Either solution is correct - but one possible problem with purely algebraic solutions is that you might accidentally translate the problem into the wrong equation. Even if you solve it correctly, your answer could be incorrect.
The most intuitive way for me is, $10 off $50 is a 20% discount, and whenever the price before discount increases, the percent discount decreases. So 20% is the answer.
More formally, letting x be the price before discount, the percent discount is (10/x)*100 = 1000/x, which is strictly decreasing in x.