AMC12 2005 A

AMC12 2005 A · Q6

AMC12 2005 A · Q6. It mainly tests Linear equations, Rates (speed).

Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

Josh 和 Mike 相距 $13$ 英里。昨天 Josh 开始骑自行车朝 Mike 家骑去。过了一会儿 Mike 开始骑自行车朝 Josh 家骑去。当他们相遇时，Josh 骑行的时间是 Mike 的两倍，并且速度是 Mike 的五分之四。他们相遇时 Mike 骑了多少英里？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $D_J, D_M$ be the distances traveled by Josh and Mike, respectively, and let $r,t$ be the time and rate of Mike. Using $d = rt$, we have that $D_M = rt$ and $D_J = \left(\frac{4}{5}r\right)\left(2t\right) = \frac 85rt$. Then $13 = D_M + D_J = rt + \frac{8}{5}rt = \frac{13}{5}rt$ $\Longrightarrow rt = D_M = 5\ \mathrm{(B)}$.

设 $D_J, D_M$ 分别为 Josh 和 Mike 骑行的距离，设 $r,t$ 分别为 Mike 的速度和时间。由 $d = rt$，有 $D_M = rt$ 且 $D_J = \left(\frac{4}{5}r\right)\left(2t\right) = \frac 85rt$。于是 $13 = D_M + D_J = rt + \frac{8}{5}rt = \frac{13}{5}rt$ $\Longrightarrow rt = D_M = 5\ \mathrm{(B)}$。

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