45-45-90 triangles

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on 45-45-90 triangles.

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Let me write that down.

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How come the pen--
oh, there you go.

00:00:08.300 --> 00:00:15.770
45-45-90 triangles.

00:00:15.770 --> 00:00:19.050
Or we could say 45-45-90 right
triangles, but that might be

00:00:19.050 --> 00:00:21.630
redundant, because we know any
angle that has a 90 degree

00:00:21.630 --> 00:00:24.110
measure in it is a
right triangle.

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And as you can imagine, the
45-45-90, these are actually

00:00:27.790 --> 00:00:30.910
the degrees of the
angles of the triangle.

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So why are these
triangles special?

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Well, if you saw the last
presentation I gave you a

00:00:35.720 --> 00:00:43.950
little theorem that told you
that if two of the base angles

00:00:43.950 --> 00:00:49.000
of a triangle are equal-- and
it's I guess only a base angle

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if you draw it like this.

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You could draw it like this, in
which case it's maybe not so

00:00:51.830 --> 00:00:55.410
obviously a base angle, but
it would still be true.

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If these two angles are equal,
then the sides that they don't

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share-- so this side and this
side in this example, or this

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side and this side in this
example-- then the two sides

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are going to be equal.

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So what's interesting about
a 45-45-90 triangle is that

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it is a right triangle
that has this property.

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And how do we know that it's
the only right triangle

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that has this property?

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Well, you could imagine a
world where I told you that

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this is a right triangle.

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This is 90 degrees, so
this is the hypotenuse.

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Right, it's the side opposite
the 90 degree angle.

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And if I were to tell you that
these two angles are equal to

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each other, what do those
two angles have to be?

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Well if we call these two
angles x, we know that the

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angles in a triangle
add up to 180.

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So we'd say x plus x
plus-- this is 90-- plus

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90 is equal to 180.

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Or 2x plus 90 is equal to 180.

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Or 2x is equal to 90.

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Or x is equal to 45 degrees.

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So the only right triangle in
which the other two angles are

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equal is a 45-45-90 triangle.

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So what's interesting about
a 45-45-90 triangle?

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Well other than what I just
told you-- let me redraw it.

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I'll redraw it like this.

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So we already know this is 90
degrees, this is 45 degrees,

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this is 45 degrees.

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And based on what I just told
you, we also know that the

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sides that the 45 degree
angles don't share are equal.

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So this side is
equal to this side.

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And if we're viewing it from a
Pythagorean theorem point of

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view, this tells us that the
two sides that are not the

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hypotenuse are equal.

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So this is a hypotenuse.

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So let's call this side
A and this side B.

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We know from the Pythagorean
theorem-- let's say the

00:03:11.360 --> 00:03:14.880
hypotenuse is equal to C-- the
Pythagorean theorem tells us

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that A squared plus B squared
is equal to C squared.

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Right?

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Well we know that A equals
B, because this is a

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45-45-90 triangle.

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So we could substitute
A for B or B for A.

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But let's just
substitute B for A.

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So we could say B squared
plus B squared is

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equal to C squared.

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Or 2B squared is
equal to C squared.

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Or B squared is equal
to C squared over 2.

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Or B is equal to the square
root of C squared over 2.

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Which is equal to C-- because
we just took the square root of

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the numerator and the square
root of the denominator-- C

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over the square root of 2.

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And actually, even though this
is a presentation on triangles,

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I'm going to give you a little
bit of actually information

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on something called
rationalizing denominators.

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So this is perfectly correct.

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We just arrived at B-- and we
also know that A equals B-- but

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that B is equal to C divided
by the square root of 2.

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It turns out that in most of
mathematics, and I never

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understood quite exactly why
this was the case, people

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don't like square root of
2s in the denominator.

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Or in general they don't
like irrational numbers

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in the denominator.

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Irrational numbers are numbers
that have decimal places that

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never repeat and never end.

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So the way that they get rid
of irrational numbers in the

00:04:49.870 --> 00:04:52.230
denominator is that you do
something called rationalizing

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the denominator.

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And the way you rationalize
a denominator-- let's take

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our example right now.

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If we had C over the square
root of 2, we just multiply

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both the numerator and
the denominator by the

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same number, right?

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Because when you multiply the
numerator and the denominator

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by the same number, that's just
like multiplying it by 1.

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The square root of 2 over
the square root of 2 is 1.

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And as you see, the reason
we're doing this is because

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square root of 2 times square
root of 2, what's the

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square root of 2 times
square root of 2?

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Right, it's 2.

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Right?

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We just said, something times
something is 2, well the square

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root of 2 times square root
of 2, that's going to be 2.

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And then the numerator is C
times the square root of 2.

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So notice, C times the square
root of 2 over 2 is the same

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thing as C over the
square root of 2.

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And this is important to
realize, because sometimes

00:05:39.520 --> 00:05:41.090
while you're taking a
standardized test or you're

00:05:41.090 --> 00:05:44.190
doing a test in class, you
might get an answer that looks

00:05:44.190 --> 00:05:46.320
like this, has a square root of
2, or maybe even a square root

00:05:46.320 --> 00:05:49.550
of 3 or whatever, in
the denominator.

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And you might not see your
answer if it's a multiple

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choice question.

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What you ned to do in that case
is rationalize the denominator.

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So multiply the numerator and
the denominator by square

00:05:57.990 --> 00:06:01.470
root of 2 and you'll get
square root of 2 over 2.

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But anyway, back
to the problem.

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So what did we learn?

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This is equal to B, right?

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So turns out that B is equal
to C times the square

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root of 2 over 2.

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So let me write that.

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So we know that A
equals B, right?

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And that equals the square
root of 2 over 2 times C.

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Now you might want to memorize
this, though you can always

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derive it if you use the
Pythagorean theorem and

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remember that the sides that
aren't the hypotenuse in a

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45-45-90 triangle are
equal to each other.

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But this is very good to know.

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Because if, say, you're taking
the SAT and you need to solve a

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problem really fast, and if you
have this memorized and someone

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gives you the hypotenuse, you
can figure out what are the

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sides very fast, or i8f someone
gives you one of the sides,

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you can figure out the
hypotenuse very fast.

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Let's try that out.

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I'm going to erase everything.

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So we learned just now that A
is equal to B is equal to the

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square root of 2
over 2 times C.

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So if I were to give you a
right triangle, and I were to

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tell you that this angle is 90
and this angle is 45, and that

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this side is, let's
say this side is 8.

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I want to figure out
what this side is.

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Well first of all, let's
figure out what side

00:07:34.590 --> 00:07:35.500
is the hypotenuse.

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Well the hypotenuse is the side
opposite the right angle.

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So we're trying to actually
figure out the hypotenuse.

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Let's call the hypotenuse C.

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And we also know this is a
45-45-90 triangle, right?

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Because this angle is 45, so
this one also has to be 45,

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because 45 plus 90 plus
90 is equal to 180.

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So this is a 45-45-90 triangle,
and we know one of the sides--

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this side could be A or B-- we
know that 8 is equal to the

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square root of 2
over 2 times C.

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C is what we're trying
to figure out.

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So if we multiply both sides of
this equation by 2 times the

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square root of 2-- I'm just
multiplying it by the inverse

00:08:22.010 --> 00:08:23.600
of the coefficient on C.

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Because the square root of 2
cancels out that square root

00:08:25.750 --> 00:08:28.430
of 2, this 2 cancels
out with this 2.

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We get 2 times 8, 16 over the
square root of 2 equals C.

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Which would be correct, but as
I just showed you, people don't

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like having radicals
in the denominator.

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So we can just say C is equal
to 16 over the square root of

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2 times the square root of 2
over the square root of 2.

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So this equals 16 square
roots of 2 over 2.

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Which is the same thing
as 8 square roots of 2.

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So C in this example is
8 square roots of 2.

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And we also knows, since this
is a 45-45-90 triangle,

00:09:13.790 --> 00:09:16.700
that this side is 8.

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Hope that makes sense.

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In the next presentation
I'm going to show you a

00:09:19.740 --> 00:09:20.680
different type of triangle.

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Actually, I might even start
off with a couple more examples

00:09:22.900 --> 00:09:25.080
of this, because I feel I
might have rushed it a bit.

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But anyway, I'll see you soon
in the next presentation.


Pythagorean theorem II

WEBVTT
Kind: captions
Language: en

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I promised you that I'd give
you some more Pythagorean

00:00:02.690 --> 00:00:05.720
theorem problems, so I will
now give you more Pythagorean

00:00:05.720 --> 00:00:06.780
theorem problems.

00:00:09.790 --> 00:00:12.382
And once again, this is
all about practice.

00:00:12.382 --> 00:00:28.020
Let's say I had a triangle--
that's an ugly looking right

00:00:28.020 --> 00:00:35.030
triangle, let me draw another
one --and if I were to tell

00:00:35.030 --> 00:00:40.750
you that that side is 7, the
side is 6, and I want to

00:00:40.750 --> 00:00:42.250
figure out this side.

00:00:42.250 --> 00:00:45.510
Well, we learned in the last
presentation: which of these

00:00:45.510 --> 00:00:46.990
sides is the hypotenuse?

00:00:46.990 --> 00:00:49.470
Well, here's the right angle,
so the side opposite the right

00:00:49.470 --> 00:00:51.600
angle is the hypotenuse.

00:00:51.600 --> 00:00:53.120
So what we want to do
is actually figure

00:00:53.120 --> 00:00:54.730
out the hypotenuse.

00:00:54.730 --> 00:01:00.730
So we know that 6 squared
plus 7 squared is equal to

00:01:00.730 --> 00:01:01.700
the hypotenuse squared.

00:01:01.700 --> 00:01:03.800
And in the Pythagorean theorem
they use C to represent the

00:01:03.800 --> 00:01:05.470
hypotenuse, so we'll
use C here as well.

00:01:10.930 --> 00:01:16.030
And 36 plus 49 is
equal to C squared.

00:01:21.150 --> 00:01:25.510
85 is equal to C squared.

00:01:25.510 --> 00:01:30.760
Or C is equal to the
square root of 85.

00:01:30.760 --> 00:01:32.490
And this is the part that most
people have trouble with, is

00:01:32.490 --> 00:01:34.650
actually simplifying
the radical.

00:01:34.650 --> 00:01:40.290
So the square root of 85: can I
factor 85 so it's a product of

00:01:40.290 --> 00:01:42.820
a perfect square and
another number?

00:01:42.820 --> 00:01:45.920
85 isn't divisible by 4.

00:01:45.920 --> 00:01:48.350
So it won't be divisible by 16
or any of the multiples of 4.

00:01:52.400 --> 00:01:55.940
5 goes into 85 how many times?

00:01:55.940 --> 00:01:58.340
No, that's not perfect
square, either.

00:01:58.340 --> 00:02:02.030
I don't think 85 can be
factored further as a

00:02:02.030 --> 00:02:04.230
product of a perfect
square and another number.

00:02:04.230 --> 00:02:06.980
So you might correct
me; I might be wrong.

00:02:06.980 --> 00:02:09.570
This might be good exercise for
you to do later, but as far as

00:02:09.570 --> 00:02:12.670
I can tell we have
gotten our answer.

00:02:12.670 --> 00:02:15.070
The answer here is the
square root of 85.

00:02:15.070 --> 00:02:17.250
And if we actually wanted to
estimate what that is, let's

00:02:17.250 --> 00:02:21.810
think about it: the square root
of 81 is 9, and the square root

00:02:21.810 --> 00:02:25.010
of 100 is 10 , so it's some
place in between 9 and 10, and

00:02:25.010 --> 00:02:26.445
it's probably a little
bit closer to 9.

00:02:26.445 --> 00:02:28.245
So it's 9 point something,
something, something.

00:02:28.245 --> 00:02:30.260
And that's a good reality
check; that makes sense.

00:02:30.260 --> 00:02:33.080
If this side is 6, this side
is 7, 9 point something,

00:02:33.080 --> 00:02:36.270
something, something makes
sense for that length.

00:02:36.270 --> 00:02:37.260
Let me give you
another problem.

00:02:37.260 --> 00:02:44.790
[DRAWING]

00:02:44.790 --> 00:02:49.250
Let's say that this is 10 .

00:02:49.250 --> 00:02:51.300
This is 3.

00:02:51.300 --> 00:02:53.090
What is this side?

00:02:53.090 --> 00:02:55.060
First, let's identify
our hypotenuse.

00:02:55.060 --> 00:02:57.680
We have our right angle here,
so the side opposite the right

00:02:57.680 --> 00:03:00.230
angle is the hypotenuse and
it's also the longest side.

00:03:00.230 --> 00:03:01.116
So it's 10.

00:03:01.116 --> 00:03:05.390
So 10 squared is equal to
the sum of the squares

00:03:05.390 --> 00:03:06.640
of the other two sides.

00:03:06.640 --> 00:03:10.256
This is equal to 3 squared--
let's call this A.

00:03:10.256 --> 00:03:11.890
Pick it arbitrarily.

00:03:11.890 --> 00:03:14.380
--plus A squared.

00:03:14.380 --> 00:03:23.860
Well, this is 100, is equal to
9 plus A squared, or A squared

00:03:23.860 --> 00:03:29.720
is equal to 100 minus 9.

00:03:29.720 --> 00:03:32.560
A squared is equal to 91.

00:03:38.390 --> 00:03:40.390
I don't think that can be
simplified further, either.

00:03:40.390 --> 00:03:41.710
3 doesn't go into it.

00:03:41.710 --> 00:03:43.950
I wonder, is 91 a prime number?

00:03:43.950 --> 00:03:44.880
I'm not sure.

00:03:44.880 --> 00:03:49.200
As far as I know, we're
done with this problem.

00:03:49.200 --> 00:03:51.890
Let me give you another
problem, And actually, this

00:03:51.890 --> 00:03:56.500
time I'm going to include one
extra step just to confuse you

00:03:56.500 --> 00:04:00.240
because I think you're getting
this a little bit too easily.

00:04:00.240 --> 00:04:01.805
Let's say I have a triangle.

00:04:05.130 --> 00:04:07.990
And once again, we're dealing
all with right triangles now.

00:04:07.990 --> 00:04:10.130
And never are you going to
attempt to use the Pythagorean

00:04:10.130 --> 00:04:12.780
theorem unless you know for a
fact that's all right triangle.

00:04:16.130 --> 00:04:19.810
But this example, we know
that this is right triangle.

00:04:19.810 --> 00:04:25.050
If I would tell you the length
of this side is 5, and if our

00:04:25.050 --> 00:04:32.810
tell you that this angle is 45
degrees, can we figure out the

00:04:32.810 --> 00:04:36.410
other two sides of
this triangle?

00:04:36.410 --> 00:04:38.220
Well, we can't use the
Pythagorean theorem directly

00:04:38.220 --> 00:04:40.830
because the Pythagorean theorem
tells us that if have a right

00:04:40.830 --> 00:04:43.750
triangle and we know two of the
sides that we can figure

00:04:43.750 --> 00:04:45.140
out the third side.

00:04:45.140 --> 00:04:47.320
Here we have a right
triangle and we only

00:04:47.320 --> 00:04:48.870
know one of the sides.

00:04:48.870 --> 00:04:51.080
So we can't figure out
the other two just yet.

00:04:51.080 --> 00:04:54.330
But maybe we can use this extra
information right here, this 45

00:04:54.330 --> 00:04:57.120
degrees, to figure out another
side, and then we'd be able

00:04:57.120 --> 00:04:59.280
use the Pythagorean theorem.

00:04:59.280 --> 00:05:01.810
Well, we know that the
angles in a triangle

00:05:01.810 --> 00:05:03.860
add up to 180 degrees.

00:05:03.860 --> 00:05:05.610
Well, hopefully you know
the angles in a triangle

00:05:05.610 --> 00:05:06.630
add up to 180 degrees.

00:05:06.630 --> 00:05:08.320
If you don't it's my fault
because I haven't taught

00:05:08.320 --> 00:05:09.720
you that already.

00:05:09.720 --> 00:05:14.310
So let's figure out what
the angles of this

00:05:14.310 --> 00:05:15.080
triangle add up to.

00:05:15.080 --> 00:05:17.410
Well, I mean we know they add
up to 180, but using that

00:05:17.410 --> 00:05:20.790
information, we could figure
out what this angle is.

00:05:20.790 --> 00:05:23.590
Because we know that this angle
is 90, this angle is 45.

00:05:23.590 --> 00:05:30.340
So we say 45-- lets call this
angle x; I'm trying to make it

00:05:30.340 --> 00:05:35.870
messy --45 plus 90--
this [INAUDIBLE]

00:05:35.870 --> 00:05:40.720
is a 90 degree angle --plus
is equal to 180 degrees.

00:05:40.720 --> 00:05:43.520
And that's because the
angles in a triangle always

00:05:43.520 --> 00:05:46.740
add up to 180 degrees.

00:05:46.740 --> 00:05:55.970
So if we just solve for x, we
get 135 plus x is equal to 180.

00:05:55.970 --> 00:05:57.550
Subtract 135 from both sides.

00:05:57.550 --> 00:06:01.190
We get x is equal
to 45 degrees.

00:06:01.190 --> 00:06:02.680
Interesting.

00:06:02.680 --> 00:06:06.800
x is also 45 degrees.

00:06:06.800 --> 00:06:11.380
So we have a 90 degree angle
and two 45 degree angles.

00:06:11.380 --> 00:06:13.710
Now I'm going to give you
another theorem that's not

00:06:13.710 --> 00:06:16.920
named after the head
of a religion or the

00:06:16.920 --> 00:06:17.560
founder of religion.

00:06:17.560 --> 00:06:19.730
I actually don't think this
theorem doesn't have a name at.

00:06:19.730 --> 00:06:26.920
All It's the fact that if I
have another triangle --I'm

00:06:26.920 --> 00:06:31.980
going to draw another triangle
out here --where two of the

00:06:31.980 --> 00:06:34.840
base angles are the same-- and
when I say base angle, I just

00:06:34.840 --> 00:06:39.890
mean if these two angles are
the same, let's call it a.

00:06:39.890 --> 00:06:44.770
They're both a --then the sides
that they don't share-- these

00:06:44.770 --> 00:06:46.610
angles share this side, right?

00:06:46.610 --> 00:06:49.560
--but if we look at the sides
that they don't share, we know

00:06:49.560 --> 00:06:53.240
that these sides are equal.

00:06:53.240 --> 00:06:54.810
I forgot what we call
this in geometry class.

00:06:54.810 --> 00:06:57.270
Maybe I'll look it up in
another presentation;

00:06:57.270 --> 00:06:57.960
I'll let you know.

00:06:57.960 --> 00:07:00.040
But I got this far without
knowing what the name

00:07:00.040 --> 00:07:01.370
of the theorem is.

00:07:01.370 --> 00:07:04.170
And it makes sense; you don't
even need me to tell you that.

00:07:07.080 --> 00:07:10.480
If I were to change one of
these angles, the length

00:07:10.480 --> 00:07:11.660
would also change.

00:07:11.660 --> 00:07:14.310
Or another way to think about
it, the only way-- no, I

00:07:14.310 --> 00:07:15.350
don't confuse you too much.

00:07:15.350 --> 00:07:18.820
But you can visually see that
if these two sides are the

00:07:18.820 --> 00:07:21.670
same, then these two angles
are going to be the same.

00:07:21.670 --> 00:07:25.430
If you changed one of these
sides' lengths, then the angles

00:07:25.430 --> 00:07:28.660
will also change, or the angles
will not be equal anymore.

00:07:28.660 --> 00:07:31.120
But I'll leave that for
you to think about.

00:07:31.120 --> 00:07:34.320
But just take my word for it
right now that if two angles in

00:07:34.320 --> 00:07:39.400
a triangle are equivalent, then
the sides that they don't share

00:07:39.400 --> 00:07:41.690
are also equal in length.

00:07:41.690 --> 00:07:43.820
Make sure you remember: not the
side that they share-- because

00:07:43.820 --> 00:07:46.920
that can't be equal to anything
--it's the side that they don't

00:07:46.920 --> 00:07:49.410
share are equal in length.

00:07:49.410 --> 00:07:52.990
So here we have an example
where we have to equal angles.

00:07:52.990 --> 00:07:55.020
They're both 45 degrees.

00:07:55.020 --> 00:07:58.910
So that means that the sides
that they don't share-- this is

00:07:58.910 --> 00:08:00.230
the side they share, right?

00:08:00.230 --> 00:08:03.210
Both angle share this side --so
that means that the side that

00:08:03.210 --> 00:08:05.080
they don't share are equal.

00:08:05.080 --> 00:08:08.460
So this side is
equal to this side.

00:08:08.460 --> 00:08:10.520
And I think you might be
experiencing an ah-hah

00:08:10.520 --> 00:08:12.020
moment that right now.

00:08:12.020 --> 00:08:15.380
Well this side is equal to this
side-- I gave you at the

00:08:15.380 --> 00:08:18.050
beginning of this problem that
this side is equal to 5 --so

00:08:18.050 --> 00:08:20.320
then we know that this
side is equal to 5.

00:08:20.320 --> 00:08:23.920
And now we can do the
Pythagorean theorem.

00:08:23.920 --> 00:08:25.750
We know this is the
hypotenuse, right?

00:08:28.940 --> 00:08:35.180
So we can say 5 squared plus 5
squared is equal to-- let's say

00:08:35.180 --> 00:08:38.950
C squared, where C is the
length of the hypotenuse --5

00:08:38.950 --> 00:08:42.010
squared plus 5 squared-- that's
just 50 --is equal

00:08:42.010 --> 00:08:44.110
to C squared.

00:08:44.110 --> 00:08:48.370
And then we get C is equal
to the square root of 50.

00:08:48.370 --> 00:08:56.250
And 50 is 2 times 25, so C is
equal to 5 square roots of 2.

00:08:56.250 --> 00:08:57.220
Interesting.

00:08:57.220 --> 00:09:00.110
So I think I might have given
you a lot of information there.

00:09:00.110 --> 00:09:02.840
If you get confused, maybe you
want to re-watch this video.

00:09:02.840 --> 00:09:05.630
But on the next video I'm
actually going to give you more

00:09:05.630 --> 00:09:08.095
information about this type of
triangle, which is actually a

00:09:08.095 --> 00:09:11.550
very common type of triangle
you'll see in geometry and

00:09:11.550 --> 00:09:14.470
trigonometry 45,
45, 90 triangle.

00:09:14.470 --> 00:09:15.930
And it makes sense why it's
called that because it has

00:09:15.930 --> 00:09:19.930
45 degrees, 45 degrees,
and a 90 degree angle.

00:09:19.930 --> 00:09:22.460
And I'll actually show you
a quick way of using that

00:09:22.460 --> 00:09:25.920
information that it is a 45,
45, 90 degree triangle to

00:09:25.920 --> 00:09:29.520
figure out the size if you're
given even one of the sides.

00:09:29.520 --> 00:09:31.870
I hope I haven't confused you
too much, and I look forward

00:09:31.870 --> 00:09:33.195
to seeing you in the
next presentation.

00:09:33.195 --> 00:09:35.120
See you later.


Introduction to the Pythagorean theorem

WEBVTT
Kind: captions
Language: en

00:00:01.030 --> 00:00:04.470
Welcome to the presentation
on the Pythagorean theorem.

00:00:04.470 --> 00:00:06.820
I apologize if my voice
sounds a little horsie.

00:00:06.820 --> 00:00:08.112
A little hoarse, not horsie.

00:00:08.112 --> 00:00:10.070
I was singing a little
bit too much last night,

00:00:10.070 --> 00:00:11.460
so please forgive me.

00:00:11.460 --> 00:00:13.710
Well, anyway, we
will now teach you

00:00:13.710 --> 00:00:15.330
about the Pythagorean theorem.

00:00:15.330 --> 00:00:17.730
And you might have
heard of this before.

00:00:17.730 --> 00:00:21.020
As far as I know, it is the
only mathematical theorem

00:00:21.020 --> 00:00:23.681
named after the
founder of a religion.

00:00:23.681 --> 00:00:25.680
Pythagoras, actually, I
think his whole religion

00:00:25.680 --> 00:00:26.790
was based on mathematics.

00:00:26.790 --> 00:00:28.040
But I'm no historian here.

00:00:28.040 --> 00:00:30.340
So I'll leave that
to the historians.

00:00:30.340 --> 00:00:33.070
But anyway, let's get started on
what the Pythagorean theorem is

00:00:33.070 --> 00:00:34.540
all about.

00:00:34.540 --> 00:00:37.790
If I were to give
you a triangle--

00:00:37.790 --> 00:00:44.030
let me give you a
triangle-- and I

00:00:44.030 --> 00:00:46.820
were to tell you that it's
not a normal triangle.

00:00:46.820 --> 00:00:50.150
It is a right triangle.

00:00:50.150 --> 00:00:53.060
And all a right triangle
is is a triangle

00:00:53.060 --> 00:00:58.060
that has one side
equal to 90 degrees.

00:00:58.060 --> 00:00:59.640
And I'll leave
you to think about

00:00:59.640 --> 00:01:01.490
whether it's ever
possible for a triangle

00:01:01.490 --> 00:01:04.330
to have more than one
side that's 90 degrees.

00:01:04.330 --> 00:01:06.570
But anyway, just granted
that a right triangle

00:01:06.570 --> 00:01:08.270
is a side that has
at least-- well,

00:01:08.270 --> 00:01:10.410
let me say a right
triangle is a triangle that

00:01:10.410 --> 00:01:13.980
has only one side
that's at 90 degrees.

00:01:13.980 --> 00:01:16.740
And if you have a right
triangle, what the Pythagorean

00:01:16.740 --> 00:01:21.160
theorem allows you to do is
if I give you a right triangle

00:01:21.160 --> 00:01:22.960
and I give you two
of the sides, we

00:01:22.960 --> 00:01:25.209
can figure out the third side.

00:01:25.209 --> 00:01:26.750
So before I throw
the theorem at you,

00:01:26.750 --> 00:01:29.320
let me actually give you a
couple of more definitions.

00:01:29.320 --> 00:01:30.950
Actually, just one more.

00:01:30.950 --> 00:01:34.740
So if this is the right
angle in a right triangle--

00:01:34.740 --> 00:01:36.130
it's at 90 degrees.

00:01:36.130 --> 00:01:38.511
And we symbolize that by
drawing the angles like this,

00:01:38.511 --> 00:01:40.760
kind of like a box instead
of drawing it like a curve,

00:01:40.760 --> 00:01:41.472
like that.

00:01:41.472 --> 00:01:43.430
I hope I'm not messing
up the drawing too much.

00:01:48.650 --> 00:01:53.216
The side opposite the right
angle is called the hypotenuse.

00:01:55.740 --> 00:01:58.490
And I really should look up
where this word comes from.

00:01:58.490 --> 00:02:01.830
Because I think it's a
large and unwieldy word,

00:02:01.830 --> 00:02:04.527
and it's a little
daunting at first.

00:02:04.527 --> 00:02:06.610
My sister told me that she
had a math teacher once

00:02:06.610 --> 00:02:10.389
who made people memorize it's
a high pot that is in use.

00:02:10.389 --> 00:02:12.200
So I don't know if
that helps you or not.

00:02:12.200 --> 00:02:14.170
But over time, you'll
use the term hypotenuse

00:02:14.170 --> 00:02:16.170
so much that it'll seem
just like a normal word.

00:02:16.170 --> 00:02:18.003
Although when you look
at it, it really does

00:02:18.003 --> 00:02:19.300
look kind of strange.

00:02:19.300 --> 00:02:21.320
Anyway, going back
to definitions,

00:02:21.320 --> 00:02:26.310
the hypotenuse is the side
opposite the 90-degree angle.

00:02:26.310 --> 00:02:28.400
And if you look at
any right triangle,

00:02:28.400 --> 00:02:31.800
you'll also quickly realize that
the hypotenuse is the longest

00:02:31.800 --> 00:02:33.930
side of the right triangle.

00:02:33.930 --> 00:02:35.900
So I think we're done
now with definitions.

00:02:35.900 --> 00:02:39.760
So what does the
Pythagorean theorem tell us?

00:02:39.760 --> 00:02:44.775
Well, let's call C is equal to
the length of the hypotenuse.

00:02:48.040 --> 00:02:51.860
And let A be the
length of this side.

00:02:51.860 --> 00:02:55.000
And let B equal the
length of this side.

00:02:55.000 --> 00:02:56.540
What the Pythagorean
theorem tells

00:02:56.540 --> 00:03:09.760
us is that A squared plus B
squared is equal to C squared.

00:03:09.760 --> 00:03:11.650
Now, that very
simple formula might

00:03:11.650 --> 00:03:14.200
be one of the most powerful
formulas in mathematics.

00:03:14.200 --> 00:03:16.680
From this, you go into
Euclidean geometry.

00:03:16.680 --> 00:03:18.622
You go into trigonometry.

00:03:18.622 --> 00:03:20.205
You can do anything
with this formula,

00:03:20.205 --> 00:03:23.620
but we'll leave that
to future lectures.

00:03:23.620 --> 00:03:25.610
Let's actually
test this formula.

00:03:25.610 --> 00:03:27.370
Or not test it--
let's use the formula.

00:03:27.370 --> 00:03:28.828
Maybe in another
presentation, I'll

00:03:28.828 --> 00:03:32.750
actually do a proof or, at
minimum, a visual proof of it.

00:03:32.750 --> 00:03:35.470
I apologize ahead of time that
I'm a bit scatterbrained today.

00:03:35.470 --> 00:03:36.990
It's been a while since
I last did a video.

00:03:36.990 --> 00:03:39.050
And once again, I told you
I sang a little bit too much

00:03:39.050 --> 00:03:39.560
last night.

00:03:39.560 --> 00:03:41.870
So my throat is sore.

00:03:41.870 --> 00:03:43.836
OK, so we have a triangle.

00:03:47.960 --> 00:03:50.446
And remember, it has
to be a right triangle.

00:03:50.446 --> 00:03:52.570
So let's say that this is
a right angle right here.

00:03:52.570 --> 00:03:54.390
It's 90 degrees.

00:03:54.390 --> 00:04:01.010
And if I were to tell you
that this side is of length 4.

00:04:01.010 --> 00:04:02.490
And actually, let
me change that.

00:04:02.490 --> 00:04:04.320
This side is of length 3.

00:04:04.320 --> 00:04:06.310
This side is of length 4.

00:04:06.310 --> 00:04:08.720
And we want to figure out
the side of this length.

00:04:08.720 --> 00:04:11.650
So the first thing I do when
I look at a right triangle

00:04:11.650 --> 00:04:13.440
is I figure out what
the hypotenuse is.

00:04:13.440 --> 00:04:15.360
Which side is the hypotenuse?

00:04:15.360 --> 00:04:16.760
Well, there's two ways to do it.

00:04:16.760 --> 00:04:17.801
There's actually one way.

00:04:17.801 --> 00:04:19.380
You look at where
the right angle is.

00:04:19.380 --> 00:04:21.420
And it's the side
opposite to that.

00:04:21.420 --> 00:04:23.590
So this is the hypotenuse.

00:04:23.590 --> 00:04:27.134
This would be C in our formula,
the Pythagorean theorem.

00:04:27.134 --> 00:04:28.550
We could call it
whatever we want.

00:04:28.550 --> 00:04:31.680
But just for simplicity,
remember A squared plus B

00:04:31.680 --> 00:04:33.490
squared is equal to C squared.

00:04:33.490 --> 00:04:37.170
So in this case, we see that the
other two sides, each of them

00:04:37.170 --> 00:04:39.520
squared, when added together
will equal C squared.

00:04:39.520 --> 00:04:44.860
So we get 3 squared
plus 4 squared

00:04:44.860 --> 00:04:50.390
is equal to C squared,
where C is our hypotenuse.

00:04:50.390 --> 00:04:57.120
So 3 squared is 9, plus
16 is equal to C squared.

00:04:57.120 --> 00:05:02.360
25 is equal to C squared.

00:05:02.360 --> 00:05:05.490
And C could be plus or minus 5.

00:05:05.490 --> 00:05:10.730
But we know that you can't have
a minus 5 length in geometry.

00:05:10.730 --> 00:05:16.690
So we know that C is equal to 5.

00:05:16.690 --> 00:05:18.310
So using the
Pythagorean theorem,

00:05:18.310 --> 00:05:21.887
we just figured out that if we
know the sides-- if one side is

00:05:21.887 --> 00:05:23.345
3, the other side
is 4, then we can

00:05:23.345 --> 00:05:24.886
use Pythagorean
theorem to figure out

00:05:24.886 --> 00:05:30.034
that the hypotenuse of this
triangle has the length 5.

00:05:30.034 --> 00:05:31.075
Let's do another example.

00:05:35.140 --> 00:05:42.170
Let's say, once again,
this is the right angle.

00:05:42.170 --> 00:05:45.260
This side is of length 12.

00:05:45.260 --> 00:05:48.720
This slide is of length 6.

00:05:48.720 --> 00:05:51.610
And I want to figure
out what this side is.

00:05:51.610 --> 00:05:53.830
So let's write down the
Pythagorean theorem.

00:05:53.830 --> 00:05:59.010
A squared plus B squared is
equal to C squared, where

00:05:59.010 --> 00:06:01.602
C is that length
of the hypotenuse.

00:06:01.602 --> 00:06:04.060
So the first thing I want to
do when I look at our triangle

00:06:04.060 --> 00:06:07.380
that I just drew is which
side is the hypotenuse.

00:06:07.380 --> 00:06:10.710
Well, this right here
is the right angle.

00:06:10.710 --> 00:06:14.615
So the hypotenuse is
this side right here.

00:06:14.615 --> 00:06:16.240
And we can also
eyeball it and say, oh,

00:06:16.240 --> 00:06:18.750
that's definitely the longest
side of this triangle.

00:06:18.750 --> 00:06:23.170
So we know that A
squared plus B squared

00:06:23.170 --> 00:06:27.110
is equal to 12
squared, which is 144.

00:06:27.110 --> 00:06:30.960
Now we know we have
one side, but we

00:06:30.960 --> 00:06:32.220
don't have the other side.

00:06:32.220 --> 00:06:33.636
So I've got to ask
you a question.

00:06:33.636 --> 00:06:36.210
Does it matter which side
we substitute for A or B?

00:06:36.210 --> 00:06:38.491
Well, no, just
because A or B-- they

00:06:38.491 --> 00:06:40.240
kind of do the same
thing in this formula.

00:06:40.240 --> 00:06:43.087
So we could pick any side to
be A other than the hypotenuse.

00:06:43.087 --> 00:06:44.670
And we'll pick the
other side to be B.

00:06:44.670 --> 00:06:48.320
So let's just say
that this side is B,

00:06:48.320 --> 00:06:51.700
and let's say that
this side is A.

00:06:51.700 --> 00:06:52.990
So we know what A is.

00:06:52.990 --> 00:07:00.490
So we get 6 squared plus
B squared is equal to 144.

00:07:00.490 --> 00:07:08.790
So we get 36 plus B
squared is equal to 144.

00:07:08.790 --> 00:07:16.320
B squared is equal
to 144 minus 36.

00:07:16.320 --> 00:07:20.870
B squared is equal to 112.

00:07:20.870 --> 00:07:23.690
Now we've got to simplify what
the square root of 112 is.

00:07:23.690 --> 00:07:25.820
And what we did in
those radical modules

00:07:25.820 --> 00:07:27.459
probably is helpful here.

00:07:27.459 --> 00:07:28.000
So let's see.

00:07:28.000 --> 00:07:32.246
B is equal to the
square root of 112.

00:07:32.246 --> 00:07:33.120
Let's think about it.

00:07:33.120 --> 00:07:34.940
How many times
does 4 go into 112?

00:07:34.940 --> 00:07:39.780
4 goes into 120 five times,
so it'll go into it 28 times.

00:07:39.780 --> 00:07:42.420
And then 4 goes
into 28 seven times.

00:07:42.420 --> 00:07:48.780
So I actually think that
this is equal to 16 times 7.

00:07:48.780 --> 00:07:49.400
Am I right?

00:07:49.400 --> 00:07:53.600
7 times 10 is 70,
plus 42 is 112.

00:07:53.600 --> 00:07:54.350
Right.

00:07:54.350 --> 00:07:56.770
So B equals the square
root of 16 times 7.

00:07:56.770 --> 00:07:59.190
See, I just factored
that as a product

00:07:59.190 --> 00:08:01.239
of a perfect square
and a prime number.

00:08:01.239 --> 00:08:03.530
Or actually, it doesn't have
to be a prime number, just

00:08:03.530 --> 00:08:05.630
a non-perfect square.

00:08:05.630 --> 00:08:12.680
And then I get B is equal
to 4 square roots of 7.

00:08:12.680 --> 00:08:14.120
So there we go.

00:08:14.120 --> 00:08:23.360
If this is 12, this is 6,
this is 4 square roots of 7.

00:08:23.360 --> 00:08:26.590
I think that's all the time I
have now for this presentation.

00:08:26.590 --> 00:08:28.590
Right after this, I'll
do one more presentation

00:08:28.590 --> 00:08:32.460
where I give a couple of more
Pythagorean theorem problems.

00:08:32.460 --> 00:08:34.030
See you soon.


More percent problems

WEBVTT
Kind: captions
Language: en

00:00:01.210 --> 00:00:07.750
Let's say I go to a store and
I have $50 in my pocket.

00:00:07.750 --> 00:00:11.070
$50 in my wallet.

00:00:14.110 --> 00:00:18.670
And at the store that day
they say it is a 25%

00:00:18.670 --> 00:00:24.680
off marked price sale.

00:00:24.680 --> 00:00:28.880
So 25% off marked price means
that if the marked price is

00:00:28.880 --> 00:00:31.670
$100 the price I'm going
to pay is going to be

00:00:31.670 --> 00:00:34.390
25% less than $100.

00:00:34.390 --> 00:00:40.830
So my question to you is if I
have $50, what is the highest

00:00:40.830 --> 00:00:42.600
marked price I can afford?

00:00:42.600 --> 00:00:45.270
Because I need to know that
before I go finding something

00:00:45.270 --> 00:00:47.270
that I might like.

00:00:47.270 --> 00:00:49.920
So let's do a little
bit of algebra.

00:00:49.920 --> 00:01:05.240
So let x be the highest marked
price that I can afford.

00:01:05.240 --> 00:01:10.290
So if the sale is 25% off of x,
we could say that the new

00:01:10.290 --> 00:01:20.680
price, the sale price will be x
minus 25% of x is equal

00:01:20.680 --> 00:01:22.790
to the sale price.

00:01:22.790 --> 00:01:26.730
And I'm assuming that I'm in
a state without sales tax.

00:01:26.730 --> 00:01:29.760
Whatever the sale price is, is
what I have to pay in cash.

00:01:29.760 --> 00:01:33.820
So x minus 25% x is equal
to the sale price.

00:01:33.820 --> 00:01:35.860
The discount is going
to be 25% of x.

00:01:35.860 --> 00:01:44.490
But we know that this is the
same thing as x minus 0.25x.

00:01:44.490 --> 00:01:46.530
And we know that that's the
same thing as-- well, because

00:01:46.530 --> 00:01:49.600
we know this is 1x, x is
the same thing is 1x.

00:01:49.600 --> 00:01:50.910
1x minus 0.25x.

00:01:50.910 --> 00:02:00.050
Well, that means that 0.75x is
equal to the sale price, right?

00:02:00.050 --> 00:02:07.180
All I did is I rewrote x minus
25% of x as 1x minus 0.25x.

00:02:07.180 --> 00:02:09.300
And that's the same
thing as 0.75x.

00:02:09.300 --> 00:02:12.180
Because 1 minus 0.25 is 0.75.

00:02:12.180 --> 00:02:15.290
So 0.75x is going to
be the sale price.

00:02:15.290 --> 00:02:17.980
Well, what's the sale
price that I can afford?

00:02:17.980 --> 00:02:22.440
Well, the sale price
I can afford is $50.

00:02:22.440 --> 00:02:28.725
So 0.75x is going to equal $50.

00:02:31.430 --> 00:02:34.820
If x is any larger number than
the number I'm solving for,

00:02:34.820 --> 00:02:37.330
then the sale price is going
to be more than $50 and I

00:02:37.330 --> 00:02:38.360
won't be able to afford it.

00:02:38.360 --> 00:02:41.520
So that's how we set the--
the highest I can pay is $50

00:02:41.520 --> 00:02:42.950
and that's the sale price.

00:02:42.950 --> 00:02:45.170
So going back to how we did
these problems before.

00:02:45.170 --> 00:02:47.710
We just divide both
sides by 0.75.

00:02:47.710 --> 00:02:52.120
And we say that the highest
marked price that I can afford

00:02:52.120 --> 00:02:58.010
is $50 divided by 0.75.

00:02:58.010 --> 00:03:00.410
And let's figure
out what that is.

00:03:00.410 --> 00:03:07.490
0.75 goes into 50-- let's
add some 0's in the back.

00:03:07.490 --> 00:03:09.815
If I take this decimal
2 to the right.

00:03:09.815 --> 00:03:15.220
Take this decimal, move it 2 to
the right, goes right there.

00:03:15.220 --> 00:03:18.930
So 0.75 goes into 50 the
same number of times

00:03:18.930 --> 00:03:22.950
that 75 goes into 5,000.

00:03:22.950 --> 00:03:23.790
So let's do this.

00:03:23.790 --> 00:03:25.870
75 goes into 50 zero times.

00:03:25.870 --> 00:03:29.710
75 goes into 500-- so let
me think about that.

00:03:29.710 --> 00:03:32.065
I think it goes
into it six times.

00:03:34.730 --> 00:03:36.110
Because seven times is
going to be too much.

00:03:36.110 --> 00:03:40.040
So it goes into it six times.

00:03:40.040 --> 00:03:44.840
6 times 5 is 30.

00:03:44.840 --> 00:03:46.890
6 times 7 is 42.

00:03:46.890 --> 00:03:50.040
Plus 3 is 45.

00:03:50.040 --> 00:03:53.150
So the remainder is 50.

00:03:53.150 --> 00:03:54.270
I see a pattern.

00:03:54.270 --> 00:03:55.640
Bring down the 0.

00:03:55.640 --> 00:03:56.640
Well, same thing again.

00:03:56.640 --> 00:04:00.620
75 goes into 500 six times.

00:04:00.620 --> 00:04:03.670
6 times 75 is going
to be 450 again.

00:04:03.670 --> 00:04:05.730
We're going to keep having
that same pattern over

00:04:05.730 --> 00:04:06.600
and over and over again.

00:04:06.600 --> 00:04:12.100
It's actually 66.666-- I hope
you don't think I'm an evil

00:04:12.100 --> 00:04:15.500
person because of this number
that happened to show up.

00:04:15.500 --> 00:04:19.090
But anyway, so the highest sale
price that I can afford or the

00:04:19.090 --> 00:04:24.270
highest marked price I can
afford is $66 dollars.

00:04:24.270 --> 00:04:28.420
And if I were to around up,
and $0.67 if I were to

00:04:28.420 --> 00:04:31.070
round to the nearest penny.

00:04:31.070 --> 00:04:34.350
If I were to write this kind of
as a repeating decimal, I could

00:04:34.350 --> 00:04:38.800
write this as 66.66 repeating.

00:04:38.800 --> 00:04:42.150
Or I also know that 0.6666
going on forever is

00:04:42.150 --> 00:04:43.230
the same thing as 2/3.

00:04:43.230 --> 00:04:46.580
So it's 66 and 2/3.

00:04:46.580 --> 00:04:48.290
But since we're working with
money and we're working with

00:04:48.290 --> 00:04:50.280
dollars, we should just
round to the nearest penny.

00:04:50.280 --> 00:04:55.320
So the highest marked price
that I can afford is $66.67.

00:04:55.320 --> 00:04:58.830
So if I go and I see a
nice pair of shoes for

00:04:58.830 --> 00:05:02.360
$55, I can afford it.

00:05:02.360 --> 00:05:05.580
If I see a nice tie for $70,
I can't afford it with

00:05:05.580 --> 00:05:07.350
the $50 in my pocket.

00:05:07.350 --> 00:05:10.620
So hopefully not only will this
teach you a little bit of math,

00:05:10.620 --> 00:05:13.680
but it'll help you do a
little bit of shopping.

00:05:13.680 --> 00:05:15.430
So let me ask you
another problem, a very

00:05:15.430 --> 00:05:17.760
interesting problem.

00:05:17.760 --> 00:05:22.480
Let's say I start with an
arbitrary-- let's put

00:05:22.480 --> 00:05:23.210
a fixed number on it.

00:05:23.210 --> 00:05:27.260
Let's say I start with $100.

00:05:27.260 --> 00:05:35.630
And after one year
it grows by 25%.

00:05:38.690 --> 00:05:42.000
And then the next year,
let's call that year

00:05:42.000 --> 00:05:45.300
two, it shrinks by 25%.

00:05:45.300 --> 00:05:47.100
So this could have happened
in the stock market.

00:05:47.100 --> 00:05:49.370
The first year I have a
good year, my portfolio

00:05:49.370 --> 00:05:51.070
grows by 25%.

00:05:51.070 --> 00:05:52.990
The second year I have a
bad year and my portfolio

00:05:52.990 --> 00:05:54.730
shrinks by 25%.

00:05:54.730 --> 00:05:57.540
So my question is how much
money do I have at the

00:05:57.540 --> 00:05:59.930
end of the two years?

00:05:59.930 --> 00:06:02.070
Well a lot of people might
say, oh, this is easy, Sal.

00:06:02.070 --> 00:06:06.270
If I grow by 25% and then I
shrink by 25% I'll end up with

00:06:06.270 --> 00:06:08.320
the same amount of money.

00:06:08.320 --> 00:06:13.780
But I'll show you it's actually
not that simple because the 25%

00:06:13.780 --> 00:06:16.520
in either case or in both cases
is actually a different

00:06:16.520 --> 00:06:18.100
amount of money.

00:06:18.100 --> 00:06:19.590
So let's figure this out.

00:06:19.590 --> 00:06:27.100
If I start with $100 and I grow
it by 25%-- 25% of $100 is $25.

00:06:27.100 --> 00:06:28.260
So I grew it by $25.

00:06:28.260 --> 00:06:34.300
So I go to $125.

00:06:34.300 --> 00:06:38.590
So after one year of growing
by 25% I end up with $125.

00:06:38.590 --> 00:06:44.670
And now this $125 is
going to shrink by 25$.

00:06:44.670 --> 00:06:48.320
So if something shrinks by 25%,
that means it's just going to

00:06:48.320 --> 00:06:52.680
be 0.75 or 75% of what
it was before, right?

00:06:52.680 --> 00:06:55.850
1 minus 25%.

00:06:55.850 --> 00:06:59.500
0.75 times $125.

00:06:59.500 --> 00:07:01.830
So let's work that out here.

00:07:01.830 --> 00:07:07.325
$125 times 0.75.

00:07:07.325 --> 00:07:10.770
And just in case you're
confused, I don't want to

00:07:10.770 --> 00:07:13.800
repeat it too much, but if
something shrinks by 25% it is

00:07:13.800 --> 00:07:16.790
now 75% of its original value.

00:07:16.790 --> 00:07:23.140
So if $125 shrinks by 25% it's
now 75% of $125 or 0.75.

00:07:23.140 --> 00:07:24.440
Let's do the math.

00:07:24.440 --> 00:07:27.980
5 times 5 is 25.

00:07:27.980 --> 00:07:31.810
2 times 5 is 10 plus 2 is 12.

00:07:31.810 --> 00:07:34.800
1 times 5-- 7.

00:07:34.800 --> 00:07:38.510
7 times 5 is 35.

00:07:38.510 --> 00:07:40.100
7 times 2 is 14.

00:07:40.100 --> 00:07:43.750
Plus 3 is 17.

00:07:43.750 --> 00:07:45.560
Sorry.

00:07:45.560 --> 00:07:46.880
7 times 1 is 7.

00:07:46.880 --> 00:07:48.980
Plus 1 is 8.

00:07:48.980 --> 00:07:53.800
So it's 5, 7, and then
this is 7 actually.

00:07:53.800 --> 00:07:55.130
14.

00:07:55.130 --> 00:07:56.490
9.

00:07:56.490 --> 00:07:58.260
94.75, right?

00:07:58.260 --> 00:08:00.230
Two decimal points.

00:08:00.230 --> 00:08:03.410
94.75.

00:08:03.410 --> 00:08:05.750
So it's interesting.

00:08:05.750 --> 00:08:11.390
If I start with $100 and it
grows by 25%, and then it

00:08:11.390 --> 00:08:16.450
shrinks by 25% I end up with
less than I started with.

00:08:16.450 --> 00:08:18.890
And I want you to think
about why that happens.

00:08:18.890 --> 00:08:24.400
Because 25% on $100 is the
amount that I'm gaining.

00:08:24.400 --> 00:08:26.790
That's a smaller number than
the amount that I'm losing.

00:08:26.790 --> 00:08:31.330
I'm losing 25% on $125.

00:08:31.330 --> 00:08:33.280
That's pretty interesting,
don't you think?

00:08:33.280 --> 00:08:35.100
That's actually very
interesting when a lot of

00:08:35.100 --> 00:08:39.480
people compare-- well, actually
I won't go into stock

00:08:39.480 --> 00:08:40.340
returns and things.

00:08:40.340 --> 00:08:41.825
But I think that should be a
pretty interesting thing.

00:08:41.825 --> 00:08:43.290
You should try that out
with other examples.

00:08:43.290 --> 00:08:46.700
Another interesting thing is
for any percentage gain, you

00:08:46.700 --> 00:08:49.050
should think about how much you
would have to lose-- what

00:08:49.050 --> 00:08:51.260
percentage you would
have to lose to end

00:08:51.260 --> 00:08:52.050
up where you started.

00:08:52.050 --> 00:08:54.080
That's another
interesting project.

00:08:54.080 --> 00:08:56.700
Maybe I'll do that in a
future presentation.

00:08:56.700 --> 00:08:59.430
Anyway, I think you're now
ready to do some of those

00:08:59.430 --> 00:09:02.200
percent madness problems.

00:09:02.200 --> 00:09:03.330
Hope you have fun.

00:09:03.330 --> 00:09:04.630
Bye.


Taking percentages

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.820
Let's get started
with some problems.

00:00:03.820 --> 00:00:04.620
Let's see.

00:00:04.620 --> 00:00:19.806
First problem: what
is 15% of 40?

00:00:19.806 --> 00:00:22.610
The way I do percent problems
is I just convert the

00:00:22.610 --> 00:00:25.630
percentage to a decimal and
then I multiply it times the

00:00:25.630 --> 00:00:27.720
number that I'm trying to
get the percentage of.

00:00:27.720 --> 00:00:32.820
So 15% as a decimal is 0.15.

00:00:32.820 --> 00:00:36.360
You learned that from the
percent to decimal conversion

00:00:36.360 --> 00:00:37.640
video, hopefully.

00:00:37.640 --> 00:00:39.550
And we just multiply
this times 40.

00:00:39.550 --> 00:00:47.990
So let's say 40 times 0.15.

00:00:47.990 --> 00:00:49.510
5 times 0 is 0.

00:00:49.510 --> 00:00:53.100
5 times 4 is 20.

00:00:53.100 --> 00:00:54.620
Put a 0 there.

00:00:54.620 --> 00:00:56.660
And then 1 times 0 is 0.

00:00:56.660 --> 00:00:59.266
1 times 4 is 4.

00:00:59.266 --> 00:01:02.720
And you get 6 0 0.

00:01:02.720 --> 00:01:04.490
Then you count the
decimal spots.

00:01:04.490 --> 00:01:05.670
1, 2.

00:01:05.670 --> 00:01:08.740
No decimals up there,
so you go 1, 2 and you

00:01:08.740 --> 00:01:10.230
put the decimal there.

00:01:10.230 --> 00:01:18.870
So 15% of 40 is equal to 0.15
times 40, which equals 6.00.

00:01:18.870 --> 00:01:22.480
Well, that's just the
same thing as 6.

00:01:22.480 --> 00:01:23.640
Let's do another problem.

00:01:23.640 --> 00:01:25.710
Hopefully, that didn't
confuse you too much.

00:01:25.710 --> 00:01:27.960
And I'm going to try to
confuse you this time just

00:01:27.960 --> 00:01:31.050
in case you weren't properly
confused the last time.

00:01:31.050 --> 00:01:47.860
What is 0.2% of-- let me
think of a number-- of 7.

00:01:47.860 --> 00:01:50.060
So a lot of people's
inclinations would just

00:01:50.060 --> 00:01:55.010
say, oh, 0.2%, that's
the same thing as 0.2.

00:01:55.010 --> 00:01:58.950
And if that was your
inclination you would be wrong.

00:01:58.950 --> 00:02:00.610
Because remember,
this isn't 0.2.

00:02:00.610 --> 00:02:03.290
This is 0.2%.

00:02:03.290 --> 00:02:04.760
So there's two ways of
thinking about this.

00:02:04.760 --> 00:02:10.350
You could say that this is
0.2/100, which is, if you

00:02:10.350 --> 00:02:13.270
multiply the numerator and
denominator by 10, is the

00:02:13.270 --> 00:02:17.830
same thing as 2/1,000.

00:02:17.830 --> 00:02:19.260
Or you can just do the
technique where you

00:02:19.260 --> 00:02:22.260
move the decimal space
over 2 to the left.

00:02:22.260 --> 00:02:26.360
In which case, if you're
starting with 0.2 and you

00:02:26.360 --> 00:02:29.620
move the decimal space 2
to the left, you go bam.

00:02:29.620 --> 00:02:30.300
Whoops!

00:02:30.300 --> 00:02:32.090
Bam, bam.

00:02:32.090 --> 00:02:33.170
That's where the decimal goes.

00:02:33.170 --> 00:02:37.370
So it's 0.002.

00:02:37.370 --> 00:02:38.410
This is key.

00:02:38.410 --> 00:02:43.370
0.2% is the same
thing as 0.002.

00:02:43.370 --> 00:02:46.110
This can always trip you up and
I've made this careless mistake

00:02:46.110 --> 00:02:47.890
all the time, so don't feel
bad if you ever do it.

00:02:47.890 --> 00:02:49.460
But just always pay careful
attention if you see a

00:02:49.460 --> 00:02:52.050
decimal and a percentage
at the same time.

00:02:52.050 --> 00:02:55.400
So now that we've figured out
how to write this percentage

00:02:55.400 --> 00:02:58.210
as a decimal we just have to
multiply it times the number

00:02:58.210 --> 00:03:00.000
that we want to take
the percentage of.

00:03:00.000 --> 00:03:09.190
So we say 0.002 times 7.

00:03:09.190 --> 00:03:10.490
Well, this is pretty
straightforward.

00:03:10.490 --> 00:03:15.070
7 times 2 is 14.

00:03:15.070 --> 00:03:17.710
And how many total numbers do
we have or how many total

00:03:17.710 --> 00:03:19.830
digits do we have behind
the decimal point?

00:03:19.830 --> 00:03:20.240
Let's see.

00:03:20.240 --> 00:03:22.930
It's 1, 2, 3.

00:03:22.930 --> 00:03:28.600
So we need 1, 2, 3 digits
behind the decimal point.

00:03:28.600 --> 00:03:36.175
So 0.2% of 7 is equal to 0.014.

00:03:36.175 --> 00:03:38.200
And you're probably thinking,
boy, that's a really,

00:03:38.200 --> 00:03:39.360
really small number.

00:03:39.360 --> 00:03:44.545
And it makes sense because
0.2%, if you want to

00:03:44.545 --> 00:03:46.860
think about it, that's
smaller than even 1%.

00:03:46.860 --> 00:03:48.585
So that's even
smaller than 1/100.

00:03:48.585 --> 00:03:52.450
And actually, if you think
about it, 0.2% is 1/500.

00:03:52.450 --> 00:03:55.760
And if you do the math,
1/500 of 7 will turn

00:03:55.760 --> 00:03:58.190
out to be this number.

00:03:58.190 --> 00:03:59.500
And that's an important
thing to do.

00:03:59.500 --> 00:04:01.240
It's always good to do a
reality check because when

00:04:01.240 --> 00:04:04.560
you're doing these decimal and
these percent problems, it's

00:04:04.560 --> 00:04:09.280
very easy to kind of lose a
factor of 10 here or there.

00:04:09.280 --> 00:04:10.240
Or gain a factor of 10.

00:04:10.240 --> 00:04:14.620
So always do a reality check to
see if your answer makes sense.

00:04:14.620 --> 00:04:18.630
So now I'm going to
confuse you even further.

00:04:18.630 --> 00:04:27.940
What if I were to ask you
4 is 20% of what number?

00:04:31.600 --> 00:04:33.730
So a lot of people's
reflex might just be,

00:04:33.730 --> 00:04:34.850
oh, let me take 20%.

00:04:34.850 --> 00:04:36.360
It becomes 0.20.

00:04:36.360 --> 00:04:37.850
And multiply it times 4.

00:04:37.850 --> 00:04:41.570
And in that case, again,
you may be wrong.

00:04:41.570 --> 00:04:42.080
Because think about it.

00:04:42.080 --> 00:04:44.700
I'm not saying
what is 20% of 4?

00:04:44.700 --> 00:04:49.560
I'm saying that 20%
of some number is 4.

00:04:49.560 --> 00:04:51.470
So now we're going to be doing
a little bit of algebra.

00:04:51.470 --> 00:04:54.090
I bet you didn't expect that
in the percent module.

00:04:54.090 --> 00:04:59.080
So let x equal the number.

00:05:05.760 --> 00:05:14.630
And this problem says that
20% of x is equal to 4.

00:05:14.630 --> 00:05:17.500
I think now it's in a form
that you might recognize.

00:05:17.500 --> 00:05:19.240
So how do we write
20% as a decimal?

00:05:19.240 --> 00:05:21.970
Well, that's just 0.20 or 0.2.

00:05:21.970 --> 00:05:23.890
And we just multiply
it by x to get 4.

00:05:23.890 --> 00:05:28.080
So 20%, that's the
same thing as 0.2.

00:05:28.080 --> 00:05:30.450
It's the same thing as 0.20,
but that last trailing

00:05:30.450 --> 00:05:31.720
0 doesn't mean much.

00:05:31.720 --> 00:05:36.160
0.2 times x is equal to 4.

00:05:36.160 --> 00:05:38.080
And now we have a level
one linear equation.

00:05:38.080 --> 00:05:39.670
I bet you didn't
expect to see that.

00:05:39.670 --> 00:05:42.540
So what do we do?

00:05:42.540 --> 00:05:43.550
Well there's two
ways to view it.

00:05:43.550 --> 00:05:45.820
You can just divide both
sides of this equation

00:05:45.820 --> 00:05:47.700
by the coefficient on x.

00:05:47.700 --> 00:05:54.010
So if you divide 0.2 here
and you divide by 0.2 here.

00:05:54.010 --> 00:05:59.460
So you get x is equal
to 4 divided by 0.2.

00:05:59.460 --> 00:06:03.350
So let's figure out what
4 divided by 0.2 is.

00:06:03.350 --> 00:06:05.120
I hope I have enough space.

00:06:05.120 --> 00:06:12.500
0.2 goes into 4-- I'm going
to put a decimal point here.

00:06:12.500 --> 00:06:14.300
And the way we do these
problems, we move the

00:06:14.300 --> 00:06:16.310
decimal point here one
over to the right.

00:06:16.310 --> 00:06:18.520
So we just get a 2 and then we
can move the decimal point

00:06:18.520 --> 00:06:20.420
here one over to the right.

00:06:20.420 --> 00:06:23.680
So this 0.2 goes into 4
the same number of times

00:06:23.680 --> 00:06:26.012
that 2 goes into 40.

00:06:26.012 --> 00:06:27.530
And this is easy.

00:06:27.530 --> 00:06:29.670
2 goes into 40 how many times?

00:06:29.670 --> 00:06:32.152
Well, 2 goes into 4 two
times and then 2 goes

00:06:32.152 --> 00:06:32.980
into 0, zero times.

00:06:32.980 --> 00:06:34.050
You could've done
that in your head.

00:06:34.050 --> 00:06:36.530
2 into 40 is twenty times.

00:06:36.530 --> 00:06:40.580
So 4 divided by 0.2 is 20.

00:06:40.580 --> 00:06:44.570
So the answer is
4 is 20% of 20.

00:06:47.120 --> 00:06:48.280
And does that make sense?

00:06:48.280 --> 00:06:49.640
Well, there's a couple of
ways to think about it.

00:06:49.640 --> 00:06:53.610
20% is exactly 1/5.

00:06:53.610 --> 00:06:55.810
And 4 times 5 is 20.

00:06:55.810 --> 00:06:56.810
That makes sense.

00:06:56.810 --> 00:06:59.430
If you're still not sure
we can check the problem.

00:06:59.430 --> 00:07:02.630
Let's take 20% of 20.

00:07:02.630 --> 00:07:12.880
So 20% of 20 is equal
to 0.2 times 20.

00:07:12.880 --> 00:07:16.182
And if you do the math
that also will equal 4.

00:07:16.182 --> 00:07:18.410
So you made sure you
got the right answer.

00:07:18.410 --> 00:07:19.490
Let's do another one like that.

00:07:24.000 --> 00:07:25.810
I'm picking numbers randomly.

00:07:25.810 --> 00:07:36.140
Let's say 3 is 9% of what?

00:07:40.440 --> 00:07:52.320
Once again, let's let x equal
the number that 3 is 9% of.

00:07:52.320 --> 00:07:53.680
You didn't have to
write all that.

00:07:53.680 --> 00:08:00.980
Well, in that case we know that
0.09x-- 0.09, that's the same

00:08:00.980 --> 00:08:04.430
thing as 9% of x--
is equal to 3.

00:08:04.430 --> 00:08:10.770
Or that x is equal to
3 divided by 0.09.

00:08:10.770 --> 00:08:15.440
Well, if we do the decimal
division, 0.09 goes into 3.

00:08:15.440 --> 00:08:17.290
Let's put a decimal point here.

00:08:17.290 --> 00:08:19.260
I don't know how many
0's I'm going to need.

00:08:19.260 --> 00:08:22.220
So if I move this decimal over
to the right twice, then I'll

00:08:22.220 --> 00:08:24.980
move this decimal over
to the right twice.

00:08:24.980 --> 00:08:27.580
So 0.09 goes into 3 the
same number of times

00:08:27.580 --> 00:08:29.960
that 9 goes into 300.

00:08:29.960 --> 00:08:32.070
So 9 goes into 30 three times.

00:08:32.070 --> 00:08:33.820
3 times nine is 27.

00:08:33.820 --> 00:08:36.050
I think I see a
pattern here already.

00:08:36.050 --> 00:08:41.310
30, 3, 3 times 9 is 27.

00:08:41.310 --> 00:08:45.010
You're going to keep getting
33-- the 3's are just

00:08:45.010 --> 00:08:46.490
going to go on forever.

00:08:46.490 --> 00:08:50.960
So it turns out that 3 is 9%
of-- you can either write it as

00:08:50.960 --> 00:08:58.870
33.3 repeating or we all know
that 0.3 forever is the

00:08:58.870 --> 00:09:00.310
same thing as 1/3.

00:09:00.310 --> 00:09:05.380
So 3 is 9% of 33 and 1/3.

00:09:05.380 --> 00:09:08.220
Either one of those would
be an acceptable answer.

00:09:08.220 --> 00:09:09.700
And a lot of times when you're
doing percentages you're

00:09:09.700 --> 00:09:11.885
actually just trying
to get a ballpark.

00:09:11.885 --> 00:09:15.790
The precision might not always
be the most important thing,

00:09:15.790 --> 00:09:18.060
but in this case we
will be precise.

00:09:18.060 --> 00:09:19.390
And obviously, on tests
and things you need to

00:09:19.390 --> 00:09:20.940
be precise as well.

00:09:20.940 --> 00:09:23.020
Hopefully, I didn't go too
fast and you have a good

00:09:23.020 --> 00:09:24.340
sense of percentage.

00:09:24.340 --> 00:09:26.770
The important thing for these
type of problems is pay

00:09:26.770 --> 00:09:28.740
attention to how the
problem is written.

00:09:28.740 --> 00:09:32.480
If it says find 10% of 100.

00:09:32.480 --> 00:09:33.050
That's easy.

00:09:33.050 --> 00:09:36.010
You just convert 10% to a
decimal and multiply it by 100.

00:09:36.010 --> 00:09:39.570
But if I were to ask you
100 is 10% of what?

00:09:39.570 --> 00:09:41.340
You have to remember that
that's a different problem.

00:09:41.340 --> 00:09:45.440
In which case, 100 is 10% of--
and if you did the math,

00:09:45.440 --> 00:09:46.850
it would be 1,000.

00:09:46.850 --> 00:09:50.730
I think I spoke very quickly on
this problem on this module, so

00:09:50.730 --> 00:09:51.980
I hope you didn't
get too confused.

00:09:51.980 --> 00:09:54.440
But I will record more.


Growing by a percentage

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:03.540
Let's do some more
percentage problems.

00:00:03.540 --> 00:00:08.130
Let's say that I start
this year in my stock

00:00:08.130 --> 00:00:13.520
portfolio with $95.00.

00:00:13.520 --> 00:00:26.010
And I say that my portfolio
grows by, let's say, 15%.

00:00:30.350 --> 00:00:31.480
How much do I have now?

00:00:40.430 --> 00:00:40.780
OK.

00:00:40.780 --> 00:00:42.900
I think you might be able to
figure this out on your own,

00:00:42.900 --> 00:00:45.426
but of course we'll do some
example problems, just in case

00:00:45.426 --> 00:00:46.710
it's a little confusing.

00:00:46.710 --> 00:00:49.440
So I'm starting with $95.00,
and I'll get rid of

00:00:49.440 --> 00:00:50.090
the dollar sign.

00:00:50.090 --> 00:00:51.910
We know we're working
with dollars.

00:00:51.910 --> 00:00:55.220
95 dollars, right?

00:00:55.220 --> 00:00:58.410
And I'm going to earn, or I'm
going to grow just because

00:00:58.410 --> 00:01:02.170
I was an excellent stock
investor, that 95 dollars

00:01:02.170 --> 00:01:04.730
is going to grow by 15%.

00:01:04.730 --> 00:01:11.090
So to that 95 dollars, I'm
going to add another 15% of 95.

00:01:11.090 --> 00:01:18.880
So we know we write 15% as a
decimal, as 0.15, so 95 plus

00:01:18.880 --> 00:01:23.540
0.15 of 95, so this is
times 95-- that dot

00:01:23.540 --> 00:01:24.780
is just a times sign.

00:01:24.780 --> 00:01:26.890
It's not a decimal, it's a
times, it's a little higher

00:01:26.890 --> 00:01:31.790
than a decimal-- So 95 plus
0.15 times 95 is what

00:01:31.790 --> 00:01:33.390
we have now, right?

00:01:33.390 --> 00:01:36.410
Because we started with 95
dollars, and then we made

00:01:36.410 --> 00:01:40.630
another 15% times what
we started with.

00:01:40.630 --> 00:01:41.940
Hopefully that make sense.

00:01:41.940 --> 00:01:46.330
Another way to say it, the 95
dollars has grown by 15%.

00:01:46.330 --> 00:01:47.940
So let's just work this out.

00:01:47.940 --> 00:01:55.460
This is the same thing as 95
plus-- what's 0.15 times 95?

00:01:55.460 --> 00:01:56.480
Let's see.

00:01:56.480 --> 00:01:58.420
So let me do this, hopefully
I'll have enough space here.

00:01:58.420 --> 00:02:03.640
95 times 0.15-- I don't
want to run out of space.

00:02:03.640 --> 00:02:05.130
Actually, let me do it up here,
I think I'm about to run out

00:02:05.130 --> 00:02:11.780
of space-- 95 times 0.15.

00:02:11.780 --> 00:02:21.730
5 times 5 is 25, 9 times 5 is
45 plus 2 is 47, 1 times 95 is

00:02:21.730 --> 00:02:29.580
95, bring down the 5,
12, carry the 1, 15.

00:02:29.580 --> 00:02:30.880
And how many decimals
do we have?

00:02:30.880 --> 00:02:32.300
1, 2.

00:02:32.300 --> 00:02:35.666
15.25.

00:02:35.666 --> 00:02:37.230
Actually, is that right?

00:02:37.230 --> 00:02:39.250
I think I made a mistake here.

00:02:39.250 --> 00:02:44.100
See 5 times 5 is 25.

00:02:44.100 --> 00:02:49.045
5 times 9 is 45, plus 2 is 47.

00:02:49.045 --> 00:02:55.500
And we bring the 0 here, it's
95, 1 times 5, 1 times 9, then

00:02:55.500 --> 00:03:00.500
we add 5 plus 0 is 5,
7 plus 5 is 12-- oh.

00:03:00.500 --> 00:03:00.790
See?

00:03:00.790 --> 00:03:01.465
I made a mistake.

00:03:01.465 --> 00:03:06.260
It's 14.25, not 15.25.

00:03:06.260 --> 00:03:08.100
So I'll ask you an
interesting question?

00:03:08.100 --> 00:03:12.580
How did I know that
15.25 was a mistake?

00:03:12.580 --> 00:03:14.310
Well, I did a reality check.

00:03:14.310 --> 00:03:18.830
I said, well, I know in my head
that 15% of 100 is 15, so if

00:03:18.830 --> 00:03:25.800
15% of 100 is 15, how can
15% of 95 be more than 15?

00:03:25.800 --> 00:03:27.500
I think that might
have made sense.

00:03:27.500 --> 00:03:29.790
The bottom line is 95
is less than 100.

00:03:29.790 --> 00:03:33.460
So 15% of 95 had to be less
than 15, so I knew my

00:03:33.460 --> 00:03:35.560
answer of 15.25 was wrong.

00:03:35.560 --> 00:03:37.940
And so it turns out that I
actually made an addition

00:03:37.940 --> 00:03:40.750
error, and the answer is 14.25.

00:03:40.750 --> 00:03:44.980
So the answer is going to be 95
plus 15% of 95, which is the

00:03:44.980 --> 00:03:57.570
same thing as 95 plus 14.25,
well, that equals what?

00:03:57.570 --> 00:04:05.350
109.25.

00:04:05.350 --> 00:04:07.560
Notice how easy I made
this for you to read,

00:04:07.560 --> 00:04:08.920
especially this 2 here.

00:04:08.920 --> 00:04:10.490
109.25.

00:04:10.490 --> 00:04:13.990
So if I start off with $95.00
and my portfolio grows-- or the

00:04:13.990 --> 00:04:17.430
amount of money I have-- grows
by 15%, I'll end

00:04:17.430 --> 00:04:21.432
up with $109.25.

00:04:21.432 --> 00:04:22.660
Let's do another problem.

00:04:25.560 --> 00:04:31.490
Let's say I start off with some
amount of money, and after a

00:04:31.490 --> 00:04:47.260
year, let's says my portfolio
grows 25%, and after growing

00:04:47.260 --> 00:04:56.320
25%, I now have $100.

00:04:56.320 --> 00:04:58.810
How much did I originally have?

00:04:58.810 --> 00:05:02.750
Notice I'm not saying that
the $100 is growing by 25%.

00:05:02.750 --> 00:05:07.280
I'm saying that I start with
some amount of money, it grows

00:05:07.280 --> 00:05:13.780
by 25%, and I end up with
$100 after it grew by 25%.

00:05:13.780 --> 00:05:16.010
To solve this one, we
might have to break out

00:05:16.010 --> 00:05:17.640
a little bit of algebra.

00:05:17.640 --> 00:05:22.480
So let x equal what
I start with.

00:05:29.610 --> 00:05:34.070
So just like the last problem,
I start with x and it grows by

00:05:34.070 --> 00:05:43.450
25%, so x plus 25% of x is
equal to 100, and we know this

00:05:43.450 --> 00:05:52.190
25% of x we can just rewrite as
x plus 0.25 of x is equal to

00:05:52.190 --> 00:05:56.900
100, and now actually we have a
level-- actually this might be

00:05:56.900 --> 00:06:01.820
level 3 system, level 3 linear
equation-- but the bottom

00:06:01.820 --> 00:06:05.075
line, we can just add the
coefficients on the x.

00:06:05.075 --> 00:06:07.350
x is the same thing
as 1x, right?

00:06:07.350 --> 00:06:12.790
So 1x plus 0.25x, well that's
just the same thing as 1 plus

00:06:12.790 --> 00:06:16.310
0.25, plus x-- we're just doing
the distributive property

00:06:16.310 --> 00:06:18.960
in reverse-- equals 100.

00:06:18.960 --> 00:06:21.090
And what's 1 plus 0.25?

00:06:21.090 --> 00:06:23.360
That's easy, it's 1.25.

00:06:23.360 --> 00:06:31.960
So we say 1.25x
is equal to 100.

00:06:31.960 --> 00:06:32.650
Not too hard.

00:06:32.650 --> 00:06:34.770
And after you do a lot of these
problems, you're going to

00:06:34.770 --> 00:06:39.370
intuitively say, oh, if some
number grows by 25%, and it

00:06:39.370 --> 00:06:42.615
becomes 100, that means that
1.25 times that number

00:06:42.615 --> 00:06:44.480
is equal to 100.

00:06:44.480 --> 00:06:46.640
And if this doesn't make sense,
sit and think about it a little

00:06:46.640 --> 00:06:50.110
bit, maybe rewatch the video,
and hopefully it'll, over time,

00:06:50.110 --> 00:06:51.610
start to make a lot
of sense to you.

00:06:51.610 --> 00:06:53.740
This type of math is
very very useful.

00:06:53.740 --> 00:06:55.900
I actually work at a hedge
fund, and I'm doing

00:06:55.900 --> 00:06:58.960
this type of math in my
head day and night.

00:06:58.960 --> 00:07:06.660
So 1.25 times x is equal
to 100, so x would equal

00:07:06.660 --> 00:07:10.680
100 divided by 1.25.

00:07:10.680 --> 00:07:11.730
I just realized you
probably don't know

00:07:11.730 --> 00:07:12.910
what a hedge fund is.

00:07:12.910 --> 00:07:14.950
I invest in stocks
for a living.

00:07:14.950 --> 00:07:16.640
Anyway, back to the math.

00:07:16.640 --> 00:07:19.900
So x is equal to 100
divided by 1.25.

00:07:19.900 --> 00:07:25.570
So let me make some space
here, just because I

00:07:25.570 --> 00:07:28.430
used up too much space.

00:07:28.430 --> 00:07:31.060
Let me get rid of my
little let x statement.

00:07:31.060 --> 00:07:34.500
Actually I think we know
what x is and we know

00:07:34.500 --> 00:07:36.620
how we got to there.

00:07:36.620 --> 00:07:38.570
If you forgot how we got
there, you can I guess

00:07:38.570 --> 00:07:39.590
rewatch the video.

00:07:42.150 --> 00:07:43.110
Let's see.

00:07:43.110 --> 00:07:47.600
Let me make the pen thin
again, and go back to

00:07:47.600 --> 00:07:49.600
the orange color, OK.

00:07:49.600 --> 00:07:55.590
X equals 100 divided by 1.25,
so we say 1.25 goes into

00:07:55.590 --> 00:07:59.410
100.00-- I'm going to add a
couple of 0's, I don't know how

00:07:59.410 --> 00:08:01.720
many I'm going to need,
probably added too many-- if I

00:08:01.720 --> 00:08:06.150
move this decimal over two to
the right, I need to move this

00:08:06.150 --> 00:08:07.450
one over two to the right.

00:08:10.390 --> 00:08:14.410
And I say how many times does
100 go into 100-- how many

00:08:14.410 --> 00:08:16.660
times does 125 go into 100?

00:08:16.660 --> 00:08:17.530
None.

00:08:17.530 --> 00:08:19.420
How many times does
it go into 1000?

00:08:19.420 --> 00:08:21.010
It goes into it eight times.

00:08:21.010 --> 00:08:24.130
I happen to know that in my
head, but you could do trial

00:08:24.130 --> 00:08:25.580
and error and think about it.

00:08:25.580 --> 00:08:28.490
8 times-- if you want to think
about it, 8 times 100 is

00:08:28.490 --> 00:08:32.600
800, and then 8 times 25 is
200, so it becomes 1000.

00:08:32.600 --> 00:08:34.480
You could work out if you like,
but I think I'm running out of

00:08:34.480 --> 00:08:36.370
time, so I'm going
to do this fast.

00:08:36.370 --> 00:08:39.920
8 times 125 is 1000.

00:08:39.920 --> 00:08:42.600
Remember this thing isn't here.

00:08:42.600 --> 00:08:48.370
1000, so 1000 minus 1000 is 0,
so you can bring down the 0.

00:08:48.370 --> 00:08:52.900
125 goes into 0 zero times,
and we just keep getting 0's.

00:08:52.900 --> 00:08:55.680
This is just a decimal
division problem.

00:08:55.680 --> 00:08:59.500
So it turns out that if your
portfolio grew by 25% and

00:08:59.500 --> 00:09:05.120
you ended up with $100.00
you started with $80.00.

00:09:05.120 --> 00:09:09.650
And that makes sense, because
25% is roughly 1/4, right?

00:09:09.650 --> 00:09:13.590
So if I started with $80.00 and
I grow by 1/4, that means I

00:09:13.590 --> 00:09:17.660
grew by $20, because
25% of 80 is 20.

00:09:17.660 --> 00:09:21.020
So if I start with 80
and I grow by 20,

00:09:21.020 --> 00:09:22.740
that gets me to 100.

00:09:22.740 --> 00:09:24.020
Makes sense.

00:09:24.020 --> 00:09:28.070
So remember, all you have to
say is, well, some number times

00:09:28.070 --> 00:09:32.950
1.25-- because I'm growing
it by 25%-- is equal to 100.

00:09:32.950 --> 00:09:36.240
Don't worry, if you're still
confused, I'm going to add at

00:09:36.240 --> 00:09:38.500
least one more presentation
on a couple of more

00:09:38.500 --> 00:09:40.440
examples like this.


Percent and decimals

WEBVTT
Kind: captions
Language: en

00:00:01.170 --> 00:00:03.080
Let's get started and
learn how to convert

00:00:03.080 --> 00:00:04.430
percentages to decimals.

00:00:04.430 --> 00:00:07.000
And if we have time maybe we'll
also learn how to convert

00:00:07.000 --> 00:00:08.840
decimals into percentages.

00:00:08.840 --> 00:00:11.700
So let's get started with what
I think is a problem you

00:00:11.700 --> 00:00:13.060
probably already
know how to do.

00:00:13.060 --> 00:00:14.930
If I said I have 50%.

00:00:18.270 --> 00:00:19.520
I don't know if I wanted
to write that thick,

00:00:19.520 --> 00:00:20.630
but we'll go with it.

00:00:20.630 --> 00:00:22.420
Actually, let me change
it to a thinner one.

00:00:22.420 --> 00:00:24.450
And I want to turn
that into a decimal.

00:00:24.450 --> 00:00:26.580
Well you probably already
have a sense of what

00:00:26.580 --> 00:00:28.940
decimal represents 50%.

00:00:28.940 --> 00:00:31.790
If I told you we're having a
sale and it's 50% off you know

00:00:31.790 --> 00:00:34.270
that's roughly half off, or
another way, how do you

00:00:34.270 --> 00:00:35.870
say half as a decimal?

00:00:35.870 --> 00:00:44.240
Well that's the same thing as
0.5 So you might have known

00:00:44.240 --> 00:00:48.150
that in your head, but is there
a system for being able to

00:00:48.150 --> 00:00:49.990
convert this 50% to 0.5?

00:00:49.990 --> 00:00:52.770
Well, it turns out it's
pretty straightforward.

00:00:52.770 --> 00:00:56.590
All you do is you say--
whatever percentage it is,

00:00:56.590 --> 00:01:03.760
that's the same thing
as the number over 100.

00:01:03.760 --> 00:01:13.772
And 50/100 is the same
thing as 5/10 or 0.5.

00:01:13.772 --> 00:01:17.250
Now even a simpler way of
converting percentage to

00:01:17.250 --> 00:01:19.620
decimals and, I think, you're
going to realize converting a

00:01:19.620 --> 00:01:22.170
percentage to a decimal or the
other way around, you can

00:01:22.170 --> 00:01:23.800
pretty much do it in your head.

00:01:23.800 --> 00:01:28.180
If I say, let's say 50-- and
I'm just going to add one

00:01:28.180 --> 00:01:34.170
decimal of accuracy here just
to show you a point-- 50.0%.

00:01:34.170 --> 00:01:38.640
If I want to convert that 50.0%
into a decimal, all I do is I

00:01:38.640 --> 00:01:40.900
get rid of the percent sign.

00:01:40.900 --> 00:01:42.320
So I'll do it here.

00:01:42.320 --> 00:01:45.500
50.0%.

00:01:45.500 --> 00:01:49.100
I get rid of the percent sign
and I take the decimal point

00:01:49.100 --> 00:01:51.960
and I move it over two
spaces to the left.

00:01:51.960 --> 00:01:55.590
So I say 1, 2.

00:01:55.590 --> 00:01:57.580
So this is where the
new decimal is.

00:01:57.580 --> 00:02:05.260
So this equals we
could say 0.500.

00:02:05.260 --> 00:02:10.560
So 50.0% is equal to 0.500.

00:02:10.560 --> 00:02:14.170
And of course, these last two
0's really don't mean anything

00:02:14.170 --> 00:02:16.640
for our purposes, so
we'll get rid of them.

00:02:16.640 --> 00:02:22.310
So that's the same thing
as 0.5 or just 0.5.

00:02:22.310 --> 00:02:24.290
50% equals 0.5.

00:02:24.290 --> 00:02:26.810
So you're probably
saying, well, sure.

00:02:26.810 --> 00:02:28.200
That looks easy, but what
if the problem gets

00:02:28.200 --> 00:02:28.700
a little harder?

00:02:28.700 --> 00:02:30.110
50% I could have
done in my head.

00:02:30.110 --> 00:02:35.080
So let's try some, I would say,
slightly harder problems.

00:02:35.080 --> 00:02:46.240
If I were to tell you
that-- let's say 16.32%.

00:02:46.240 --> 00:02:49.100
Well, let's just do it the
way I just showed you.

00:02:49.100 --> 00:02:50.200
I'll rewrite it down here.

00:02:50.200 --> 00:02:56.520
16.32%.

00:02:56.520 --> 00:02:59.610
So if we get rid of the percent
sign, scratch it out, we just

00:02:59.610 --> 00:03:03.740
have to move the decimal over
two spaces to the left.

00:03:03.740 --> 00:03:07.830
So 1, 2.

00:03:07.830 --> 00:03:09.250
This is the new place
for the decimal.

00:03:09.250 --> 00:03:10.580
That decimal goes away.

00:03:10.580 --> 00:03:23.340
So it's 0.1632 is
equal to 16.32%.

00:03:23.340 --> 00:03:24.850
I think you might be
getting the idea now.

00:03:24.850 --> 00:03:28.310
Let me do another one in green.

00:03:28.310 --> 00:03:31.910
Let's say I had-- and this
one actually confuses

00:03:31.910 --> 00:03:32.540
a lot of people.

00:03:32.540 --> 00:03:41.830
Let's say I had 0.25%.

00:03:41.830 --> 00:03:45.950
So the important thing to
remember is, I'll rewrite here.

00:03:45.950 --> 00:03:53.170
0.25-- and maybe I'll write
00.-- And you're probably

00:03:53.170 --> 00:03:55.260
wondering why I'm doing this,
but I think you'll see in a

00:03:55.260 --> 00:03:58.100
second why I wrote that leading
0 there even though it doesn't

00:03:58.100 --> 00:04:00.020
seem to add much to it.

00:04:00.020 --> 00:04:01.630
00.25%.

00:04:01.630 --> 00:04:03.250
Well, what's the system
I just showed you?

00:04:03.250 --> 00:04:05.570
You get rid of the percent
sign and you move the

00:04:05.570 --> 00:04:10.000
decimal over 1, 2 spaces.

00:04:10.000 --> 00:04:15.600
So that equals 0.0025.

00:04:15.600 --> 00:04:20.380
So 0.25% is equal to 0.0025.

00:04:20.380 --> 00:04:24.480
And you could put a leading
0 here if you want.

00:04:24.480 --> 00:04:27.090
Actually, I should probably
tell you to always do that

00:04:27.090 --> 00:04:29.260
because it makes it
easier to read.

00:04:29.260 --> 00:04:34.050
So 0.25% is equal to 0.0025.

00:04:34.050 --> 00:04:37.270
And I want to just
contrast that with 25%.

00:04:41.200 --> 00:04:44.250
25%, what do you
think that equals?

00:04:44.250 --> 00:04:44.920
Well.

00:04:44.920 --> 00:04:47.970
you do the same thing
that we've been doing.

00:04:47.970 --> 00:04:51.680
You get rid of the percent sign
and you move the decimal space.

00:04:51.680 --> 00:04:53.220
In this case-- actually,
let me leave that there.

00:04:53.220 --> 00:04:53.800
I'll just rewrite it here.

00:04:53.800 --> 00:04:54.130
25%.

00:04:56.800 --> 00:04:58.320
And you're probably saying,
where is the decimal in this?

00:04:58.320 --> 00:05:02.213
Well, the decimal is after the
number because that 25% is

00:05:02.213 --> 00:05:04.560
the same thing as 25.0%.

00:05:04.560 --> 00:05:08.510
So if we get rid of the percent
sign, we move the decimal over

00:05:08.510 --> 00:05:16.550
two spaces to the left, and
that equals 0.25 as a decimal,

00:05:16.550 --> 00:05:19.220
or 0.250, but we can
ignore that last 0.

00:05:19.220 --> 00:05:29.280
So 25% equals 0.25 while
0.25% is equal to 0.0025.

00:05:29.280 --> 00:05:31.970
And I want you to maybe sit
and think about how small

00:05:31.970 --> 00:05:35.390
of a number 0.25% is.

00:05:35.390 --> 00:05:37.910
Let's do a couple more and
maybe we'll convert going

00:05:37.910 --> 00:05:38.510
the other direction.

00:05:41.400 --> 00:05:48.820
Let's say I have the decimal
0.01 and I wanted to convert

00:05:48.820 --> 00:05:49.860
that into a percent.

00:05:56.090 --> 00:05:58.120
Well, here we just
do it the opposite.

00:05:58.120 --> 00:06:00.550
We could look at it two ways.

00:06:00.550 --> 00:06:03.870
We could say well, whatever
number this is, we multiply it

00:06:03.870 --> 00:06:06.180
by 100 and add a percent sign.

00:06:06.180 --> 00:06:12.770
So if you say 0.01 times
100, and then we'll

00:06:12.770 --> 00:06:15.590
add a percent sign.

00:06:15.590 --> 00:06:15.750
So a.

00:06:15.750 --> 00:06:17.150
0.01 times 100?

00:06:17.150 --> 00:06:18.740
Well that's just 1.

00:06:18.740 --> 00:06:20.420
You could do the math.

00:06:20.420 --> 00:06:21.380
You add the percent sign.

00:06:21.380 --> 00:06:23.550
Well, that equals 1%.

00:06:23.550 --> 00:06:27.930
Or an even easier way, when we
go from a percent to a decimal,

00:06:27.930 --> 00:06:31.770
we move the decimal place
over two to the left.

00:06:31.770 --> 00:06:33.930
So when you go from a
decimal to a percent,

00:06:33.930 --> 00:06:35.160
we'll do the opposite.

00:06:35.160 --> 00:06:37.250
We move the decimal
two to the right.

00:06:37.250 --> 00:06:40.680
So if we do that, let
me just rewrite it.

00:06:40.680 --> 00:06:42.420
0.01.

00:06:42.420 --> 00:06:44.960
Just go 1, 2.

00:06:44.960 --> 00:06:46.180
The new decimal place is here.

00:06:48.860 --> 00:06:54.530
If I get rid of that decimals
that's 01 decimal 00 whatever.

00:06:54.530 --> 00:06:56.610
Obviously, this leading 0
means nothing, so that's

00:06:56.610 --> 00:06:58.830
the same thing as 1.00.

00:06:58.830 --> 00:07:02.260
Which is the same thing as 1.

00:07:02.260 --> 00:07:07.080
And does it make sense that
moving the decimal space two to

00:07:07.080 --> 00:07:09.580
the right-- that's really just
the same thing as multiplying

00:07:09.580 --> 00:07:11.720
it by 100, right?

00:07:11.720 --> 00:07:15.070
If I multiply something by 10
it's like moving the decimal

00:07:15.070 --> 00:07:16.990
space one to the right.

00:07:16.990 --> 00:07:20.720
If I divide something by
10 it's like moving the

00:07:20.720 --> 00:07:22.230
decimal space one to left.

00:07:22.230 --> 00:07:24.320
Let's do a couple more
while I have time.

00:07:24.320 --> 00:07:27.630
I think I have three
more minutes.

00:07:27.630 --> 00:07:36.390
Let's say I had 1.25 and
I wanted to convert

00:07:36.390 --> 00:07:37.810
that to a percent.

00:07:37.810 --> 00:07:41.380
Well, the easiest way is just
to take-- I'll rewrite it here.

00:07:41.380 --> 00:07:43.200
1.25.

00:07:43.200 --> 00:07:47.120
Take the decimal point,
move it two to the right.

00:07:47.120 --> 00:07:49.190
That's here.

00:07:49.190 --> 00:07:50.220
And then I'll add a percent.

00:07:50.220 --> 00:07:55.460
So that equals 125%.

00:07:55.460 --> 00:07:57.100
And if you think about it,
the way people talk about

00:07:57.100 --> 00:07:58.260
percent it makes sense.

00:07:58.260 --> 00:08:04.700
If I told you that I'm going
to pay 1.25 times the

00:08:04.700 --> 00:08:06.150
price of something.

00:08:06.150 --> 00:08:07.860
That makes sense that that's
also I'm going to pay

00:08:07.860 --> 00:08:09.960
125% of the price.

00:08:09.960 --> 00:08:12.330
Or if it doesn't make sense
hopefully if you do these

00:08:12.330 --> 00:08:14.630
problems enough it will
start to make sense.

00:08:14.630 --> 00:08:15.690
Let's do a couple of more.

00:08:15.690 --> 00:08:17.940
And you can go back and pause
this if you think I'm going too

00:08:17.940 --> 00:08:19.380
fast, which I might be doing.

00:08:26.300 --> 00:08:27.680
Let me think.

00:08:27.680 --> 00:08:35.920
If I were to say 0.003
and I want to write

00:08:35.920 --> 00:08:39.520
this as a percent.

00:08:39.520 --> 00:08:43.450
Well, once again, we can
move the decimal space

00:08:43.450 --> 00:08:45.010
two to the right.

00:08:45.010 --> 00:08:46.510
So 1, 2.

00:08:46.510 --> 00:08:49.830
And that's analogous to
multiplying it by 100.

00:08:49.830 --> 00:08:53.350
So if we multiply the decimal
two to the right we get

00:08:53.350 --> 00:08:57.120
00 decimal point 3.

00:08:57.120 --> 00:09:00.080
And then we add the percent.

00:09:00.080 --> 00:09:01.860
At least this first leading 0
doesn't mean anything, so

00:09:01.860 --> 00:09:03.150
that's the same thing as 0.3%.

00:09:06.810 --> 00:09:09.180
The important thing to realize
is when you're converting from

00:09:09.180 --> 00:09:12.550
a percent to a decimal or a
decimal to a percent, you're

00:09:12.550 --> 00:09:15.000
really just moving where
that decimal point is.

00:09:15.000 --> 00:09:18.490
And if you run out of spaces
you just have to add or get

00:09:18.490 --> 00:09:20.460
rid of 0's accordingly.

00:09:20.460 --> 00:09:23.070
And the important thing to
always have in your mind is,

00:09:23.070 --> 00:09:27.260
when I convert from a decimal
to a percent, the number in

00:09:27.260 --> 00:09:29.230
front of percent signs
going to get bigger.

00:09:29.230 --> 00:09:31.590
And when I go from a percent
sign to a decimal, I'm going

00:09:31.590 --> 00:09:33.920
to get a smaller number.

00:09:33.920 --> 00:09:41.880
If I say 25%, that's the
same thing as 0.25.

00:09:41.880 --> 00:09:44.430
So this is a percent,
and this is a decimal.

00:09:47.120 --> 00:09:52.080
So I went from a bigger number,
25, to a smaller number, 0.25.

00:09:52.080 --> 00:09:54.570
25% is equal to 0.25.

00:09:54.570 --> 00:09:59.840
Similarly, if I had a
decimal, let's say 0.1.

00:09:59.840 --> 00:10:01.660
When I convert it to a
percentage it's going to

00:10:01.660 --> 00:10:03.300
be a larger percentage.

00:10:03.300 --> 00:10:08.060
So 0.1 is the same
thing as 10%.

00:10:08.060 --> 00:10:10.430
And how did I do that again?

00:10:10.430 --> 00:10:14.310
Well I said 0.1, I added an
extra 0 because I'm going to

00:10:14.310 --> 00:10:16.850
have to move the decimal space
over to the right twice.

00:10:16.850 --> 00:10:20.790
So I go 1, 2, and I get a 10.

00:10:20.790 --> 00:10:22.730
10%.

00:10:22.730 --> 00:10:24.940
Hopefully that answers all
your questions for now.

00:10:24.940 --> 00:10:26.450
Have fun.


Ordering numeric expressions

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:04.360
Welcome to the presentation
on ordering numbers.

00:00:04.360 --> 00:00:06.940
Let's get started with some
problems that I think, as you

00:00:06.940 --> 00:00:09.270
go through the examples
hopefully, you'll understand

00:00:09.270 --> 00:00:10.910
how to do these problems.

00:00:10.910 --> 00:00:11.700
So let's see.

00:00:11.700 --> 00:00:23.200
The first set of numbers that
we have to order is 35.7%,

00:00:23.200 --> 00:00:44.590
108.1% 0.5, 13/93,
and 1 and 7/68.

00:00:44.590 --> 00:00:46.590
So let's do this problem.

00:00:46.590 --> 00:00:48.810
The important thing to remember
whenever you're doing this type

00:00:48.810 --> 00:00:52.820
of ordering of numbers is to
realize that these are all just

00:00:52.820 --> 00:00:56.940
different ways to represent--
these are all a percent or a

00:00:56.940 --> 00:01:00.270
decimal or a fraction or a
mixed-- are all just different

00:01:00.270 --> 00:01:02.680
ways of representing numbers.

00:01:02.680 --> 00:01:05.110
It's very hard to compare when
you just look at it like this,

00:01:05.110 --> 00:01:07.130
so what I like to do is I
like to convert them

00:01:07.130 --> 00:01:08.190
all to decimals.

00:01:08.190 --> 00:01:11.100
But there could be someone who
likes to convert them all to

00:01:11.100 --> 00:01:14.220
percentages or convert them all
to fractions and then compare.

00:01:14.220 --> 00:01:16.920
But I always find decimals to
be the easiest way to compare.

00:01:16.920 --> 00:01:19.370
So let's start with this 35.7%.

00:01:19.370 --> 00:01:21.940
Let's turn this into a decimal.

00:01:21.940 --> 00:01:25.090
Well, the easiest thing to
remember is if you have a

00:01:25.090 --> 00:01:27.490
percent you just get rid of
the percent sign and

00:01:27.490 --> 00:01:28.580
put it over 100.

00:01:28.580 --> 00:01:38.970
So 35.7% is the same
thing as 35.7/100.

00:01:38.970 --> 00:01:43.020
Like 5%, that's the same thing
as 5/100 or 50% is just

00:01:43.020 --> 00:01:45.050
the same thing as 50/100.

00:01:45.050 --> 00:01:53.990
So 35.7/100, well, that
just equals 0.357.

00:01:53.990 --> 00:01:55.730
If this got you a little
confused another way to think

00:01:55.730 --> 00:02:01.970
about percentage points is if I
write 35.7%, all you have to do

00:02:01.970 --> 00:02:05.540
is get rid of the percent sign
and move the decimal to the

00:02:05.540 --> 00:02:10.140
left two spaces and
it becomes 0.357.

00:02:10.140 --> 00:02:11.870
Let me give you a couple of
more examples down here.

00:02:11.870 --> 00:02:16.050
Let's say I had 5%.

00:02:16.050 --> 00:02:20.020
That is the same
thing as 5/100.

00:02:20.020 --> 00:02:22.670
Or if you do the decimal
technique, 5%, you could just

00:02:22.670 --> 00:02:24.730
move the decimal and you
get rid of the percent.

00:02:24.730 --> 00:02:28.630
And you move the decimal over 1
and 2, and you put a 0 here.

00:02:28.630 --> 00:02:30.370
It's 0.05.

00:02:30.370 --> 00:02:33.280
And that's the same
thing as 0.05.

00:02:33.280 --> 00:02:36.380
You also know that 0.05 and
5/100 are the same thing.

00:02:36.380 --> 00:02:37.620
So let's get back
to the problem.

00:02:37.620 --> 00:02:40.772
I hope that distraction didn't
distract you too much.

00:02:40.772 --> 00:02:43.190
Let me scratch out all this.

00:02:43.190 --> 00:02:49.050
So 35.7% is equal to 0.357.

00:02:49.050 --> 00:02:51.870
Similarly, 108.1%.

00:02:51.870 --> 00:02:54.080
Let's to the technique where we
just get rid of the percent and

00:02:54.080 --> 00:02:59.350
move the decimal space over
1, 2 spaces to the left.

00:02:59.350 --> 00:03:08.600
So then that equals 1.081.

00:03:08.600 --> 00:03:11.570
See we already know that
this is smaller than this.

00:03:11.570 --> 00:03:14.140
Well the next one is easy,
it's already in decimal form.

00:03:14.140 --> 00:03:16.040
0.5 is just going to
be equal to 0.5.

00:03:18.820 --> 00:03:21.050
Now 13/93.

00:03:21.050 --> 00:03:24.340
To convert a fraction into
a decimal we just take the

00:03:24.340 --> 00:03:27.320
denominator and divide
it into the numerator.

00:03:27.320 --> 00:03:29.350
So let's do that.

00:03:29.350 --> 00:03:33.020
93 goes into 13?

00:03:36.530 --> 00:03:39.760
Well, we know it goes
into 13 zero times.

00:03:39.760 --> 00:03:43.990
So let's add a
decimal point here.

00:03:43.990 --> 00:03:47.550
So how many times
does 93 go into 130?

00:03:47.550 --> 00:03:49.530
Well, it goes into it one time.

00:03:49.530 --> 00:03:51.410
1 times 93 is 93.

00:03:55.061 --> 00:03:56.550
Becomes a 10.

00:03:56.550 --> 00:03:58.960
That becomes a 2.

00:03:58.960 --> 00:04:03.700
Then we're going to
borrow, so get 37.

00:04:03.700 --> 00:04:06.590
Bring down a 0.

00:04:06.590 --> 00:04:10.010
So 93 goes into 370?

00:04:10.010 --> 00:04:10.470
Let's see.

00:04:10.470 --> 00:04:14.790
4 times 93 would be 372,
so it actually goes into

00:04:14.790 --> 00:04:15.695
it only three times.

00:04:19.390 --> 00:04:22.880
3 times 3 is 9.

00:04:22.880 --> 00:04:25.270
3 times 9 is 27.

00:04:30.110 --> 00:04:31.605
So this equals?

00:04:31.605 --> 00:04:38.050
Let's see, this equals-- if we
say that this 0 becomes a 10.

00:04:38.050 --> 00:04:39.620
This become a 16.

00:04:39.620 --> 00:04:42.440
This becomes a 2.

00:04:42.440 --> 00:04:45.210
81.

00:04:45.210 --> 00:04:48.120
And then we say, how many
times does 93 go into 810?

00:04:48.120 --> 00:04:50.860
It goes roughly 8 times.

00:04:50.860 --> 00:04:52.860
And we could actually keep
going, but for the sake of

00:04:52.860 --> 00:04:55.640
comparing these numbers, we've
already gotten to a pretty

00:04:55.640 --> 00:04:57.580
good level of accuracy.

00:04:57.580 --> 00:05:00.740
So let's just stop this problem
here because the decimal

00:05:00.740 --> 00:05:02.720
numbers could keep going on,
but for the sake of comparison

00:05:02.720 --> 00:05:04.410
I think we've already got a
good sense of what this

00:05:04.410 --> 00:05:05.360
decimal looks like.

00:05:05.360 --> 00:05:10.330
It's 0.138 and then
it'll just keep going.

00:05:10.330 --> 00:05:13.010
So let's write that down.

00:05:13.010 --> 00:05:15.340
And then finally, we have
this mixed number here.

00:05:15.340 --> 00:05:18.070
And let me erase some of
my work because I don't

00:05:18.070 --> 00:05:18.840
want to confuse you.

00:05:18.840 --> 00:05:22.700
Actually, let me keep it
the way it is right now.

00:05:22.700 --> 00:05:26.120
The easiest way to convert a
mixed number into a decimal is

00:05:26.120 --> 00:05:29.630
to just say, OK, this is 1
and then some fraction

00:05:29.630 --> 00:05:32.920
that's less than 1.

00:05:32.920 --> 00:05:36.420
Or we could convert it to a
fraction, an improper fraction

00:05:36.420 --> 00:05:38.790
like-- oh, actually there are
no improper fractions here.

00:05:38.790 --> 00:05:39.640
Actually, let's do it that way.

00:05:39.640 --> 00:05:41.630
Let's convert to an improper
fraction and then convert

00:05:41.630 --> 00:05:44.110
that into a decimal.

00:05:44.110 --> 00:05:46.060
Actually, I think I'm going to
need more space, so let me

00:05:46.060 --> 00:05:48.740
clean up this a little bit.

00:05:58.240 --> 00:05:58.595
There.

00:05:58.595 --> 00:06:01.040
We have a little more
space to work with now.

00:06:04.260 --> 00:06:08.570
So 1 and 7/68.

00:06:08.570 --> 00:06:13.700
So to go from a mixed number to
an improper fraction, what you

00:06:13.700 --> 00:06:18.760
do is you take the 68 times 1
and add it to the

00:06:18.760 --> 00:06:19.720
numerator here.

00:06:19.720 --> 00:06:21.000
And why does this make sense?

00:06:21.000 --> 00:06:26.120
Because this is the same
thing as 1 plus 7/68.

00:06:26.120 --> 00:06:29.680
1 and 7/68 is the same
thing as 1 plus 7/68.

00:06:29.680 --> 00:06:32.800
And that's the same thing as
you know from the fractions

00:06:32.800 --> 00:06:40.330
module, as 68/68 plus 7/68.

00:06:40.330 --> 00:06:47.650
And that's the same thing
as 68 plus 7-- 75/68.

00:06:47.650 --> 00:06:51.790
So 1 and 7/68 is
equal to 75/68.

00:06:51.790 --> 00:06:54.870
And now we convert this to a
decimal using the technique

00:06:54.870 --> 00:06:56.350
we did for 13/93.

00:06:56.350 --> 00:06:58.570
So we say-- let me
get some space.

00:06:58.570 --> 00:07:05.000
We say 68 goes into 75--
suspicion I'm going

00:07:05.000 --> 00:07:07.360
to run out of space.

00:07:07.360 --> 00:07:09.160
68 goes into 75 one time.

00:07:09.160 --> 00:07:13.290
1 times 68 is 68.

00:07:13.290 --> 00:07:16.460
75 minus 68 is 7.

00:07:16.460 --> 00:07:17.350
Bring down the 0.

00:07:17.350 --> 00:07:20.490
Actually, you don't have to
write the decimal there.

00:07:20.490 --> 00:07:21.100
Ignore that decimal.

00:07:21.100 --> 00:07:24.400
68 goes into 70 one time.

00:07:24.400 --> 00:07:28.150
1 times 68 is 68.

00:07:28.150 --> 00:07:30.990
70 minus 68 is 2,
bring down another 0.

00:07:30.990 --> 00:07:33.240
68 goes into 20 zero times.

00:07:33.240 --> 00:07:36.550
And the problem's going to keep
going on, but I think we've

00:07:36.550 --> 00:07:38.990
already once again, gotten to
enough accuracy that

00:07:38.990 --> 00:07:40.040
we can compare.

00:07:40.040 --> 00:07:48.320
So 1 and 7/68 we've now figured
out is equal to 1.10-- and if

00:07:48.320 --> 00:07:51.000
we kept dividing we'll keep
getting more decimals of

00:07:51.000 --> 00:07:53.510
accuracy, but I think we're
now ready to compare.

00:07:53.510 --> 00:07:56.550
So all of these numbers I just
rewrote them as decimals.

00:07:56.550 --> 00:08:00.550
So 35.7% is 0.357.

00:08:00.550 --> 00:08:05.000
108.1%-- ignore this for
now because we just used

00:08:05.000 --> 00:08:05.720
that to do the work.

00:08:05.720 --> 00:08:09.660
It's 108.1% is equal to 1.081.

00:08:09.660 --> 00:08:11.260
0.5 is 0.5.

00:08:11.260 --> 00:08:15.770
13/93 is 0.138.

00:08:15.770 --> 00:08:20.850
And 1 and 7/68 is 1.10
and it'll keep going on.

00:08:20.850 --> 00:08:23.010
So what's the smallest?

00:08:23.010 --> 00:08:26.320
So the smallest is
0.-- actually, no.

00:08:26.320 --> 00:08:28.300
The smallest is right here.

00:08:28.300 --> 00:08:31.450
So I'm going to rank them
from smallest to largest.

00:08:31.450 --> 00:08:36.250
So the smallest is 0.138.

00:08:36.250 --> 00:08:40.640
Then the next largest
is going to be 0.357.

00:08:40.640 --> 00:08:44.470
Then the next largest
is going to be 0.5.

00:08:44.470 --> 00:08:47.460
Then you're going to have 1.08.

00:08:47.460 --> 00:08:54.620
And then you're going
to have 1 and 7/68.

00:08:54.620 --> 00:08:56.840
Well, actually, I'm going to do
more examples of this, but for

00:08:56.840 --> 00:08:59.890
this video I think this is the
only one I have time for.

00:08:59.890 --> 00:09:01.910
But hopefully this gives you a
sense of doing these problems.

00:09:01.910 --> 00:09:04.370
I always find it easier
to go into the decimal

00:09:04.370 --> 00:09:05.280
mode to compare.

00:09:05.280 --> 00:09:06.680
And actually, the hints
on the module will

00:09:06.680 --> 00:09:08.670
be the same for you.

00:09:08.670 --> 00:09:11.040
But I think you're ready at
least now to try the problems.

00:09:11.040 --> 00:09:13.170
If you're not, if you want to
see other examples, you might

00:09:13.170 --> 00:09:16.830
just want to either re-watch
this video and/or I might

00:09:16.830 --> 00:09:20.530
record some more videos with
more examples right now.

00:09:20.530 --> 00:09:22.560
Anyway, have fun.