There are three runners Deepak, Sandeep and Pawan with their respective speeds of 10 km/h, 20 km/h. They are initially at P and they have to run between the two points P and Q which are 10 km apart from each other. They start their race at 6 am and end at 6 pm on the same day. If they run between P and Q without any break, then how many times they will be together either at P and Q during the given time period?

Answer

To solve the problem of how many times Deepak, Sandeep, and Pawan will meet at points P and Q during their race from 6 AM to 6 PM, we can follow these steps:

### Step 1: Determine the speeds and distances
- **Deepak's speed**: 10 km/h
- **Sandeep's speed**: 20 km/h
- **Pawan's speed**: 30 km/h
- **Distance between P and Q**: 10 km

### Step 2: Calculate the time taken for each runner to complete one round trip
- **Deepak's time for one round trip (P to Q and back to P)**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{10 \text{ km} + 10 \text{ km}}{10 \text{ km/h}} = 2 \text{ hours} \]

- **Sandeep's time for one round trip**:
\[ \text{Time} = \frac{10 \text{ km} + 10 \text{ km}}{20 \text{ km/h}} = 1 \text{ hour} \]

- **Pawan's time for one round trip**:
\[ \text{Time} = \frac{10 \text{ km} + 10 \text{ km}}{30 \text{ km/h}} = \frac{2}{3} \text{ hours} \approx 40 \text{ minutes} \]

### Step 3: Determine how many round trips each runner can complete from 6 AM to 6 PM
- **Total time of the race**: 12 hours (from 6 AM to 6 PM)

- **Deepak's round trips**:
\[ \text{Number of trips} = \frac{12 \text{ hours}}{2 \text{ hours/trip}} = 6 \text{ trips} \]

- **Sandeep's round trips**:
\[ \text{Number of trips} = \frac{12 \text{ hours}}{1 \text{ hour/trip}} = 12 \text{ trips} \]

- **Pawan's round trips**:
\[ \text{Number of trips} = \frac{12 \text{ hours}}{\frac{2}{3} \text{ hours/trip}} = 18 \text{ trips} \]

### Step 4: Calculate the meeting times at points P and Q
- They will meet at point P every time they complete a round trip:
- **Deepak**: 6 times (at the end of each trip)
- **Sandeep**: 12 times
- **Pawan**: 18 times

### Step 5: Find the least common multiple (LCM) of their round trip times to find common meeting points
- **Deepak**: 2 hours
- **Sandeep**: 1 hour
- **Pawan**: 40 minutes (or \(\frac{2}{3}\) hours)

The LCM of their round trip times will give us the time interval at which they all meet together at either P or Q.

- **Convert all times to hours**:
- Deepak: 2 hours
- Sandeep: 1 hour
- Pawan: \(\frac{2}{3}\) hours

### Step 6: Calculate the LCM
- The LCM of 2, 1, and \(\frac{2}{3}\) is 2 hours.

### Step 7: Determine how many times they meet during the 12-hour period
- They will meet every 2 hours:
- From 6 AM, they will meet at:
- 6 AM (start)
- 8 AM
- 10 AM
- 12 PM
- 2 PM
- 4 PM
- 6 PM

This gives us a total of **7 times** they will be together at either P or Q during the race.

### Final Answer:
**They will meet together 7 times.**

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