Question

AIME 铁人三项包括半英里游泳、 $30$ 英里自行车骑行和八英里跑步。Tom 游泳、骑车和跑步的速度都恒定。他跑步速度是游泳速度的五倍，骑车速度是跑步速度的两倍。Tom 用四又四分之一小时完成 AIME 铁人三项。求他骑车用了多少分钟。

The AIME Triathlon consists of a half-mile swim, a $30$ -mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

答案

答案：150

知识点：路程、速度与时间比与比例单位换算

难度评级：1840

解答：

设 Tom 的游泳速度为每小时 $s$ 英里。则他跑步速度为 $5s$ ，骑车速度为 $10s$ 。总时间为所以 $s = \frac{5.1}{4.25} = 1.2$ 英里每小时。 $\begin{aligned} &\frac{0.5}{s} + \frac{30}{10s} + \frac{8}{5s} \\ &= \frac{0.5 + 3 + 1.6}{s} \\ &= \frac{5.1}{s} = 4.25, \end{aligned}$

他骑车速度为每小时 $12$ 英里，因此骑车用时 $\frac{30}{12} = 2.5$ 小时，也就是 $150$ 分钟。

Let Tom's swimming speed be $s$ miles per hour. Then he runs at $5s$ and bicycles at $10s.$ The total time in hours is $\begin{aligned} &\frac{0.5}{s} + \frac{30}{10s} + \frac{8}{5s} \\ &= \frac{0.5 + 3 + 1.6}{s} \\ &= \frac{5.1}{s} = 4.25, \end{aligned}$ so $s = \frac{5.1}{4.25} = 1.2$ miles per hour.

He bicycles at $12$ miles per hour, so the ride takes $\frac{30}{12} = 2.5$ hours, which is $150$ minutes.

2013 AIME I 第 1 题

答案：150

难度评级：1840

解答：

其他年份的第 1 题