If 4 is added to the numerator of fraction, it becomes `1/3` and if 3 is added to the denominator of the same fraction it becomes `1/6` then the sum of the numerator and denominator is :

Options

A. 32
B. 7
C. 4
D. 3

Answer

Let's solve the problem step by step.

Let the fraction be represented as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Set up the equations based on the problem statement

1. According to the first condition, if 4 is added to the numerator, the fraction becomes \( \frac{1}{3} \):
\[ \frac{x + 4}{y} = \frac{1}{3} \]
Cross-multiplying gives:
\[ 3(x + 4) = y \]
Simplifying this, we get:
\[ y = 3x + 12 \quad \text{(Equation 1)} \]

2. According to the second condition, if 3 is added to the denominator, the fraction becomes \( \frac{1}{6} \):
\[ \frac{x}{y + 3} = \frac{1}{6} \]
Cross-multiplying gives:
\[ 6x = y + 3 \]
Rearranging this, we have:
\[ y = 6x - 3 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations

Now we have two equations:
1. \( y = 3x + 12 \) (Equation 1)
2. \( y = 6x - 3 \) (Equation 2)

We can set these equations equal to each other:
\[ 3x + 12 = 6x - 3 \]

### Step 3: Rearranging the equation

Rearranging gives:
\[ 12 + 3 = 6x - 3x \]
\[ 15 = 3x \]

### Step 4: Solve for \( x \)

Dividing both sides by 3:
\[ x = 5 \]

### Step 5: Substitute \( x \) back to find \( y \)

Now substitute \( x = 5 \) into either Equation 1 or Equation 2 to find \( y \). Let's use Equation 1:
\[ y = 3(5) + 12 \]
\[ y = 15 + 12 = 27 \]

### Step 6: Find the sum of the numerator and denominator

Now that we have \( x = 5 \) and \( y = 27 \), we can find the sum:
\[ x + y = 5 + 27 = 32 \]

Thus, the sum of the numerator and denominator is **32**.