Here is the question:
As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of 0.800 c , relative to you. At the instant the spaceracer passes you, both of you start timers at zero.
a) At the instant when you measure that the spaceracer has traveled $1.30×10^8m$ past you, what does the race pilot read on her timer?
I used the time dilation formula and got .325 seconds. This makes sense, because time is running slower for the race pilot spaceship.
b) When the race pilot reads the value calculated in the previous part on her timer, what does she measure to be your distance from her?
Relative to her, you are moving at .8c in the opposite direction. Travelling at this speed for .325 seconds gives $7.8×10^7 m$
c) At the instant when the race pilot reads the value calculated in part A on her timer, what do you read on yours?
NOW HERE IS MY PROBLEM
I realize that from my original calculation in A, that when the faster spacecraft is at .325 seconds, the slower one is at .52. However, I do not understand this. Wouldn't you have to do a second time dilation calculation? With respect to the race pilot, after .325 seconds shouldn't the space utility vehicle have time run slower because the race pilot observes the vehicle moving at .8c in the opposite direction? After applying this calculation, I got .195 seconds. However, it was wrong. Why is the second time dilation calculation wrong for this situation?