Why is multiplying or dividing the numerator and denominator by the same nonzero numberthe same as multiplying or dividing the fraction by 1?

Answer 1

That is how equivalent fractions are made.

When fractions are multiplied together, the numerators are multiplied together and the denominators are multiplied together.

When the numerator and denominator of a fraction are both multiplied by the same non-zero number, the effect is of multiplying the original fraction by the non-zero number over the non-zero number, but any number over itself is 1, thus it is the same as multiplying by 1:

let a/b be a fraction and let n be the non-zero number, then:

(a x n)/(b x n) = a/b x n/n = a/b x 1 since any number divided by itself is 1.

Dividing by the same non-zero number is the same effect. However, in this case, it is a slightly different proof:

For this, it needs to be remembered that a fraction a/b is the same as a ÷ b.

And to divide by a fraction, the divisor is inverted (numerator and denominator swap places) and then multiplied.

let a/b be a fraction and n be the non-zero number, then:

(a ÷ n)/(b ÷ n) = (a ÷ n) ÷ (b ÷ n)

= a/n ÷ b/n

= a/n x n/b

= (a x n)/(n x b)

= (a x n)/(b x n) (since multiplication is commutative)

= a/b x n/n

= a/b x 1 (as before).