Three friends J, K, and L jog around a circular stadium. They complete one round in 12 seconds, 18 seconds, and 20 seconds respectively. After how many minutes will all three of them meet again together at the starting point?

Introduction / Context:
This question tests your ability to use the concept of Least Common Multiple (LCM) in a real life style problem. When multiple people are moving around a circular track with different times for completing one round, they will all meet again at the starting point after a time equal to the LCM of their individual times. Such questions are common in aptitude exams to check understanding of synchronization and periodic events.

Given Data / Assumptions:

• Time taken by J to complete one round = 12 seconds.
• Time taken by K to complete one round = 18 seconds.
• Time taken by L to complete one round = 20 seconds.
• We need to find after how many minutes all three meet again at the starting point.
• All of them start together from the same point at the same time.

Concept / Approach:
When dealing with repeated actions over time, the moment when all events coincide again is determined by the Least Common Multiple of their time periods. Here, each friend's jogging cycle is an event: J returns to the start every 12 seconds, K every 18 seconds, and L every 20 seconds. The time after which they all meet together again at the starting point is the LCM of 12, 18, and 20. Once we find the LCM in seconds, we convert that value into minutes as required by the question.

Step-by-Step Solution:
Step 1: Prime factorize each time period. 12 = 2^2 * 3. 18 = 2 * 3^2. 20 = 2^2 * 5. Step 2: Identify the distinct primes: 2, 3, and 5. Step 3: Take the highest power of each prime. For 2, highest power is 2^2. For 3, highest power is 3^2. For 5, highest power is 5^1. Step 4: Multiply to obtain the LCM. LCM = 2^2 * 3^2 * 5 = 4 * 9 * 5 = 36 * 5 = 180 seconds. Step 5: Convert seconds to minutes. 180 seconds = 180 / 60 = 3 minutes. Thus, all three friends will meet again at the starting point after 3 minutes.

Verification / Alternative check:
We can verify by checking how many rounds each person completes in 180 seconds. J: 180 / 12 = 15 rounds. K: 180 / 18 = 10 rounds. L: 180 / 20 = 9 rounds. Each result is an integer, which means each friend completes a whole number of laps and returns exactly to the starting point at 180 seconds. Because 180 seconds is derived from the LCM, it is the smallest such positive time when this happens, confirming that 3 minutes is indeed the first meeting time after the start.

Why Other Options Are Wrong:
5, 8, 12, and 6 minutes: These correspond to 300, 480, 720, and 360 seconds respectively. While 360 seconds (6 minutes) is also a common multiple of 12, 18, and 20, it is not the smallest such positive time. The question asks for the first time they meet again, so we must choose the minimum time, which is 180 seconds, that is 3 minutes.

Common Pitfalls:
Some students mistakenly add the times or average them instead of taking the LCM. Others may find a common multiple but forget to ensure it is the least common multiple. Errors can also occur during prime factorization or when converting between seconds and minutes. To avoid these issues, always factor carefully, use the highest powers of primes for LCM, and read the units in the question very attentively.

Final Answer:
All three friends will meet again at the starting point after 3 minutes.