Bike is faster. Assume walk speed = 1mph Assume bike speed = 2mph Assume lamp posts are 10 miles apart At 11 hours, it is clear that if both walk, they will be 11 miles away. Let's take the bike case. Assume Fred rides first. At 5 hours, Fred will be 10 miles away at the first post, John will be 5 miles away. John walks for 5 more hours (5 more miles) and gets the bike. During the 5 hours of walking, Fred also was walking for 5 miles. So now we're at 10 hours. Fred is 15 miles away and John is 10 miles away at the first post. John gets the bike and starts riding (@ 2mph). In one hour, Fred is still walking and has walked one mile. John has now ridden 2 miles. So Fred is at 16 miles and John is at 12 miles. This is now hour 11. Remember, at 11 hours walking they were only 11 miles. However, riding one is 16 miles away and the other is 12 miles away. Given that they will NEVER go SLOWER by walking, they are already ahead and will only get further ahead.