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.
In part one of To Kill a Mockingbird, Scout tells us that she was six and Jem was nearly ten. So at the beginning of the story, Jem is nine. The story is told to us by Scout, and goes to great lengths to describe the relationship between her and Jem. Although they are close in age, the two couldn't be more different. Jem is a serious boy, while Scout is not afraid to say whatever is on her mind. The story spans over a three year period in the lives of Jem and Scout.
We learn early on all about the Ewells of Maycomb county. We also learn of Boo Radley and Dill. Scout details all of the people who are or will become important in her life. Over the course of the novel, we see the bother and sister bond between the two of them. They fight terribly, but what brother and sister don't fight? Although they fight and Scout can get on Jem's nerves, the two of them are extremely close and will always be there for each other. We see this bond by the end, when Jem and Scout are put to the ultimate test.
Jem is nine years old at the beginning of the novel. Within a few pages of the beginning of Chapter One, Scout begins to reveal details that occured when she was "almost six and Jem was nearly ten." Also, Jem is twelve years old at the beginning of Part Two.
When the novel first begins, Jem is ten years old.
Nine, almost ten at the start.
Twelve in Part Two.