Ram and Shyam run a 10km race on a circular track of length 1000m. They complete one round in 200 seconds and 400 seconds respectively. After how much time from start will the faster person meet the slower person for the last time?

Answer

To solve the problem of when Ram and Shyam will meet for the last time during their 10 km race on a circular track, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- Ram and Shyam are running a 10 km race on a circular track of length 1000 m.
- Ram completes one round in 200 seconds, and Shyam completes one round in 400 seconds.

2. **Calculate the Speed of Each Runner**:
- **Ram's Speed**:
\[ \text{Speed of Ram} = \frac{\text{Distance}}{\text{Time}} = \frac{1000 \text{ m}}{200 \text{ s}} = 5 \text{ m/s} \]
- **Shyam's Speed**:
\[ \text{Speed of Shyam} = \frac{1000 \text{ m}}{400 \text{ s}} = 2.5 \text{ m/s} \]

3. **Determine the Relative Speed**:
- Since they are running in the same direction, the relative speed between Ram and Shyam is:
\[ \text{Relative Speed} = \text{Speed of Ram} - \text{Speed of Shyam} = 5 \text{ m/s} - 2.5 \text{ m/s} = 2.5 \text{ m/s} \]

4. **Calculate the Time to Meet for the First Time**:
- The distance that needs to be covered for Ram to meet Shyam for the first time is the length of the track (1000 m):
\[ \text{Time to meet} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{1000 \text{ m}}{2.5 \text{ m/s}} = 400 \text{ seconds} \]

5. **Determine the Total Distance Covered Until They Meet for the Last Time**:
- Since Ram completes 10 km (10,000 m) in the race, and he meets Shyam every time he completes 2 rounds (2000 m), we can find out how many times they meet:
- Total distance for Ram to run is 10,000 m.
- Distance per meeting is 2000 m (2 rounds).
- Number of meetings = Total distance / Distance per meeting = \( \frac{10,000 \text{ m}}{2000 \text{ m}} = 5 \) times.

6. **Calculate the Time for the Last Meeting**:
- Since they meet every 400 seconds, the time for the last meeting is:
\[ \text{Time for last meeting} = 5 \times 400 \text{ seconds} = 2000 \text{ seconds} \]

### Final Answer:
Ram and Shyam will meet for the last time after **2000 seconds**.

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