When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes 1/4.

Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$

If 3 is added to the denominator and 2 is subtracted from the numerator, the fraction becomes $\frac{1}{4}$. Thus, we have

$\frac{x-2}{y+3}=\frac{1}{4}$

$\Rightarrow 4(x-2)=y+3$

$\Rightarrow 4 x-8=y+3$

$\Rightarrow 4 x-y-11=0$

If 6 is added to the numerator and the denominator is multiplied by 3 , the fraction becomes $\frac{2}{3}$. Thus, we have

$\frac{x+6}{3 y}=\frac{2}{3}$

$\Rightarrow 3(x+6)=6 y$

$\Rightarrow 3 x+18=6 y$

$\Rightarrow 3 x-6 y+18=0$

$\Rightarrow 3(x-2 y+6)=0$

$\Rightarrow x-2 y+6=0$

So, we have two equations

$4 x-y-11=0$

$x-2 y+6=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{(-1) \times 6-(-2) \times(-11)}=\frac{-y}{4 \times 6-1 \times(-11)}=\frac{1}{4 \times(-2)-1 \times(-1)}$

$\Rightarrow \frac{x}{-6-22}=\frac{-y}{24+11}=\frac{1}{-8+1}$

$\Rightarrow \frac{x}{-28}=\frac{-y}{25}=\frac{1}{-7}$

$\Rightarrow \frac{x}{28}=\frac{y}{35}=\frac{1}{7}$

$\Rightarrow x=\frac{28}{7}, y=\frac{35}{7}$

$\Rightarrow x=4, y=5$

Hence, the fraction is $\frac{4}{5}$.