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This is related to the fact that dividing by a number is the same as multiplying with the number's reciprocal.
Dividing by a rational number (other than zero) is simply multiplication by its reciprocal.
Because multiplying is the inverse of dividing
Because multiplying or dividing them by the same NON-ZERO number does not alter their ratio.
Because doing so is equivalent to multiplying or dividing by x/x, which can be cancelled down to 1.