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Distance, Rate, and Time

Let's continue using our equation solving skills to tackle some other problems. This time we're going to try motion problems. The primary concept we're dealing with, with motion problems is distance equals rate times time. For any object that moves at a constant rate. We can find the distance that it travels by multiplying its rate by its time. The time is how long the object is traveling at that given rate. So, let's say a car is traveling at 60 miles per hour and it traveled for 2 hours. Well, that means in 1 hour it goes 60 miles, so in 2 hours it goes double that, which is 120 miles. This formula should make sense, and the units down here should also make sense. We know if we multiply miles per hour times the number of hour, our hours will cancel leaving us with just miles. So if we know that distance equals rate times time. What equations could we write for the rate and for the time? In other words, I want you to solve this equation for r and write the expression here. And then solve this equation for t, and write that expression here.

Distance, Rate, and Time

Well, we know the rate has to be the distance divided by the time. An easy way to think about it is just a rate, like miles per hour. We know miles is a distance, per, which means we'll use division, an hour, which is time. If we want to solve using algebra, we'll take this equation and we'll divide both sides By the variable t. This will isolate the r. So we know the rate is distance divided by time. If we want to solve for time, then instead of dividing both sides by t, let's divide both sides by r. When we divide this side by r and this side by r, we'll get d divided by r is equal to 1 t. So we know the amount of time an object traveled would be its distance traveled divided by its rate.

Identify the Variable

Let's see if we can tackle a motion problem together. Janet and Rohan agree to meet at LAX, the Los Angeles airport. Janet's going to travel 250 miles, while Rohan is going to travel 300 miles. If Rohan travels 20 miles faster than Janet, and they both spend the same amount of time traveling let's see if we can find Janet's speed. Now this looks like a complicated problem but we're going to break it up into steps and see if we can tackle it together. Now for any of these motion problems I think it's best to use a table. A table like this helps us organize our information so we can figure out the distance, the rate and the time for each of the objects. In this case we're wondering about Janet's speed and Rohan's speed. So read through this problem and then when you think you understand it try to identify what should be the variable. Which of these amounts should be our variable so that way we can set up the right expressions to begin solving our problem.

Identify the Variable

In our problem, we're looking for Janet's speed. So, I think it's best if we let that be our variable. So, we can use a variable like x to represent her rate or her speed.

Distance Traveled and Rohans Speed

So now that we know Janet's speed, let's see if we can find some of this other missing information. Let's start with distance. How far did Janet travel and how far did Rohan travel? Enter those answers here. And don't worry about entering the units, I've already included them in our table.

Distance Traveled and Rohans Speed

Well, we know Janet travels 250 miles, since that's stated in the problem. And Rohan's distance, it's 300, so we put 300 here.

Rohans Rate

This is great. We've filled in half of our table already, and we're almost ready to set up some equations. But there is something else that we can figure out. Let's see if we can figure out Rohan's rate. What expression do you think we could write here? You'll need to think about the relationship between Janet's rate and Rohan's rate to answer this question. Good luck.

Rohans Rate

We know Rohan traveled 20 miles per hour faster than Janet. We don't know Janet's speed, since it's just x, but we do know that Rohan would be 20 more than this. So Rohan's speed must be x plus 20. We simply just add on our 20 to x, which is Janet's speed.

Expressions for Time

Alright, this is great. Now, we have Janet's distance and rate, and Rohan's distance and rate. We just need to find expressions for time for Janet and for Rohan. We know that distance equals rate times time. That was the problem we saw at the beginning of this lesson. And in fact, we saw that we could take this equation and manipulate it to have these other equations. So if we want to find the time, we can simply divide the distance by the rate. This is what we're going to use in our problem. Knowing this, I want you to use this information in the table to find an expression for Janet's time and for Rohan's time.

Expressions for Time

Janet's time would be 250 divided by x, and Rohan's time would be 300 divided by x plus 20. Remember, time equals distance divided by rate. So we just take Janet's distance 250, and divide it by her rate, which is x. We do the same thing for Rohan. We take his distance of 300 and divide it by his rate, which is x plus 20.

Motion Problem Equation

All right, great. We've completed a table full of information. We have information about Janet and information about Rohan. Now, there's one last critical piece to this motion problem. And it's right here. We know Janet and Rohan spend the same amount of time traveling. This means Janet's time has to equal Rohan's time. It means that we can set up an equation. So what equation can we write to solve for Janet's speed? Put that equation here.

Motion Problem Equation

Well, we know Janet's time is equal to Rohan's time. So we just set these two expressions equal to each other. Now this is a really powerful concept. We were able to use a variable for Janet's speed, relate all this other information, to come up with an equation relating the time for both Janet and Rohan. This is why it's so important when you tackle motion problems, it's best to organize your information in a chart. See if you can find relationships between the variables and any numbers that you are given. In our case, we knew that Janet and Rohan took the same amount of time to get to LAx. So we can set this time equal to this time.

Motion Problem Solved

So, now that we have an equation, we're ready to solve. I want you to find x and then tell me, what's Janet's speed and what's Rohan's speed. Good luck.

Motion Problem Solved

what we saw for x janet is traveling at 100 miles per hr.(mph). and rohan travels at 120 mph. great work if we found these 2 answers. To start solving these equation we will cross multiply . this is really the same as multiplying by the lcd of x(x+20) on both sides. if we multiply this side by x(x+20) we will be left with 250(x+20) . and if we multiply this side by x(x+20) we will be left with 300x. since the x+20 reduced to one. now I won't show that for the rest of the lesson but whenever we cross multiply we should understand thats what we are really doing. so 250(x+20)=300x we distribute 250 to both of the terms to get 250x+5000=300x we subtract 250xfrom both sides to began to isolate x. so 5000=50x. and then finally we divide both sides by 50 to get x is equal to 100(100=x).

Whats x

so we solve for x.but what is x? we want to think back to our problem.where could we originally set x to equal. choose the best choice from each of these .

Whats x

well we know x is janet's speed. when we first started solving problem we wanted to find janet's speed. so we let janet's speed or her rate equal x. we didn't know it.from there we were able to fill in the other pieces of information. we knew janet's distance and rohan's distance and we also knew that rohan traveled 20 mph faster than janet. once we have these 4 pieces of information we were able to write expressions for each of the person's time. we knew janet's time 250/x.and rohan's time 300/x+20. and since they both spend the same amount of time traveling we can say janet's time equal to rohan's time. we took the equation we saw for x which is janet's speed. he traveled 20 mph faster than janet. so we just add on 20 to janet's speed which equals 120 mph for rohan.

Motion Problem 2

Let's try a second problem on motion. This time, we have a bus traveling 10 miles per hour slower than a passenger van. If the bus can travel 200 miles in the same amount of time that the van can travel 240 miles, find the speed of each vehicle. Now, we're not going to do this entire problem in one step. But if you can, great. We're going to break this problem into steps just like we did before. So we have a bus and we have a van. Which of these do you think would be the best choice for our variable?

Motion Problem 2

In this case, I think it's best to make the speed of the van be the variable. We're going to let this equal x. Now, we could let the bus be the variable x, and I know it's not that clear since we need to find the speed of each of the vehicles. But we do know that the bus travels ten miles per hour slower than the van. So, if we let x be the van, we can easily write an expression for the rate of our bus.

Motion Problem 2 Table

So, now that we have x to be the speed of the van, I want you to fill in these other missing parts of our table. We can use this statement to figure out what the speed of the bus would be and then we could fill in the distances, and then the times.

Motion Problem 2 Table

Since the bus travels 10 miles per hour slower than the van, we can use x minus is 200. Our van on the other hand travels 240 miles in that same amount of time, and if we remember from before. We know distance equals rate times time. So we know time is equal to the distance divided by the rate. So to find the amount of time it takes the bus to travel 200 miles, we takes 200 and we divide it by the rate, x minus 10. This is our expression for time. And for the van, we take its distance, 240, and we divide it by its rate which is x. And here's our complete table. Great work if you found most of these answers. Even more amazing if you got them all correct, even if it wasn't your first try.

Motion Problem 2 Equation

So now that we have all the information we need, let's see if we can solve. I want you to set up the equation that we'll need in order to solve for x.

Motion Problem 2 Equation

Here, the equation we can write is 200 divided by x minus 10 equals 240 divided by x. Great thinking if you found this equation. The key is that the bus can travel 200 miles in the same amount of time that the van can travel 240 miles. So, this time, for the bus to travel 200 miles has to be equal to this time for the van to travel 240 miles.

Motion Problem 2 Solved

So, use your equation to figure out the speed of the bus and the speed of the van. Enter the numbers for each of those here.

Motion Problem 2 Solved

It turns out that the bus travels at 50 miles per hour, and the van travels at start solving this equation by multiplying both sides by the lowest common denominator. That would be x times x minus 10. We're really just cross multiplying, so we get 200x equal 240 times x minus 10. Next, we distribute the us to this equation which we can then use to solve for x. We just divide both sides by negative 40. So we'll have 1x is equal to 60. Now that we've solved for x, we want to remember what it actually represents. In this case we know x was the speed or the rate of the van. So, we know the van speed is really 60 mph, and the bus's speed would have to be 10 less than that. So the bus traveled at 50 mph.

Motion Problem 3

Here's our last motion problem for this lesson. Olivia can ride her bike 4 mph faster than Terrance. If Olivia can go 30 miles in the same time that Terrance can go 15 miles, what are their speeds? Again I think its best to use a table for a problem like this. So let's organize our information. We'll have Olivia and her distance, rate and time. And then we'll have Terrance and his distance, his rate and his time. So let's see if you can fill in all the missing parts to our table. Put in the correct number or the correct expression in each part. Also be sure that you use x for your variable. Think carefully about which item should be the variable here. And good luck!

Motion Problem 3

We know Olivia can travel 30 miles, so we'll have her distance as 30. And Terrance can go 15 miles, so his distance would be 15. And here's where we want to be careful. We can either let x be Olivia's rate or Terrance's rate. We know Olivia can go 4 miles per hour faster than Terrance. So we'll just add four to whatever Terrance's rate is. We don't know Terrance's rate, so I think it's best to use x here. If we do this, Terrance will be x and Olivia will be four more than that, four x plus four. And finally, to find the time, we'll take the distance of Olivia and divide it by her rate. So the time is 30 divided by x plus 4. We do the same thing for Terrance, since his time would be distance divided by rate, or 15 divided by x.

Motion Problem 3 Equation

Now that we have all of our information, let's write one equation to solve for x. What do you think it is?

Motion Problem 3 Equation

Our equation would be 30 divided by x plus 4 equals 15 divided by x. Great thinking if you found this. Again the key works here are in the same time. We know Olivia can bike 30 miles in the same time that Terrance can go 15. So, since we have expressions for time, for those two distances, we can set these times equal to one another.

Motion Problem 3 Solved

So use this equation and figure out what's Olivia's speed and what's Terrance's speed.

Motion Problem 3 Solved

Olivia bikes at e miles per hour and Terrance bikes at 4 miles per hour. Great solving for getting these two correct. We start solving the equation by cross multiplying. So we'll have 30 times x equals x plus 4 times 15. Then we distribute 15 to both of these terms to get 30 x equals 15 x plus 60. We subtract 15 x from both sides to get 15 x equals 60. And finally, we take our equation and we divide each side by 15 to get x is equal to 4. We know x really represents Terrance's speed. So Terrance's speed is 4 miles per hour. Olivia bikes 4 miles per hour faster than Terrance. So we know she travels at 8 miles per hour.