One of the minor amusements XKCD
upon the world recently was the equation "16/64 = 1/4", derived by
cancelling the sixes from the left hand side. Deriving the formula for
generating other such misleading fractions is relatively banal but I
did it the other day: if the fraction is to be ab/bc = a/c, then the
equation 10ac + bc = 10ab + ac needs to be true. Creating a nine-by-nine
matrix of a and b allows us to derive integer value for b = 9ac / (10a - c).
Thus the only fractions for which this holds are: 16/64 = 1/4, 19/95 = 1/5,
26/65 = 2/5 and the reducible fraction 49/98 = 4/8; and the uninteresting
series where a = b = c beginning 11/11 = 1/1.