Problem Description

The fraction ⁴⁹/₉₈ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that ⁴⁹/₉₈ = ⁴/₈, which is correct, is obtained by cancelling the 9s.

We shall consider fractions like, ³⁰/₅₀ = ³/₅, to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

Analysis

This is curious problem and was able to solve it in just a few lines and 84 iterations. Just generated all fractions with 2 digit numerators and denominators and ignored ‘trivial’ examples. Used symmetry to consider distinct fractions – which was really not necessary, but…why not.

Solution

Runs < 1 second in Python.

d = 1
for i in xrange(1, 10):
  for j in xrange(1, i):
    for k in xrange(1, j):
      ki = k*10 + i
      ij = float(i)*10 + j
      if (ij/ki == float(j)/k):
        d *= ij/ki
print "Answer to PE33 = ",d

Comments

Added the float function to keep calculations from converting to integers.