In a fraction the denominator is 2 more than 3 times the numerator. If 1 is added in both numerator and denominator, the fraction becomes 1/3. What is the fraction.

Options

A. a)`4/13`
B. b)`3/11`
C. c)`5/13`
D. d)`5/11`

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \). According to the problem, the denominator is 2 more than 3 times the numerator. Therefore, we can express the denominator as:
\[ \text{Denominator} = 3x + 2 \]

### Step 2: Write the Fraction
The fraction can be written as:
\[ \text{Fraction} = \frac{x}{3x + 2} \]

### Step 3: Set Up the Condition
The problem states that if 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{1}{3} \). Therefore, we can set up the equation:
\[ \frac{x + 1}{(3x + 2) + 1} = \frac{1}{3} \]
This simplifies to:
\[ \frac{x + 1}{3x + 3} = \frac{1}{3} \]

### Step 4: Cross-Multiply
To eliminate the fraction, we can cross-multiply:
\[ 3(x + 1) = 1(3x + 3) \]
Expanding both sides gives:
\[ 3x + 3 = 3x + 3 \]

### Step 5: Solve the Equation
Now, we simplify the equation:
\[ 3x + 3 = 3x + 3 \]
This equation is always true, which means we need to check for specific values of \( x \) that satisfy the original conditions.

### Step 6: Check Possible Values for \( x \)
We can try integer values for \( x \) to find a valid fraction:
1. If \( x = 1 \):
- Denominator = \( 3(1) + 2 = 5 \) → Fraction = \( \frac{1}{5} \)
2. If \( x = 2 \):
- Denominator = \( 3(2) + 2 = 8 \) → Fraction = \( \frac{2}{8} = \frac{1}{4} \)
3. If \( x = 3 \):
- Denominator = \( 3(3) + 2 = 11 \) → Fraction = \( \frac{3}{11} \)

### Step 7: Verify the Condition
Now, we check if adding 1 to both the numerator and denominator gives \( \frac{1}{3} \):
- For \( x = 3 \):
\[ \frac{3 + 1}{11 + 1} = \frac{4}{12} = \frac{1}{3} \]
This condition holds true.

### Conclusion
Thus, the fraction is:
\[ \frac{3}{11} \]