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20 Apr 2020, 06:25
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C
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Two friends A and B simultaneously start running around a circular track in the same direction, from the same point. A runs at 6 m/s and B runs at b m/s, where b is a positive integer. If they cross each other at exactly two points on the circular track, how many values can b take?
A. 1
B. 2
C. 3
D. 4
E. 5
Are You Up For the Challenge: 700 Level Questions
A. 1
B. 2
C. 3
D. 4
E. 5
Are You Up For the Challenge: 700 Level Questions
08 Jun 2020, 13:48
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Bunuel
Solution:
If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.
Since b can be either greater than 6 or less than 6, let’s first assume that b < 6. If b < 6, we can let the reduced ratio be 6/b = r/s such r - s = 2. Notice that after being reduced, r < 6. So (r, s) can be (3, 1), (5, 3).
If (r, s) = (3, 1), we have 6/b = 3/1 and this gives us b = 2.
If (r, s) = (5, 3), we have 6/b = 5/3 and this gives us b = 18/5.
However, since b is a positive integer, b can’t be 18/5. So we only have one value for b when b < 6. Next, let’s assume now b > 6. Again we can let the reduced ratio be 6/b = r/s such s - r = 2. Notice that after being reduced, r < 6. So (r, s) can be (1, 3), (3, 5).
If (r, s) = (1, 3), we have 6/b = 1/3 and this gives us b = 18.
If (r, s) = (3, 5), we have 6/b = 3/5 and this gives us b = 10.
Since both values of b are integers, we have 2 integer values for b when b > 6. Therefore, altogether, there are 3 integer values for b.
Answer: C
04 Jun 2020, 18:04
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If two runners moves in same direction with simplified ratio of speed a:b where a>b then they meet a-b times in the circular track.
And if the runners moves in opposite direction then they meet a+b times on a circular track.
Now,
6:b can be 3:1, 1:3, 3:5
So b can be 2, 18 and 10 respectively.
Don't take 2:4 because it will reduced to 1:2 so only three values of b are possible.
Posted from my mobile device
And if the runners moves in opposite direction then they meet a+b times on a circular track.
Now,
6:b can be 3:1, 1:3, 3:5
So b can be 2, 18 and 10 respectively.
Don't take 2:4 because it will reduced to 1:2 so only three values of b are possible.
Posted from my mobile device
