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Distance, Rate, and Time

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Today, we're going to talk about distance, rate, and time.

Rate in the context of distance and time is better known as speed. For example, miles per hour is a rate. So we're going to look at the mathematical relationship between distance, rate, and time, and then we'll do some real-world examples.

So let's look at the mathematical relationship for distance, rate, and time. We're going to be looking at some formulas where D will be our distance, R will be our rate, and T will be your time. We're going to manipulate the formula so that we can find the mathematical relationship in terms of each of these three variables. So let's start with distance.

I know the distance that I've traveled is going to be equal to my rate times the time. So for example, if I was driving down the road at 60 miles per hour for 2 hours, the distance that I traveled would be 60 miles per hour times 2 hours, or 120 miles. So we know that our mathematical relationship in terms of distance is D equals R times T.

Now, I can manipulate this same equation to find the relationship in terms of R and in terms of T. So if I want to find it in terms of my rate, I want to cancel out the multiplying by T. So I'm going to divide both sides by T. So now I've isolated my R-variable and I found that my rate is equal to my distance over my time.

So if I have gone 100 miles and I'm going for 2 hours, then my speed was 50, or 50 miles per hour.

Next, we find that our formula for our rate is equal to the distance over the time. So let's go back to our original formula and manipulate it so that we have the relationship in terms of time.

So now if I want to isolate my T-variable, this time I'm going to divide both sides by my R. These cancel out and I see that my time is equal to my distance divided by my rate.

So for example, if I am driving for 50 miles and I'm going 50 miles per hour, , then it makes sense that it would take me 1 hour, or 50 divided by 50-- 1 hour to get there.

So we have our last formula or relationship which tells us that our time is equal to the distance over the rate. So let's use these three formulas, relationships, to solve some real-world problems.

So here's my first example. Jason ran for 45 minutes, or 0.75 hours, at a speed of 7 miles per hour. How far did he run?

So we want to know how far he ran. That's going to be our distance. And we can see that our rate is going to be 7 miles per hour and our time is going to be 0.75 hours.

So using my formula, I'm going to multiply my rate, which is 7 miles per hour, or 7 miles in 1 hour, and multiply that by my time, which is 0.75 hours. And I can just put this over 1.

Now, if you remember from doing unit conversions when we want to figure out our units, I can see that my hours in the denominator and here in the numerator are going to cancel out. And so my answer is going to be in terms of miles, which is good. That's what I want because I'm trying to find the distance.

And then I just need to multiply or simplify my fractions. Multiply 7 times 0.75, which will give me 5.25. And 1 times 1 is just 1. So we could have a denominator of 1 here in the bottom of a fraction, but we know that's just equal to 5.25 anyways. And 0.7 hours at a speed of 7 miles per hour, Jason ran a total distance of 5.25 miles.

So here's my second example. Lee boked to his friend's house in 2 hours. He knows that his friend's house is 42 miles from where he started and he wants to know, how fast did he bike?

So how fast you bike is going to be a rate. And we know that our distance is 42 miles and our time is 2 hours. So we can use this formula for rate being equal to distance over time.

So I'm going to start by substituting my value for my distance, 42 miles. And my time is going to be 2 hours.

I can simplify my two numbers. So 42 divided by 2 is going to give me 21. And none of my units cancel out, so the units of my answer are going to be in terms of miles per hour. So my rate is going to be 21 miles per hour.

Here's my third example. Shira wants to rollerblade to the corner store. She knows she can rollerblade at 12 miles per hour and that the store is 4 miles away. How long will it take for her to get there?

So we want to find how long it will take, which means we want to find the time. We know the distance is 4 miles and the rate is 12 miles per hour. So we can use this formula for time, which again time is equal to the distance over the rate divided by the rate.

So my distance is going to be 4 miles and my rate is going to be 12 miles per hour, or 12 miles in 1 hour.

Now, I know that dividing by 12 is the same as multiplying by 1/12, so I'm going to rewrite this as 4 miles times 1/12. And this will be 1 hour for 12 miles. This is still our speed. So I can put this over 1 and see how my units cancel.

So the miles up in the numerator, in this fraction, will cancel with the miles in the denominator of this fraction. So now I'm left with hours as my unit, which is good because I'm trying to find time. And I'm just going to simplify the numbers in my fraction.

So 4 times 1 is 4. 1 times 12 is 12. So this reduces to 1/3. 1/3 hours, which is the same as 20 minutes.

So let's go over our key points from today. As usual, make sure you get these in your notes if you don't have them already.

So we defined some variables where D equals distance, R equals rate, and T equals time. And we wrote some equations for each of those variables.

So we saw that our distance is equal to the rate times the time, our rate is equal to the distance divided by the time, and our time is equal to the distance divided by the rate. So I hope that these examples and key points helped you understand a little bit more about the relationship between distance, rate, and time. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.