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?

Answer: Two pies!

f

 

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?

 f

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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