多个旅行者问题
Transcript
多个旅行者和多次旅行。一些基于动作的问题涉及多个旅行者，或者
trips of more than one segment. We already saw a little of this in the previous video on average speed.
基本策略是每个旅行者，每次旅行都会获得自己的d = RT等式。有时我们必须设置多个方程，然后使用这些技术
that we've learned for solving two equations with two unknowns.
这是一个练习问题。暂停视频，然后我们会谈谈这个。 All right, Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 miles an hour, and 保罗以恒定的速度为40.当玛莎抵达B时，保罗距离酒店仍然有50英里。
What is the distance? So notice first of all that the time. We'll just say that time is, the time from the starting point to when Martha arrived. And so, Martha of course, traveled that whole distance D, but what about Paul? 保罗并没有一路走到D.他仍然有50英里，所以这是D 50。
And so the distance that we're going to use for Martha is D. The distance we're going to use for Paul is D minus 50. 所以现在我们有东西要插入距离和速率和时间。因此，对于玛莎来说，距离d，速率为60次T. For Paul, D minus 50, because at that same time he was 50 miles short of the city of B.
So he went D minus 50, and his speed was 40, and the same time T. Well notice now we've got two equations with two unknowns. The first one is solve for D, so just plug it into the second one. We get 60T minus 50 equals 40T. Add 50 to both sides, subtract 40T from both sides. We get 20T equals 50, divide by 20, we get T equals 5 over 2, or 2.5 hours.
Now we have to solve for D, now that we have the value of T. Plug this in, it makes sense to plug it into the first equation. D equals 60 times 2.5. Well 2 times 60 is 120. 0.5, or one half times 60 is 30. 120 plus 30 is 150, and that's the distance.
这是一个练习问题。暂停视频，然后我们会谈谈这个。 好的，弗兰克和格鲁吉亚同时开始从A到B旅行。格鲁吉亚的恒定速度为1.5倍坦率的速度速度。 When Georgia arrived at B, she turned around immediately and returned by the same route.
She crossed paths with Frank who was coming toward B when they were 60 miles away from B. How far away are A and B? So I'm gonna use the variable D for the distance between A and B. 所以这是我们正在寻找的变量。对于弗兰克和格鲁吉亚，弗兰克有R.的速度。
Georgia has a rate of 1.5 times that, so 1.5R. Georgia goes all the way from A to B, and 然后她回来了60英里。所以她封面的总距离是D加60。 弗兰克从一开始,但他没有成功,他做n't get to B. He's short 60 miles.
So the distance that he covers is D minus 60. And of course the time that we'll use, T, 将从起点到他们交叉路径再次出发。所以在那个时候，格鲁吉亚覆盖了D Plus 60，Frank Coves D mitus 60。 所以现在我们可以设置我们的两个等于RT方程。d minus 60等于RT，即坦率。
D plus 60 equals 1.5R times T, that's for Georgia. I'm going to rewrite that second equation as the right side is 1.5 times R times T. And now, since the first equation is solve for R times T, I can just plug in. Notice even though there are three unknowns here, I can eliminate two of them with one substitution. And I'm just left with an equation with D.
So that's very convenient, So I plug that in. I replace RT by D minus 60. That's the substitution from the first equation. So I get D plus 60 equals 1.5 times the quantity D minus 60. 我们得到1.5d减去，然后1.5倍60，1次60,0.5是30。
30 plus 60 is 90, so 1.5 times 60 is 90. Then what we'll do is subtract D from both sides. We'll add 90 to both sides. 0.5 is one half, we cancel that by multiplying both sides by 2, and we get D equals 300.
So the distance is 300. That's the distance between A and B. 这是一个稍微难的问题。暂停视频，然后我们会谈谈这个。 Kevin drove from A to B at a constant speed of 60 miles an hour, turned around and returned at a constant speed of 80 miles an hour.
Exactly 4 hours before the end of his trip, he was still approaching B, and only 15 miles away from it. What is the distance between A and B? Let's think about this for a minute. 所以在第一个部分，从A到B开始，他每小时60英里的旅行。回去，他在一个小时的时间里旅行了。
And then starting from B, so from P all the way to B, and then all the way back to A, that whole segment took 4 hours. 所以p到b，回到a，是4小时。很好地注意到，我只是很感兴趣 this little interval right here, from P to B. We know the distance and we know the speed, so we could figure out the time.
The time would equal the distance, 15 miles divided by the speed, 60 miles an hour, 15 over 60 is one quarter. 所以这是一个小时的四分之一。我们可以将这一点写为15分钟，但让我们实际把它留成一小部分。 Well if P to B to A was 4 hours, and P to B is a quarter of an hour. Then the route from B back to A has to be, the time has to be the difference.
So that is 4 minus one quarter. So I'm gonna change 4 into an improper fraction. 16岁以上，减去一个四分之一，是15岁，但我现在只是将此作为一个不正确的分数。 That is the time, 15 over 4 hours is the time of the trip from be back to A. Well now we know the speed as well as the time, so we can figure out the distance.
当然，那个距离是我们正在寻找的距离，距离A到B的距离。 So the speed is 80, the time is 15 over 4. Remember the great trick, cancel before you multiply. 80 divided by 4 is 20. 20 times 15, well 2 times 15 is 30, 20 times 15 is 300.
And so that's actually the distance between A and B. In summary, when a word problem involves multiple travelers, 多次旅行或带有多条腿的行程，记住每个旅行者，每个行程和/或每条腿都应该等于RT等式。 Sometimes you will be able to solve for all quantities in one equation, and use those numbers to help solve for other equations.
更常见的是，您必须使用用于解决两个未知数的两个方程的技术。 我们谈到替代和消除。如果这些对您来说是新的，您可以在代数模块中了解更多信息。
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这是一个练习问题。暂停视频，然后我们会谈谈这个。 All right, Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 miles an hour, and 保罗以恒定的速度为40.当玛莎抵达B时，保罗距离酒店仍然有50英里。
What is the distance? So notice first of all that the time. We'll just say that time is, the time from the starting point to when Martha arrived. And so, Martha of course, traveled that whole distance D, but what about Paul? 保罗并没有一路走到D.他仍然有50英里，所以这是D 50。
And so the distance that we're going to use for Martha is D. The distance we're going to use for Paul is D minus 50. 所以现在我们有东西要插入距离和速率和时间。因此，对于玛莎来说，距离d，速率为60次T. For Paul, D minus 50, because at that same time he was 50 miles short of the city of B.
So he went D minus 50, and his speed was 40, and the same time T. Well notice now we've got two equations with two unknowns. The first one is solve for D, so just plug it into the second one. We get 60T minus 50 equals 40T. Add 50 to both sides, subtract 40T from both sides. We get 20T equals 50, divide by 20, we get T equals 5 over 2, or 2.5 hours.
Now we have to solve for D, now that we have the value of T. Plug this in, it makes sense to plug it into the first equation. D equals 60 times 2.5. Well 2 times 60 is 120. 0.5, or one half times 60 is 30. 120 plus 30 is 150, and that's the distance.
这是一个练习问题。暂停视频，然后我们会谈谈这个。 好的，弗兰克和格鲁吉亚同时开始从A到B旅行。格鲁吉亚的恒定速度为1.5倍坦率的速度速度。 When Georgia arrived at B, she turned around immediately and returned by the same route.
She crossed paths with Frank who was coming toward B when they were 60 miles away from B. How far away are A and B? So I'm gonna use the variable D for the distance between A and B. 所以这是我们正在寻找的变量。对于弗兰克和格鲁吉亚，弗兰克有R.的速度。
Georgia has a rate of 1.5 times that, so 1.5R. Georgia goes all the way from A to B, and 然后她回来了60英里。所以她封面的总距离是D加60。 弗兰克从一开始,但他没有成功,他做n't get to B. He's short 60 miles.
So the distance that he covers is D minus 60. And of course the time that we'll use, T, 将从起点到他们交叉路径再次出发。所以在那个时候，格鲁吉亚覆盖了D Plus 60，Frank Coves D mitus 60。 所以现在我们可以设置我们的两个等于RT方程。d minus 60等于RT，即坦率。
D plus 60 equals 1.5R times T, that's for Georgia. I'm going to rewrite that second equation as the right side is 1.5 times R times T. And now, since the first equation is solve for R times T, I can just plug in. Notice even though there are three unknowns here, I can eliminate two of them with one substitution. And I'm just left with an equation with D.
So that's very convenient, So I plug that in. I replace RT by D minus 60. That's the substitution from the first equation. So I get D plus 60 equals 1.5 times the quantity D minus 60. 我们得到1.5d减去，然后1.5倍60，1次60,0.5是30。
30 plus 60 is 90, so 1.5 times 60 is 90. Then what we'll do is subtract D from both sides. We'll add 90 to both sides. 0.5 is one half, we cancel that by multiplying both sides by 2, and we get D equals 300.
So the distance is 300. That's the distance between A and B. 这是一个稍微难的问题。暂停视频，然后我们会谈谈这个。 Kevin drove from A to B at a constant speed of 60 miles an hour, turned around and returned at a constant speed of 80 miles an hour.
Exactly 4 hours before the end of his trip, he was still approaching B, and only 15 miles away from it. What is the distance between A and B? Let's think about this for a minute. 所以在第一个部分，从A到B开始，他每小时60英里的旅行。回去，他在一个小时的时间里旅行了。
And then starting from B, so from P all the way to B, and then all the way back to A, that whole segment took 4 hours. 所以p到b，回到a，是4小时。很好地注意到，我只是很感兴趣 this little interval right here, from P to B. We know the distance and we know the speed, so we could figure out the time.
The time would equal the distance, 15 miles divided by the speed, 60 miles an hour, 15 over 60 is one quarter. 所以这是一个小时的四分之一。我们可以将这一点写为15分钟，但让我们实际把它留成一小部分。 Well if P to B to A was 4 hours, and P to B is a quarter of an hour. Then the route from B back to A has to be, the time has to be the difference.
So that is 4 minus one quarter. So I'm gonna change 4 into an improper fraction. 16岁以上，减去一个四分之一，是15岁，但我现在只是将此作为一个不正确的分数。 That is the time, 15 over 4 hours is the time of the trip from be back to A. Well now we know the speed as well as the time, so we can figure out the distance.
当然，那个距离是我们正在寻找的距离，距离A到B的距离。 So the speed is 80, the time is 15 over 4. Remember the great trick, cancel before you multiply. 80 divided by 4 is 20. 20 times 15, well 2 times 15 is 30, 20 times 15 is 300.
And so that's actually the distance between A and B. In summary, when a word problem involves multiple travelers, 多次旅行或带有多条腿的行程，记住每个旅行者，每个行程和/或每条腿都应该等于RT等式。 Sometimes you will be able to solve for all quantities in one equation, and use those numbers to help solve for other equations.
更常见的是，您必须使用用于解决两个未知数的两个方程的技术。 我们谈到替代和消除。如果这些对您来说是新的，您可以在代数模块中了解更多信息。
