Trains M and N are traveling west on parallel tracks. At exactly noon, the front of train M, which is...

Phase 1: Owning the Dataset

Let's draw a timeline showing the trains' positions:

At noon (12:00):
Location X                                     
    |-------- 30 km -------->
    M                        N
    ↓                        ↓
    (80 km/h west)          (65 km/h west)

Given information:

• Train M: Starts at location X, travels west at 80 km/h
• Train N: Starts 30 km west of X, travels west at 65 km/h
• Both trains eventually meet at location Y

Phase 2: Understanding the Question

We need to find:

1. Distance Train M travels from noon to 12:45 p.m.
2. Distance Train N travels from noon to 1:00 p.m.

This is a direct application of: \(\mathrm{Distance} = \mathrm{Speed} \times \mathrm{Time}\)

Phase 3: Finding the Answer

For Train M (noon to 12:45 p.m.):

• Time elapsed: 12:45 - 12:00 = 45 minutes = 0.75 hours
• Speed: 80 km/h
• Distance = \(80 \times 0.75 = 60\) km

For Train N (noon to 1:00 p.m.):

• Time elapsed: 1:00 - 12:00 = 1 hour
• Speed: 65 km/h
• Distance = \(65 \times 1 = 65\) km

Phase 4: Solution

Final Answer:

• Front of Train M: 60 km
• Front of Train N: 65 km

Both calculations directly apply the distance formula to find how far each train travels in the given time period.