Why does the Denominator of the Denominator go to the Numerator?

Question

Quite blindly I've learnt a basic rule about fractions:

The Denominator of the Denominator goes to the numerator.

I'm confused about it and I'll give an example as to why. Imagine the following:

1/2/2

Now, if the denominator of the denominator went to the numerator this would be 2/2 which is 1
and that's the what everyone agrees upon.

But what I feel is that since 1/2 is 0.5 ,

1/2/2 = 0.5/2 = 0.25

So, I hope you can see why I feel confused.
_{^{I apologize if this question seems stupid}}

Answer 1

There are two confusions here. One is that the expression 1/2/3, when there are no parentheses, is defined to be (1/2)/3, which is different from the expression 1/(2/3), and that is why you got the wrong answer in your post. The second is why "the denominator of the denominator goes to the numerator", and it is not something you need to memorize, instead you can do the following

$$\dfrac1{\left(\frac23\right)} = \dfrac1{\left(\frac23\right)}\cdot 1 = \dfrac1{\left(\frac23\right)}\cdot \dfrac33 = \dfrac{1\cdot 3}{\left(\frac23\right)\cdot 3} = \dfrac{3}{\frac{2}3\cdot 3} = \dfrac32.$$

That is, if the denominator in the denominator is 3, just multiply the entire fraction by $\frac33$, and this will cancel the denominator in the denominator.

Answer 2

You need to be careful with the order of operations:

$$ 1/2/2 = (1/2)/2 = 1/4 $$ while $$ 1/(2/2) = 2/2 = 1 $$

Answer 3

Just convert such divisions into multiplication like this, to avoid confusion.

$$\dfrac{\dfrac{k}{m}}{\dfrac{p}{q}}=\dfrac{k}{m} \cdot \dfrac{q}{p}$$

Answer 4

Hmmm. I don't have much to add. My precursors have covered it pretty good. But, "in a word", multiplication and division are inverse operations. So, division by x and multiplication by the reciprocal of x, $\frac {1}{x} $, are the same. Finally, the reciprocal of $\frac {a}{b} $ is $\frac {b}{a} $. It can be a little confusing. Just follow it through. ..