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Two runners are running around a circular track in the opposite direction such that they cross each other at six different points on the track. By what percentage the slower runner should increase his speed such that the number of points of crossings reduces to 2?
- a)500%
- b)250%
- c)300%
- d)400%
Correct answer is option 'D'. Can you explain this answer?
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Two runners are running around a circular track in the opposite direct...
Solution:
Let the speeds of the slower and faster runners be S and F respectively.
Let the length of the track be L.
When they cross each other, their relative speed is S + F.
Hence, the time taken to cross each other is L / (S + F).
Since they cross each other 6 times, the total time they spend crossing each other is 6L / (S + F).
Now, in order to reduce the number of points of crossings to 2, the relative speed of the two runners should be such that they cross each other at opposite ends of the track.
Let the distance between the two points of crossing be x.
Then, the time taken to cover this distance by the slower runner is x / S.
Similarly, the time taken to cover this distance by the faster runner is x / F.
Since they are crossing each other at opposite ends of the track, the time taken by the slower runner to cover the entire track is L / S.
Similarly, the time taken by the faster runner to cover the entire track is L / F.
Hence, we have the equation:
(L / S) - (x / S) = (x / F)
Solving for x, we get:
x = (L / 2) * (S / F + 1)
Now, in order to reduce the number of points of crossings to 2, the time taken to cover the distance x should be equal to the time taken to cross each other once.
Hence, we have the equation:
x / S = L / (S + F)
Solving for F, we get:
F = 2S
Hence, the faster runner is running at twice the speed of the slower runner.
Now, in order to reduce the number of points of crossings to 2, the slower runner should increase his speed by a factor of 4.
Hence, the percentage increase in speed is:
((4S - S) / S) * 100 = 300%
Therefore, the correct answer is option (d) 400%.
Let the speeds of the slower and faster runners be S and F respectively.
Let the length of the track be L.
When they cross each other, their relative speed is S + F.
Hence, the time taken to cross each other is L / (S + F).
Since they cross each other 6 times, the total time they spend crossing each other is 6L / (S + F).
Now, in order to reduce the number of points of crossings to 2, the relative speed of the two runners should be such that they cross each other at opposite ends of the track.
Let the distance between the two points of crossing be x.
Then, the time taken to cover this distance by the slower runner is x / S.
Similarly, the time taken to cover this distance by the faster runner is x / F.
Since they are crossing each other at opposite ends of the track, the time taken by the slower runner to cover the entire track is L / S.
Similarly, the time taken by the faster runner to cover the entire track is L / F.
Hence, we have the equation:
(L / S) - (x / S) = (x / F)
Solving for x, we get:
x = (L / 2) * (S / F + 1)
Now, in order to reduce the number of points of crossings to 2, the time taken to cover the distance x should be equal to the time taken to cross each other once.
Hence, we have the equation:
x / S = L / (S + F)
Solving for F, we get:
F = 2S
Hence, the faster runner is running at twice the speed of the slower runner.
Now, in order to reduce the number of points of crossings to 2, the slower runner should increase his speed by a factor of 4.
Hence, the percentage increase in speed is:
((4S - S) / S) * 100 = 300%
Therefore, the correct answer is option (d) 400%.
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Community Answer
Two runners are running around a circular track in the opposite direct...
The number of points of crossing on the track = 6.
Let the speed of the two runners be a & b.
The points of crossing = 6
We know that if two players run at the speed in the ratio a:b(in reduced ratio) in the same direction around the circular track. The number of distinct meeting points is |a-b|.
And if two players run at the speed in the ratio a:b in the opposite direction around the circular track. The number of distinct meeting points is |a+b|.
a+b=6 where a,b are coprime.
Thus, a & b are 1,5.
The number of points of crossing = 2, when both the runners are running at the same speed in the opposite direction.
Percentage by which the slower runner should increase his speed = (5-1)/1 x100= 400%
Let the speed of the two runners be a & b.
The points of crossing = 6
We know that if two players run at the speed in the ratio a:b(in reduced ratio) in the same direction around the circular track. The number of distinct meeting points is |a-b|.
And if two players run at the speed in the ratio a:b in the opposite direction around the circular track. The number of distinct meeting points is |a+b|.
a+b=6 where a,b are coprime.
Thus, a & b are 1,5.
The number of points of crossing = 2, when both the runners are running at the same speed in the opposite direction.
Percentage by which the slower runner should increase his speed = (5-1)/1 x100= 400%
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Two runners are running around a circular track in the opposite direction such that they cross each other at six different points on the track. By what percentage the slower runner should increase his speed such that the number of points of crossings reduces to 2?a)500%b)250%c)300%d)400%Correct answer is option 'D'. Can you explain this answer?
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