Wouldn't the dozenal system have their own fractions with this same kind of problem? Overall, in the grand scheme of things, base 10 and base 12, whichever you use, gets you to the same answers. That said, this guy right here explains the dozenal system so charismatically. Even though the video is almost ten minutes (HA! BASE TEN! EAT THAT!), it goes by so quickly.
I think I was a little unclear above. Let me try this again: The article tries to make a statement about the superiority of base-12 by pointing out that you can divide 12 by 2, 3 and 4 and get integers as result. Then it tries to show how bad base-10 is by dividing ONE (not ten) by 2, 3 and 4. And while that would be a decent argument had the author used ten (only 10 / 2 = 5, the others are still rationals), I think it's WTF worthy to use one instead here.
I'm unconvinced that a base-12 numbering system is particularly better than base-10, but I can explain what the author was trying to say about fractions and decimals. In base-12, one half would be represented as 0.6, one third as 0.4, and one quarter as 0.3. In base-10, the number immediately following the decimal mark is the tenths place, so one half is 0.5, or five tenths; in base-12, that first digit after the decimal mark would be the twelfths place, so one half would be 0.6, for six twelfths.