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Solution 1:

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

$\begin{align*} \sqrt{(8x)^2+(6x)^2}&=60\quad\Rightarrow\\ \sqrt{100x^2}&=60\quad\Rightarrow\\ 10x&=60\quad\Rightarrow\\ x&=6. \end{align*}$

They are 60 miles apart after 6 hours.

Solution 2:

We see that 6 and 8 are part of a multiple of the Pythagorean triple (3, 4, 5). We have 2(3, 4, 5) = (6, 8, 10). So after Adam and Simon travel $x$ hours, the right triangle has sidelengths (6x, 8x, 10x). For the hypotenuse (their distance apart) to be 60, we have to multiply (6, 8, 10) by 6, so x = 6 and they have to travel 6 hours.