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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!

IAmAlsoLearning
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    I believe you mean 1/(5/3); as you've written it, it's (1/5)/3, which is equal to 1/15. And yes, it is a very general rule that 1/(p/q) = q/p; try multiplying both sides by (p/q) to show this. – Steven Stadnicki Mar 12 '18 at 02:42
  • Yes thanks, I see now why it is equal to 3/5. Thank you very much! Sorry about the wrongly worded equation. – IAmAlsoLearning Mar 12 '18 at 02:48
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    Why is this downvoted? It's a sincere question that does not appear to be homework. OP has made an interesting observation and is trying to understand it. It may have been wrongly worded but that was an honest error. – littleO Mar 12 '18 at 02:50

2 Answers2

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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.

Bram28
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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 $1/(p/q) = q/p$.

littleO
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