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# There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?  Verified
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Hint: Here we use the concept of Least Common Multiple to find the answer for the above question because both Ravi and Sonia start from the same point but they will take different time to complete one round. So, they will meet at the starting point at a particular time such that, these duration of time should be divisible by both of their times taken to drive.

From the question it is given that Ravi and Sonia start from the same point but take different time to complete the round. So, now we have to find the time taken by both of them to meet again at the starting point.
That means we have to find the LCM of 18 and 12.
Now we will find the LCM of 18 and 12 in prime factorisation method:
18 can be expressed as $18=2\times {{3}^{2}}$
12 can be expressed as $12={{2}^{2}}\times 3$
Now prime factor 2 is common here. So, take prime factor with highest power i.e. ${{2}^{2}}$
Now prime factor 3 is common here. So, take prime factor with highest power i.e. ${{3}^{2}}$
Then there is no common prime factor in 18 and 12, so there is no effect on LCM.
Now, by prime factorisation LCM of a and b is ${{2}^{2}}\times {{3}^{2}}=36$
So, LCM of 18 and 12 is 36.
That means the time taken by Ravi and Sonia to meet again at the starting point is 36 minutes.

Note: There is another way to solve this problem in an alternative way.
Ravi takes 12 minutes to drive one round around the field.
Sonia takes 18 minutes to drive one round around the field.
Ravi takes 24 minutes to drive two rounds around the field.
Sonia takes 36 minutes to drive two rounds around the field.
Ravi takes 36 minutes to drive three rounds around the field.