If $1$ is added to the denominator of a fraction, the fraction becomes $\frac{1}{2}$. If $1$ is added to the numerator of the fraction. the fraction becomes $1$. Then the fraction is

Let the numerator be n and the denominator be d. Then
$\frac{n}{d+2}=\frac{1}{2}$ ........... (1)
or $2n=d+1$
$\frac{n+1}{d}=1$
or $n+1=d$ ............(2)
Substituting for d in equation (1), we get
$2n=n+1+1$
or $2n-n=2$
or $n=2$
$\therefore d=n+1=2+1=3$
Hence, $\frac{n}{d}=\frac{2}{3}$