special relativity - Time dilation all messed up!

There is a problem with my logic and I cannot seem to point out where. There's a rocket ship travelling at close-to-c speed v without any acceleration (hypothetically), and there is an observer AA with a clock A on earth, and there's another observer on the rocket BB with a clock B and these two clocks were initially in sync when the rocket was at rest with a FoR (frame of reference) attached to the earth. Now, this rocket's moving and AA tells b is running slower than A, and is running slower by a factor of $ \gamma $ where $$ \gamma = 1 /(1-v^2/c^2)^{1/2} $$ $$ t_a/t_b = \gamma $$ where v is the relative velocity between the two i.e the earth and the rocket! That would mean the time elapsed on A is greater than that on B but this will happen only in the FoR of AA? So $t_b$ in this equation must be the time on B as observed by AA? Is this correct? What do the terms mean in the equations? If the symmetry holds and BB doesn't accelerate, then BB could say that $$ t_b/t_a = \gamma $$ right? where $t_b$ and $t_a$ are the times on B and A with respect to FoR of BB? but I was solving this problem and I took the earth FoR of A, but the prof took the rocket FoR of B? Like how will I know which FoR to solve the problem from? It'd greatly help if the terms in all the above equations were laid down neatly! DO we even need these FoRs?? Because in all the solved problems the prof is't specifying any and is using random ones! Please help!!!!

This is the question where i messed up. The first rocket bound for Alpha Centauri leaves Earth at a velocity (3/5)c. To commemorate the ten year anniversary of the launch, the nations of Earth hold a grand celebration in which they shoot a powerful laser, shaped like a peace sign, toward the ship.

1. According to Earth clocks, how long after the launch(of the rocket) does the rocket crew first see the celebratory laser light?

This must be 25 years. My reasoning is: If v = 3c/5 10v+ vt= ct where t is time taken by the light to reach the rocket from earth as calculated from earth.. and I solved that for t. and added 10 years to that because the time starts at the launch of the rocket!

2. According to clocks on the rocket, how long after the launch does the rocket crew first see the celebratory laser light?

This is 20 years. Here, I say: If it takes 25 years as observed by clocks on earth for the laser to reach the rocket, what should be the corresponding time as seen on a clock on the rocket? Using the formula:

25 = $\gamma$t where $\gamma$= 5/4

solved for t!

3. According to the rocket crew, how many years had elapsed on the rocket's clocks when the nations of Earth held the celebration? That is, based on the rocket crews' post-processing to determine when the events responsible for their observations took place, how many years have passed on the rocket's clocks when the nations of Earth hold the celebration?

For this, I did the following: 10 years on earth = T years on rocket ship where T must be lesser than 10 as observed from Earth FoR! Therefore, T= 4(10)/5 years = 8 years! But, prof says, 10 years in earth = T years on rocket ship where T must be GREATER than 10 as observed from the Rocket FoR??? Therefore, T = 10(5/4) years = 12.5 years!!

What does this question actually want?