Averages

Welcome to the presentation on averages. Averages is probably a concept that you've already used before, maybe not in a mathematical way. But people will talk in terms of, the average voter wants a politician to do this, or the average student in a class wants to get out early. So you're probably already familiar with the concept of an average. And you probably already intuitively knew that an average is just a number that represents the different values that a group could have. But it can represent that as one number as opposed to giving all the different values. And let's give a couple of examples of how to compute an average, and you might already know how to do this. So let's say I had the numbers 1, 3, 5, and 20. And I asked you, what is the average of these four numbers? Well, what we do is, we literally just add up the numbers. And then divide by the number of numbers we have. So we say 1 plus 3 is 4. So let me write that. 1 plus 3 plus 5 plus 20 equals, let's see, 1 plus 3 is 4. 4 plus 5 is 9. 9 plus 20 is 29. And we had 4 numbers; one, two, three, four. So 4 goes into 29. And it goes, 7, 7, 28. And then we have 10, I didn't have to do that decimal there, oh well. 2, 8, 25. So 4 goes into 29 7.25 times. So the average of these four numbers is equal to 7.25. And that might make sense to you because 7.25 is someplace in between these numbers. And we can kind of view this, 7.25, as one way to represent these four numbers without having to list these four numbers. There are other representations you'll learn later on. Like the mode. You'll also the mean, which we'll talk about later, is actually the same thing as the average. But the average is just one number that you can use to represent a set of numbers. So let's do some problems which I think are going to be close to your heart. Let's say on the first four tests of an exam, I got a-- let's see, I got an 80, an 81. An 87, and an 88. What's my average in the class so far? Well, all I have to do is add up these four numbers. So I say, 80 plus 81 plus 87 plus 88. Well, zero plus 1 is 1. 1 plus 7 is 8. 8 plus 8 is 16. I just ran eight miles, so I'm a bit tired. And, 4/8, so that's 32. Plus 1 is 33. And now we divide this number by 4. 4 goes into 336. Goes into 33, 8 times. 8 times 4 is 32. 33 minus 32 is 1, 16. 4. So the average is equal to 84. So depending on what school you go to that's either a B or a C. So, so far my average after the first four exams is an 84. Now let's make this a little bit more difficult. We know that the average after four exams, at four exams, is equal to 84. If I were to ask you what do I have to get on the next test to average an 88, to average an 88 in the class. So let's say that x is what I get on the next test. So now what we can say is, is that the first four exams, I could either list out the first four exams that I took. Or I already know what the average is. So I know the sum of the first four exams is going to 4 times 84. And now I want to add the, what I get on the 5th exam, x. And I'm going to divide that by all five exams. So in other words, this number is the average of my first five exams. We just figured out the average of the first four exams. But now, we sum up the first four exams here. We add what I got on the fifth exam, and then we divide it by 5, because now we're averaging five exams. And I said that I need to get in an 88 in the class. And now we solve for x. Let me make some space here. So, 5 times 88 is, let's see. 5 times 80 is 400, so it's 440. 440 equals 4 times 84, we just saw that, is 320 plus 16 is 336. 336 plus x is equal to 440. Well, it turns out if you subtract 336 from both sides, you get x is equal to 104. So unless you have a exam that has some bonus problems on it, it's probably impossible for you to get ah an 88 average in the class after just the next exam. You'd have to get 104 on that next exam. And let's just look at what we just did. We said, after 4 exams we had an 84. What do I have to get on that next exam to average an 88 in the class after 5 exams? And that's what we solved for when we got x. Now, let's ask another question. I said after four exams, after four exams, I had an 84 average. If I said that there are 6 exams in the class, and the highest score I could get on an exam is 100, what is the highest average I can finish in the class if I were to really study hard and get 100 on the next 2 exams? Well, once again, what we'll want to do is assume we get 100 on the next 2 exams and then take the average. So we'll have to solve all 6 exams. So we're going to have the average of 6, so in the denominator we're going to have 6. The first four exams, the sum, as we already learned, is 4 exams times the 84 average. And this dot is just times. Plus, and there's going to be 2 more exams, right? Because there's 6 exams in the class. And I'm going to get 100 in each. So that's 200. And what's this average? Well, 4 times 84, we already said, is 336. Plus 200 over 6. So that's 536 over 6. 6 goes into 5 36. I don't know if if I gave myself enough space. But 6 goes into 53, 8 times. 48. 56. 9 times. 9 times 6 is 54. 6 minus is 20 6 goes into-- so we'll see it's actually 89.333333, goes on forever. So 89.3 repeating. So no matter how hard I try in this class, the best I can do. Because I only have two exams left, even if I were to get 100 on the next two exams. I can finish the class with an 89.333 average. Hopefully, I think some of this might have been a little bit of a review for you. You already had kind of a sense of what an average is. And hopefully these last two problems not only taught you how to do some algebra problems involving average, but they'll also help you figure out how well you have to do on your exams to get an A in your math class. I think you're now ready for the average module. Have fun.