Two towns A and B are 100 km apart. At 9 a.m. a car drives from A towards B at an average speed of 80 km/h. At 9:30 a motorcyclist drives from B towards A at a speed of 120 km/h.
Calculate at what distance from A they meet each other.

To determine the distance from Town A at which the car and the motorcyclist meet, we can solve the problem step by step.

### Step 1: Calculate the time taken by the car to meet the motorcyclist.
- The car starts at 9:00 a.m., and the motorcyclist starts at 9:30 a.m.
- By the time the motorcyclist starts, the car has already been traveling for 30 minutes.

Since the car travels at an average speed of 80 km/h, in 30 minutes (which is 0.5 hours), the car covers:
\[
\text{Distance covered by car in 30 minutes} = 80 \times 0.5 = 40 \text{ km}
\]
So, at 9:30 a.m., the car is 40 km closer to B, leaving 60 km between the car and the motorcyclist.

### Step 2: Determine the combined speed of the car and the motorcyclist.
- The car is moving towards B at 80 km/h.
- The motorcyclist is moving towards A at 120 km/h.

Therefore, the combined speed at which the distance between the car and the motorcyclist is closing is:
\[
\text{Combined speed} = 80 + 120 = 200 \text{ km/h}
\]

### Step 3: Calculate the time taken for them to meet.
- The distance they need to close between them is 60 km.
- At a combined speed of 200 km/h, the time taken to close this distance is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{60}{200} = 0.3 \text{ hours} = 18 \text{ minutes}
\]

### Step 4: Calculate the distance from A where they meet.
- The car continues to travel during these 18 minutes at 80 km/h.
- The distance the car covers in 18 minutes is:
\[
\text{Distance covered by car in 18 minutes} = 80 \times 0.3 = 24 \text{ km}
\]

Since the car had already covered 40 km by 9:30 a.m., the total distance from A where they meet is:
\[
\text{Total distance from A} = 40 + 24 = 64 \text{ km}
\]

### Final Answer:
The car and the motorcyclist meet **64 km from Town A**.

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. If the motorcyclist had started at 9:00 a.m. instead, where would they have met?
2. What if the speed of the car was 90 km/h? How would that change the meeting point?
3. How much time would it take for the motorcyclist to reach Town A if they did not meet the car?
4. What if the distance between A and B was 150 km instead of 100 km?
5. How would the solution change if both vehicles traveled at constant acceleration rather than constant speed?

**Tip:** When solving problems involving relative motion, always account for the time difference between the start times of the moving objects to correctly calculate the distances covered.

Learn how to calculate the distance from Town A where a car and motorcyclist meet. This problem involves relative motion, speed calculations, and time calculations. Suitable for high school students.

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