Too Busy For Words - the PaulWay Blog

Thu 18th Nov, 2010

Fractional Distillation

One of the minor amusements XKCD unleashed 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.

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