What you and your instructor, not to mention a lot of folks I've come across, is you've confused what happens in a transformer. When Power stays the same, and in a transformer it does, and you increase the Voltage, then your Current must decrease proportionally (P=E*I). Voltage and Current then become indirectly proportional.
You are confusing the primary to secondary ratio of a transformer with how a transformer responds to a change in voltage. Yes, a transformer follows the laws of conservation of energy and the power out equals the power in (minus losses), but when examining a change in voltage on the windings, the transformer still follows Ohm's law. (Ohm's law remains true--hence its called a "law"--even when dealing with both resistive and reactive circuits. It just takes a slightly more complex form.)
So no, when you reduce the voltage on a transformer, the current through the windings (on both sides) will also be reduced--unless you change the load, but that is beyond the scope of the question--however, it is related to the second part of this posting.
a motor is anything but simple, it is inductive, and may often be inductive-capacitive. the motor has counter emf which resists the change of the source, and the answer that the current goes down when voltage goes up by your instructor was intended to apply to an inductive circuit like a motor.
As you did, I too suspected that the question was related to a motor. However, your explanation for the current and voltage being inversely related in a motor due to it being an inductive element is incorrect. If that were true, then all inductive devices would violate Ohm's Law--which they don't.
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The issue with motors getting reduced voltage has confused many people I have met that are nevertheless pretty knowledgeable with electricity. The problem is that the increase in current is not directly related to the reduction in voltage, per se.
When you power a motor in an under-voltage condition, as long as the RPM of the motor does not change, then the current will reduce proportionally--following Ohm's Law. However, in reality, when the voltage to a motor is reduced, the motor loses power and its RPM drops. The current in an induction motor winding is related to the RPM of the motor's rotor, as the result of back emf.
So the bottom line of giving a motor an under-voltage condition is that the motor will spin slower, and it is this lower RPM that results in a lower impedance, and therefore, a higher current. It still follows Ohm's Law, but the impedance is dependent on the frequency of the rotor, and is not fixed like we are accustomed to seeing in other circuits.
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Oh, by the way. I have come across some people that are so accustomed to seeing this inverse proportionality with motors, that they have mistakenly come to believe that Ohm's Law, and all circuits, behave this way. It is entirely possible that this test question was derived from this common mistake in thought.
You have drawn the assumption that the motor output is constant and it's speed doesn't change. What then is the mechanism that causes the current to increase? Do the electrons have little minds that tell them, "Hey this guy put in a Form-1.602 Request for constant power, so we gotta hurry up." 
