There is a circular path around a sports field, Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they start at the same point and at the same time and go in the same direction, after how much time they will meet at starting point?

Question

Accepted Answer

The correct option is 1:

To solve this problem, we need to find the least common multiple (LCM) of the times taken by Priya and Harish to complete one round. When they meet again at the starting point, it will be after a time that is a common multiple of their individual times, and specifically, the smallest such positive time will be the LCM.

**Key Concept:** The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of them.

**Step-by-step explanation:**1.  **Identify the given times:**
    
      * Time taken by Priya = 18 minutes
      * Time taken by Harish = 12 minutes

2.  **Find the prime factorization of each number:**
    
      * For 18: $18 = 2 	imes 9 = 2 	imes 3 	imes 3 = 2 	imes 3^2$
      * For 12: $12 = 2 	imes 6 = 2 	imes 2 	imes 3 = 2^2 	imes 3$

3.  **Calculate the LCM:** To find the LCM, we take the highest power of all prime factors present in the factorizations.
    
      * The prime factors are 2 and 3.
      * The highest power of 2 is $2^2$.
      * The highest power of 3 is $3^2$.
      * LCM $(18, 12) = 2^2 	imes 3^2 = 4 	imes 9 = 36$Therefore, Priya and Harish will meet again at the starting point after 36 minutes.

The incorrect option is 2: 18 minutes
Explanation: While 18 is a multiple of 18 (Priya's time), it is not a multiple of 12 (Harish's time). At 18 minutes, Harish would have completed $18/12 = 1.5$ rounds and would not be at the starting point.

The incorrect option is 3: 8 minutes
Explanation: 8 minutes is neither a multiple of 18 nor 12. At 8 minutes, neither Priya nor Harish would have completed a full round, so they would not be at the starting point.

The incorrect option is 4: They will not meet
Explanation: This is incorrect because as long as they are moving at a constant speed around a circular path and start at the same time and point, they will always meet again at the starting point after a time equal to the LCM of their individual times.