AMC 8 2013 Problem 11

AMC8 2013 年第 11 题

行程问题★★★☆☆

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?

Ted 的爷爷这周有 3 天使用跑步机跑步。他每天跑 2 英里。周一他跑步的速度是 5 英里每小时，周三是 3 英里每小时，周五是 4 英里每小时。若他爷爷这 3 天跑步的速度都保持在 4 英里每小时的话，他在跑步机上花费的时间会更少一些，那么少多少分钟？

正确答案 · Correct Answer

解析 · Solution

Solution 1

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,

\displaystyle 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,

\displaystyle 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,

\displaystyle 4x = 2 \text{ miles}\implies x= \frac{2}{4} = \frac{1}{2} \text{ hours}

. Adding up the hours yields

\displaystyle \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,

\displaystyle 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:

\displaystyle \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=4

Short Solution In Two Short Steps! 1. Calculate the minutes he used: Monday - 1 mile = 12 minutes. Thus, he used 12x2 = 24 minutes. Wednesday - 1 mile = 20 minutes. Thus, he used 20x2 = 40 minutes. Friday - 1 mile= 15 minutes. Thus, he use 15x2 = 30 minutes. 24+40+30 = 94 2. Minus the minutes he could've used: From Friday's experience, we know two miles takes 30 minutes. 30x3 = 90 minutes. 94-90 = 4. Thus, the answer is

4{\text{}}

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