A fraction becomes `9/(11)` , if 2 is added to both the numerator and the denominator . If 3 is added to both the numerator and the denominator it becomes `5/6`. What is the fraction ?

Options

A. `7/9`
B. `3/7`
C. `5/9`
D. `7/(10)`

Answer

To solve the problem step by step, we will denote the fraction as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Set up the equations based on the problem statement.
According to the problem:
1. When 2 is added to both the numerator and the denominator, the fraction becomes \( \frac{9}{11} \).
\[ \frac{x + 2}{y + 2} = \frac{9}{11} \]
Cross-multiplying gives us:
\[ 11(x + 2) = 9(y + 2) \]
Expanding this, we have:
\[ 11x + 22 = 9y + 18 \]
Rearranging gives us the first equation:
\[ 11x - 9y = -4 \quad \text{(Equation 1)} \]

2. When 3 is added to both the numerator and the denominator, the fraction becomes \( \frac{5}{6} \).
\[ \frac{x + 3}{y + 3} = \frac{5}{6} \]
Cross-multiplying gives us:
\[ 6(x + 3) = 5(y + 3) \]
Expanding this, we have:
\[ 6x + 18 = 5y + 15 \]
Rearranging gives us the second equation:
\[ 6x - 5y = -3 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.
Now we have a system of linear equations:
1. \( 11x - 9y = -4 \)
2. \( 6x - 5y = -3 \)

We can solve these equations using substitution or elimination. Here, we will use the elimination method.

### Step 3: Multiply the equations to align coefficients.
To eliminate \( y \), we can multiply Equation 1 by 5 and Equation 2 by 9:
\[ 5(11x - 9y) = 5(-4) \implies 55x - 45y = -20 \quad \text{(Equation 3)} \]
\[ 9(6x - 5y) = 9(-3) \implies 54x - 45y = -27 \quad \text{(Equation 4)} \]

### Step 4: Subtract the equations.
Now we subtract Equation 4 from Equation 3:
\[ (55x - 45y) - (54x - 45y) = -20 + 27 \]
This simplifies to:
\[ x = 7 \]

### Step 5: Substitute \( x \) back to find \( y \).
Now that we have \( x = 7 \), we can substitute this value back into either Equation 1 or Equation 2 to find \( y \). We'll use Equation 1:
\[ 11(7) - 9y = -4 \]
This simplifies to:
\[ 77 - 9y = -4 \]
Rearranging gives:
\[ -9y = -4 - 77 \implies -9y = -81 \implies y = 9 \]

### Step 6: Write the final fraction.
Now we have \( x = 7 \) and \( y = 9 \). Therefore, the fraction is:
\[ \frac{x}{y} = \frac{7}{9} \]

### Conclusion
Thus, the fraction is \( \frac{7}{9} \).