Answer
The earth will be about 2680 years older at the end of the astronauts journey.
Work Step by Step
We can use the time dilation equation to solve for the time elapsed on earth because we know the trip length was 120 years and the given relative velocity is 0.9990c. So;
$\Delta t=\Delta t_0 / \sqrt{1-(v/c)^2 }$
$\Delta t=120 years / \sqrt{1-(0.9990c/c)^2 }$
$\Delta t \approx 2680 years$
The earth will be about 2680 years older at the end of the astronauts journey.
