A fraction becomes 4 when 1 is added to both the numerator and denominator and it becomes 7 when 1 is substracted from both the numerator and denominator. The numerator of the given fraction is :

Options

A. 2
B. 3
C. 7
D. 15

Answer

To solve the problem step by step, let's denote the numerator of the fraction as \( N \) and the denominator as \( D \).

### Step 1: Set up the equations based on the problem statement.
According to the problem:
1. When 1 is added to both the numerator and denominator, the fraction becomes 4:
\[ \frac{N + 1}{D + 1} = 4 \]
Multiplying both sides by \( D + 1 \):
\[ N + 1 = 4(D + 1) \]
Expanding the right side:
\[ N + 1 = 4D + 4 \]
Rearranging gives us the first equation:
\[ N - 4D = 3 \quad \text{(Equation 1)} \]

2. When 1 is subtracted from both the numerator and denominator, the fraction becomes 7:
\[ \frac{N - 1}{D - 1} = 7 \]
Multiplying both sides by \( D - 1 \):
\[ N - 1 = 7(D - 1) \]
Expanding the right side:
\[ N - 1 = 7D - 7 \]
Rearranging gives us the second equation:
\[ N - 7D = -6 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.
Now we have two equations:
1. \( N - 4D = 3 \)
2. \( N - 7D = -6 \)

We can solve these equations by elimination. Subtract Equation 1 from Equation 2:
\[ (N - 7D) - (N - 4D) = -6 - 3 \]
This simplifies to:
\[ -3D = -9 \]
Dividing both sides by -3:
\[ D = 3 \]

### Step 3: Substitute the value of \( D \) back to find \( N \).
Now that we have \( D \), we can substitute it back into Equation 1 to find \( N \):
\[ N - 4(3) = 3 \]
This simplifies to:
\[ N - 12 = 3 \]
Adding 12 to both sides:
\[ N = 15 \]

### Final Answer:
The numerator of the given fraction is \( \boxed{15} \).

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