#### Answer

Aaron catches up with Andy after $\frac{2}{3}$ hours.

#### Work Step by Step

Let t represent the time it takes Aaron to catch up with Andy.
Since Aaron starts walking one-half hour later than Andy; this means that Andy walks one-half hour more than Aaron. Since Aaron walks t hours, Andy walks t + 0.5 hours.
Andy walks at 2 miles per hour, for a distance of (t + 0.5) $\times$ 2 = 2t + 1 miles.
Aaron walks at 3.5 miles per hour, for a distance of t$\times$ 3.5 = 3.5t miles.
Since Aaron catch up with Andy, the distances they travel must be equal. So we can set up the following equation:
2t + 1 = 3.5t
1.5t = 1
Divide both sides by 1.5
t = $\frac{2}{3}$
Aaron catches up with Andy after $\frac{2}{3}$ hours.