If there are four places on the same meridian 500 km. apart and the local time at one place is 12.00 noon, what will be the time at the three other places?

Local time is determined by the longitude of a place. The Earth rotates 360° in 24 hours, meaning it passes through 15° in one hour or 1° every 4 minutes [c1][c2]. A meridian is a line of longitude that runs from the North Pole to the South Pole [t1]. All places located on the same meridian share the same longitude [c5]. Since local solar time is calculated based on the position of the sun relative to a specific meridian, every point on that meridian experiences noon (when the sun is at its highest point) at the exact same moment [t1][t4]. Therefore, if four places are on the same meridian, even if they are 500 km apart (which implies a change in latitude, not longitude), their local time will be identical [c5][t2]. Thus, if it is 12:00 noon at one place, it is 12:00 noon at all other places on that same meridian.

Sources

1. [1] Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.) > Chapter 18: Latitudes and Longitudes > Longitude and Time > p. 243
2. [2] Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.) > Chapter 2: The Earth's Crust > Longitude and Time > p. 11
3. [3] Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.) > Chapter 2: The Earth's Crust > Sample Objective Ouestions > p. 16
4. [4] https://oceanservice.noaa.gov/facts/longitude.html