A fraction becomes `(1)/(3)` when 1 is subtracted from both the numerator and the denominator. The same fraction becomes `(1)/(2)` when 1 is added to both the numerator and the denominator. The sum of numerator and denominator of the fraction is

Options

A. 10
B. 18
C. 7
D. 16

Answer

To solve the problem, we need to find a fraction \( \frac{x}{y} \) that meets the conditions given in the question. Let's break it down step by step.

### Step 1: Set up the equations
Let the fraction be \( \frac{x}{y} \).

1. When 1 is subtracted from both the numerator and the denominator, the fraction becomes \( \frac{1}{3} \):
\[ \frac{x - 1}{y - 1} = \frac{1}{3} \]
Cross-multiplying gives us:
\[ 3(x - 1) = y - 1 \]
Simplifying this, we get:
\[ 3x - 3 = y - 1 \implies 3x - y = 2 \quad \text{(Equation 1)} \]

2. When 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{1}{2} \):
\[ \frac{x + 1}{y + 1} = \frac{1}{2} \]
Cross-multiplying gives us:
\[ 2(x + 1) = y + 1 \]
Simplifying this, we get:
\[ 2x + 2 = y + 1 \implies 2x - y = -1 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations
Now we have two equations:
1. \( 3x - y = 2 \) (Equation 1)
2. \( 2x - y = -1 \) (Equation 2)

We can subtract Equation 2 from Equation 1 to eliminate \( y \):
\[ (3x - y) - (2x - y) = 2 - (-1) \]
This simplifies to:
\[ 3x - 2x = 2 + 1 \implies x = 3 \]

### Step 3: Substitute \( x \) back to find \( y \)
Now that we have \( x = 3 \), we can substitute it back into either equation to find \( y \). We'll use Equation 1:
\[ 3(3) - y = 2 \implies 9 - y = 2 \implies y = 9 - 2 = 7 \]

### Step 4: Find the sum of the numerator and denominator
Now we have \( x = 3 \) and \( y = 7 \). The sum of the numerator and denominator is:
\[ x + y = 3 + 7 = 10 \]

### Final Answer
Thus, the sum of the numerator and the denominator of the fraction is \( \boxed{10} \).

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