Two cars start simultaneously from opposite ends of a straight main road that is 150 km long. The first car travels 25 km along the main road, then turns right and goes 15 km, then turns left and travels another 25 km before finally returning back to the main road. In the meantime, due to a minor breakdown, the second car has managed to cover only 35 km along the main road. At that moment, what is the distance between the two cars?

Introduction:
This is a direction–sense cum distance problem involving two cars starting from opposite ends of a straight main road. One car leaves the main road, moves along side roads with right and left turns, and then returns to the main road. The other car travels only a short distance along the main road. We must track the path of each car in a coordinate-style way and then compute the distance between their positions when the first car has just returned to the main road and the second car has covered 35 km.

Given Data / Assumptions:
• Length of the main road between the two starting points is 150 km.• Car 1: moves 25 km along the main road, then 15 km after a right turn, then 25 km after a left turn, then returns to the main road.• Car 2: due to a breakdown, travels only 35 km along the main road from its end towards the other car.• We assume the main road is straight and runs East–West, and both cars start at the same time from opposite ends, heading towards each other.

Concept / Approach:
The best method is to place the main road on a coordinate axis. Let the western end be at x = 0 and the eastern end at x = 150. Both cars start respectively at these points and initially travel along the x-axis. Car 1 then leaves the road, moves in a perpendicular direction, and later returns to the road at a certain x-coordinate. Car 2 continues only along the main road. Once we know the final x-coordinates of both cars on the main road at the required instant, the distance between them is simply the difference of those coordinates.

Step-by-Step Solution:
Step 1: Place Car 1 at the western end of the road at (0, 0). It travels 25 km towards the east along the main road to (25, 0).Step 2: At (25, 0), Car 1 turns right. Since it has been travelling east, a right turn takes it towards the south. It travels 15 km to (25, -15).Step 3: From there it turns left. Facing south, a left turn takes it towards the east. It travels a further 25 km to (50, -15).Step 4: It then returns to the main road. The main road is along the x-axis (y = 0), so it must travel 15 km due north (from y = -15 to y = 0). Thus it rejoins the main road at (50, 0).Step 5: Meanwhile, Car 2 starts from the eastern end (150, 0) and runs 35 km west along the same road. Its position is therefore (115, 0).Step 6: At the moment described, Car 1 is at x = 50, y = 0 and Car 2 is at x = 115, y = 0.Step 7: The distance between them is the difference in x-coordinates: 115 - 50 = 65 km.

Verification / Alternative check:
Note that the “out and back” southward detour of Car 1 (down 15 km then back up 15 km) does not change its final position on the main road, which only depends on the east–west components. Also, Car 2 has not left the main road at all. Hence the distance must be along the road and equals 65 km, matching the simple coordinate difference we calculated.

Why Other Options Are Wrong:
Values such as 75 km, 80 km or 85 km would require miscalculating the east–west displacement of Car 1 or Car 2, or mistakenly including the temporary southward movement of Car 1 in the final separation. The distance between the two points on a straight line is always the absolute difference between their coordinates, which is 65 km here.

Common Pitfalls:
Many students get distracted by the extra turns of Car 1 and incorrectly add the 15 km detour to the separation between the cars. Others assume that both cars travel the full 150 km, forgetting that Car 2 moves only 35 km due to a breakdown. Drawing a simple diagram with coordinates helps keep track of each movement without confusion.

Final Answer:
At the described moment, the two cars are separated by a distance of 65 km.