Understanding Circular Track Meeting Points

This problem involves two runners, Munaf and Surya, starting at the same point on a circular track and running in the same direction. We are given information about when they meet at the same point on the track and the ratio of their speeds.

When two individuals run on a circular track starting from the same point in the same direction, they meet when the difference in the distance covered by them is an integer multiple of the track length. The point where they meet depends on the ratio of their speeds.

Speed Ratio and Distinct Meeting Points

Let the speed of the faster runner (say, Munaf) be $v_f$ and the speed of the slower runner (say, Surya) be $v_s$. The problem states that the ratio of the speed of the faster boy to that of the slower one is $n : 1$. So, we have:

$$ \frac{v_f}{v_s} = \frac{n}{1} $$

Since the faster runner is mentioned, $v_f > v_s$, which means $n > 1$. Also, 'n' is a natural number.

When two runners start from the same point and run in the same direction on a circular track with speeds in the ratio $v_f : v_s = n : 1$, the number of distinct points on the track where they can meet is given by $|n - 1|$. Since $n$ is a natural number and $n > 1$, the number of distinct meeting points is $n - 1$.

Analyzing the Meeting Pattern

The problem states that they meet at the same point on the track for the 31st time and the 43rd time. The meeting points on a circular track repeat in cycles. If the number of distinct meeting points is $P$, then the $(k)$-th meeting point is the same as the $(k + m \times P)$-th meeting point for any positive integer $m$.

In this case, the 31st meeting point is the same as the 43rd meeting point. This means that the difference between the meeting numbers, $43 - 31$, must be a multiple of the number of distinct meeting points.

The difference in meeting numbers is:

$$ 43 - 31 = 12 $$

This difference (12) must be a multiple of the number of distinct meeting points, which we found to be $(n - 1)$.

So, we can write:

$$ 12 = k \times (n - 1) $$

where $k$ is a positive integer.

This implies that $(n - 1)$ must be a divisor of 12.

Finding Possible Values of n

Let's find the divisors of 12. The positive divisors of 12 are 1, 2, 3, 4, 6, and 12.

Since $(n - 1)$ must be a divisor of 12, the possible values for $(n - 1)$ are 1, 2, 3, 4, 6, or 12.

We can find the corresponding possible values for $n$ by adding 1 to each of these divisors:

So, the possible values for $n$ are 2, 3, 4, 5, 7, and 13.

Comparing with Given Options

The given options for the value of $n$ are:

• 5
• 4
• 3
• 6

We need to find which of these options is NOT a possible value of $n$. Let's compare the given options with the list of possible values (2, 3, 4, 5, 7, 13):

• Is 5 a possible value? Yes, 5 is in the list.
• Is 4 a possible value? Yes, 4 is in the list.
• Is 3 a possible value? Yes, 3 is in the list.
• Is 6 a possible value? No, 6 is NOT in the list of possible values for $n$.

Conclusion on the Impossible Value

Based on our analysis, for the 31st and 43rd meeting points to be the same, the value of $(n-1)$ must be a divisor of 12. This restricts the possible values of $n$. Out of the given options, 6 is the only value that does not appear in the list of possible values for $n$. Therefore, 6 is NOT a possible value of 'n'.

Revision Table: Circular Track Concepts

Additional Information on Circular Track Problems

Understanding relative speed is crucial for circular track problems. When runners move in the same direction, the relative speed is the difference between their speeds. When they move in opposite directions, the relative speed is the sum of their speeds.

If runners start at the same point and run in opposite directions with a speed ratio of $n:1$, the number of distinct meeting points is $n+1$. If they start at different points, the analysis becomes slightly more complex but still relies on relative speed and the fraction of the track length between them.

The first time runners meet at the starting point when running in the same direction is after the faster runner has gained a full lap over the slower runner. This happens after a time equal to Track Length / (Relative Speed). Subsequent meetings at the starting point also occur at intervals of this time.