Intuitively, why does $\dfrac{a}{c} = \dfrac{1}{\frac{c}{a}}$?

Question

For intuition, I reference objects. Imagine making a dessert with: $a$ as apples and $c$ as chestnuts.
Question. How and why is

$\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{a}} \qquad (4) \qquad ?$

I ask only about intuition; please omit formal arguments and proofs (eg: Intuition is not generated by the explanation that rationalising (4)'s denominator produces (3)).

Answer 1

The fraction $\frac{a}c$ is the number of apples per chestnut, and $\frac{c}a$ is the number of chestnuts per apple.

Suppose that you have $x$ chestnuts per apple, where $x$ can be any positive real number. No matter what $x$ is, each one of those $x$ chestnuts must be getting $\frac1x$ of an apple, so there are $\frac1x$ apples per chestnut.

Thus, in the particular case that $x=\frac{c}a$, it must be the case that

$$\frac{a}c=\frac1{\frac{c}a}\;.$$

Answer 2

I think the best way to think about this is as a nested fraction. Let's let $k=c/a$.Your number is

$$a/c=1/(c/a)=1/k=1/\text{chestnuts per apple}$$

Intuitively, you can think about this in one of two ways as:

1. the flip (i.e. Inverse) of the thing on the bottom (making it the number of apples per chestnut) or

2. you have one apple and you can feed people at a rate of $k$ chestnuts per apple, and so if you have to feed $1$ chestnut, how much of an apple do you need? Reminder that the equation is for feeding people is $c=ka$.

Answer 3

You're making it less intuitive by introducing apples and chestnuts.

Generally, "$\frac 1x$" is that number by which you need to multiply $x$ in order to obtain the product $1$.

So, isn't it clear that $\frac bc$ is the correct candidate for $\frac 1{(\frac cb)}$, since $\frac cb \times \frac bc=1$?

Addendum: There's nothing special about the "$1$". So, even more generally, "$\frac px$" is that number by which you need to multiply $x$ in order to obtain the product $p$.

This is the very definition of division.

Answer 4

I believe that your ratio in (3) represents how many apples per child. Then $\frac{c}{a}$ is the ratio of how many children eat the same apple. Finally, your (4) I guess it is the ratio when comparing the situation: $1$ apple is eaten by $1$ child, which is $\frac{1}{1}$, with your current ratio of children per apple. Does that make sense? hm..