سرعت متوسطفصل: بخش ریاضی / درس: مسائل کلمه / درس 7
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The first thing I’ll say is, as you study for the test, it’s important to reflect from time to time on the perspective of the person who writes the test. The more you can understand the perspective of that person, the more insight you’ll have into the questions. In particular, people who write the questions for the test love questions that have very predictable traps, because those predictable traps make it very easy to separate the folks who know their stuff from the folks who don’t.
So this is a good example here in the topic of average speed. This is one of the test’s favorite kinds of word problems because it contains some very predictable traps. Normally, if we need the average of two numbers, we simply add them and divide by two. That’s normally how you’d find an average.
That is not how to find the average velocity, but that is the tempting trap into which so many test takers fall. Here’s an example. Bob drove 120 miles at 60 miles per hour and then another 120 at 40 miles per hour. What was his average speed for the trip? In a moment we’ll solve this problem but first let me point out the trap.
So many test takers will simply average 40 and 60. Of course the average of 40 and 60 is 50. So they’re gonna say that the average speed has to be 50 miles per hour. That trap will always be listed as an answer choice. Avoid that trap. So 50 miles an hour is the wrong answer.
It is not the answer to this question, but so many people fall into the trap. They average the two numbers, and they think that’s how you find the average velocity. And that is not how you do it. To solve a question of this sort, we use D equals RT, which is true for each leg of the trip and true for the trip overall.
So there’s a D equals RT for the first leg. There’s another D equals RT for the second leg. There’s also a D equals RT for the trip as a whole. In particular, average velocity equals total distance over total time. So the R of the total trip, that is the average velocity. And that is the total distance divided by the total time.
That is the formula for average velocity. Often we have to find the total distance by adding the distances of the individual legs and find the total time by adding the individual times. Now, now we’re ready to do this problem. So pause the video and then we’ll talk about this. Okay.
So in the first leg he drove 120 miles, he drove at a rate of 60 miles an hour so that would be a time of two hours. So now we have D R and T for the first leg. Let’s calculate this for the second leg. For the second leg, 120 miles divided by 40 miles an hour, so that took three hours.
The total distance is 120 plus 120, that is 240. The total time is five hours. So 240 divided by 5. I’m actually gonna multiply numerator and denominator by 2 so that I’m dividing by 10 instead of dividing by 5. That makes it much easier.
And this divides to 48 miles per hour. 48 miles per hour, not 50 miles an hour, is the correct answer. Sometimes the problem won’t give you all the numbers. For example, Cassandra drove from A to B at a constant 60 miles an hour speed. Then she returned on the same route from B to A at a constant speed of 20 miles an hour.
What was her average speed for the whole trip? Of course the trap answer is just averaging 60 and 20, that would be 40. So, it is true that 60 plus 20 divided by 2 is 40, that’s how you’d find the average of those two numbers. But for speeds, we cannot find an average speed just by averaging the individual speeds.
So that is a trap answer. And again, that trap answer will always be listed as an answer choice. And there will always be large numbers of test takers who fall into this trap. If you can avoid that trap alone that sets you apart from so many other test takers. All right, so now try this, and then we’ll talk about it.
Okay, so this is tricky. One approach is to just call the distance a variable. D equals the distance from A to B. So the first leg, the time would be D over R, that’s D over 60. For the second leg, the time again is D over R, that’s D over 20.
Those are the two times. The total time is the sum of those two times, we’ll find the common denominator, combine them, we can simplify a little bit, so the total time of the trip is D over 15. The total distance is 2D. So the average velocity is 2 D over D divided by 15, which is 2 D times the reciprocal of the fraction in the denominator, 15 over D.
Notice the Ds cancel. And this would always be true, if you pick a variable for the distance, it will always cancel. And what we get is 30 miles an hour. And that is the correct average speed. Now here’s another approach, another perfectly valid approach.
We can just pick a value for the distance. So let’s say that the distance is something nice and round like 60 miles an hour. Well, she travels at 60 miles an hour going there, she covers 60 miles in one hour. Coming back at 20 miles an hour will take her three hours to go 60 miles an hour.
Well that means the whole trip took four hours, and she covered 120 miles, 120 miles in four hours. That gives us an average of 30 miles an hour. Here’s a test-like practice question. Pause the video and then we’ll talk about this.
An airplane has a 3,600 mile trip. It covers the first 1,800 miles of the trip at 40 miles an hour. Which of the following is closest to the constant speed the plane would have to follow in the last 1,800 miles so that the average speed of the whole trip is 450 miles per hour?
So 450 miles per hour is the average of the trip. 400 is the speed in the first half. So, of course, what, what are people going to fall into a trap about? They’re going to think, well, what two thinks do you have to average to get 450? Well, you average 400 and 500, they would average to 450. Therefore 500 must be the speed in the second half of the trip.
So that is our suspected trap answer, again it’s listed, we expect it to be listed, and we expect it not to be correct. So let’s go about this. In the first leg, we know the distance, we know the rate, we can figure out the time. For the whole trip we know the distance, the rate.
We divide, multiply top and bottom by 2, cancel. We know the time. That allows us to figure out the time of the second leg. The time of the second leg, we subtract, that has to be three and a half hours. So the speed in that second leg has to be the distance, the distance of 1,800 miles, divided by 3.5 hours.
Okay, well that would be hard to do without a calculator. But we can use a little bit of estimation here. First of all, we’ll multiply top and bottom by 2, so we get 3,600 over 7. Notice that 36 hundred over 7 has to be bigger than 35 hundred over 7. 35 hundred over 7 is 500 miles an hour.
So, clearly the speed is larger than 500 miles an hour. Also, 3,600 over 7 has to be smaller than 3600 over 6. Well that equals 600 mph so that means that the speed, the answer, has to be between 500 and 600 mph. Well, estimating like this, we can eliminate enough to narrow down to a single answer choice.
And in fact, that alone eliminates down to the answered choice D. So that would allow us to solve the question. Alternately, we could calculate this. We could do a quick calculation without a calculator, very easily, just by saying, let’s think about this, 3,600 over 7. Well, that’s 3500 plus 100.
We’ll split the fractions up. That first fraction is clearly 500 and how many times does 7 go into 100? Well it goes into 100 about 14 times. So that’s about 514 and that’s enough to get us our answer. In summary, when finding the average velocity, do not fall into the trap of thinking it’s a simple numerical average of two things.
You have to find D and T of each leg of the trip and add across the legs to find the total distance and the total time. The average velocity equals the total distance divided by the total time.
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