Why do you have to multiply or divide the numerator or denominator of the fraction to find an equivalent fraction?

Answer 1

Equivalent fraction have the same value.

Any whole number divided by itself is one

Multiplying any number by 1 results in the same number.

Thus when you multiply the numerator and denominator of a fraction by the same number, it leaves the value of the fraction the same, but the numerator and denominator will be different, creating an equivalent fraction.

For example if you have a Pizza and cut it in half, and take one of those pieces you will have 1/2 of the pizza. However, if you now cut the half you have and the half left on the plate into half again (each piece will be exactly the same size, a quarter of the pizza), you will still have half the pizza, but it will be made up of 2 of the 4 pieces of the pizza - you will have 2/4 of the pizza; thus 1/2 and 2/4 are equivalent fractions.

Now, looking at those fractions again, what happens if you multiply the 1/2 by 2/2 (which is the same as multiplying by 2 ÷ 2 = 1):

1/2 x 2/2 = (1x2)/(2x2) = 2/4

so 1/2 and 2/4 are equivalent fractions.

When dividing the numerator and denominator by the same number, this is really the same, but just going from a fraction with large numerator and denominator to one with smaller numerator and denominator - the reverse of the example above (sticking the pieces of pizza together again).