Bunuel wrote:
Which of the following cannot be the ratio of speeds of two joggers running on a circular jogging track if while running they meet at a diametrically opposite point to the point from where both of them started?
(A) 3 : 5
(B) 1 : 3
(C) 1 : 5
(D) 2 : 5
(E) 5 : 3
Are You Up For the Challenge: 700 Level QuestionsFirst, a point that is diametrically opposite the starting point means that the runners will meet at the other side of the diameter of the circle (i.e., 180 degrees from where they started the race).
Now, let’s assume that the runners start at position A. We see that position B is diametrically opposite to their starting point , and it is equivalent to a half-circle (since it is 180 degrees from the starting point). Now, if both of them run at the same speed (a ratio of 1 : 1), then they will, of course start at point A and will also meet at point B (and actually, they will meet at all points on the circle, since they are traveling at the same speed). Let’s look at each answer choice.
Choice A: If they have a speed ratio of 3 : 5, then the first runner will run from point A to point B, then from B to A, and then from point A to point B again, having run 3 half-circles around the track. In the same amount of time, the second runner will have run 5 half-circles, from A to B, then B to A, then A to B, then B to A, and finally A to B. In this case, they will meet at point B. Thus, the ratio 3 : 5 is possible.
Choice B: If they have a speed ratio of 1 : 3, then we see that the first runner advances from point A to point B (one half-circle). In the same time, the second runner will have gone three half-circles (from A to B, then B to A, and then A to B). Again, they will meet at point B, and so a ratio of 1 : 3 is possible.
Choice C: If they have a speed ratio of 1 : 5, then we see that the first runner advances from point A to point B (one half-circle). In the same amount of time, the second runner will have gone five half-circles (from A to B, then B to A, then A to B, then B to A, and, finally, from A to B). The ratio 1 : 5 is possible.
Choice D: If they have a speed ratio of 2 : 5, then we see that the first runner will go from A to B and then from B to A, while the second runner will go A to B, then B to A, then A to B, then B to A and finally A to B. They will not meet at point B. In fact, they will be on opposite sides of the diameter. Thus, a ratio of 2 : 5 is not possible.
By this point, you may have noticed that, in order for the two runners to meet at a point that is diametrically opposite from their starting point, the ratio must consist of two odd numbers (or be reducible to lowest terms as a ratio of two odd numbers). This is why choice D’s ratio of 2 : 5 will not work. In this case, when one runner is at point B (an odd number of half-circles from the starting point), the other runner will always be at point A (an even number of half-circles from the starting point), and vice versa.
Thus, we see that
choice E, a ratio of 5 : 3, consists of two odd numbers, and thus the runners will also meet at point B. Thus, a ratio of 5 : 3 is possible.
Answer: D _________________
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