If 2 is added to both the numerator and the denominator of a fraction, it becomes $\frac{2}{3}$. If 3 is subtracted from both, it becomes $\frac{1}{2}$. What is the fraction?

Explanation

1. 01

   Let the fraction be $\frac{n}{d}$. First condition: $3(n + 2) = 2(d + 2)$, so $3n - 2d = -2$.

2. 02

   Second condition: $2(n - 3) = d - 3$, so $d = 2n - 3$.

3. 03

   Substitute: $3n - 2(2n - 3) = -2$, so $-n + 6 = -2$ and $n = 8$; then $d = 13$.

4. 04

   Check: $\frac{10}{15} = \frac{2}{3}$ and $\frac{5}{10} = \frac{1}{2}$.