Final answer:
The cranial nerve responsible for transmitting gustatory information to the brain from the posterior two-thirds of the tongue is the glossopharyngeal (cranial nerve IX).
Step-by-step explanation:
Gustatory information, or taste perception, reaches the brain via several cranial nerves. Among these, the cranial nerve responsible for transmitting taste sensations from the posterior two-thirds of the tongue is the glossopharyngeal cranial nerve (cranial nerve IX). Moreover, the facial nerve (cranial nerve VII) carries taste sensations from the anterior third of the tongue. To a lesser extent, the vagus nerve (cranial nerve X) is involved with taste sensations in the extreme posterior part of the tongue.
In the context of the student's question about which cranial nerve transmits gustatory information to the brain, the correct answer among the provided options is number 3) cranial nerve IX, the glossopharyngeal nerve. Although both the facial (VII) and glossopharyngeal (IX) nerves are involved in taste sensation, the question seems to focus on the primary nerve associated with the gustatory pathway described.