To explain why $\frac{1}{2}=\frac{2}{4}$ I use slices of pizza and show how eating one slice of a pizza cut in half is the same thing as eating two slices of a pizza cut in quarters.
Is there a way to show why $\frac{1}{(1/2)}=2$ using an example like the one above?
On
I know you are looking for something about pizzas but I think fractions in general work better if you work with a definition more akin to the axiomatic definition.
I mean what is $\displaystyle\frac{1}{\pi}$ in terms of pizzas?
I would define $\displaystyle \frac{1}{x}$ as the number, which when multiplied by $x$, gives me 1. This works fine for pizzas because, e.g. $2\times\frac{1}{2}=1$ --- two half slices is the whole.
Now whatever $\displaystyle \frac{1}{\frac{1}{2}}$ is, if I multiply it by $\displaystyle \frac{1}{2}$ I get one so that
$$\frac{1}{\frac{1}{2}}\cdot\frac{1}{2} =1...$$
but this is nothing other than two... (or multiply both sides of the equation on the left by two).
On
Use ratios.
Pizza will be consumed at a rate of one person per one-half pizza.
$$\frac{dP}{d\Pi} = \frac{1\ \textrm{person}}{\frac12\ \textrm{pizza}}.$$
Pizza is delivered at a rate of 1 pizza per day
$$\frac{d\Pi}{dt} = \frac{1\ \textrm{pizza}}{1\ \textrm{day}}.$$
To avoid an excess of pizza, we require at least
$$\frac{dP}{d\Pi}\frac{d\Pi}{dt} = \frac{1\ \textrm{person}}{\frac12\ \textrm{pizza}}\cdot \frac{1\ \textrm{pizza}}{1\ \textrm{day}} = 2 \frac{\textrm{person}}{\textrm{day}}.$$
On
To get over compound fractions extend your first example to $=\frac{\frac12}1$ since it is the same serving as as eating $1/2$ of an entire pizza.
Looked at from the perspective of you feeding me: $\frac42=\frac21$ because if you previously cut the pizza in $4$ slices and give me $2$ I have the same as if you cut it into $2$ slices and give me $1$ and both are the same as taking one entire pizza and giving me $1/2.$ Maybe candy bars are better.
If I eat one pizza in half an hour, that is the same rate as eating two pizzas per hour.