AMC12 2017 A

AMC12 2017 A · Q13

AMC12 2017 A · Q13. It mainly tests Word problems (algebra), Rates (speed).

Driving at a constant speed, Sharon usually takes 180 minutes to drive from her house to her mother’s house. One day Sharon begins the drive at her usual speed, but after driving $\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by 20 miles per hour. This time the trip takes her a total of 276 minutes. How many miles is the drive from Sharon’s house to her mother’s house?

Sharon 以恒定速度驾驶，通常从家到她妈妈家的车程需要 180 分钟。一天 Sharon 以平常速度开始驾驶，但行驶了 $\frac{1}{3}$ 的路程后，遇到大雪暴，速度降低了 20 英里每小时。这次总车程用了 276 分钟。Sharon 从家到她妈妈家的车程有多少英里？

(A) 132 132

(B) 135 135

(D) 141 141

(E) 144 144

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let $d$ be the requested distance in miles, and suppose that Sharon usually drives at speed $r$ in miles per hour. Then $\frac{d}{r}=3$. The total time in hours for Sharon’s trip with the snowstorm is then $\frac{1}{3}\frac{d}{r}+\frac{2}{3}\frac{d}{r-20}=\frac{23}{5}$. Because $\frac{d}{r}=3$, this reduces to $$ 1+\frac{\frac{2}{3}}{\frac{r}{d}-\frac{20}{d}}=1+\frac{\frac{2}{3}}{\frac{1}{3}-\frac{20}{d}}=\frac{23}{5}. $$ Solving for $d$ gives $d=135$.

答案（B）：设$d$为所求距离（英里），并假设Sharon通常以每小时$r$英里的速度行驶。则$\frac{d}{r}=3$。暴风雪情况下Sharon行程的总时间（小时）为$\frac{1}{3}\frac{d}{r}+\frac{2}{3}\frac{d}{r-20}=\frac{23}{5}$。由于$\frac{d}{r}=3$，可化为 $$ 1+\frac{\frac{2}{3}}{\frac{r}{d}-\frac{20}{d}}=1+\frac{\frac{2}{3}}{\frac{1}{3}-\frac{20}{d}}=\frac{23}{5}. $$ 解得$d=135$。

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