For part a, I got half of the answer which is 222 contracts. However, I don’t understand why they complete the second part and arrive at 242 contracts for a total of 464. Don’t get me wrong, I understand both of the calculations. What I don’t understand is the two components and when we should use the first equation, when to use the second and when to use the combination.

there are two parts to the question … one modifies the amount of the portfolio - so you buy some futures to first get the amount (reallocated) - then you buy some more to adjust the duration to the new desired duration.

First you change the portfolio’s beta, then you change the portfolio’s weight. First target Beta = .9 V = 180000 (Note numbers aren’t exactly the one from the question). Your portfolio now has a beta of .9. Now you want to reduce dollar exposure. target beta 0, V = -28000.