Solve the following problems using two variables :
If the numerator of a fraction is increased by 1 , its value becomes `3/4` If its denominator is invreased by 2 , its value becomes `1/2` Find the fraction .

Answer

To solve the problem, we will use two variables to represent the numerator and denominator of the fraction. Let's denote the numerator as \( x \) and the denominator as \( y \).

### Step-by-Step Solution:

1. **Set up the equations**:
According to the problem, we have two conditions:
- If the numerator is increased by 1, the value of the fraction becomes \( \frac{3}{4} \).
- If the denominator is increased by 2, the value of the fraction becomes \( \frac{1}{2} \).

From these conditions, we can write the following equations:
\[ \frac{x + 1}{y} = \frac{3}{4} \]
\[ \frac{x}{y + 2} = \frac{1}{2} \]

2. **Cross-multiply to eliminate the fractions**:
For the first equation:
\[ 4(x + 1) = 3y \]
Expanding this gives:
\[ 4x + 4 = 3y \quad \text{(Equation 1)} \]

For the second equation:
\[ 2x = y + 2 \]
Rearranging gives:
\[ 2x - y = 2 \quad \text{(Equation 2)} \]

3. **Solve the system of equations**:
We now have a system of two equations:
\[ 4x - 3y = -4 \quad \text{(from Equation 1)} \]
\[ 2x - y = 2 \quad \text{(from Equation 2)} \]

We can express \( y \) in terms of \( x \) from Equation 2:
\[ y = 2x - 2 \]

4. **Substitute \( y \) into Equation 1**:
Substitute \( y = 2x - 2 \) into Equation 1:
\[ 4x - 3(2x - 2) = -4 \]
Simplifying this gives:
\[ 4x - 6x + 6 = -4 \]
\[ -2x + 6 = -4 \]
\[ -2x = -4 - 6 \]
\[ -2x = -10 \]
\[ x = 5 \]

5. **Find \( y \)**:
Now substitute \( x = 5 \) back into the equation for \( y \):
\[ y = 2(5) - 2 = 10 - 2 = 8 \]

6. **Conclusion**:
The fraction is:
\[ \frac{x}{y} = \frac{5}{8} \]

### Final Answer:
The fraction is \( \frac{5}{8} \).