Fraction is A/B. (A-1)/(B-2)=2/3, so 3A-3=2B-4, 3A+1=2B so B=(3A+1)/2. A must be an odd number so that 3A+1 is divisible by 2. B>2 so that we don't divide by zero. This means that A can't be 1. A/B<1 to be a proper fraction, so A<B. But B=(3A+1)/2 so A<(3A+1)/2, 2A<3A+1, A+1>0, A>-1. Well, we already had A>1 and odd.
A few examples:
A=3, fraction=3/5
A=5, fraction=5/8
A=7, fraction=7/11
There are many fractions satisfying the requirement, but 3/5 gives 2/3 (no cancelling necessary).