In a rational number, twice the numerator is 2 more than the denominator.

Let the denominator be $\mathrm{x}$.

$\therefore$ The numerator $=\frac{x+2}{2}$

$\therefore$ The rational number $=\frac{\mathrm{x}+2}{2 \mathrm{x}}$

According to the question,

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

or $\frac{x+2+6}{2(x+3)}=\frac{2}{3}$

or $\frac{x+8}{2 x+6}=\frac{2}{3}$

or $3 x+24=4 x+12$

or $x=24-12$

or $x=12$

$\therefore$ T he rational number $=\frac{12+2}{2 \times 12}=\frac{14}{24}=\frac{7}{12}$