Since you posted this on PS forum, could you give the answer choices as well?
The reason why I ask is because I can get a range for the possible answer, but not an exact number.
To get the upper limit, we can use the fact that the perimeters of the two triangles are equal and since they share the side AD, we can see that AB+BD = AC + DC
and since AB has to be > AC, BD has to be < DC so BD/DC has to be < 1.
for the lower limit, since AD = 3, DC has to be < 3. and since BD + DC = 4, when DC = 3 BD = 1 and BD/DC has to be > 1/3.
Therefore, 1/3 < BD/DC < 1. This is the best I can do with the givens.
If anyone else could shine some light on this problem, I would appreciate it.