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Beg Algebra: Percent

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About this Lesson

  • Type: Video Tutorial
  • Length: 10:41
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 115 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Beg Algebra: Decimals (4 lessons, $5.94)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/beginningalgebra. The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Percent
One of the most useful and important concepts from this class that we'll actually infuse into our everyday lives once we finish with this class is the notion of percents. These are really important. You see them all the time, and I just wanted to remind you what these things are all about.
So what is a percent? It's sort of a really weird thing, mathematically speaking. In terms of everyday thinking, it's a perfectly fine thing. So what do I mean when I say, for example, 20%? Or 25% since that's what I'm writing here. Okay. So this is of course how we denote it. This is the symbol that we use to denote percentage; 25%. Well, what does that mean? Well, what it means, actually, is a division. So there's two ways to think about it. One way to think about it is if you have 25%, what you do is take 25 and divide it by the special number 100. So, actually, a percent is a fraction, and it always is this number divided by 100. How can you remember that? Very easy; % = divide by 100. What could be easier. I just made that up. Isn't that great? So it's very easy to remember. You're dividing by 100, and what's the idea of percent? What does percent sort of represent here? Well, it represents a portion of something. Now remember, one of the most important lessons--"of" is times, "of" means multiplication. So, usually, a percent is in relation to something else. For example, if you notice, this fraction equals a quarter if you simplify it, because 100 is 4 o 25, and so I can cancel the 25's. The idea is that if, for example, an object in a store was 25% off, what does that mean? Well, what it means is that the actual price equals 25% of the original price. So 25% of--that means 25% times the price. So, in this case, it's a fraction of the price. So it basically means 1/4 of the price. 25% of the price means 1/4 the price.
Now let's take a look at even more of a real world example because things aren't usually 25% of the price. That's really, really rare. Usually it's 25% off, right, off or something like that. Let me tell you what actually happened to me. This is great. I was in the store, and I saw in one of these high tech stores, one of these really cool MP3 players. Have you seen one of these things? They're so cool and they're so exciting, and you can listen to music, you know, and you can put them in your ears and you can hear it, and it's wild man. The kids are just loving it. They're great. So I wanted one of these things. I said, ooh, it's so cool and you can scroll down. Oh, in fact, look at these songs that are already on it. I can't read them to you because--so anyway. So I got really excited because this one only costs, as you can see, $125.00, and so I quickly checked into my pants, and let's see what I got doing here. I've got $120. I cannot afford the MP3 player. So that was sort of sad, until I turned it over, and I realized it's actually on sale--5% off. Now what does that mean, 5% off? Well, 5% off means the actual sales price is not this, but it's actually this with that off. What does off mean? It means that I subtract. So I take 125 and I subtract 5% of the price, and we all know what "of" means. "Of," of course, is times. So what does that mean? It means that to figure out the sale price, what I first have to do is find 5% of this price and then take that amount and remove it from the actual price. So, as sort of a real world skill here, what do we know? We know that the price is $125. We also know that it's 5% off. So to find the appropriate actual sales price, we have to put these two together. We have to bridge them together through algebra. So let's try it right now live.
So the first step--there's two steps here--first find 5% of $125.00 and then subtract it off of the price. So, first the 5%. So we take $125.00 and we want to 5% of it. So that means we multiply by 5%. Right? That's the same thing as saying 5% of--because "of" means multiply--$125.00. Well, what's a percent? Now I remind you that a percent means that number divided by 100. So this equals that number, 5, divided by 100 times $125.00. Now we can simplify this a little teeny bit. So you can simplify and notice that this actually is what? Well, this is 5 times something, 5 o 20 in fact. So, in fact, this is 5 ÷ 5 o 20, and then I'm multiplying it by 125 divided by the invisible 1. So these actually cancel, and so I discover that 5% is the same thing as multiplying by 1/20. So what I see here is 125 ÷ 20, and what does that equal? Well, uh, gee, you can do a lot of things here I guess. You could, you could, I don't know. How would you want to do this? You can either factor the top, pull out a common factor of 5. So it's 5 times what? It would be a 1 and a 5 divided by 20. Oh, wait a minute. I don't like that. It's 5 o 25. Sorry, and now if I cancel this 5 away with one of these 5's then I'm left with just a 4, so what do I have? I have 25/4, and now you can actually long divide that out if you want to. You can plug it into the calculator. In fact, just for fun, I'll use a calculator since we've been long dividing so much, it might be sort of fun just to show you that, in fact, you can actually do this on a calculator too. You can always check your answer, by the way, on a calculator. It's always good to do that, and here what we get is 6.25, but what does that mean exactly? What is 6.25? Well, what were the units that we started with here? We found 5% of $125.00. So, in fact, it's green folks. It's green, and that unit stays with us the whole way. In the background, in terms of the mathematics, but at the end of the day, when you get to the checkout, it's $6.25. So what we just figured out was what 5% of the original price of $125.00 is, but 5% off means that that's how much we deduct from the actual price. So, now, to find out the actual price, what I've got to do is take $125.00 and subtract off $6.25. Now that subtraction requires lots of carrying and stuff, but somehow, with money, we can do it a lot easier I think because obviously it's going to end in a 75, and then what's going to happen here? Well here, I would have actually borrowed, so I have a 4, and so here I've got a 4 and I've got a 124, and I'm going to subtract 6, and so that's 14, and so it's going to be--I have no idea. In fact, let's try it. Just to show you that you an actually do this on a calculator. $125.00 - 6.25 =, can you see that? It equals $118.75. And you can certainly check your answer. How would you check? Just add these two up and see what you get. 5 + 5 = 10, so 0 carry a 1, 7 + 2 = 9 + 1 = 0, carry a 1, and then here you have 8 + 6 = 14 + 1 = 15, 1 + 1 = 2, so it works out. So that is the price, the sale price of this exciting, potentially mine, MP3 player, and if you notice, look what I've got. I've got $120.00. The thing only costs $118.75, and so this looks really, really good, except what if like tax was really, really high? Well, happily, I am buying this on a tax free day, and so, in fact, I can purchase this. I even get some change in my pocket, which is just enough to buy my first long-playing record. In fact, where do you put the record on this thing? Hmm, I don't know. Anyway, you can certainly see how to deal with percentages. It's just that number divided by 100, and if you happen to know where to put the record in, would you please just e-mail me? Thanks. See you soon.
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