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...My question now is, what is the best way to evaluate the turns ratio? Is it Ns/Np or Np/ Ns?

Consider the primary as the “input” and the secondary as the “output”:-The turns ratio is 500/300 = 1.666666667:1. With 220Volts on the primary (500 turns), the secondary voltage is 220/ratio = 132.In this case the transformer is a “step down” transformer.If used as a “step up” transformer, the ratio would be 300/500 = 0.6:1, and if 132 Volts is applied to the secondary (300 turns), the voltage on the 500 turn coil will be 132/0.6 = 220Volts.This seems a simpler way of calculating.In this case of course the “secondary” is now being used as the “primary”, or input. BE AWARE!! that if 220Volts were applied to the secondary (300 turns) the voltage on the 500 turn winding would be 367 Volts. Also the current will exceed the rating for the coil. It is dangerous to connect mains voltage to the secondary of a step down mains transformerAnother way of calculating is:-The voltage per turn on the primary is 500/220, or 2.273 turns per volt, therefore the secondary voltage will be 300/2.273 = 132volts, ie the “turns per Volt” is the same for both (or even additional) windings on the core.Needless to say, current follows similar rules except in the opposite sense, so current available on the secondary of a “step down” transformer will be greater than that on the primary. I have never heard of a “primary” or a “secondary” transformer

I have never heard of a “primary” or a “secondary” transformer

Sorry, are you asking if "is better" to write "3/5" or to write "5/3"?