Let p/q be the original fraction. We know that:
[tex]\dfrac{p}{q+1}=\dfrac{1}{2},\quad \dfrac{p+1}{q}=1[/tex]
We deduce that:
[tex]2p=q+1,\quad p+1=q[/tex]
If we plug q=p+1 in the first equation, we have
[tex]2p=p+1+1 \iff p=2[/tex]
and we deduce q=3.
In fact, if we start from 2/3, adding 1 to the denominator gives 2/4=1/2
Adding 1 to the numerator gives 3/3=1.
