AMC 8 2015 Problem 17
AMC8 2015 年第 17 题
行程问题★★★☆☆
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?
Jeremy 的爸爸在交通高峰期把他送到学校需要 20 分钟。这一天路上车不多，因此他爸爸开车的速度提高了 18 英里每小时，把他送到学校花了 12 分钟，那么从他家到学校的距离有多少英里？
- A.4
- B.6
- C.8
- D.9
- E.12
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
For starters, we identify as distance and as velocity (speed). Writing the equations gives us and . This gives , which gives v=27, which then gives d=9.
d = rt, . , 10r = 270 so r = 27, plug into the first one and it's miles to school.
We set up an equation in terms of the distance and the speed In miles per hour. We have , giving (5)(x)=(3)(x+18) 5x=3x+54 2x=54 x=27 Hence, .
Since it takes of the original time for him to get to school when there is no traffic, the speed must be of the speed in no traffic or more. Letting be the rate and we know that , so we have miles per hour. Solving for gives us 27 miles per hour. Because 20 minutes is a third of an hour, the distance would then be 9 miles.
When driving in rush hour traffic, he drives 20 minutes for one distance ( ) to the school. It means he drives 60 minutes for 3 distances ( ) to the school. When driving with no traffic, he drives 12 minutes for one distance ( ) to the school. It means he drives 60 minutes for 5 distances ( ) to the school. Subtracting these two situations gives us 5d-3d = 18 = 2d, so . So the distance to the school is 9 miles.
(Ratios) In rush hour traffic, we can create this ratio for each distance we are going to try supposing the distance is : : 20 hours In no traffic, we can do the same: : 12 minutes We want the ratio to be the distance to 60 minutes: : 20 minutes = 3 d: 60 minutes : 12 minutes = 5 d: 60 minutes This shows the speed in miles per hour. This means that the speed in the 2nd ratio has to be 18 faster than the first ratio. We get , or. Distance equals speed multiplied by time. Let x/60 represent the speed when Jeremy's father is driving through traffic. Let (x+18)/60 represent the speed when there is no traffic. Because distance equals speed multiplied by time, 20x/60 would represent the distance to school. The "20" in this expression represents the 20 minutes it takes Jeremy's father to drive to school through the traffic, and "60" means 60 minutes. Anyway, 20x/60 simplifies to x/3. Using the same distance equation, the expression 12(x+18)/60 also represents the distance to school. (x+18)/60 represents the speed when there is no traffic, and the "12" means 12 minutes it takes to get to school when there is no traffic. 12(x+18)/60 simplifies to (x+18)/5. Now, since both expressions equate to the distance to school, you get the equation x/3=(x+18)/5. Solving for x gives you 27. This means that Jeremy's father drives 27 miles per hour in traffic. To find the distance to school, simply take 27 and plug it into the expressions (x+18)/5 or x/3. Both give you 9, which is the distance to school.žen D.
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