Coincidence of 1/5/3 Being Equivalent to 3/5?

Question

I was doing a math problem and realised that;

$\frac 1{\frac 53}$ was equal to $\frac35$

Is this purely a coincidence or is there some way to prove that $\frac 1{\frac 53}$ is equal to $\frac35$ ?

Any help would be appreciated!

Answer 1

Assume that $p$ and $q$ are nonzero numbers. What is $1/(p/q)$?

In other words, what number should we multiply by $p/q$ to obtain $1$?

Clearly, $q/p$ works: $$ \frac{p}{q}\cdot \frac{q}{p} =1. $$

So $1/(p/q) = q/p$.

Answer 2

First,

$$\frac{\frac 15}3=\frac{1}{15}$$

But I suppose you meant:

$$\frac{1}{\frac{5}{3}}=\frac{3}{5}$$

And yes, that is no coincidence, since:

$$\frac{3}{5}\cdot \frac{5}{3}=1$$

And I think you can see how this works for any ratio.