The denominator of a fraction is 2 more than thrice its numerator. If the numerator as well as denominator is increased by one, the fraction becomes `1//3`. What was the original fraction?

Options

A. `(4)/(13)`
B. `(3)/(11)`
C. `(5)/(13)`
D. cannot be determined

Answer

To solve the problem step by step, let's 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 \( n \). According to the question, the denominator \( d \) is given as:
\[ d = 3n + 2 \]

### Step 2: Set Up the Equation After Increasing Both Numerator and Denominator
When both the numerator and denominator are increased by 1, the new fraction becomes \( \frac{1}{3} \). Therefore, we can write the equation:
\[ \frac{n + 1}{d + 1} = \frac{1}{3} \]

### Step 3: Substitute the Denominator in the Equation
Substituting \( d \) from Step 1 into the equation gives us:
\[ \frac{n + 1}{(3n + 2) + 1} = \frac{1}{3} \]
This simplifies to:
\[ \frac{n + 1}{3n + 3} = \frac{1}{3} \]

### Step 4: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 3(n + 1) = 1(3n + 3) \]
This simplifies to:
\[ 3n + 3 = 3n + 3 \]

### Step 5: Analyze the Result
The equation \( 3n + 3 = 3n + 3 \) is always true, which means that there are infinitely many solutions. This indicates that the original fraction cannot be determined uniquely based on the information provided.

### Conclusion
Since we cannot find a unique solution for the original fraction, we conclude that the answer is:
**Option D: Can't be determined.**