Step 1: Formulate the first equation

Let the fraction be $\frac{x}{y}$. Adding 1 to the numerator and subtracting 1 from the denominator gives 1: $\frac{x + 1}{y - 1} = 1$ This simplifies to $x + 1 = y - 1$, or $y - x = 2$ (Equation 1).

$y - x = 2$

Step 2: Formulate the second equation

If 1 is added to the denominator while the numerator remains unchanged, the fraction becomes $\frac{2}{3}$: $\frac{x}{y + 1} = \frac{2}{3}$ Cross-multiplying gives $3x = 2(y + 1)$, which simplifies to $3x - 2y = 2$ (Equation 2).

$3x - 2y = 2$

Step 3: Solve the system of equations

From Equation 1, $y = x + 2$. Substitute this into Equation 2: $3x - 2(x + 2) = 2 \implies 3x - 2x - 4 = 2 \implies x = 6$ Then find $y$: $y = 6 + 2 = 8$

$x = 6, \quad y = 8$

Step 4: Find the positive difference

The numerator is $6$ and the denominator is $8$. The positive difference between the numerator and denominator is: $|6 - 8| = 2$

$|6 - 8| = 2$