To mark up in retail by $20$%, do I add $0.20$ times the original cost, or divide by $0.80$? Why is it that when I take a cost of say $\$15.60$ and want to mark the item up at retail 20% that I'm being told two different ways with two different answers? 
The first way (my way) would be to multiply the original cost by $.20$ to get $20$%, then simply add that number to original cost. The second way is to take the original cost and divide it by $.80$ and the number you get ends up being the cost. But this second way is more than the first. The second way almost adds an extra $5$%. 
Why are these different, and which is the proper way to mark up by a percentage?

Simply because adding $0.2x$ to $x$, giving you $1.2x$, is not the same thing as $x/0.8$. This is because
$$1/0.8 = 1.25$$
You're not increasing by the same factor, the second option would be $1.25x$ instead of the very correct $1.2x$.
