AMC10 2021 A

AMC10 2021 A · Q6

AMC10 2021 A · Q6. It mainly tests Rates (speed), Arithmetic misc.

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

Chantal 和 Jean 从步道起点向消防塔开始徒步。Jean 背着沉重的背包，走得较慢。Chantal 以 4 英里/小时的速度开始行走。到塔的中途，步道变得非常陡峭，Chantal 减速到 2 英里/小时。到达塔后，她立即转身，以 3 英里/小时的速度下行陡峭部分。她在中间点遇到了 Jean。他们相遇时 Jean 的平均速度是多少英里/小时？

(A) \frac{12}{13} \frac{12}{13}

(B) 1 1

(D) \frac{24}{13} \frac{24}{13}

(E) 2 2

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let $2d$ miles be the distance from the trailhead to the fire tower, where $d>0.$ When Chantal meets Jean, the two have traveled for \[\frac d4 + \frac d2 + \frac d3 = d\left(\frac 14 + \frac 12 + \frac 13\right) =d\left(\frac{3}{12} + \frac{6}{12} + \frac{4}{12}\right)=\frac{13}{12}d\] hours. At that point, Jean has traveled for $d$ miles, so his average speed is $\frac{d}{\frac{13}{12}d}=\boxed{\textbf{(A)} ~\frac{12}{13}}$ miles per hour.

设从步道起点到消防塔的距离为 $2d$ 英里，其中 $d>0$。Chantal 遇见 Jean 时，两人已经行走了 \[\frac d4 + \frac d2 + \frac d3 = d\left(\frac 14 + \frac 12 + \frac 13\right) =d\left(\frac{3}{12} + \frac{6}{12} + \frac{4}{12}\right)=\frac{13}{12}d\] 小时。此时 Jean 已行走了 $d$ 英里，因此他的平均速度是 $\frac{d}{\frac{13}{12}d}=\boxed{\textbf{(A)} ~\frac{12}{13}}$ 英里/小时。

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