The Hobbit shows that the events of the Hobbit took place sixty years before Bilbo's grand birthday party which took place in the 'Lord of the Rings: The Fellowship of the Ring' as clearly the openings of both the movies cross each other.
From the movie Bilbo seems to be very young too. However he himself announced that it was his 111th birthday on the party. That leaves him 51 years old when Gandalf meets him for the first time. But on the contrary he shows no sign of old age.
Am I missing something from the book?