Writing equations for relationships between quantities

- [Narrator] We're told Amad is going to walk 20 kilometers for a charity fundraiser. In the first part of this question, they say, write an equation that represents how many hours, t, the walk will take if Amad walks at a constant rate of r kilometers per hour. Pause this video and see if you could have a go at that. All right, now let's work through this together. So you might be familiar with the notion that distance is equal to rate times time. For example, if you were to walk at a rate of five kilometers per hour for two hours, you would say five times two, five kilometers per hour, times two hours, would give you 10 kilometers. Now in this situation, they've given us the number of kilometers or the distance in this situation, so in this situation d is equal to 20. So 20 is going to be equal to our rate, which we are told is going to be r kilometers per hour, times our time, which is t hours. Now they're asking us for an equation that represents how many hours the walk will take if Amad walks at a constant rate of r. So the way that it's phrased, it sounds like they want us to solve for t, where t is going to be equal to some expression here that deals with r and probably some other things, so if we put in any r here, then we can get the time. So if we know what the rate is, if you put that in here, because it's already solved for t, we'll be able to solve for that time. You could think of r as the independent variable that you could try different values out for, and that t is the dependent variable. It's the thing that we have solved for. So let's do that. Let's rewrite this expression here by solving for t and I could do it right over here. If I have 20 is equal to rt, if I wanna solve for t, what can I do? Well, I could divide both sides by r. If I do that on the right-hand side, then I'm just left with a t here, because an r divided by r is just one. And on the left, I have 20 over r, so I have t is equal to 20 over r and we're done. This will tell us how many hours Amad will take to walk based on the rate. You give me a rate. I'm just gonna divide 20 by that and I'm going to give you t. You might say, why is this useful? Well, this is useful because now that we have it written this way, any time someone gives an r to you, you just take 20 divided by that and it essentially is already gonna solve for what your time is, how long Amad's gonna have to walk. Question two, how many hours will the walk take if Amad walks at a constant rate of 6 kilometers per hour? Well, here is an example of that, where they are giving us the actual rate and they want the time. So we just take the 6 and replace it in for r, so we get t is equal to 20 over 6, which is 3 and 1/3 hours, which would be the same thing as 3 hours and 20 minutes, depending on how you wanna view it. Let's do another example here. So here we're told at the end of each day, a restaurant makes soup with whatever amount of vegetable stock is unused that day. Let me re-center this a little bit. The soup recipe calls for 400 milliliters of water for every 500 milliliters of vegetable stock. Write an equation that represents how much water the restaurant should use, and we'll use the variable w, with any amount of vegetable stock, v. All right, and then we'll do part two right after that, so let's look at this, 400 milliliters of water for every 500 milliliters of vegetable stock. And so to get my head around this, I like just to think about, let's put a little table here, and so you could say amount of water, let me write it this way, water and vegetable stock, vegetable stock. So for every 500 milliliters of vegetable stock, so if you had 500 milliliters of vegetable stock, and I won't write the milliliters, then you're going to have 400 milliliters of water. If you had 1,000 here, which is two times that, well, you're gonna have twice as much water, which is going to be 800. And so you can see this relationship that's forming. No matter what the vegetable stock is, if you essentially take 4/5 of that, that is the amount of water. You take 4/5, that is the amount of water. So if you had only 5 milliliters of vegetable stock, you take 4/5, you get the amount of water. So another way to think about it is the water that you need to use is going to be 4/5 of the amount of vegetable stock that you are going to be using, and so actually we just did part one. We wrote an equation that represents how much water the restaurant should use with any amount of vegetable stock and the way that they phrased it, we're solving for w given some v that you might have. And since we're solving for w here, we would consider w the dependent variable and v as the independent variable. You can give me different vs and then I can put that into this little equation here, and I can solve for the w, so we've done the first part. If there are 800 milliliters of unused vegetable stock, how much water should the restaurant use to make soup? Well, we can just take this 800 and substitute it in for v to figure that out. In this situation, the amount of water to use is 4/5 times 800, and that's going to be, let's see, 800 divided by 5, 100 divided by 5 is 20 and so, and then you're gonna have 8 of those, so it's 160, so this was 160 times 4 is equal to 640 milliliters, and we are done.