AMC8 2013

AMC8 2013 · Q11

AMC8 2013 · Q11. It mainly tests Fractions, Rates (speed).

Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?

泰德的祖父这周3天使用了跑步机。每天他走了2英里。周一他以5英里每小时的速度慢跑。周三和周五他分别以3英里每小时和4英里每小时的速度行走。如果祖父总是以4英里每小时的速度行走，他会在跑步机上花费更少的时间。他节省了多少分钟？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (D)

正确答案：（D）

Solution

We use that fact that $d=rt$. Let d= distance, r= rate or speed, and t=time. In this case, let $x$ represent the time. On Monday, he was at a rate of $5 \text{ m.p.h}$. So, $5x = 2 \text{ miles}\implies x = \frac{2}{5} \text { hours}$. For Wednesday, he walked at a rate of $3 \text{ m.p.h}$. Therefore, $3x = 2 \text{ miles}\implies x = \frac{2}{3} \text { hours}$. On Friday, he walked at a rate of $4 \text{ m.p.h}$. So, $4x = 2 \text{ miles}\implies x=\frac{2}{4}=\frac{1}{2} \text {hours}$. Adding up the hours yields $\frac{2}{5} \text { hours}$ + $\frac{2}{3} \text { hours}$ + $\frac{1}{2} \text { hours}$ = $\frac{47}{30} \text { hours}$. We now find the amount of time Grandfather would have taken if he walked at $4 \text{ m.p.h}$ per day. Set up the equation, $4x = 2 \text{ miles} \times 3 \text{ days}\implies x = \frac{3}{2} \text { hours}$. To find the amount of time saved, subtract the two amounts: $\frac{47}{30} \text { hours}$ - $\frac{3}{2} \text { hours}$ = $\frac{1}{15} \text { hours}$. To convert this to minutes, we multiply by $60$. Thus, the solution to this problem is $\dfrac{1}{15}\times 60=\boxed{\textbf{(D)}\ 4}$

我们使用距离 = 速度 × 时间的事实。令 d= 距离，r= 速度，t= 时间。在这种情况下，令 x 表示时间。周一，他以 5 m.p.h. 的速度。所以，$5x = 2$ 英里 $\implies x = \frac{2}{5}$ 小时。周三，他以 3 m.p.h. 的速度行走。因此，$3x = 2$ 英里 $\implies x = \frac{2}{3}$ 小时。周五，他以 4 m.p.h. 的速度行走。所以，$4x = 2$ 英里 $\implies x=\frac{2}{4}=\frac{1}{2}$ 小时。总时间为 $\frac{2}{5}$ 小时 + $\frac{2}{3}$ 小时 + $\frac{1}{2}$ 小时 = $\frac{47}{30}$ 小时。现在计算如果每天以 4 m.p.h. 行走所需时间。设置方程，$4x = 2$ 英里 × 3 天 $\implies x = \frac{3}{2}$ 小时。节省的时间为 $\frac{47}{30}$ 小时 - $\frac{3}{2}$ 小时 = $\frac{1}{15}$ 小时。将此转换为分钟，乘以 60。因此，答案为 $\dfrac{1}{15}\times 60=\boxed{\textbf{(D)}\ 4}$

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