AMC10 2012 A

AMC10 2012 A · Q16

AMC10 2012 A · Q16. It mainly tests Rates (speed), Geometry misc.

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?

三名跑步者同时从500米圆形跑道上的同一点开始顺时针跑步，他们的速度分别是4.4、4.8和5.0米/秒。他们一直跑到再次在圆形跑道上的某处同时到达为止。他们跑了多少秒？

(A) 1,000 1000

(B) 1,250 1250

(D) 5,000 5000

(E) 10,000 10000

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Label the runners $A$, $B$, and $C$ in increasing order of speed. After the start, runner $B$ and runner $C$ will be together again once runner $C$ has run an extra 500 meters. Hence it takes $\frac{500}{5.0-4.8}=2500$ seconds for runners $B$ and $C$ to be together again. Similarly, it takes $\frac{500}{4.8-4.4}=1250$ seconds for runner $A$ and runner $B$ to be together again. Runners $A$ and $B$ will also be together at $2\cdot1250=2500$ seconds, at which time all three runners will be together.

答案（C）：按速度从慢到快给跑者 $A$、$B$、$C$ 排序。出发后，当跑者 $C$ 比跑者 $B$ 多跑 500 米时，$B$ 与 $C$ 会再次相遇。因此，$B$ 与 $C$ 再次相遇所需时间为 $\frac{500}{5.0-4.8}=2500$ 秒。类似地，跑者 $A$ 与跑者 $B$ 再次相遇所需时间为 $\frac{500}{4.8-4.4}=1250$ 秒。跑者 $A$ 与 $B$ 也会在 $2\cdot1250=2500$ 秒时再次相遇，此时三名跑者将同时相遇。

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