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Multiple Traveler Questions

Multiple traveler problems. Some motion based word problems involve more than one traveler and or trips of more than one segment. We already saw a little of this in the previous video on average speed. The basic strategy in a multiple prob-, multiple traveler problem is that each traveler and each leg of the trip gets its own D equals RT equation.

That’s the big idea right there. Here’s a practice problem. Pause the video, and then we’ll talk about this. Okay, Martha and Paul started traveling from A to B at a constant speed. Martha traveled at a constant speed of 60 miles an hour, and Paul at a constant speed of 40 miles an hour.

When Martha arrived at B, Paul was still 50 miles away. What is the distance between A and B? Well, lets think about this. We don’t know the distance. And we don’t know the time it took, say the the time that it took for Martha to travel that whole distance.

But for Martha we could certainly say that D equals 60 times T. Because Martha traveled the entire distance D in a time T. In that same time, Paul was traveling at 40 miles an hour. And he didn’t travel the whole distance T. He was still 50 miles away. So he traveled the distance D minus 50, and that equals 40T.

Well by far, the easiest thing to do here, the first equation is already solved for D. We’ll just plug that into the second equation. Solve for T, divide by 20. We get time equals two and a half hours. So the distance is how far someone would travel at 60 miles an hour, in two and a half hours.

And of course that would be 150 miles. That’s the distance. Here’s another practice problem. Pause the video and then we’ll talk about this. Okay, Frank and Georgia started traveling from A to B at the same time.

Georgia’s constant speed was 1.5 times Frank’s constant speed. When Georgia arrived at B, she turned around and immediately returned by the same route. She crossed paths with Frank, who was coming toward B, when they were both 60 miles away from B. So, in other words, here’s A, here’s B.

Frank, poor, old Frank, he was just plodding along. He did this, this was Frank. Meanwhile, Georgia went all the way to B, turned around. And then they met each other. They crossed paths right here. And this point is 60 miles outside of B.

And so now we want this whole distance, D, the entire distance from A to B. Well, this is a little tricky here because we don’t know the times. We don’t know the distance, and we don’t even know the speeds. We know the ratio of the speeds, but we don’t know the speeds. So I’ll just call Frank’s speed R. And then we can call Georgia’s speed 1.5 times R.

Well okay, that’s fine but that’s still three unknowns. And we’re a bit suspicious here because I don’t think we’re gonna get three equations. So how are we gonna solve something with three unknowns? Well let’s see how this plays out. Well first of all, clearly, for good old Frank, he traveled the distance D minus 60.

He was still 60 miles away from D. So he traveled that distance in that time. Here the time is the time it takes for them to meet. So in the time it takes for them to meet, traveling at a speed R, he traveled the distance D minus 60. Meanwhile, Georgia was traveling at a speed of 1.5 R.

She traveled the whole distance D and then came back 60 miles. So she traveled D plus 60 in that same time. So that second equation I’m just gonna rewrite it a little bit. And notice that I can, I can get that term RT. Well this is beautiful. The first equation is solved for RT.

If I substitute that in I can eliminate two variables at once. And the only variable I’m left with is the distance. So this is a really slick move. Eliminate two variables at once and then be able to solve for the distance. So D plus 60 equals 1.5. And for RT I’m subbing in D minus 60.

Well now multiply by 1.5. Course 60 times 1.5 is 90. Subtract D from both sides I get 0.5. 60 equals 0.5 D minus 90. Add 90 to both sides. Multiply both sides by two.

D equals 300. In summary, when a word problem involves multiple travelers, multiple trips, or a trip with multiple legs, remem, remember that each traveler, each trip, and or each leg deserves its own D = RT equation. Sometimes, you will be able to solve for all the quantities in one equation, and use those numbers to solve for other equations.

More often you will have to use the techniques for solving two or more equations with two or more unknowns. And these are the techniques of substitution and elimination, which we used in this video.