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  • Let . Suppose that there are 440 Metro stations in a city located along a circular path. Each pair of stations is

Let n> 2. Suppose that there are 440 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 100 times the number of blue lines, then find the number of blue lines and red lines?

 

Option: 1

954,95400


Option: 2

662,66200


Option: 3

820,82000


Option: 4

260,26000


Answers (1)

best_answer

Let n > 2. In a city with 440 Metro stations located along a circular path, each pair of nearest stations is connected by a blue line, while all remaining pairs of stations are connected by a red line. It is known that the number of red lines is 100 times the number of blue lines. We need to determine the number of blue lines and red lines.

Let's assume the number of blue lines is x. According to the given information, the number of red lines is 100 times the number of blue lines, so the number of red lines is 100x.

To find the total number of lines, we need to consider the connections between all pairs of stations. For n stations, the number of ways to choose 2 stations is given by \mathrm{n C 2} , which is equal to \mathrm{n ! /(2 !(n-2) !)}.

Therefore  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 equal to \mathrm{100 x+x=101 x},
we can set up the equation: \mathrm{101 x=n(n-1) / 2}.

To simplify this equation, we can multiply both sides by 2: \mathrm{202 x=n(n-1)}.
Since there are 440 Metro stations in the city, we can substitute this value into the equation
 

\begin{aligned} & 202 x=440(440-1) . \\ & 202 x=440(439) . \\ & 202 x=192560 . \end{aligned}

Dividing both sides by 202, we find:
x=192560 / 202 .
x=954 .

Therefore, there are 954 blue lines.
To find the number of red lines, we can substitute the value of x  into the equation:
Number of red lines =100 x=100 \times 954=95400.

Therefore, there are 954 blue lines and 95400 red lines in this scenario.

In conclusion, in the city with 440 Metro stations, there are 954 blue lines and 95400 red lines, with each pair of nearest stations connected by a blue line and all remaining pairs of stations connected by a red line.


 

Posted by

Devendra Khairwa

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