Quebec went to 9.975 because they used to have "stacked" taxes, where you would charge GST first, then charge QST on the combined total as a second step. (yes, they QST taxed the GST) A few years back they went to a "sane" system of only charging GST and QST on the base amount, however they didn't want to lose money so they adjusted QST to 9.975 so the final tax collected would be the same as before.

Source: I write point of sale systems in Canada