Question

连续四天，Linda 每天旅行一小时，并且她的速度使她每行进一英里所需分钟数都是整数。第一天之后，每天她的速度都降低，使得每行进一英里所需分钟数比前一天多 $5$ 分钟。四天中每天她行进的距离也都是整数英里。这四次旅行一共多少英里？

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?

$10$

$15$

$25$

$50$

$82$

答案：C

知识点：因数等差数列

难度评级：1610

视频讲解：

Click to load, then click again to play

文字解答：

Linda 每天旅行 $60$ 分钟。因为每天每英里所需分钟数是整数，且每天行进英里数也是整数，所以每天的每英里分钟数必须是 $60$ 的因数。 $60$ 的因数为 $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30$ 和 $60$ 其中唯一一组四个相邻相差 $5$ 的数是 $5, 10, 15$ 和 $20$ 。四天总路程为 $\dfrac{60}{5} + \dfrac{60}{10} + \dfrac{60}{15} + \dfrac{60}{20} = 25$ 英里。

所以正确答案是 C。

Linda traveled for $60$ minutes every day. Since one mile was traveled in an integer amount of minutes each day, her minutes per mile every day must be a factor of $60.$ The factors of $60$ are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,$ and $60.$ The only sequence of four of these numbers that differ by $5$ are $5, 10, 15,$ and $20.$ For the four days, she traveled $\dfrac{60}{5} + \dfrac{60}{10} + \dfrac{60}{15} + \dfrac{60}{20} = 25$ miles.

Thus, C is the correct answer.

2017 AMC 8 第 23 题

答案：C

难度评级：1610

视频讲解：

文字解答：

其他年份的第 23 题