Spacecraft I, containing students taking a physics exam, approaches the Earth with a speed of 0.7c (relative to the Earth), while spacecraft II, containing professors proctoring the exam, moves at 0.2c (relative to the Earth) directly toward the students. The professors stop the exam after 88.0 min have passed on their clock.

For what time interval does the exam last as measured by the students?

The issue is how do we find the ||\gamma||. In this question, both the students and the professor are moving with velocities relative to the earth. The velocity of the professor relative to the students was not given directly. We need to find that velocity. Set up the reference frame shown below:

The earth is the rest frame set as O and both the professor and the students are traveling relative to the earth. Set the students' frame as O', which is moving with v=0.7c in +x relative to the earth. Now we need to find the velocity (u') of the professor traveling relative to the students. Use Lorentz transformation on velocity (shown in the diagram), we can find u'.

With u' found, we can then take the students' frame and make observation of the professor's time:

Here with u' known, we can use time dilation to find the students' observation of the professor's time. The professor holds the clock and has the proper time. We then have: $$\Delta t'=\gamma \Delta t_p$$