Question

Abe 可以在 $15$ 小时内粉刷完一个房间，Bea 的粉刷速度比 Abe 快 $50$ %，Coe 的速度是 Abe 的两倍。Abe 开始粉刷房间，前一个半小时独自工作。然后 Bea 加入 Abe，两人一起工作，直到房间粉刷了一半。接着 Coe 加入 Abe 和 Bea，三人一起工作，直到整个房间粉刷完成。求从 Abe 开始到三人完成粉刷经过了多少分钟。

Abe can paint the room in $15$ hours, Bea can paint $50$ percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.

答案

答案：334

知识点：速率分数

难度评级：1890

解答：

Abe 每分钟粉刷 $\frac{1}{900}$ 个房间，所以 Bea 每分钟粉刷 $\frac{3}{2} \cdot \frac{1}{900} = \frac{1}{600}$ ，Coe 每分钟粉刷 $\frac{2}{900} = \frac{1}{450}$ 。前 $90$ 分钟 Abe 粉刷了 $\frac{90}{900} = \frac{1}{10}$ 个房间。

Abe 和 Bea 合作的速率为 $\frac{1}{900} + \frac{1}{600} = \frac{1}{360}$ ，他们需要把总量从 $\frac{1}{10}$ 提高到 $\frac{1}{2}$ ，所需时间为 $\frac{2}{5} \cdot 360 = 144$ 分钟。三人合速率为 $\frac{1}{360} + \frac{1}{450} = \frac{1}{200}$ ，所以剩下半个房间需要 $\frac{1}{2} \cdot 200 = 100$ 分钟。

总时间为 $90 + 144 + 100 = 334$ 分钟。

Abe paints $\frac{1}{900}$ of the room per minute, so Bea paints $\frac{3}{2} \cdot \frac{1}{900} = \frac{1}{600}$ per minute and Coe paints $\frac{2}{900} = \frac{1}{450}$ per minute. In the first $90$ minutes Abe paints $\frac{90}{900} = \frac{1}{10}$ of the room.

Abe and Bea together paint $\frac{1}{900} + \frac{1}{600} = \frac{1}{360}$ per minute, and they must bring the total from $\frac{1}{10}$ up to $\frac{1}{2},$ which takes $\frac{2}{5} \cdot 360 = 144$ minutes. All three together paint $\frac{1}{360} + \frac{1}{450} = \frac{1}{200}$ per minute, so the remaining half of the room takes $\frac{1}{2} \cdot 200 = 100$ minutes.

The total time is $90 + 144 + 100 = 334$ minutes.

2014 AIME II 第 1 题

答案：334

难度评级：1890

解答：

其他年份的第 1 题