##### adam, Noodle Person

So, if the track is 1/8 of a mile, let's call every lap a "one-eighth mile" run. We know John ran 24 laps, or that he ran 24 "one-eighth miles," just consecutive, one right after another. Let's stop worrying about rates or tricks or math for a second, and just ask: how many real miles is 24 "one-eighth" miles? We know it's less than 24---a lot less, since you have to go around 8 times just to get to 1 mile. Well wait, if we go around 8 times, we get 1 mile. That means if we go around 2*8, or 16 times, we get 2 miles; And let's just think to the next full mile---if we go 3*8, or 24 times, we get 3 miles. He did go around 24 times, so he must have run 3 miles on a 1/8 track.

Division and multiplication are inverses of each other. So we solved this by looking for an intuition for how many full miles corresponded to how many laps, with a bunch of steps of multiplication. But you can cut right to the chase and solve it faster with division---24 laps * 1 mile per 8 laps, means:

total distance = 24 Lap (1 mi / 8 Lap) total distance = 24/8 total distance = 3