The answer to this question can be confusing because the very first line of the novel is, "When he was nearly thirteen, my brother got his arm badly broken at the elbow" (3). Hence, one could argue that Jem is thirteen years old at the beginning of the story, but this could be simply a device which foreshadows what happens to Jem as revealed at the end of the book. After mentioning Jem's broken elbow, Scout then jumps as far as she can go to give the reader the history of the Finch name in Maycomb County. It's almost like she starts with the end of the story and then diverts the attention of the reader back to where the story really begins.
After discussing her ancestor Simon Finch, Scout brings the reader up to date by describing Atticus's life, Finch's Landing, and their cook Calpurnia. Then she mentions that their mother died when Scout was two, which would have made Jem six at the time of her death. That is the end of the Finch history lesson and where the story begins:
"When I was almost six and Jem was nearly ten, our summertime boundaries (within calling distance of Calpurnia) were Mrs. Henry Lafayette Dubose's house two doors to the north of us, and the Radley Place three doors to the south" (6).
The timeline continues from this point forward, so many would think that Jem is nine, nearly ten, at the beginning of the book, even though two other ages were discussed previous to this pivotal paragraph. It is from this point on that a present time period is established as Jem and Scout meet Dill, and an uninterrupted narration follows chronologically thereafter. Therefore, Jem is mentioned to have had a broken arm at age thirteen in the opening line of the book, and he is then mentioned to be four years older than Scout when their mother dies when she is two years old, but the full narration of the story, other than background information, starts with Jem being nine (almost ten) years old.