I was merrily ninjaing away in class the other day, teaching binomial expansion my favourite way, and found that one of the rows gave the slightly awful \(\\frac{-5}{81} \\cdot \\frac{1}{256} \\cdot 27\)

My student, quite understandably, reached for a calculator. And because one of the main attributes of a mathematical ninja is arseholery, I tried to beat her to the answer.

Naturally, I said “-5 over 768” at about the same time as she said “that’s a crazy big number on the bottom… 20,736…” and I realised there was a Fundamental Difference in the way ninjas and regular people process fractions.

What regular people do is, type the fractions in one at a time.

Cancelling fractions - the ninja way

What ninjas do is, simplify before anything else. One of the many nice things about fractions (and numbers in general) is, it doesn’t matter what order you multiply things in — which means, when you have a big times sum, you get to cancel any factor on top with the same factor on the bottom. And once a ninja spots that 81 is $3 \times 27$, it becomes comparatively easy — just -5 on top and $3 \times 256$ on the bottom (($250 \times 3 = 750; 6 \times 3 = 18$))

The reason I bring it up: if you want a simple way to flabberghast your classmates, try cross-cancelling fractions and churn these numbers out in your head.

(You can estimate the decimal as well — multiply top and bottom by 4 to get $\frac{20}{3072}$, which is $\frac{2}{300}$ less 2.4% — which is 0.06 recurring less 0.0016, making about 0.0065. The calculator says… 0.00651).