Question

Suppose the a mountain bike that sells for $419 is on sale for 20.5% off.

Answer 1

Regular price:  $419
Sale price:  (1-0.205)($419) = $333.11

The bike will cost $333.11 before tax.  You haven't shared the tax rate, but I can at least tell you that the cost of the bike after tax will be 

(1.00 + rate as a decimal fraction)*$333.11.

Example:  If the tax rate is 7.5%, then the bike will cost (1.00+0.075)*($333.11).