Suppose the initial distance between the bike riders was 35 miles. The closing speed is 15 mph + 20 mph = 35 mph. From this we see that the time required for the riders to meet is 1 hour. The faster rider will cover 20 miles in that time. Thus the meeting point will be 20 miles from the starting point of faster rider. Of course the riders will meet the dog at that point also. The ratio of the distance covered by the faster rider to the initial separation distance is 20/35 = 4/7. The corresponding ratio for the slower rider is 3/7. From this we see that the meeting point is further from the starting point of the faster rider than it is from the starting point of the slower rider. The ratio of these distances is (4/7) / (3/7) = 4/3. This last result holds for any initial separation distance.
This problem is a variation of a classic problem. In the classic problem the posed question is: How much distance did the dog cover in all that running back and forth? The answer is simple. Since the total time is 1 hour, the dog covers 38 miles. The ratio of the dog's travel to that of the faster rider is 38/20 = 19/10 = 1.9. This result holds for any initial separation distance.
