AMC12 2002 B

AMC12 2002 B · Q17

AMC12 2002 B · Q17. It mainly tests Rates (speed), Work problems.

Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?

安迪的草坪面积是贝丝草坪的两倍，也是卡洛斯草坪的三倍。卡洛斯的割草机割草速度是贝丝割草机的一半，也是安迪割草机的三分之一。如果他们同时开始割草，谁会最先完成？

(A) Andy Andy

(B) Beth Beth

(D) Andy and Carlos tie for first. Andy和Carlos并列第一。

(E) All three tie. 三者并列。

Answer

Correct choice: (B)

正确答案：（B）

Solution

We say Andy's lawn has an area of $x$. Beth's lawn thus has an area of $\frac{x}{2}$, and Carlos's lawn has an area of $\frac{x}{3}$. We say Andy's lawn mower cuts at a speed of $y$. Carlos's cuts at a speed of $\frac{y}{3}$, and Beth's cuts at a speed $\frac{2y}{3}$. Each person's lawn is cut at a time of $\frac{\text{area}}{\text{rate}}$, so Andy's is cut in $\frac{x}{y}$ time, as is Carlos's. Beth's is cut in $\frac{3}{4}\times\frac{x}{y}$, so the first one to finish is $\boxed{\mathrm{(B)}\ \text{Beth}}$.

设安迪的草坪面积为$x$。则贝丝的草坪面积为$\frac{x}{2}$，卡洛斯的草坪面积为$\frac{x}{3}$。设安迪的割草机割草速度为$y$。则卡洛斯的割草机速度为$\frac{y}{3}$，贝丝的割草机速度为$\frac{2y}{3}$。每个人割完草所需时间为$\frac{\text{面积}}{\text{速度}}$，因此安迪割完需要$\frac{x}{y}$时间，卡洛斯也需要$\frac{x}{y}$时间。贝丝割完需要$\frac{3}{4}\times\frac{x}{y}$时间，所以最先完成的是$\boxed{\mathrm{(B)}\ \text{Beth}}$。

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