Jane and Mary live 40 km apart. Jane starts at 8 00 am to Mary's home at a speed of 16km/hr. At 8 30am, Mary started towards Jane's home at a speed of 8km/hr. At what time and spot did they meet?

To solve this problem, let's determine how far each person travels before they meet.

### Step 1: Determine the time taken by Jane before Mary starts
- Jane starts at 8:00 AM.
- Mary starts at 8:30 AM.
- Jane has been traveling for 30 minutes (or 0.5 hours) before Mary starts.

In this time, Jane travels:
\[
\text{Distance} = \text{Speed} \times \text{Time} = 16 \, \text{km/hr} \times 0.5 \, \text{hr} = 8 \, \text{km}
\]

So, by 8:30 AM, Jane is 8 km closer to Mary.

### Step 2: Calculate the remaining distance between them
The total distance between them is 40 km. After Jane's 8 km, the remaining distance is:
\[
40 \, \text{km} - 8 \, \text{km} = 32 \, \text{km}
\]

### Step 3: Determine the relative speed
When both start moving towards each other at 8:30 AM:
- Jane's speed = 16 km/hr
- Mary's speed = 8 km/hr
- Relative speed = 16 km/hr + 8 km/hr = 24 km/hr

### Step 4: Calculate the time taken to meet
The time taken to cover the remaining 32 km at a relative speed of 24 km/hr is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{32 \, \text{km}}{24 \, \text{km/hr}} = \frac{4}{3} \, \text{hours} \approx 1.33 \, \text{hours}
\]
This is approximately 1 hour and 20 minutes.

### Step 5: Determine the meeting time
Since they start moving towards each other at 8:30 AM, the time of meeting will be:
\[
8:30 \, \text{AM} + 1 \, \text{hour} \, 20 \, \text{minutes} = 9:50 \, \text{AM}
\]

### Step 6: Determine the meeting spot
To find the distance Jane travels in this time:
\[
\text{Distance traveled by Jane} = \text{Speed} \times \text{Time} = 16 \, \text{km/hr} \times \frac{4}{3} \, \text{hr} = \frac{64}{3} \, \text{km} \approx 21.33 \, \text{km}
\]

Jane travels approximately 21.33 km from her starting point.

### Final Answer:
- **They meet at 9:50 AM.**
- **The meeting spot is approximately 21.33 km from Jane's home and 18.67 km from Mary's home.**

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. What if Jane and Mary started at the same time? Where would they meet?
2. How would the meeting point change if Mary’s speed increased to 10 km/hr?
3. If they both started at 9:00 AM, at what time would they meet?
4. What distance would Mary have covered by the time they meet?
5. How long would it take Jane to reach Mary’s home if she continued after meeting?

**Tip:** When solving relative motion problems, understanding relative speed is key to simplifying the solution.

Learn how Jane and Mary, starting 40 km apart, meet at 9:50 AM. Solve using principles of relative motion, speed, distance, and time calculation. Suitable for high school students.

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