AMC8 2017

AMC8 2017 · Q23

AMC8 2017 · Q23. It mainly tests Rates (speed), Divisibility & factors.

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?

连续四天，每天琳达以某种速度旅行一小时，使得她每英里旅行的时间是整数分钟。从第二天起，她的每英里旅行时间比前一天增加 5 分钟。每天她旅行的距离也是整数英里。四次旅行的总英里数是多少？

(A) 10 10

(B) 15 15

(D) 50 50

(E) 82 82

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Her time for each trip was $60$ minutes. The factors of $60$ are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,$ and $60$. Her daily number of minutes for a mile form a sequence of four numbers where each number is $5$ more than the previous number. Also, these numbers must each be a factor of $60$ since the number of miles traveled must be an integer. The only such sequence from among the factors of $60$ is $5, 10, 15, 20$. So her rates in miles per minute for the four days were $\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}$, and multiplying each by $60$ minutes gives her distances in miles as $12, 6, 4,$ and $3$, for a total distance of $25$ miles.

答案 (C)：她每次往返的时间是 $60$ 分钟。$60$ 的因数是 $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30$ 和 $60$。她每天“每英里所用的分钟数”构成一个长度为 4 的数列，其中每一项都比前一项多 $5$。此外，这些数都必须是 $60$ 的因数，因为每天行进的英里数必须是整数。在 $60$ 的因数中唯一符合条件的序列是 $5, 10, 15, 20$。因此，她四天的配速（英里/分钟）分别是 $\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}$；将它们分别乘以 $60$ 分钟，得到每天的路程（英里）为 $12, 6, 4, 3$，总路程为 $25$ 英里。

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