If 1 is added to the denominator of a fraction it becomes `(1)/(2)` . If 1 is added to the numerator it becomes 1. The product of numerator and denominator of the fraction is

Options

A. 6
B. 10
C. 12
D. 14

Answer

Let's solve the problem step by step.

### Step 1: Define the Variables
Let the numerator of the fraction be \( n \) and the denominator be \( d \).

### Step 2: Set Up the First Equation
According to the problem, if 1 is added to the denominator, the fraction becomes \( \frac{1}{2} \). This can be expressed as:
\[ \frac{n}{d + 1} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2n = d + 1 \]
This is our **first equation**.

### Step 3: Set Up the Second Equation
The problem states that if 1 is added to the numerator, the fraction becomes 1. This can be expressed as:
\[ \frac{n + 1}{d} = 1 \]
This simplifies to:
\[ n + 1 = d \]
This is our **second equation**.

### Step 4: Substitute and Solve
Now we have two equations:
1. \( 2n = d + 1 \)
2. \( n + 1 = d \)

We can substitute the value of \( d \) from the second equation into the first equation. From the second equation, we can express \( d \) as:
\[ d = n + 1 \]

Substituting this into the first equation:
\[ 2n = (n + 1) + 1 \]
This simplifies to:
\[ 2n = n + 2 \]
Subtracting \( n \) from both sides gives:
\[ n = 2 \]

### Step 5: Find the Denominator
Now that we have \( n \), we can find \( d \) using the second equation:
\[ d = n + 1 = 2 + 1 = 3 \]

### Step 6: Calculate the Product
Now we need to find the product of the numerator and denominator:
\[ n \times d = 2 \times 3 = 6 \]

### Final Answer
The product of the numerator and denominator of the fraction is \( 6 \).

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