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  • Let n be a natural number. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of neares

Let n be a natural number. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by a blue line, whereas all remaining pairs of stations are connected by a red line. If the number of red lines is 56 times the number of blue lines, then what is the value of n?

 

Option: 1

4


Option: 2

6


Option: 3

8


Option: 4

2


Answers (1)

best_answer

To solve this problem, let's set up the equations based on the given information.

Let's assume that the number of blue lines is x.

According to the problem, the number of red lines is 56 times the number of blue lines, so the number of red lines is 56x.

Now, let's consider the total number of lines. The total number of lines is the sum of the blue lines and the red lines.

Since each pair of stations is connected by a straight track, we can use the combination formula to determine the total number of lines.

The number of ways to choose 2 stations out of n stations is given by \mathrm{n C 2} , which is equal to \mathrm{n ! /(2 !(n-2) !)}  .

So, the total number of lines is \mathrm{n C 2=n ! /(2 !(n-2) !)=n(n-1) / 2} .

Given that the total number of lines is \mathrm{56 x+x=57 x}, we can set up the equation:
\mathrm{57 x=n(n-1) / 2}

To simplify this equation, we can multiply both sides by 2:
\mathrm{114 x=n(n-1)}

Now, let's consider the possible values of n:

- If n is even, then n(n-1) is divisible by 4, but not necessarily by 57.

- If n is odd, then n(n-1) is divisible by 57, but not necessarily by 4.

Considering these conditions, we can test different values of n to find the solution.

After testing a few values, we find that n = 8 satisfies the condition, as n(n-1) = 8(7) = 56, which is divisible by 4 and 57.

Therefore, the value of n is 8.

 

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Gaurav

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