1957 AMC 12 第 20 题

Question

一名男子驾车出行，去程平均速度为

50

英里每小时，沿原路返回时平均速度为

45

英里每小时。他全程的平均速度为：

A man makes a trip by automobile at an average speed of

50

mph. He returns over the same route at an average speed of

45

mph. His average speed for the entire trip is:

47\frac7{19}

英里每小时

47\frac7{19}

mph

47\frac14

英里每小时

47\frac14

mph

47\frac12

英里每小时

47\frac12

mph

47\frac{11}{19}

英里每小时

47\frac{11}{19}

mph

以上都不是

none of these

选取一个方便的单程距离，用总路程除以总时间

Use a convenient one-way distance and divide total distance by total time

往返路程相等时，两种速度的算术平均数并不是全程平均速度

For equal distances the arithmetic mean of the two speeds is not the average speed

设每一程的距离为

d

。平均速度（单位为英里每小时）为

\begin{aligned} \frac{2d}{\frac{d}{50}+\frac{d}{45}} &=\frac{2}{\frac{1}{50}+\frac{1}{45}}\\ &=\frac{900}{19}\\ &=47\frac7{19} \end{aligned}\text{。}

因此，正确答案是 A。

Let each leg have distance

d.

The average speed, in miles per hour, is

\begin{aligned} \frac{2d}{\frac{d}{50}+\frac{d}{45}} &=\frac{2}{\frac{1}{50}+\frac{1}{45}}\\ &=\frac{900}{19}\\ &=47\frac7{19}. \end{aligned}

Thus, the correct answer is A.

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