In a city, the local shuttle trains start from and arrive at the station at fixed intervals and run at a uniform speed. A boy was walking down the railway track at a certain speed. Every half an hour, a local train overtook him and every 20 minutes, a local train passed him in the opposite direction. Find the time interval between a local train passing a certain point on the railway track and the immediately next local train passing that point in the same direction.

To solve this problem, we need to understand that the frequency of trains passing the boy in both directions is determined by the interval at which the trains leave the station.

1. First, let's consider the trains overtaking the boy. They do so every 30 minutes, which means that a train leaves the station in the same direction as the boy every 30 minutes.

2. Now, let's consider the trains passing the boy in the opposite direction. They do so every 20 minutes, which means that a train leaves the station in the opposite direction every 20 minutes.

3. Since the trains are leaving the station in both directions at these intervals, a train is leaving the station either in the boy's direction or the opposite direction every 10 minutes. This is because the least common multiple (LCM) of 20 and 30 minutes is 60 minutes, and there are 6 intervals of 10 minutes in an hour.

4. Therefore, a train passes a certain point on the track in the same direction every 20 minutes. This is because a train leaves the station in the same direction as the boy every 30 minutes, but a train is leaving the station (either in the same direction or the opposite direction) every 10 minutes. So, the boy will see a train going in the same direction every 20 minutes (30 minutes - 10 minutes = 20 minutes).

So, the time interval between a local train passing a certain point on the railway track and the immediately next local train passing that point in the same direction is 20 minutes.