The punchline rears its worn out head

Thanks, Quick. Your tip about the single digit number for *c* sent me in the right direction. Actually, I see numerous errors now. I also substituted 16 for *b* in my step 2. And steps 1 and 2 are themselves erroneous. I repeatedly followed Soroban's steps in the order that he presented them in, rather than the order that was intended to be followed.

To solve for *c* we use

$\displaystyle \frac{10ab}{9a+b}$

where *a* and *b* are any single digits we want to test. We want the result of the test to be a whole number, rather than a fraction. For instance, going through various combinations, if

*a* = 1 and *b* = 2

$\displaystyle \frac{10 \cdot 1 \cdot 2}{9 \cdot 1 + 2} = \frac{20}{11}$

or if *a* = 1 and *b* = 3

$\displaystyle \frac{10 \cdot 1 \cdot 3}{9 \cdot 1 + 3} = \frac{30}{12} = \frac{\not2 \cdot \not3 \cdot 5}{\not2 \cdot 2 \cdot \not3} = \frac{5}{2}$

or if *a* = 1 and *b* = 4

$\displaystyle \frac{10 \cdot 1 \cdot 4}{9 \cdot 1 + 4} = \frac{40}{13}$

and so on, then our formula fails the test. However, if *a* = 1 and *b* = 6, then

$\displaystyle c = \frac{10 \cdot 1 \cdot 6}{9 \cdot 1 + 6} = \frac{60}{15}$

$\displaystyle = \frac{2 \cdot 2 \cdot \not3 \cdot \not5}{\not3 \cdot \not5} = 4$

and our formula passes the test, because we have a single digit for *c*.

We then apply our solution for *c* to the formula

$\displaystyle \frac{10a+b}{10b+c} = \frac{a}{c}$

This formula results in the butt of the joke, because

$\displaystyle \frac{10 \cdot 1 + 6}{10 \cdot 6 + 4} = \frac{16}{64} = \frac{\not2 \cdot \not2 \cdot \not2 \cdot \not2}{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot 2 \cdot 2} = \frac{1}{4}$

Which means that

$\displaystyle \frac{1\not6}{\not64} = \frac{1}{4}$

might be an incorrect method of simplifying the fraction, but it certainly does not give the wrong answer.

Another fraction in Soroban's list,

$\displaystyle \frac{1\!\!\!\not{9}}{\not{9}5} = \frac{1}{5}$

might show an incorrect way of simplifying, but if we solve for *c*

$\displaystyle \frac{10 \cdot 1 \cdot 9}{9 \cdot 1 + 9} = \frac{90}{18}$

$\displaystyle = \frac{\not2 \cdot 5 \cdot \not3 \cdot \not3}{\not2 \cdot \not3 \cdot \not3} = 5$

and apply the formula

$\displaystyle \frac{10 \cdot 1 + 9}{10 \cdot 9 + 5} = \frac{19}{95}$

$\displaystyle \frac{19}{5 \cdot 19} = \frac{1}{5}$

And so on. Well I really drilled the humor out of the joke, but at least I get the joke now. :)

Don't all clap at once. :rolleyes: