When professor Levik was very young he didn't care too much for mathematics, especially fractions. One day his teacher asked him to find the smaller of 2/5 and 3/7 and he jumped at what he though was a shortcut in solving the problem. He replaced 2/5 with 2/3 (2/(5-2)) and replaced 3/7 with 3/4 (3/(7-3)). He then replaced each of the two new fractions with 2/1 (2/(3-2)) and 3/1 (3/(4-3)), respectively, and concluded that the first fraction, 2/5, was the smaller of the two.

Was young Levik's method valid or was this case a lucky fluke?

This method works fine, as long as we stick to positive numerators and denominators.

if a/b < c/d
then b/a > d/c  (if a,b,c,d > 0)
then (b/a) - 1 > (d/c -1)
then (b-a)/a > (d-c)/c
then a/(b-a) < c/(d-c) (if (b-a) and (d-c) > 0)

so the method works, and it is ok to subtract the numerator from the denominator before comparing!

Posted by Steve Herman on 2005-04-01 15:28:26