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. When will they meet again at the starting point?
36 minutes
We are told that Sonia and Ravi both start from the same point on a circular path at the same time. Sonia takes \(18\) minutes for one round, and Ravi takes \(12\) minutes for one round.
We want to find after how many minutes they will both be together again at the starting point.
This type of question is solved using the LCM (Least Common Multiple) of their times.
Idea: Each time Sonia completes a round, a multiple of \(18\) minutes has passed. Each time Ravi completes a round, a multiple of \(12\) minutes has passed. They will both be at the starting point together when the time passed is a common multiple of \(18\) and \(12\).
The earliest such time is the LCM of \(18\) and \(12\).
Step 1: Write prime factorisation
Factorise \(18\) into primes.
\(18 = 2 \times 9\)
\(9 = 3 \times 3\)
So,
\(18 = 2 \times 3^2\)
Now factorise \(12\) into primes.
\(12 = 2 \times 6\)
\(6 = 2 \times 3\)
So,
\(12 = 2^2 \times 3\)
Step 2: Find the LCM using highest powers
List all distinct prime factors: \(2\) and \(3\).
For \(2\):
Highest power is \(2^2\) (from 12).
For \(3\):
Highest power is \(3^2\) (from 18).
So,
\(\text{LCM} = 2^2 \times 3^2\)
Now calculate step by step.
\(2^2 = 4\)
\(3^2 = 9\)
Multiply these:
\(4 \times 9 = 36\)
So,
\(\text{LCM}(18, 12) = 36\)
Step 3: Interpret the result
The LCM \(36\) tells us that after \(36\) minutes:
– Sonia will have completed a whole number of rounds.
– Ravi will have completed a whole number of rounds.
So both will be back together at the starting point.
Final answer: They will meet again at the starting point after \(36\) minutes.