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Because multiplying or dividing them by the same NON-ZERO number does not alter their ratio.

Q: Why is multiplying or dividing the numerator and the denominator by the same number the same as multiplying or dividing the fraction by 1?

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Dividing a friction is multiplying by its reciprocal. The reciprocal of a fraction is the numerator and the denominator switched around.

Because doing so is equivalent to multiplying or dividing by x/x, which can be cancelled down to 1.

by dividing the numerator by the denominator

Yes. One method for dividing fractions is to multiply the numerator fraction by the reciprocal of the denominator fraction.

123.456 is changed into fraction by multiplying and dividing it by a number such that both denominator and numerator don't contain decimal. Multiplying and dividing by 100, we get 123.456 as 123456/100.

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Dividing a friction is multiplying by its reciprocal. The reciprocal of a fraction is the numerator and the denominator switched around.

Because doing so is equivalent to multiplying or dividing by x/x, which can be cancelled down to 1.

Yes. One method for dividing fractions is to multiply the numerator fraction by the reciprocal of the denominator fraction.

by dividing the numerator by the denominator

123.456 is changed into fraction by multiplying and dividing it by a number such that both denominator and numerator don't contain decimal. Multiplying and dividing by 100, we get 123.456 as 123456/100.

because of mathematical equivalence: it doesn't change the result

Dividing the numerator and denominator by a common factor is how you simplify a fraction.

The same way as multiplying a whole number by an ordinary fraction: Multiply the numerator (top number) by the whole number and put it over the original denominator (bottom number). This can then be simplified by dividing the numerator and denominator by their highest common factor; or it can be converted into a decimal (which could be a whole number) by dividing the numerator by the denominator.

by dividing both numerator and denominator

You will get a rational fraction.

That is correct. It is easier to simplify the fraction before multiplying all the factors in the numerator and the denominator.That is correct. It is easier to simplify the fraction before multiplying all the factors in the numerator and the denominator.That is correct. It is easier to simplify the fraction before multiplying all the factors in the numerator and the denominator.That is correct. It is easier to simplify the fraction before multiplying all the factors in the numerator and the denominator.

By dividing both the numerator and denominator by the same number is like dividing by 1.