Are There Other Strange Division Patterns Besides Cancelling 6's?

[madah12]

Homework Statement

how come that 16/64=.25
166/664=.25
1666/6664=.25
and any 1then n number of sixes / the same number of sixes then 4 = .25
same thing with 19 / 95

is there other strange division patterns?

 

[statdad]

(1 followed by n sixes) * 4 = (string of n sixes followed by 4)

 

[Mark44]

$$\frac{166}{664} = \frac{16 * 13.75}{64 * 13.75} = \frac{16}{64} = \frac{1}{4}$$
$$\frac{1666}{6664} = \frac{16 * 104.375}{64 * 104.375} = \frac{16}{64} = \frac{1}{4}$$

You could probably show the same sort of thing is happening with 19/95, 199/995, and so on.

In all these examples the numerator and denominator of 16/64 are being multiplied by the same number, yielding a fraction that is equal to 16/64 = 1/4. Same with 19/95 and the others.

 

[sjb-2812]

Several things like this. Consider 26/65 (or maybe even 49/98). Consider fractions like (10a+b)/(10b+c) = a/c or similar

 

[Mark44]

Here's one I always liked-just cancel the 6's.
$$\frac{1\rlap{/}6}{\rlap{/}64} \;=\;\frac{1}{4}$$