## Fractions are Hard!

### 2.4 Getting quirky

Okay! Let’s now be wild.

Let’s make sense of the sharing problem $$\dfrac{1}{\left(\dfrac{1}{2}\right)}$$, distributing one pie among half a boy!

Now $$\dfrac{a}{b}$$, in general, represents the amount of pie each boy receives in a sharing problem. That is, the amount of pie a full individual boy receives.

In $$\dfrac{1}{1/2}$$ we are providing one pie for half a boy. So if each half is assigned one pie, how much pie per (whole) boy is that?

We have:

$$\dfrac{1}{\left(\dfrac{1}{2}\right)}=2$$.

Whoa!

In the same way, distributing one pie to each third of a boy yields  pies for an individual boy:

$$\dfrac{1}{\left(\dfrac{1}{3}\right)}=3$$.

And distributing five pies for every seventh of a boy yields a total of 35 pies for a full boy:

$$\dfrac{5}{\left(\dfrac{1}{7}\right)}=35$$.

 EXERCISE: Make sense of $$\dfrac{2}{\left(\dfrac{2}{3}\right)}$$.

 Challenge question: Two-and-a-half pies are to be shared equally among four-and-a-half boys! How much pie does an individual (whole) boy receive?   It is possible to think our way through this right now, but it is tricky. (If you are up for it, can you develop a philosophically swift way to see your way through this?)

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