There are three runners Tom, Dick and Harry with their respective speeds of 10 km/h, 20 km/h and 30 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?

Options

A. 5
B. 7
C. 4
D. 12

Answer

To solve the problem, we need to determine how many times Tom, Dick, and Harry will meet at points P and Q during their race from 6 AM to 6 PM.

### Step-by-Step Solution:

1. **Determine the speeds of the runners:**
- Tom's speed = 10 km/h
- Dick's speed = 20 km/h
- Harry's speed = 30 km/h

2. **Calculate the time taken by each runner to complete one round trip (P to Q and back to P):**
- The distance from P to Q = 10 km.
- Time taken by Tom for one round trip:
- Time to go from P to Q = Distance / Speed = 10 km / 10 km/h = 1 hour.
- Time to return from Q to P = 10 km / 10 km/h = 1 hour.
- Total time for Tom = 1 + 1 = 2 hours.

- Time taken by Dick for one round trip:
- Time to go from P to Q = 10 km / 20 km/h = 0.5 hours.
- Time to return from Q to P = 10 km / 20 km/h = 0.5 hours.
- Total time for Dick = 0.5 + 0.5 = 1 hour.

- Time taken by Harry for one round trip:
- Time to go from P to Q = 10 km / 30 km/h = 1/3 hours (20 minutes).
- Time to return from Q to P = 10 km / 30 km/h = 1/3 hours (20 minutes).
- Total time for Harry = 1/3 + 1/3 = 2/3 hours (40 minutes).

3. **Determine the least common multiple (LCM) of their round trip times to find when they meet:**
- Tom completes a round trip in 2 hours.
- Dick completes a round trip in 1 hour.
- Harry completes a round trip in 2/3 hours (or 40 minutes).

- Convert all times to minutes:
- Tom: 120 minutes
- Dick: 60 minutes
- Harry: 40 minutes

- The LCM of 120, 60, and 40 is 120 minutes (2 hours).

4. **Calculate the total time of the race:**
- The race lasts from 6 AM to 6 PM, which is 12 hours or 720 minutes.

5. **Determine how many times they meet:**
- Since they meet every 120 minutes, we divide the total race time by the time interval:
- Number of meetings = Total time / Time interval = 720 minutes / 120 minutes = 6 times.

6. **Include the initial meeting at the start (6 AM):**
- They start together at point P at 6 AM, so we add 1 to the number of meetings.
- Total meetings = 6 + 1 = 7 times.

### Final Answer:
They will be together at points P and Q a total of **7 times** during the race.