Question

A space traveller leaves Earth for 10 years at .85c. According to an observer on Earth, how much time has passed?

Answer 1

First of all, you didn't tell us WHO measured the "10 years".

If it was the people on Earth, then 10 years passed according to them.

If it was 10 years on the space traveler's clock,  then the clock in the
OTHER place, like on Earth, is subject to the relativistic 'time dilation'.

If the clocks are moving relative to each other, then the time interval measured
on either clock is equal to the interval measured on the other clock, divided by

       √(1 - v²/c²) .

You said that  v/c  = 0.85 .

v²/c² = (0.85)² = 0.7225

1 - v²/c² =  1 - 0.7225 = 0.2775

√(1 - v²/c²)  =  √0.2775 = 0.5268

If one clock counts up 10 years, then the other one counts up

(10years) / 0.5268 =  18.983 years


I believe that's the way to do this, and I'll gladly take your points,
but let me recommend that you get a second opinion before you
actually take off on your 10-year interstellar mission.