new fraction.
We can set up the equation:
\[ \frac{x + 2}{2y} = \frac{1}{2} \]

### Step 3: Cross-multiply to eliminate the fractions.
Cross-multiplying gives us:
\[ 2(x + 2) = 1 \cdot (2y) \]
This simplifies to:
\[ 2x + 4 = 2y \]

### Step 4: Rearrange the equation.
Now, we can rearrange this equation to isolate \( y \):
\[ 2x - 2y + 4 = 0 \]
Dividing the entire equation by 2 gives:
\[ x - y + 2 = 0 \]
Thus, we can express \( y \) in terms of \( x \):
\[ y = x + 2 \]

### Step 5: Find the sum of the numerator and denominator.
Now, we need to find the sum of the numerator and denominator:
\[ x + y = x + (x + 2) = 2x + 2 \]

### Step 6: Determine the possible values.
Since \( x \) and \( y \) must be positive integers, \( x \) must be greater than 0. Therefore, \( 2x + 2 \) must also be greater than 2. The smallest integer value for \( x \) is 1, which gives:
\[ 2(1) + 2 = 4 \]
If \( x = 2 \):
\[ 2(2) + 2 = 6 \]
If \( x = 3 \):
\[ 2(3) + 2 = 8 \]
And so on. Thus, the sum \( x + y \) can take on values such as 4, 6, 8, etc.

### Conclusion
The sum of the numerator and denominator of the original fraction can be any even integer greater than or equal to 4.

---