If 4 is added to the numerator of a 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

To solve the problem step by step, let's denote the numerator of the fraction as \( N \) and the denominator as \( D \). The fraction can be represented as \( \frac{N}{D} \).

### Step 1: Set up the first equation
According to the problem, if 4 is added to the numerator, the fraction becomes \( \frac{1}{3} \):
\[ \frac{N + 4}{D} = \frac{1}{3} \]
Cross-multiplying gives us:
\[ 3(N + 4) = D \]
This simplifies to:
\[ 3N + 12 = D \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation
The problem also states that if 3 is added to the denominator, the fraction becomes \( \frac{1}{6} \):
\[ \frac{N}{D + 3} = \frac{1}{6} \]
Cross-multiplying gives us:
\[ 6N = D + 3 \]
This simplifies to:
\[ 6N - D = 3 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 3N + 12 = D \)
2. \( 6N - D = 3 \)

We can substitute Equation 1 into Equation 2. From Equation 1, we can express \( D \) as:
\[ D = 3N + 12 \]
Substituting this into Equation 2:
\[ 6N - (3N + 12) = 3 \]
This simplifies to:
\[ 6N - 3N - 12 = 3 \]
\[ 3N - 12 = 3 \]
Adding 12 to both sides gives:
\[ 3N = 15 \]
Dividing by 3 gives:
\[ N = 5 \]

### Step 4: Find the value of \( D \)
Now substitute \( N = 5 \) back into Equation 1 to find \( D \):
\[ D = 3(5) + 12 = 15 + 12 = 27 \]

### Step 5: Calculate the sum of \( N \) and \( D \)
Now we can find the sum of the numerator and denominator:
\[ N + D = 5 + 27 = 32 \]

### Conclusion
The sum of the numerator and denominator is \( \boxed{32} \).