# Averages

## Video transcript

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.