If the numerator of a fraction is increased by 2 and denominator is decreased by 1, it becomes `(2)/(3)` . If the numerator is increased by 1 and denominator is increased by 2 it becomes `(1)/(3)` Find the fraction.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the numerator of the fraction be \( x \) and the denominator be \( y \).

### Step 2: Set up the equations based on the problem statement
1. According to the first condition, if the numerator is increased by 2 and the denominator is decreased by 1, the fraction becomes \( \frac{2}{3} \):
\[ \frac{x + 2}{y - 1} = \frac{2}{3} \]
Cross-multiplying gives:
\[ 3(x + 2) = 2(y - 1) \]
Expanding this:
\[ 3x + 6 = 2y - 2 \]
Rearranging gives us the first equation:
\[ 3x - 2y = -8 \quad \text{(Equation 1)} \]

2. According to the second condition, if the numerator is increased by 1 and the denominator is increased by 2, the fraction becomes \( \frac{1}{3} \):
\[ \frac{x + 1}{y + 2} = \frac{1}{3} \]
Cross-multiplying gives:
\[ 3(x + 1) = 1(y + 2) \]
Expanding this:
\[ 3x + 3 = y + 2 \]
Rearranging gives us the second equation:
\[ 3x - y = -1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have the system of equations:
1. \( 3x - 2y = -8 \) (Equation 1)
2. \( 3x - y = -1 \) (Equation 2)

We can solve these equations simultaneously.

From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = 3x + 1 \]

### Step 4: Substitute \( y \) in Equation 1
Substituting \( y \) in Equation 1:
\[ 3x - 2(3x + 1) = -8 \]
Expanding this:
\[ 3x - 6x - 2 = -8 \]
Combining like terms:
\[ -3x - 2 = -8 \]
Adding 2 to both sides:
\[ -3x = -6 \]
Dividing by -3:
\[ x = 2 \]

### Step 5: Find \( y \)
Now substitute \( x = 2 \) back into the expression for \( y \):
\[ y = 3(2) + 1 = 6 + 1 = 7 \]

### Step 6: Write the fraction
Thus, the fraction is:
\[ \frac{x}{y} = \frac{2}{7} \]

### Final Answer
The fraction is \( \frac{2}{7} \).

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