Averages Introduction to averages and algebra problems involving averages.
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- 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.
- 34 00:01:36,91 --> 00:01:40,88 So 4 goes into 29 7.25 times.
- So the average of these four numbers is equal to 7.25.
- 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.
- 62 00:03:16,95 --> 00:03:20,75 And now we divide this number by 4.
- 4 goes into 336.
- 67 00:03:31,85 --> 00:03:34 33 minus 32 is 1, 16.
- 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.
- 78 00:04:20,31 --> 00:04:23,49 So let's say that x is what I get on the next test.
- 80 00:04:28,18 --> 00:04:31,99 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.
- 97 00:05:22,71 --> 00:05:25,08 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.
- 117 00:06:34,67 --> 00:06:39,05 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
- 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.
- 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.
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