Question

Mariam, peter and Kelly take 30, 40 and 50 minutes respectively to run around a circular track. If they started their race at 8:00 am from the same starting point; What is the earliest time they will be at the starting point together? After how many hours will they be at the starting point together?​

Answer 1

To find the earliest time when Mariam, Peter, and Kelly will be at the starting point together, we need to find the time when each of them completes a whole number of laps. Since they are running on a circular track, we can assume that the track has a circumference of 1 unit (e.g. 1 mile, 1 kilometer, etc.) to simplify the calculations.

Let's start by finding the number of laps each person completes in one hour:

- Mariam: 1 lap in 30 minutes, so 2 laps in 1 hour
- Peter: 1 lap in 40 minutes, so 1.5 laps in 1 hour (since 40 minutes is 2/3 of an hour)
- Kelly: 1 lap in 50 minutes, so 1.2 laps in 1 hour (since 50 minutes is 5/6 of an hour)

The only way for all three runners to be at the starting point together is if they have completed the same number of laps. The smallest number of laps that all three runners can complete in a whole number of hours is the least common multiple (LCM) of their lap counts.

The LCM of 2, 1.5, and 1.2 is 12, since:

- 2 laps = 12/6 laps
- 1.5 laps = 12/8 laps
- 1.2 laps = 12/10 laps

So, the earliest time when they will be at the starting point together is when they have completed 12 laps each, which will take:

- Mariam: 6 hours (2 laps per hour)
- Peter: 8 hours (1.5 laps per hour)
- Kelly: 10 hours (1.2 laps per hour)

To find the time when they will be at the starting point together, we need to add the time they started (8:00 am) to the time it takes each person to complete 12 laps:

- Mariam: 8:00 am + 6 hours = 2:00 pm
- Peter: 8:00 am + 8 hours = 4:00 pm
- Kelly: 8:00 am + 10 hours = 6:00 pm

Therefore, the earliest time when they will be at the starting point together is 2:00 pm, and they will be at the starting point together after 6 hours have passed since that time, at 8:00 pm.

Answer 2

• 6:00 pm
• 10 hours

Explanation:

You want to know when 3 runners starting at 8:00 am and running a track in 30 minutes, 40 minutes, and 50 minutes will all be at the starting point again.

LCM

The least common multiple of the times for one circuit is ...

(10 min) × LCM(3, 4, 5) = (10 min)(3·4·5) = 600 min = 10 hours

After 10 hours, the runners will all be at the starting point. That time is 6:00 pm.

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Additional comment

When numbers have no common factors (are mutually prime), their least common multiple is their product.

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