Three cars A, B, and C start from a point at 5:00 p.m., 6:00 p.m., and 7:00 p.m. respectively and travel in the same direction at uniform speeds of 60 km/h, 80 km/h, and x km/h respectively. If all three cars meet at another point at the same instant during their journeys, what is the value of x (in km/h)?

Introduction / Context:
This problem involves three cars starting from the same point at different times and travelling in the same direction at constant speeds. Despite the staggered start times and different speeds, all three meet at some other point at the same moment. We are asked to find the speed of the third car, C. This is a relative time and distance question that uses the idea that the distances travelled by all three cars to the meeting point must be equal at the instant of meeting.

Given Data / Assumptions:
- Car A starts at 5:00 p.m. with speed 60 km/h.
- Car B starts at 6:00 p.m. with speed 80 km/h.
- Car C starts at 7:00 p.m. with speed x km/h (unknown).
- All cars travel in the same direction along a straight road at constant speeds.
- They meet at some point at the same instant during their journeys.

Concept / Approach:
Let the meeting happen at time T, measured in hours after 5:00 p.m. By that time, each car has been in motion for a different duration: A for T hours, B for T - 1 hours (since B starts an hour later), and C for T - 2 hours (since C starts two hours later). The distance travelled by each car to the meeting point must be equal, so we set their distance expressions equal and solve. First we use A and B to find T, then use A and C (or B and C) to find x.

Step-by-Step Solution:
Step 1: Let T be the time in hours after 5:00 p.m. when they all meet.Distance travelled by A: D = 60 * T.Distance travelled by B: D = 80 * (T - 1), since B starts at 6:00 p.m.Step 2: Set A’s and B’s distances equal:60T = 80(T - 1).Solve: 60T = 80T - 80 ⇒ 20T = 80 ⇒ T = 4 hours.Thus, the meeting occurs at 5:00 p.m. + 4 hours = 9:00 p.m.Step 3: Distance to the meeting point is D = 60 * 4 = 240 km.Step 4: Use Car C’s data to find x.C starts at 7:00 p.m., so its travel time by 9:00 p.m. is T - 2 = 2 hours.Distance of C to the meeting point is also D, so D = x * 2.Therefore, 240 = 2x ⇒ x = 120 km/h.

Verification / Alternative check:
Check all three cars: Car A runs from 5:00 p.m. to 9:00 p.m. (4 hours) at 60 km/h → 240 km. Car B runs from 6:00 p.m. to 9:00 p.m. (3 hours) at 80 km/h → 240 km. Car C runs from 7:00 p.m. to 9:00 p.m. (2 hours) at 120 km/h → 240 km. All three distances match, confirming that the computed x is correct and that they meet at the same point at the same time.

Why Other Options Are Wrong:
- Values like 110, 105, or 100 km/h would lead to distances for Car C that are not 240 km in 2 hours. For example, at 100 km/h for 2 hours, Car C would travel only 200 km, which is less than the 240 km travelled by A and B.

Common Pitfalls:
Students sometimes forget to adjust the time for each car correctly (for example, using T instead of T - 1 or T - 2), or they attempt to equate speeds instead of distances. The crucial point is that at the meeting instant, all cars have travelled the same distance from the starting point, even though they have been moving for different lengths of time. Carefully tracking start times and travel durations prevents such errors.

Final Answer:
The value of x, the speed of car C, is 120 km/h.