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# A circular field has a circumference of 360km. Two cyclists Sumeet and Johan start together and can cycle at speeds of 12 km/hr and 15 km/hr, respectively, round the circular field. After how many hours will they meet again at the starting point?

Last updated date: 09th Apr 2024
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Hint – In this question use the relation between distance, time and speed that is ${\text{distance = speed }} \times {\text{ time}}$. Use the concept that the time after which they will meet at the starting point will be the L.C.M of the individual times they take to cover the given circumference at their respective speeds.

Given data
Circumference of a circular field = 360 km.
Speed of cyclist Sumeet = 12 km/hr.
And the speed of cyclist Johan = 15 km/hr.
Now as we know that time is the ratio of distance to speed.
So the time taken by Sumeet to complete one revolution of a circular field = $\dfrac{{360}}{{12}} = 30$ hr.
And the time taken by Johan to complete one revolution of a circular field = $\dfrac{{360}}{{15}} = 24$ hr.
Now it is given that they start at the same point so they meet again at the starting point is the L.C.M of respective times.
So we have to find out the L.C.M of 30 and 24.
So first factorize 24 and 30.
So factors of 30 are
$\Rightarrow 30 = 2 \times 3 \times 5$
And the factors of 24 are
$\Rightarrow 24 = 2 \times 2 \times 2 \times 3$
As we know L.C.M is the product of common factors and remaining factors together.
So the L.C.M of 24 and 30 is
$\Rightarrow L.C.M = 2 \times 2 \times 2 \times 3 \times 5 = 120$
So they will meet again after 120 hours at the starting point.
So this is the required answer.

Note – There can be another method to solve this problem, we can use the concept of relative speed, in this we will be making one cyclist stationary and will be giving its speed to another cyclist but exactly in the opposite direction, then using the relationship between distance, time and speed, the time can be calculated.