Problem:
A and B start running along a circular track from the same point at the same time in the same direction. To cover the whole track, A needs 5 seconds less than B. They meet for the first time after 30 seconds. How many times will they meet in a minute if they run in opposite directions?

Solution:
To solve this problem, we need to understand the concept of relative speed and the time taken to complete one round of the circular track.

Understanding the Distance:
Let's assume the length of the circular track is "d" units. Since A takes 5 seconds less than B to complete the track, we can say that the speed of A is greater than the speed of B.

Understanding the Relative Speed:
When two objects move in the same direction at different speeds, the relative speed of the faster object with respect to the slower object is the difference between their speeds.

In this case, since A is faster than B, the relative speed of A with respect to B is the difference between their speeds, which is (d/30) - (d/35) = d(1/30 - 1/35) = d/210.

Time Taken to Meet:
When two objects start from the same point and move in the same direction, they meet after the time taken by the faster object to gain a full lap over the slower object.

In this case, A and B meet for the first time after 30 seconds. Since A takes 5 seconds less than B to complete the track, B takes 35 seconds to complete one round. Therefore, A gains a full lap over B in 35 seconds.

Calculating the Number of Meetings:
To calculate the number of meetings in a minute, we need to find the time taken by A to complete one round.

Since A takes 30 seconds to meet B for the first time and A gains a full lap over B in 35 seconds, the time taken by A to complete one round is the least common multiple (LCM) of 30 and 35, which is 210 seconds.

Calculating the Number of Meetings in a Minute:
To calculate the number of meetings in a minute, we need to convert the time taken to complete one round into minutes.

There are 60 seconds in a minute, so the time taken to complete one round in minutes is 210/60 = 3.5 minutes.

Since A and B meet once every 3.5 minutes, the number of meetings in a minute is 60/3.5 = 17.14.

Therefore, A and B will meet approximately 17 times in a minute if they run in opposite directions.

Summary:
- A and B start running along a circular track from the same point at the same time in the same direction.
- A takes 5 seconds less than B to complete the whole track.
- A and B meet for the first time after 30 seconds.
- The relative speed of A with respect to B is (d/30) - (d/35) = d/210.
- A gains a full lap over B in 35 seconds.
- The time taken by A to complete one round is the LCM of 30 and 35, which is 210 seconds.
- The time taken to complete one