Fake Cancellations that Work

fakecancellations.md

Did you ever notice that $\frac{16}{64} = \frac{1}{4}$? In fact, $\frac{1\not 6}{\not 64} = \frac{1}{4}$.

What other such examples are there?

In [7]: nummaker = lambda x,y: 10*x+y

In [8]: nummakers = [lambda x,y: nummaker(x,y), lambda x,y: nummaker(y,x)]

In [9]: [(a,b,c,top(a,b), bottom(b,c)) for a in range(1,10) for b in range(1,10) for c in range(1,10) for top in nummakers for bottom in nummakers if top(a,b) != bottom(b,c) if top(a,b)*c == bottom(b,c)*a]
Out[9]:
[(1, 6, 4, 16, 64),
 (1, 9, 5, 19, 95),
 (2, 6, 5, 26, 65),
 (4, 6, 1, 64, 16),
 (4, 9, 8, 49, 98),
 (5, 6, 2, 65, 26),
 (5, 9, 1, 95, 19),
 (8, 9, 4, 98, 49)]

So the complete list of such fake cancellations of two-digit divisions is:

$$ \begin{align*} \frac{1\not6}{\not64} &= \frac14 & \frac{1\not9}{\not95} &= \frac15 & \frac{2\not6}{\not65} &= \frac25 & \frac{4\not9}{\not98} &= \frac48 \\ \frac{\not64}{1\not6} &= \frac41 & \frac{\not95}{1\not9} &= \frac51 & \frac{\not65}{2\not6} &= \frac52 & \frac{\not98}{4\not9} &= \frac84 \\ \end{align*} $$