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Think about it
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You can call it whatever you want, as long as it equals 1/2 it's the same number.
So yes, multiplying by 2/2 to make it more intuitively obvious is perfectly valid and a good way to think about it. Most arithmetic tricks are ultimately multiplying by 1 or adding 0 just to make the problem easier to handle.
Oh yeah, I just meant that they said I multiplied by 2, which in my head is 2/1 but I was multiplying by 1. Just trying to be clear.