Remember, $~$\\frac{a}{m}$~$ is made of $~$a$~$ copies of pieces of size $~$\\frac{1}{m}$~$; so what we do is cut all of the $~$\\frac{1}{m}$~$\-chunks individually into $~$n$~$ pieces, and then give everyone $~$a$~$ of the little pieces we've made\. But "cut a $~$\\frac{1}{m}$~$\-chunk into $~$n$~$ pieces" is just "cut an apple into $~$n$~$ pieces, but instead of doing it to one apple, do it to a $~$\\frac{1}{m}$~$\-chunk": that is, it is $~$\\frac{1}{m} \\times \\frac{1}{n}$~$, or $~$\\frac{1}{m \\times n}$~$\.

This relies on a principle "other way" introduces but, in my opinion, is not explicit enough about: $~$\frac{n}{m} = n \times \frac{1}{m}$~$. Could this be made more explicit at the end of the "other way" section? e.g. "What we've really shown here is that $~$\frac{n}{m} = n \times \frac{1}{m}$~$".

## Comments

Patrick Stevens

I think this actually belongs in the Multiplication article, but you're quite right that I've not been explicit enough. I intend to have a meditation on the various ways that the notation is consistent, but this one doesn't need any division at all so it should appear earlier.