AMC12 2004 A

AMC12 2004 A · Q15

AMC12 2004 A · Q15. It mainly tests Rates (speed), Invariants.

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?

Brenda 和 Sally 在一条环形跑道上朝相反方向跑步，从直径相对的两点出发。她们第一次相遇时，Brenda 已跑了 100 米。第二次相遇时，Sally 已从第一次相遇点继续跑了 150 米。两人速度恒定。求跑道的长度（米）。

(A) 250 250

(B) 300 300

(D) 400 400

(E) 500 500

Answer

Correct choice: (C)

正确答案：（C）

Solution

Call the length of the race track $x$. When they meet at the first meeting point, Brenda has run $100$ meters, while Sally has run $\frac{x}{2} - 100$ meters. By the second meeting point, Sally has run $150$ meters, while Brenda has run $x - 150$ meters. Since they run at a constant speed, we can set up a proportion: $\frac{100}{x- 150} = \frac{\frac{x}{2} - 100}{150}$. Cross-multiplying, we get that $x = 350\Longrightarrow\boxed{\mathrm{(C)}\ 350}$.

设跑道长度为 $x$。第一次相遇时，Brenda 跑了 $100$ 米，而 Sally 跑了 $\frac{x}{2} - 100$ 米。到第二次相遇时，Sally 跑了 $150$ 米，而 Brenda 跑了 $x - 150$ 米。由于两人速度恒定，可列比例：$\frac{100}{x- 150} = \frac{\frac{x}{2} - 100}{150}$。交叉相乘得 $x = 350\Longrightarrow\boxed{\mathrm{(C)}\ 350}$。

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