Step 1: Formulate the equations from the given conditions

Let the fraction be $\frac{x}{y}$. According to the first condition, adding 2 to the numerator and 5 to the denominator gives $\frac{1}{2}$.

$\frac{x + 2}{y + 5} = \frac{1}{2} \implies 2(x + 2) = y + 5 \implies 2x - y = 1$

Step 2: Formulate the second equation

According to the second condition, subtracting 2 from both the numerator and the denominator gives $\frac{1}{3}$.

$\frac{x - 2}{y - 2} = \frac{1}{3} \implies 3(x - 2) = y - 2 \implies 3x - y = 4$

Step 3: Solve the simultaneous equations

Subtract the first equation ($2x - y = 1$) from the second equation ($3x - y = 4$) to find $x$.

$(3x - y) - (2x - y) = 4 - 1 \implies x = 3$

Step 4: Find the denominator $y$ and calculate the difference

Substitute $x = 3$ into $2x - y = 1$ to get $y = 5$. The fraction is $\frac{3}{5}$. The positive difference between the denominator and numerator is $5 - 3 = 2$.

$y - x = 5 - 3 = 2$