Common Core Algebra I.Unit 4.Lesson 2.Unit Conversions

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit four lesson two on unit conversions. Before we start this lesson, let me remind you that you can find a copy of the worksheet that goes with the video. Along with a homework assignment by clicking on the video's description. As well, don't forget about the QR codes at the top of each page. Those QR codes allow you to get right to this video by scanning them with either your smartphone or a tablet. All right, let's begin. Units are one of the most important and yet ignored parts of mathematics. They're critical in understanding the world that we live in, understanding what we do every day and how we measure things. They're all about what we consider to be the unit hole. The mile, the foot, the inch, the centimeter. Units have existed for a very, very long time. And being able to convert from one unit to another is extremely important. Critical mistakes can happen, especially in engineering when one group may be working in English systems like feet, and another group is working in a metric system, or a system internationale. Where they're using meters or centimeters. So today, we're going to look at general ways that you can convert units. And some of these problems are tough, so stick with them, try your best. And try to have fun with them. They're really all about multiplication. All right, let's get into exercise one. John has traveled a total of 4.5 miles. If there are 5280 feet in each mile, how many feet did John travel? Set up and solve a proportion for this problem. Also, do the problem by multiplying by E ratio. All right. So I'd like to really walk you through both of these, and then for the rest of the rest of the lesson, we're really going to be going with the second way that we do this. All right. So here's what we're doing. To do this problem by using a ratio, what we would say is that there are 5280 feet. In one mile. Setting up a proportion from the last lesson, we now have X feet, let's say. Divided by 4.5 miles. Cross multiplying, and what we get is one times X, which is X and 4.5, 4.5 times 5280, and that tells us because I can't do this in my head. I have to do a little cheat sheet. Is that he's traveled 23,000 760 feet. Now that's a perfectly good way of doing it. But it's too slow. When we know that there's a proportional relationship between two quantities. And by the way, this is very important. All right? Proportional relationships typically exist when two quantities are zero at the same time. So if he's traveled zero miles, he's traveled zero feet. Right? Makes sense. Watch out things that may seem proportional, maybe aren't, like, Fahrenheit and Celsius. Zero Fahrenheit doesn't correspond to zero Celsius. They are not in proportion to one another. So let's look at how we can do it with multiplication, because this is extremely important for future science courses, future work we do in math. This is how we're going to do it. We're going to do 4.5 miles. And what we're going to do is we're going to multiply by a ratio. Okay, and that ratio is going to consist of these feet and these miles. Two things that are equal. I'm going to put the 5280 feet in the numerator. And I'm going to put one mile in the denominator. And here's why. Units in the numerator and these really are in the numerator. I'm going to just put a one here. Units in the numerator will cancel units in the denominator. All right? Leaving me with the desired unit. Okay? So if you're ever wondering, well, why didn't I put the 5280 feet in the denominator, it was because I wanted to be left with units of feet in the numerator. All right? I wanted the miles to cancel out. So I had these miles in the denominator, they canceled. Now I just have 4.5 times 5280 feet, which again gives me my 23,000. 760 feet. Let me just make sure that's right. And it is. Okay? So every time we want a unit to cancel, we're going to multiply by a ratio. And we're going to want to make sure that things in the numerator cancel things in the denominator. That's going to be our big piece for today. So I'm going to clear this out, and then we're going to get a lot of practice. All right. So pause the video if you need to. Okay, here we go. Let's move on, and I think I'm going to go back to blue. I like the red. It's got some zing to it. But blue is a little bit kinder of a color. All right, there are exactly 2.5 centimeters in each inch. How many centimeters are in one foot? Show the work that leads to your answer. All right. Well, one thing that you should know is that there are 12 inches in every foot. All right. So here's how I'm going to do this. I'm going to actually do it with a double multiplication. I'm going to take one foot, which is what I'm starting with. And I'm going to multiply it by 12 inches. Divided by one foot. 12 inches in one foot. The feet cancel. I now have inches. Now what I'm going to do is I'm going to multiply by 2.54 centimeters in one inch. And now the inches cancel. What am I left with? My desired unit, the centimeter. So I take 12, I multiply it by 2.54, and I get 30.48 centimeters. This is something that a lot of engineers know. Because very often you have to convert between feet and centimeters. Feet being in an English unit, centimeters being a unit that almost the rest of the world uses. We use it a little bit. A little bit. All right, but that was pretty easy. We just have to make sure we keep setting up ratios of two things that are equal, right? 12 inches is equal to one foot, 2.54 centimeters is equal to one inch. We just keep setting up these ratios and we make sure that the units that we want to cancel are sort of an opposite places numerator denominator numerator denominator. And that the final unit that we want is in the numerator. All right, that'll modify a little bit when we have units that we want in both numerators and denominators, but for right now that gets the job done. Okay, let me clear out this text. And let's move on to exercise three. All right. A bathtub contains 14.5 cubic feet of water, right? So a volume, cubic feet. If water drains out of the bathtub at a rate of four gallons per minute, then how long will it take to the nearest minute to drain the bathtub? There are 7.5 gallons of water per cubic foot, show the work that leads to your answer. All right, so the real problem here is that we don't know the volume of the bathtub in gallons. If we did, then that four gallons per minute would be helpful for us. But it's not right now. So let's get the volume in gallons. So here we go. We've got 14.5 cubic feet. I'm going to write that down like this. You should be aware of those units. Times what do we know? 7.5 gallons per one cubic foot. Right? Those units cancel, and what I find is I find the volume of my bathtub as 108 .75 gallons. Right, 108.75 gallons. Now, hopefully what I should feel comfortable with now is let me let T equal the time to drain. All right, T is the time to drain in minutes. Then if I take my four gallons per minute right there for gallons per minute, and I multiply it by how much time it's been draining, right? That tells me how much it's drained. In gallons, if I now set that equal to one O 8, .75, right? The amount of gallons I have to drain, I should be able to find out how much time it takes. So I'm going to divide both sides by four, and ultimately this gets ugly. It's like 27.1 8 7 5. I have to cheat. And it says round to the nearest minute. So that's going to be 27. Minutes. All right. Takes about a half an hour now to drain this bathtub. Seems like it's either a really big bathtub or it's draining pretty slow. One of the two. I think it's kind of a big bathtub. All right. This is important though. Very often we're given information such as in this problem. That doesn't mesh well together, right? So the volume is in cubic feet, but the rate is in gallons per minute. So we have to do that conversion. And that's where units are really quite important. You know, in real practical problems, almost every number that you're working with has units associated with it. They're not just numbers, right? So if I'm traveling a certain distance, I can't travel 15. I can travel 15 feet, 15 miles, 15 yards, 15 centimeters. I can't travel 15, right? So units are absolutely critical. I'm going to clear this one out, and then we're going to do a ripper, one that's quite challenging. All right, here we go. Let's take a look. This is one of my absolute favorites. The mile in the kilometer are relatively close in size. I don't know if you realize that, but one mile and one kilometer aren't too far off from one another. I think that's why a lot of times if you're a track and field runner and you kind of watch track and field, you'll see that sometimes people will run the mile and sometimes they'll run the kilometer, right? Or otherwise known as the thousand meters. Okay? The question here is, can you convert one mile into an equivalent in kilometers? So how much is one mile worth in kilometers? Here's what I'm going to give you. 2.54 centimeters in an inch, you knew that. 5280 feet in a mile. We've seen that already. A hundred centimeters in a meter. I hope you knew that already. And a thousand meters in a kilometer. I also hope you knew that. All else you should be able to do for yourself. Round your answer to the nearest tenth of a kilometer. This takes a lot of multiplications, but you can do it. So what I'd like you to do is I'd like you to pause the video and see if you can get through this. By the way, if you want to check your answer, of course, it's very simple. I'm sure you've probably done this before to go into something like Google or Bing and type in mile to kilometer and see what that conversion is. Now, don't do that to get the final answer. Do that to check your final answer. We're going to work through it anyway in a minute. All right. So go ahead and pause the video now and take some time. All right, let's go through it. Well, we want to get all the way from miles to kilometers. All right, that's going to take a little bit. But let's start. Now what I always tell students, if you don't know what to do, do anything you can do. Well, one thing we can definitely do is convert miles and defeat. And we're going to do that. By multiplying by 5280 feet divided by one mile. All right? Great. But now we're in feet. All right? So where can we go? Well, I can certainly convert from feet to inches. I should know how to do this because I know that there are 12 inches. In one foot. Great. So now we're in terms of inches. Now, I can definitely get from inches to centimeters. That's what this thing is going to do for me. Now I need my inches down here, so I've got one inch, and I've got 2.54 centimeters. Great. Now here's where it might get a little bit tricky. The next thing I know is that there are a hundred centimeters in a meter, but the centimeters have to be down here and so does the hundred. So I know there's a hundred centimeters and one meter. So there's going to be a little division going on here as well, right? But now I'm in meters. And likewise, we know that there are a thousand meters in one kilometer. And that gets me to kilometers. So I'm going to take my calculator out, and I'm going to put this in. I'm going to do 5280. I'm going to multiply by 12. I'm going to multiply by 2.54. Then I'm going to divide by 100 and then I'm going to divide by 1000. Okay? And when I do that, I get 1.6 zero 9, keep in mind. We have kilometers. Rounded to the nearest tenth. 1.6. So they're relatively close in size, not too close. One mile equals 1.6 kilometers. All right? What a string of multiplications. Keep in mind that every single one of these ratios is one where the two quantities is equal, are equal, right? 5280 feet is equal to one mile. 12 inches is equal to one foot, 2.54 centimeters is equal to one inch, one meter is equal to a hundred centimeters, one kilometer is equal to a thousand meters. And in each case I made sure that my units canceled. And the left me with the unit I wanted. Kilometers. All right. I'm going to clear this one out. Let's move on to the next problem. Okay. One interesting conversion is from a speed expressed in feet per second to a speed, expressed in miles per hour, right? We sometimes think better in miles per hour because that's how the speeds of our cars are measured. You know, even my own son, who's only 9 years old, has a feeling for what it means to be moving at 45 mph, 60 mph, 20 mph, right? So what, but when we move as people, if I'm running, let's say. I'm probably going to be thinking about that in terms of feet per second. Okay? So let's take a look. Convert a speed of 45 mph. In two feet per second, given that there are 5280 feet in a mile. All right, so let's start off. 45 miles in one hour. Okay, that's what it means what that word per means. Per always means divide. All right. Now, I have to change this into phi per second, right? So I got to do two things. I have to do some multiplication to get the miles into feet, then I have to do some multiplication to get the hours into seconds. Let's take care of the feet into, I'm sorry, the miles in defeat. That's not too bad. We know that one mile is equal to 5280 feet. All right? So we've got our feet. No problem. Now hours, that's going to be a little bit tougher, right? We know that one hour, right? We need the hours up here because the hours are down here. Is equal to 60 minutes. So that's good. That cancels. But we need seconds, not minutes. So now we know that one minute equals 60 seconds. And look at that, we now have feet per second. Now remember what we're going to do here is we're going to actually on our calculator going to do 45 times 5280. Then we're going to divide by 60, and we're going to divide by 60 again, right? We have two divisions by 60. When all is said and done, what we find is that 66 feet per second. All right. So 45 mph is 66 feet per second. I love the second part. Now right now, as I'm recording this video, it's the year 2014. And right now, the fastest human being, as measured by the Olympics, I suppose, is a guy named Usain Bolt. Usain Bolt from Jamaica is just amazing. In a certain sense, one might say he's a freak of nature. He's so fast, and he's awesome. But in 2009, Usain ran 100 meters in a blazing 32.2 feet per second, 32.2 feet per second. But I don't have a great feeling for that. You know, is that like, is he like Jaguar fast? What is it? You know, I think best in miles per hour. So we're going to go in the opposite direction now. We're going to figure out how fast Usain is running compared to a typical car. So 32.2 feet per one second and what we want to do is figure out how fast he's going in miles per hour. So we're going to almost do the opposite of what we did in a right the first thing I'm going to do is maybe convert that feet into miles. Now we know that 5280 feet has to be in the denominator. And one mile has to be in the numerator. That's great because it gets me my miles. Now, we're going to do the opposite in terms of the time. We know that there are 60 seconds in one minute, that gives me minutes, and that's not what I need. And then we know that there are 60 minutes in one hour. Ah, there it is. So what is that? Well, now we're going to do 32.2, we can do the division or the multiplication first, and it's up to you. I think I do multiply by 60, multiply by 60, and then divide by 5280. We could divide by all these ones as well, but we really don't need to. All right? And when we do that, we find out where is that? That Usain is running at 21.95 or let's say 22 mph. Isn't that fascinating? If you got stuck behind somebody driving at 22 mph, you would think that person was crawling. They were just moving terribly solo, but the fastest human being on earth, saying bolt, running as fast as he can possibly run, only runs that fast. So much for superheroes. He's still darn fast. I'm going to clear the text out. All right, let's go in. Let's keep going. All right. Exercise 6. A local factory has to add a liquid ingredient to make their product at a rate of 13 quarts every 5 minutes. How many gallons per hour do they need to add? Show the work that leads to your answer. So what I'd like you to do is pause the video right now and see if you can figure this out. This is kind of a cool problem. Hopefully you know how many quarts there are in every gallon, but if not, feel free to do a quick Internet search to figure it out. All right? But let me give you a hint. It's a court. Court, like, you know, quarter. All right, pause the video now. See how many gallons per hour we have to add? All right, let's work through it. Let's start with 13 quarts per and this is important. Not one minute. But 5 minutes. All right, somehow we have to get to gallons per hour. All right, well, you can do the chords first or the minutes first, it's up to you. I'm going to take care of the courts first. All right, what I know is that there are four quarts. That's why it's called a quart. Per one gallon. All right. Simple enough. Quartz cancel. And now we have our gallons. I also know that there are 60 minutes per one hour. So now my minutes cancel, and I have my hours. Now actually on my calculator, I'm going to do 13 times 60. 13 times 60, but then I'm going to have to divide by 5 and I'm going to have to divide by four. All of that said and done gives me 39 gallons of ingredient I have to add every hour. All right. So we can convert rates like we did in the last problem with miles per hour defeat per second, but we can also look at different types of rates and convert them. All right, all about multiplying ratios and canceling units. I'm going to clear out the text, here we go. All right. Let's try the last problem. Exercise 7. A tractor can plant a field at a rate of 2.5 acres per 5 minutes. Apparently I like 5 minute time periods. If a mammoth farm measuring four square miles needs planting, how long will it take to plant the field? There are 640 acres in a square mile, determine your answer to the nearest hour. If the tractor runs 8 hours a day, what is the minimum number of days it will take to plant the farm. So this is kind of a cool cool problem. A four square mile farm is a big farm, big farm. All right. And yet, this tractor can really move quite quickly. Planting 2.5 acres every 5 minutes. So the question is, really, how many hours is it going to take? And then how many days is it going to take? And it's a little bit tricky because here we can only operate 8 hours a day. Anyhow, I'd like you to pause the video right now. Take up to ten minutes and see if you can answer really what amounts to be two different questions here. How many hours? And how many days? All right, let's work through it. Oh boy. So, what do we know? We know what we have four square miles. And yet the rate is an acres. So perhaps the first thing to do is figure out how many acres we have. We have four square miles, and we know that one square mile is going to be 640 acres. The acre is the typical area measurement for pharma. Now, I worked that out and I find that that's 2560 acres. That's a big farm, especially if you live in the east in the Midwest, or in the great plains. That might be kind of a typical farm on the large side, but sort of typical. Now, what do we know? We know that we've got 2.5 acres per 5 minutes, right? So we can set up a proportion here. 5 minutes is to 2.5 acres as X minutes is to 2560 acres. So this is going back to just the proportional work that we did yesterday. I'm sorry, not yesterday. But in the last lesson, I always think about it in terms of days. So I'll get 2.5 X is equal to 5 times 2560. So 2.5 X is equal to 12,000 800 by my calculation. Divide both sides by 2.5, and we get X equals 5120. But 5120 watt. This is very, very important. We just found the time in minutes. All right, so that's how long it's going to take in minutes. To plant this four square mile field. But now I need to convert it to hours and days. All right, so 5120 minutes. Times 60 minutes. Per one hour. The minutes cancel. And I figure out that will be 85.3 hours. And rounded to the nearest hour, that's 85. Right. Now, it would be tempting to divide that by 24 to get the number of days. And yet, we're told that the tractor can only run 8 hours a day. Right? So if I have 85 hours or 85.3, hopefully it won't make a difference. And I know that there's 8 hours in one working day. I know there's 24 hours in a day, right? But in a working day here, there's only 8 hours, and I do that. And I'll get 10.6 days, but we should really report that it will take 11 days to plant this field. Quite a ways, but it's a huge, huge farm. Quite some time that is. All right, so units are very, very important. It's important to understand them, and it's important to really kind of like nail them down and understand which units we need to use. I'm going to clear this pretty complicated problem out. So pause the video now and write down what you need to. All right. Let's finish up. So in today's lesson, what we did is we talked about units. They're important, how to view them, and how to convert them using simple multiplication by ratios of two equivalent quantities, like 5280 feet per one mile. You're going to get a lot more practice on this, obviously, in the homework. And then it's going to come up here and there in the course as we move along. All right. For now though, I'd like to thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weiler. And until next time, keep thinking, I keep solving problems.