"This relies on a principle "other way" introduc..."

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 $\frac{n}{m} = n \times \frac{1}{m}$".