|19-year old Elijah Wood on the set of The Fellowship of the Ring|
A passage from Chapter 2, "The Shadow of the Past," holds the key.
As time went on [after the party], people began to notice that Frodo also showed signs of good 'preservation': outwardly he retained the appearance of a robust and energetic hobbit just out his tweens... but it was not until Frodo approached the usually more sober age of fifty that they began to think it queer.Tweens in the Hobbit vernacular is described in Chapter 1 as "the irresponsible twenties between childhood and coming of age at thirty-three." Thus, when Frodo left home at the age of 50 he looked more like a Hobbit in his mid-thirties. Because of the power of the Ring (which he inherited from Bilbo on his 33rd birthday), Frodo's physical aging appeared to stop (at least outwardly).
Does this justify Elijah Wood's casting in the films from a book standpoint? Not quite - although it does help somewhat. Instead of appearing thirty years too young in the films Frodo only looks about fifteen years shy of the mark.
We must try to not be too critical, however. From a producer's standpoint there were other reasons to cast such a young actor. Primarily, they wanted someone who the target audience (teens and twenty-year olds) could connect with. They perhaps gained better box-office numbers, but an interesting and enlightening aspect of the rich relationship between the hobbits from the book was altered. In the movies, Frodo, Sam, Merry and Pippin are close in age, but in the book Frodo is fifty while his friends and cousins are all younger than forty. Throughout the story the three younger companions reach out to and support an older hobbit and friend that assumes the role of an elder in many ways, an element that is lost in the films when the character is both portrayed as physically and literally younger.