Remember that auto-transformers have a single winding in series. While some auto-transformers are specifically manufactured as auto-transformers, this problem is "transforming" (haha) a normal two-winding transformer to an autotransformer by connecting one high leg and one low leg together. Since its easier to draw these out, below are two ways to wire up the two-winding transformer. (I picked polarities arbitrarily to demonstrate the effect)

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So you can have any combination of 120 and 480 on one side and 360 and 600 on the other. (I'm going to leave out the 120/480 option out of this convo ... although you could still use those ratios if you really wanted to)

For the apparent power, you can solve it the way described in the the solution, you'll have to be more specific about what you don't understand ... or use Section 4.3.1.7 of the NCEES handbook to find your formula:

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Since Vc = 120V (because this is the winding being used) and Vse = 480V, the winding ratios are Nc=1 and Nse=4.

Set Sw = 45kVA because it is provided

Solve for Sio and you get Sio = 56.25kVA