Question

鲁道夫以恒定速度骑车，并且每骑完一英里就在里程点休息五分钟。詹妮弗也以恒定速度骑车，她的速度是鲁道夫的四分之三，但詹妮弗每骑完两英里才休息五分钟。詹妮弗和鲁道夫同时开始骑车，并且恰好同时到达 $50$ 英里标记处。问他们共用了多少分钟？

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$ -mile mark at exactly the same time. How many minutes has it taken them?

答案

答案：620

知识点：路程、速度与时间分式方程

难度评级：2020

小提示：

两人到达终点时都不再休息，所以鲁道夫休息 $49$ 次，詹妮弗休息 $24$ 次

Neither rider takes a break upon arriving, so Rudolph takes $49$ breaks and Jennifer takes $24$

大提示：

设鲁道夫的速度为每分钟 $r$ 英里，令 $\frac{50}{r} + 245$ 与 $\frac{200}{3r} + 120$ 相等

With Rudolph’s rate $r$ miles per minute, equate $\frac{50}{r} + 245$ and $\frac{200}{3r} + 120$

解答：

设鲁道夫每分钟骑 $r$ 英里。他在第 $1$ 英里到第 $49$ 英里后休息，所以总时间为 $\frac{50}{r} + 49 \cdot 5 = \frac{50}{r} + 245$ 分钟。詹妮弗每分钟骑 $\frac{3r}{4}$ 英里，并在第 $2, 4, \ldots, 48$ 英里后休息，所以她的总时间为 $\frac{50}{\frac{3r}{4}} + 24 \cdot 5 = \frac{200}{3r} + 120$ 分钟。

令两人的时间相等，得到 $\frac{50}{r} + 245 = \frac{200}{3r} + 120\text{，}$ 所以 $\begin{aligned} 125 &= \frac{200 - 150}{3r} \\ &= \frac{50}{3r} \end{aligned}\text{，}$ 因此 $r = \frac{2}{15}$ 。共同用时为 $\frac{50}{\frac{2}{15}} + 245 = 375 + 245 = 620$ 分钟。

Let Rudolph bike at $r$ miles per minute. He rests after each of miles $1$ through $49,$ so his total time is $\frac{50}{r} + 49 \cdot 5 = \frac{50}{r} + 245$ minutes. Jennifer bikes at $\frac{3r}{4}$ miles per minute and rests after each of miles $2, 4, \ldots, 48,$ so her total time is $\frac{50}{\frac{3r}{4}} + 24 \cdot 5 = \frac{200}{3r} + 120$ minutes.

Setting the times equal gives $\frac{50}{r} + 245 = \frac{200}{3r} + 120,$ so $\begin{aligned} 125 &= \frac{200 - 150}{3r} \\ &= \frac{50}{3r}, \end{aligned}$ and $r = \frac{2}{15}.$ The common time is $\frac{50}{\frac{2}{15}} + 245 = 375 + 245 = 620$ minutes.

2008 AIME II 第 2 题

答案：620

难度评级：2020

小提示：

大提示：

解答：

其他年份的第 2 题