Train A leaves station A for station B at 4 p.m. at an average speed of 80 km/h. At 8 p.m., another train leaves station A for station B on a parallel track at an average speed of 120 km/h. At what distance (in km) from station A will the second train overtake the first?

Introduction / Context:
This is a classic chasing problem with two trains starting from the same station at different times but travelling at different speeds. The first train leaves earlier at a lower speed, and the faster second train starts later and eventually overtakes it. We need to find the distance from the starting station where this overtaking occurs.

Given Data / Assumptions:

• First train (Train A) speed = 80 km/h.
• Second train speed = 120 km/h.
• Train A departs at 4 p.m.
• Second train departs at 8 p.m., which is 4 hours later.
• Both trains travel from station A towards station B on parallel tracks at constant speeds.

Concept / Approach:
Let T be the time in hours after 4 p.m. when the second train catches the first. Train A travels the entire T hours, while the second train travels only T minus 4 hours because it starts later. At the catching moment, the distances from A must be equal. We equate these distances to form an equation in T, solve for T, and then compute the distance from station A using Train A speed.

Step-by-Step Solution:
Step 1: Let T be the time in hours after 4 p.m. when the second train overtakes the first. Step 2: Distance travelled by Train A = 80 * T km. Step 3: The second train starts at 8 p.m., so its travel time is T - 4 hours. Step 4: Distance travelled by second train = 120 * (T - 4) km. Step 5: At overtaking, distances from A are equal, so 80 * T = 120 * (T - 4). Step 6: Simplify: 80T = 120T - 480, which gives 40T = 480. Step 7: Solve for T: T = 480 / 40 = 12 hours after 4 p.m. Step 8: Distance from A when they meet = 80 * 12 = 960 km.

Verification / Alternative check:
At T = 12 hours, time for Train A is 12 hours, and for the second train it is 12 - 4 = 8 hours. Distances are 80 * 12 = 960 km for Train A and 120 * 8 = 960 km for the second train. Both distances are equal, confirming they meet 960 km from station A. The result fully agrees with the conditions given in the question.

Why Other Options Are Wrong:

• 900 km, 940 km, and 980 km all correspond to times that do not satisfy the relation between the travel times of the two trains and their speeds.
• For example, 900 km would imply T = 900 / 80 which does not work correctly when we test it in the equation 80T = 120(T - 4).

Common Pitfalls:
Students sometimes forget that the second train travels for fewer hours. Using the same time variable for both trains without subtracting 4 hours for the later start leads to incorrect equations. Another common error is to use relative speed directly over the entire time interval instead of considering the head start that Train A has before the second train begins its journey.

Final Answer:
The second train will overtake the first train 960 km from station A.