jmc

If Adam travels east and Simon travels south, then their paths are perpendicular and their distance apart is the hypotenuse of a right triangle. Let $x$ be the number of hours it takes for Adam and Simon to be $60$ miles apart. Then Adam has traveled $8x$ miles and Simon has traveled $6x$ miles. By the Pythagorean Theorem, we have $$\begin{array}{rl}\sqrt{(8x{)}^2+(6x{)}^2}& =60\Rightarrow \\ \sqrt{100x^2}& =60\Rightarrow \\ 10x& =60\Rightarrow \\ x& =6.\end{array}$$They are $60$ miles apart after $\boxed{6}$ hours.