Multiplying a decimal by a power of 10

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.170
A single postage
stamp costs $0.44.

00:00:04.170 --> 00:00:07.145
How much would a roll
of 1,000 stamps cost?

00:00:07.145 --> 00:00:09.020
And there's really a
couple of ways to do it.

00:00:09.020 --> 00:00:11.450
And I'll do it both ways just
to show you they both work.

00:00:11.450 --> 00:00:13.184
One is kind of a
faster way, but I

00:00:13.184 --> 00:00:15.100
want to make sure you
understand why it works.

00:00:15.100 --> 00:00:17.340
And then we'll verify
that it actually gives us

00:00:17.340 --> 00:00:19.920
the right answer using maybe
the more traditional way

00:00:19.920 --> 00:00:21.600
of multiplying decimals.

00:00:21.600 --> 00:00:23.900
So we're starting at $0.44.

00:00:23.900 --> 00:00:26.540
I'll just write it as 0.44.

00:00:26.540 --> 00:00:27.700
Well, that's one stamp.

00:00:27.700 --> 00:00:29.390
So this is 1 stamp.

00:00:29.390 --> 00:00:31.010
I'll write it like
this, 1 stamp.

00:00:31.010 --> 00:00:33.270
How much would 10 stamps cost?

00:00:33.270 --> 00:00:36.900
Well, if 1 stamp is
$0.44, then 10 stamps,

00:00:36.900 --> 00:00:39.740
we could move the decimal
to the right one place.

00:00:39.740 --> 00:00:42.560
And so it would be--
and now this leading 0

00:00:42.560 --> 00:00:44.040
is not that useful.

00:00:44.040 --> 00:00:47.290
So it would now be $4.4.

00:00:47.290 --> 00:00:51.420
Or if you want to make it
clear, it would be $4.40.

00:00:51.420 --> 00:00:56.310
Now, what happens if you
want to have 100 stamps?

00:00:56.310 --> 00:00:58.250
Well, the same idea
is going to happen.

00:00:58.250 --> 00:01:00.070
We're now taking 10
times more, so we're

00:01:00.070 --> 00:01:02.500
going to move the decimal
to the right once.

00:01:02.500 --> 00:01:06.962
So 100 stamps are
going to cost $44.

00:01:06.962 --> 00:01:08.420
And this should
make sense for you.

00:01:08.420 --> 00:01:14.670
If 1 stamp is 44/100
of $1, then 100 stamps

00:01:14.670 --> 00:01:19.124
are going to be 44/100
of $100, or $44.

00:01:19.124 --> 00:01:20.540
Or you could view
it as we've just

00:01:20.540 --> 00:01:22.430
moved the decimal
over one place.

00:01:22.430 --> 00:01:26.340
So if we want 1,000 stamps,
we'd move the decimal

00:01:26.340 --> 00:01:28.034
to the right one more time.

00:01:28.034 --> 00:01:29.450
Moving the decimal
to the right is

00:01:29.450 --> 00:01:31.330
equivalent to multiplying by 10.

00:01:31.330 --> 00:01:35.250
So then it would be $440.

00:01:35.250 --> 00:01:37.670
And we could add another
trailing zero just

00:01:37.670 --> 00:01:40.482
to make clear that there's
no cents over here.

00:01:40.482 --> 00:01:42.190
So if you wanted to
do it really quickly,

00:01:42.190 --> 00:01:44.020
you could have
started with $0.44.

00:01:44.020 --> 00:01:46.510
And you say look, I'm
not multiplying by 10.

00:01:46.510 --> 00:01:48.280
I'm not multiplying by 100.

00:01:48.280 --> 00:01:50.030
I'm multiplying by 1,000.

00:01:50.030 --> 00:01:52.260
So you're going to have to
put another trailing zero

00:01:52.260 --> 00:01:53.070
over here.

00:01:53.070 --> 00:01:59.750
And you would move the decimal
from over here to over here.

00:01:59.750 --> 00:02:02.100
You've essentially
multiplied this times 10

00:02:02.100 --> 00:02:04.720
times 10 times 10,
which is 1,000.

00:02:04.720 --> 00:02:10.150
So then this would become $440.

00:02:10.150 --> 00:02:13.705
And let's verify that this works
exactly the same if we multiply

00:02:13.705 --> 00:02:16.730
it the traditional way, the
way we multiply decimals.

00:02:16.730 --> 00:02:24.980
So if you have
1,000 times $0.44.

00:02:24.980 --> 00:02:26.440
So you start over here.

00:02:26.440 --> 00:02:28.560
4 times 0 is 0.

00:02:28.560 --> 00:02:29.960
4 times 0 is 0.

00:02:29.960 --> 00:02:31.240
4 times 0 is 0.

00:02:31.240 --> 00:02:32.970
4 times 1 is 4.

00:02:32.970 --> 00:02:35.840
Or, you could just say,
hey, this was 4 times 1,000.

00:02:35.840 --> 00:02:37.830
Then we're going to
go one place over.

00:02:37.830 --> 00:02:39.830
So we're going to add a 0.

00:02:39.830 --> 00:02:42.630
And once again, we're going
to have 4 times 0 is 0.

00:02:42.630 --> 00:02:44.270
4 times 0 is 0.

00:02:44.270 --> 00:02:45.840
4 times 0 is 0.

00:02:45.840 --> 00:02:47.470
4 times 1 is 4.

00:02:47.470 --> 00:02:49.740
Or we just did 4 times 1,000.

00:02:49.740 --> 00:02:52.400
So that is 4,000 if you
don't include this 0

00:02:52.400 --> 00:02:54.760
that we added here ahead
of time because we're

00:02:54.760 --> 00:02:56.830
going one place to the left.

00:02:56.830 --> 00:02:58.160
And then we have nothing left.

00:02:58.160 --> 00:03:01.020
I haven't at all thought
about the decimals right now.

00:03:01.020 --> 00:03:03.980
So far I really just viewed
it as 1,000 times 44.

00:03:03.980 --> 00:03:06.180
I've been ignoring the decimal.

00:03:06.180 --> 00:03:10.146
So if it was 1,000 times 44,
we would get 0 plus 0 is 0.

00:03:10.146 --> 00:03:11.920
0 plus 0 is 0.

00:03:11.920 --> 00:03:12.820
0 plus 0 is 0.

00:03:12.820 --> 00:03:15.500
4 plus 0 is 4.

00:03:15.500 --> 00:03:16.857
4 plus nothing is 4.

00:03:16.857 --> 00:03:19.190
And if you ignore the decimal,
that makes a lot of sense

00:03:19.190 --> 00:03:21.040
because 1,000 times 4 is 4,000.

00:03:21.040 --> 00:03:24.290
And 1,000 times 40
would be 40,000.

00:03:24.290 --> 00:03:26.100
So you would get 44,000.

00:03:26.100 --> 00:03:28.850
But this, of
course, is not a 44.

00:03:28.850 --> 00:03:30.490
This is a 44/100.

00:03:30.490 --> 00:03:32.930
We have, between the
two numbers, two numbers

00:03:32.930 --> 00:03:34.180
behind the decimal point.

00:03:34.180 --> 00:03:35.740
So we need to have
two numbers behind

00:03:35.740 --> 00:03:38.260
or to the right of the
decimal point in our answer,

00:03:38.260 --> 00:03:40.640
so one, two right over there.

00:03:40.640 --> 00:03:46.440
So once again, we get
$440 for the 1,000 stamps.


Unit conversion word problem: yards to inches

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.736
We're asked, how many inches
are in 4 and 1/2 yards?

00:00:03.736 --> 00:00:05.110
And we'll do it
a couple of ways.

00:00:05.110 --> 00:00:06.830
One, we could just say how
many inches are in 4 yards

00:00:06.830 --> 00:00:08.621
and how many inches
are in a 1/2 of a yard?

00:00:08.621 --> 00:00:10.910
And this really
4 plus 1/2 yards.

00:00:10.910 --> 00:00:13.170
Or we could convert this
into an improper fraction

00:00:13.170 --> 00:00:14.159
first and then convert.

00:00:14.159 --> 00:00:15.700
But before I even
do that, let's just

00:00:15.700 --> 00:00:18.040
think about how many
inches there are in a yard.

00:00:18.040 --> 00:00:22.106
So if I have 1 yard,
we know that there

00:00:22.106 --> 00:00:30.020
are 3 feet for
every 1 yard, Right?

00:00:30.020 --> 00:00:32.865
And when you say, why am
I multiplying by 3 instead

00:00:32.865 --> 00:00:34.802
of saying there's 1
yard for every 3 feet?

00:00:34.802 --> 00:00:36.760
And the easiest way to
think about it is you're

00:00:36.760 --> 00:00:39.250
going to have a larger
value over here,

00:00:39.250 --> 00:00:43.340
and you're also going to want
to have these units right

00:00:43.340 --> 00:00:44.230
over here cancel out.

00:00:44.230 --> 00:00:46.610
So yard is canceling
out with yard.

00:00:46.610 --> 00:00:51.817
So you have 1 yard is
equal to 3 feet, which

00:00:51.817 --> 00:00:53.150
is kind of what we already knew.

00:00:53.150 --> 00:00:55.370
I'm just showing you how
the dimensions cancel out.

00:00:55.370 --> 00:00:58.340
And how many inches
are there per foot?

00:00:58.340 --> 00:01:04.980
Well, we know that there are
12 inches for every 1 foot.

00:01:04.980 --> 00:01:07.830
And same logic over here,
inches is a smaller unit

00:01:07.830 --> 00:01:10.574
of measurement so it makes sense
that we're multiplying by 12.

00:01:10.574 --> 00:01:11.990
3 feet is going
to be more inches,

00:01:11.990 --> 00:01:13.970
so we're multiplying by 12.

00:01:13.970 --> 00:01:15.610
And also these
units cancel out--

00:01:15.610 --> 00:01:17.970
foot in the numerator,
foot in the denominator.

00:01:17.970 --> 00:01:22.602
3 times 12 divided by 1
is equal to 36 inches.

00:01:22.602 --> 00:01:24.060
So you might have
already known it.

00:01:24.060 --> 00:01:26.143
But this is nice to have
the dimensions cancel out

00:01:26.143 --> 00:01:26.920
like this.

00:01:26.920 --> 00:01:31.970
We know that 1 yard
is equal to 36 inches.

00:01:31.970 --> 00:01:34.990
Or there are 36 inches
for every 1 yard.

00:01:34.990 --> 00:01:37.540
And so we can now
either break this down,

00:01:37.540 --> 00:01:39.610
or we can turn this into
an improper fraction.

00:01:39.610 --> 00:01:43.670
First I'll just break it down
into 4 yards plus 1/2 yards.

00:01:43.670 --> 00:01:45.790
So we could say that
this is 4 yards.

00:01:45.790 --> 00:01:49.650
So 4 yards is going
to be equal to-- well,

00:01:49.650 --> 00:01:53.030
let's just multiply
it times 36 inches,

00:01:53.030 --> 00:01:56.140
36 inches for every 1 yard.

00:01:56.140 --> 00:01:57.660
The yards cancel out.

00:01:57.660 --> 00:02:04.910
4 times 36 is 120,
plus 24, so that's 144.

00:02:04.910 --> 00:02:09.720
So this is equal to 144 inches.

00:02:09.720 --> 00:02:11.690
That's just the 4 yards.

00:02:11.690 --> 00:02:18.110
And then if we do the 1/2
yards, so 1/2 of a-- I

00:02:18.110 --> 00:02:20.410
guess I say 1/2
yard, once again,

00:02:20.410 --> 00:02:26.850
times 36 inches per yard,
the yards cancel out.

00:02:26.850 --> 00:02:32.800
1/2 times 36 is going to
be equal to 18 inches.

00:02:32.800 --> 00:02:34.510
So 4 and 1/2 yards
is the same thing

00:02:34.510 --> 00:02:38.620
as 4 yards plus 1/2 yards, which
is the same thing as 144 inches

00:02:38.620 --> 00:02:42.960
plus 18 inches, which is
going to give us-- let's just

00:02:42.960 --> 00:02:44.700
add it up over
here on the right,

00:02:44.700 --> 00:02:47.770
144 plus 18.

00:02:47.770 --> 00:02:50.240
4 plus 8 is 12.

00:02:50.240 --> 00:02:51.790
4 plus 1 is 5.

00:02:51.790 --> 00:02:55.110
You have this 1 up here, so
it's 6, and then we have a 1.

00:02:55.110 --> 00:02:59.072
So when you add them all
together, you get 162 inches.

00:02:59.072 --> 00:03:00.530
The other way to
do this would have

00:03:00.530 --> 00:03:02.680
been to convert this
into an improper fraction

00:03:02.680 --> 00:03:03.980
and then multiply by the unit.

00:03:03.980 --> 00:03:05.520
So let's do it that way.

00:03:05.520 --> 00:03:08.130
If I have 4 and 1/2 of
anything, really-- so

00:03:08.130 --> 00:03:09.580
let me write 4 and 1/2.

00:03:09.580 --> 00:03:11.410
I'm trying to find
a suitable color.

00:03:11.410 --> 00:03:15.460
So if I have 4 and 1/2,
this is the same thing--

00:03:15.460 --> 00:03:18.362
4 is the same thing as 8/2.

00:03:18.362 --> 00:03:20.570
This is the same thing-- so
let me write it this way.

00:03:20.570 --> 00:03:23.250
4 and 1/2 is the same
thing as 4 plus 1/2,

00:03:23.250 --> 00:03:25.930
which is the same
thing as-- if we

00:03:25.930 --> 00:03:28.030
want to have the same
denominator as this 2

00:03:28.030 --> 00:03:30.420
over here or as
this 1/2 over here,

00:03:30.420 --> 00:03:32.980
this is the same thing as 8/2.

00:03:32.980 --> 00:03:36.700
Or you could say 4/1 is
the same thing as 8/2,

00:03:36.700 --> 00:03:40.180
if we want to have a common
denominator, so 8/2 plus 1/2.

00:03:40.180 --> 00:03:42.430
Actually, let me write it
that way, just so you really

00:03:42.430 --> 00:03:44.000
understand what we're doing.

00:03:44.000 --> 00:03:46.680
4 is the same thing as 4/1.

00:03:46.680 --> 00:03:48.750
So it's 4/1 plus 1/2.

00:03:48.750 --> 00:03:51.700
If we want to find a
common denominator, it's 2.

00:03:51.700 --> 00:03:56.870
So 4/1 is the same
thing as 8/2 plus 1/2,

00:03:56.870 --> 00:03:59.190
which is equal to 9/2.

00:03:59.190 --> 00:04:01.090
Now, I did it this way,
which takes longer,

00:04:01.090 --> 00:04:03.310
just so you really understand
how we converted it,

00:04:03.310 --> 00:04:06.680
why it makes-- hopefully
conceptually why it just makes

00:04:06.680 --> 00:04:10.100
intuitive sense, why 4 and
1/2 is the same thing as 9/2.

00:04:10.100 --> 00:04:11.810
But if you want a
simple process for it,

00:04:11.810 --> 00:04:13.975
you could just say,
look, 4 times 2 is 8.

00:04:13.975 --> 00:04:15.700
8 plus 1 is 9.

00:04:15.700 --> 00:04:19.480
And that gives you that 9
right over there, so 9/2.

00:04:19.480 --> 00:04:22.725
So we have 9/2 yards that we
want to convert to inches.

00:04:27.760 --> 00:04:34.610
Same process-- times
36 inches per yard.

00:04:34.610 --> 00:04:36.760
Yard in the numerator,
yard in the denominator.

00:04:36.760 --> 00:04:41.400
We are left with 9/2 times 36.

00:04:41.400 --> 00:04:47.070
We could say times 36/1 if we
like, 36 Inches for every 1

00:04:47.070 --> 00:04:48.060
yard.

00:04:48.060 --> 00:04:51.470
36-- or the number 36
really is the same as 36/1.

00:04:51.470 --> 00:04:54.420
And then we're left with
just inches in our units.

00:04:54.420 --> 00:04:55.980
We're just left with inches.

00:04:55.980 --> 00:04:58.780
And over here there's several
ways that we can simplify it.

00:04:58.780 --> 00:05:01.100
Probably the easiest
way to simplify it is we

00:05:01.100 --> 00:05:03.694
can divide both our numerator
and our denominator by 2.

00:05:03.694 --> 00:05:04.860
So let me write it this way.

00:05:04.860 --> 00:05:05.985
I don't want to skip steps.

00:05:05.985 --> 00:05:11.645
So we have 9 times 36 over
2 times 1, or over 2 inches.

00:05:11.645 --> 00:05:13.770
And we can divide the
numerator and the denominator

00:05:13.770 --> 00:05:15.880
by 2 to simplify it.

00:05:15.880 --> 00:05:18.446
They're both divisible by 2.

00:05:18.446 --> 00:05:20.620
36 divided by 2 is 18.

00:05:20.620 --> 00:05:22.500
2 divided by 2 is 1.

00:05:22.500 --> 00:05:25.310
So we're really just left
with 9 times 18 inches.

00:05:25.310 --> 00:05:27.830
We can just multiply 9 times 18.

00:05:27.830 --> 00:05:29.360
Let me do it over here.

00:05:29.360 --> 00:05:31.990
18 times 9.

00:05:31.990 --> 00:05:34.430
8 times 9 is 72.

00:05:34.430 --> 00:05:41.620
1 times 9 is 9, plus 7 is
16, so we get 162 inches.

00:05:41.620 --> 00:05:47.940
So all of this simplifies to
162 inches, and we are done.


Dividing a decimal by a power of 10

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:08.090
We're asked to divide 99.061,
or 99 and 61/1000, by 100.

00:00:08.090 --> 00:00:09.470
And there's a few ways to do it.

00:00:09.470 --> 00:00:10.420
But what I want to
do in this video

00:00:10.420 --> 00:00:12.760
is focus on kind of a faster
way to think about it.

00:00:12.760 --> 00:00:14.343
And hopefully it'll
make sense to you.

00:00:14.343 --> 00:00:16.990
And that's also the focus of
it, that it makes sense to you.

00:00:16.990 --> 00:00:19.170
So 99, let's just think
about it a little bit.

00:00:19.170 --> 00:00:24.570
So if we-- so 99.061.

00:00:24.570 --> 00:00:27.000
So if we were to
divide this by 10, just

00:00:27.000 --> 00:00:29.550
to make the point clear, if
we were to divide this by 10,

00:00:29.550 --> 00:00:30.920
what would we get?

00:00:30.920 --> 00:00:34.210
Well, we would essentially
move the decimal place one spot

00:00:34.210 --> 00:00:35.570
to the left.

00:00:35.570 --> 00:00:38.690
And it should make sense because
we have a little over 99.

00:00:38.690 --> 00:00:40.940
If you took 99
divided by 10, you

00:00:40.940 --> 00:00:42.990
should have a little over 9.

00:00:42.990 --> 00:00:46.550
So essentially, you would
move the decimal place one

00:00:46.550 --> 00:00:48.760
to the left when
you divide by 10.

00:00:48.760 --> 00:00:54.710
So this would be
equal to 9.9061.

00:00:54.710 --> 00:00:59.150
If you were to divide it
by 100, which is actually

00:00:59.150 --> 00:01:04.269
the focus of this problem, so
if we divide 99.061 divided

00:01:04.269 --> 00:01:08.270
by 100, if we move the decimal
place once to the left,

00:01:08.270 --> 00:01:10.210
we're dividing by 10.

00:01:10.210 --> 00:01:12.470
To divide it by 100, we
have to divide by 10 again.

00:01:12.470 --> 00:01:13.760
So we move it over twice.

00:01:13.760 --> 00:01:15.980
So one, two times.

00:01:15.980 --> 00:01:20.790
And so now the decimal place
is out in front of that first

00:01:20.790 --> 00:01:23.250
leading 9, which also
should make sense.

00:01:23.250 --> 00:01:27.440
99 is almost 100, or a
little bit less than 100.

00:01:27.440 --> 00:01:29.420
So if you divide it
by 100, we should

00:01:29.420 --> 00:01:31.830
be a little bit less than 1.

00:01:31.830 --> 00:01:34.640
And so if you move the
decimal place two places over

00:01:34.640 --> 00:01:37.050
to the left, because we're
really dividing by 10 twice,

00:01:37.050 --> 00:01:38.640
if you want to think
of it that way,

00:01:38.640 --> 00:01:44.870
we will get the decimal in
front of the 99-- 0.99061.

00:01:44.870 --> 00:01:46.120
We should put a zero out here.

00:01:46.120 --> 00:01:48.340
Just sometimes it
clarifies things.

00:01:48.340 --> 00:01:50.100
So then we get this
right over here.

00:01:50.100 --> 00:01:51.996
Now, one way to think
about it, although I do

00:01:51.996 --> 00:01:54.620
want you to always imagine that
when you move the decimal place

00:01:54.620 --> 00:01:57.515
over to the left, you really are
dividing by 10 when you move it

00:01:57.515 --> 00:01:58.130
to the left.

00:01:58.130 --> 00:02:01.070
When you move it to the right,
you're multiplying by 10.

00:02:01.070 --> 00:02:03.200
Sometimes people
say, hey, look, you

00:02:03.200 --> 00:02:05.300
could just count the
number of zeroes.

00:02:05.300 --> 00:02:08.156
And if you're dividing, so over
here, you were dividing by 100.

00:02:08.156 --> 00:02:13.990
100 has two zeroes, and
we're dividing by it,

00:02:13.990 --> 00:02:18.100
so we could move our decimal
two spaces to the left.

00:02:18.100 --> 00:02:20.300
That's all right to do
that, especially it's

00:02:20.300 --> 00:02:21.730
kind of a fast way to do it.

00:02:21.730 --> 00:02:24.200
If this had 20 zeroes,
you would immediately

00:02:24.200 --> 00:02:26.700
say, OK, let's move the
decimal 20 places to the left.

00:02:26.700 --> 00:02:29.650
But I really want you to think
about why that's working,

00:02:29.650 --> 00:02:32.080
why that makes sense,
why it's giving you

00:02:32.080 --> 00:02:37.430
a number that seems to be the
right kind of size number,

00:02:37.430 --> 00:02:39.710
why it makes sense that if
you take something that's

00:02:39.710 --> 00:02:42.580
almost 100 and
divide by 100, you'll

00:02:42.580 --> 00:02:45.140
get something that's almost 1.

00:02:45.140 --> 00:02:47.880
And that part, frankly, is just
a really good reality check

00:02:47.880 --> 00:02:49.440
to make sure you're going
in the right direction

00:02:49.440 --> 00:02:50.170
with the decimal.

00:02:50.170 --> 00:02:52.980
Because if you tried
this 5, 10 years

00:02:52.980 --> 00:02:56.406
from now, maybe your
memory of the rule

00:02:56.406 --> 00:02:58.280
or whatever you want to
call it for doing it,

00:02:58.280 --> 00:03:00.113
you're like hey, wait,
do I move the decimal

00:03:00.113 --> 00:03:01.162
to the left or the right?

00:03:01.162 --> 00:03:03.620
It's really good to do that
reality check to say, OK, look,

00:03:03.620 --> 00:03:07.120
If I'm dividing by 100, I should
be getting a smaller value

00:03:07.120 --> 00:03:08.940
and moving the
decimal to the left

00:03:08.940 --> 00:03:10.620
gives me that smaller value.

00:03:10.620 --> 00:03:13.602
If I was multiplying by 100,
I should get a larger value.

00:03:13.602 --> 00:03:15.060
And moving the
decimal to the right

00:03:15.060 --> 00:03:17.860
would give you
that larger value.


Perimeter and unit conversion

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:06.260
The perimeter of a rectangular
fence measures 0.72 kilometers.

00:00:06.260 --> 00:00:10.480
The length of the fenced
area is 160 meters.

00:00:10.480 --> 00:00:12.260
What is it's width?

00:00:12.260 --> 00:00:14.150
Now, the first thing
that jumps out at me,

00:00:14.150 --> 00:00:15.650
and it might have
jumped out at you,

00:00:15.650 --> 00:00:17.399
is that they're giving
us different units.

00:00:17.399 --> 00:00:20.100
They're giving us the perimeter
in terms of kilometers,

00:00:20.100 --> 00:00:22.350
and they're giving us the
length in terms of meters.

00:00:22.350 --> 00:00:24.891
And I'm assuming that they want
the width in terms of meters,

00:00:24.891 --> 00:00:26.990
since that's what they're
giving us the length in.

00:00:26.990 --> 00:00:28.740
So what I want to do
right from the get go

00:00:28.740 --> 00:00:31.236
is just convert the
perimeter into meters.

00:00:31.236 --> 00:00:32.860
So we have the
perimeter, and I'll just

00:00:32.860 --> 00:00:38.770
represent that with a p, is
equal to 0.72 kilometers.

00:00:38.770 --> 00:00:40.290
Which I'll write km for short.

00:00:40.290 --> 00:00:42.300
Kilometers, 1,000 meters.

00:00:42.300 --> 00:00:45.240
That's what the
prefix kilo- means.

00:00:45.240 --> 00:00:48.590
And so we can say that
for every 1 kilometer

00:00:48.590 --> 00:00:49.780
we have 1,000 meters.

00:00:49.780 --> 00:00:54.696
Or we have 1,000 meters
per every 1 kilometer.

00:00:54.696 --> 00:00:56.070
And you might say
Sal, how do you

00:00:56.070 --> 00:00:59.255
know to multiply by 1,000,
instead of divide by 1,000?

00:00:59.255 --> 00:01:01.380
And one way to think about
it, and this is probably

00:01:01.380 --> 00:01:03.254
the best way to think
about it, is just look,

00:01:03.254 --> 00:01:06.510
a kilometer is a
bunch of meters.

00:01:06.510 --> 00:01:08.450
It's actually 1,000 meters.

00:01:08.450 --> 00:01:10.620
So if I'm converting
kilometers into meters,

00:01:10.620 --> 00:01:12.390
I should have a
much larger number.

00:01:12.390 --> 00:01:14.060
Whatever my number
is in kilometers,

00:01:14.060 --> 00:01:16.660
it must be a much
larger number of meters.

00:01:16.660 --> 00:01:18.440
So that's what tells
you to multiply.

00:01:18.440 --> 00:01:20.980
And also, if you care
about dimensional analysis,

00:01:20.980 --> 00:01:23.010
the dimensions
cancel out here too.

00:01:23.010 --> 00:01:25.350
We have kilometers in
the numerator, kilometers

00:01:25.350 --> 00:01:26.650
in the denominator.

00:01:26.650 --> 00:01:31.800
And so when you multiply it, you
have 0.72 times 1,000 meters.

00:01:31.800 --> 00:01:33.670
And to multiply
anything times 1,000,

00:01:33.670 --> 00:01:37.120
or really any power
of 10, you say, look,

00:01:37.120 --> 00:01:39.280
if I multiply it by 10,
I'll move the decimal

00:01:39.280 --> 00:01:41.060
to the right one space.

00:01:41.060 --> 00:01:43.890
That'd be multiplying it
by 10, it would be 7.2.

00:01:43.890 --> 00:01:46.800
Multiplying it by
100 would give us 72.

00:01:46.800 --> 00:01:50.420
If we're multiplying by
1,000, that would give us 720.

00:01:50.420 --> 00:01:52.870
And I'll put a trailing
0 here, just so

00:01:52.870 --> 00:01:55.630
that we have something
to move to the right of.

00:01:55.630 --> 00:02:01.230
So this is going to be
equal to 720 meters.

00:02:01.230 --> 00:02:03.320
So that is the perimeter.

00:02:03.320 --> 00:02:05.740
Now let's remind ourselves
what the perimeter even is.

00:02:05.740 --> 00:02:08.530
And then hopefully we
can figure out the width.

00:02:08.530 --> 00:02:11.009
So let me draw a
little box over here.

00:02:14.530 --> 00:02:18.500
And they tell us that
the length is 160 meters.

00:02:18.500 --> 00:02:21.050
So let's say that's this
dimension right over here.

00:02:21.050 --> 00:02:23.330
The length is 160 meters.

00:02:23.330 --> 00:02:26.340
So that would be this dimension
and this, it's a rectangle.

00:02:26.340 --> 00:02:29.110
So these sides are
both the same length.

00:02:29.110 --> 00:02:32.140
And our width is what
we need to solve for.

00:02:32.140 --> 00:02:35.640
So that's our width, and
this is also our width.

00:02:35.640 --> 00:02:39.190
And the perimeter is the
measure going around it.

00:02:39.190 --> 00:02:42.130
So the perimeter is going to be
this length, plus this width,

00:02:42.130 --> 00:02:46.300
plus that same length again,
plus that width over there.

00:02:46.300 --> 00:02:48.160
Or another way to think
about it-- and this

00:02:48.160 --> 00:02:49.490
might be a simpler
way-- so there's

00:02:49.490 --> 00:02:51.240
a couple of ways you
could think about it.

00:02:51.240 --> 00:02:54.050
You could say that
the perimeter is

00:02:54.050 --> 00:02:57.840
equal to the length
plus the width,

00:02:57.840 --> 00:03:02.260
plus the length plus the width.

00:03:02.260 --> 00:03:03.860
And then we know
what the length is.

00:03:03.860 --> 00:03:10.260
So the perimeter would be equal
to 160 meters plus the width.

00:03:10.260 --> 00:03:12.270
Actually, let me
write the units down.

00:03:12.270 --> 00:03:18.750
160 meters plus the width,
plus 160 meters plus the width.

00:03:18.750 --> 00:03:20.990
And then we know what
know the perimeter is,

00:03:20.990 --> 00:03:22.860
that's actually 720 meters.

00:03:22.860 --> 00:03:27.730
So 720 meters is the perimeter.

00:03:27.730 --> 00:03:30.140
So you would get 720
meters is equal to 160

00:03:30.140 --> 00:03:33.340
meters, plus the width, plus
160 meters plus the width.

00:03:33.340 --> 00:03:36.800
Now there's a bunch of different
ways to solve for the width.

00:03:36.800 --> 00:03:38.894
One way, you could
just say, look,

00:03:38.894 --> 00:03:40.560
if I just have the
width plus the length

00:03:40.560 --> 00:03:43.740
once, that's going to add
up to half of the perimeter.

00:03:43.740 --> 00:03:48.230
So if I just go halfway
around the rectangle,

00:03:48.230 --> 00:03:50.550
that's going to add up
to half the perimeter.

00:03:50.550 --> 00:03:55.130
So if I take my width, which
is w, plus my length, which

00:03:55.130 --> 00:04:02.170
is 160 meters, this should be
equal to 1/2 of the perimeter.

00:04:02.170 --> 00:04:04.920
This should be equal to 1/2
times our perimeter, which

00:04:04.920 --> 00:04:07.230
is 720 meters.

00:04:07.230 --> 00:04:13.100
Or you get width plus 160 meters
is equal to 1/2 times 720,

00:04:13.100 --> 00:04:21.019
or 720 divided by
2 is 360 meters.

00:04:21.019 --> 00:04:23.360
And so now you have
a situation where

00:04:23.360 --> 00:04:28.940
we have the width plus
160 meters is 360 meters.

00:04:28.940 --> 00:04:33.520
So we could now subtract
360 from both sides.

00:04:33.520 --> 00:04:36.580
Or sorry, we could subtract
160 from both sides

00:04:36.580 --> 00:04:37.260
to solve for it.

00:04:37.260 --> 00:04:38.801
Or you could even
do it in your head.

00:04:38.801 --> 00:04:42.770
If I say something plus 160 is
360, you could, in your head,

00:04:42.770 --> 00:04:44.490
say well, that
something must be 200.

00:04:44.490 --> 00:04:48.010
200 plus 160 is 360.

00:04:48.010 --> 00:04:52.390
So you could just say
your width is 200 meters.

00:04:52.390 --> 00:04:54.780
Or if you want to do it a
little bit more formally,

00:04:54.780 --> 00:04:57.280
you could say look,
subtract 160 meters

00:04:57.280 --> 00:05:01.020
from both sides
of this equation.

00:05:01.020 --> 00:05:07.236
And you are left with the
width is equal to 200 meters.

00:05:07.236 --> 00:05:08.360
So that's one way to do it.

00:05:08.360 --> 00:05:09.600
We've solved the problem.

00:05:09.600 --> 00:05:11.100
The other way is,
you could actually

00:05:11.100 --> 00:05:12.710
go straight from this equation.

00:05:12.710 --> 00:05:14.120
So we get 720.

00:05:14.120 --> 00:05:16.190
I'm just going to assume
everything is in meters.

00:05:16.190 --> 00:05:16.980
720.

00:05:16.980 --> 00:05:23.450
160 meters plus
160 meters is 320.

00:05:23.450 --> 00:05:25.690
And that's going to
be the same thing

00:05:25.690 --> 00:05:27.950
as the width plus the width.

00:05:27.950 --> 00:05:29.370
Or 2 times the width.

00:05:29.370 --> 00:05:32.910
Anything plus itself is
just 2 times that anything.

00:05:32.910 --> 00:05:38.010
Now if this plus 320 is equal to
720, you could do in your head.

00:05:38.010 --> 00:05:40.550
You could say well,
what plus 320 is 720?

00:05:40.550 --> 00:05:43.535
Well, this thing
must be equal to 400.

00:05:43.535 --> 00:05:44.910
Or you could do
it more formally,

00:05:44.910 --> 00:05:47.720
and subtract 320 from
both sides of this.

00:05:47.720 --> 00:05:49.440
And you would get--
if you subtract 320

00:05:49.440 --> 00:05:52.990
from here-- you would
get 400 is equal to,

00:05:52.990 --> 00:05:57.850
subtract 320 from this side,
you get 2 times the width.

00:05:57.850 --> 00:06:00.120
So if I have 2 times
something is equal to 400,

00:06:00.120 --> 00:06:02.169
that something must be 200.

00:06:02.169 --> 00:06:03.710
Or if we want to do
it more formally,

00:06:03.710 --> 00:06:06.830
you could divide both sides
of this equation by 2.

00:06:06.830 --> 00:06:12.240
Either way, you will get the
width is equal to 200 meters.


Multi-step inequalities 3

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.830
We're asked to solve for p.

00:00:01.830 --> 00:00:04.640
And we have the inequality
here negative 3p minus 7

00:00:04.640 --> 00:00:06.640
is less than p plus 9.

00:00:06.640 --> 00:00:09.180
So what we really want
to do is isolate the p

00:00:09.180 --> 00:00:10.937
on one side of this inequality.

00:00:10.937 --> 00:00:12.770
And preferably the
left-- that just makes it

00:00:12.770 --> 00:00:13.570
just a little easier to read.

00:00:13.570 --> 00:00:16.160
It doesn't have to be, but we
just want to isolate the p.

00:00:16.160 --> 00:00:18.020
So a good step to
that is to get rid

00:00:18.020 --> 00:00:19.490
of this p on the
right-hand side.

00:00:19.490 --> 00:00:21.410
And the best way I can
think of doing that

00:00:21.410 --> 00:00:23.247
is subtracting p from the right.

00:00:23.247 --> 00:00:24.830
But of course, if
we want to make sure

00:00:24.830 --> 00:00:27.140
that this inequality is
always going to be true,

00:00:27.140 --> 00:00:28.880
if we do anything to
the right, we also

00:00:28.880 --> 00:00:30.170
have to do that to the left.

00:00:30.170 --> 00:00:32.770
So we also have to
subtract p from the left.

00:00:32.770 --> 00:00:36.500
And so the left-hand side,
negative 3p minus p--

00:00:36.500 --> 00:00:38.000
that's negative 4p.

00:00:38.000 --> 00:00:40.760
And then we still have
a minus 7 up here-- is

00:00:40.760 --> 00:00:43.000
going to be less than p minus p.

00:00:43.000 --> 00:00:44.080
Those cancel out.

00:00:44.080 --> 00:00:46.120
It is less than 9.

00:00:46.120 --> 00:00:48.300
Now the next thing
I'm in the mood to do

00:00:48.300 --> 00:00:50.670
is get rid of this
negative 7, or this minus

00:00:50.670 --> 00:00:52.830
7 here, so that we
can better isolate

00:00:52.830 --> 00:00:54.540
the p on the left-hand side.

00:00:54.540 --> 00:00:56.960
So the best way I can think
of to get rid of a negative 7

00:00:56.960 --> 00:00:58.160
is to add 7 to it.

00:00:58.160 --> 00:00:59.750
Then it will just
cancel out to 0.

00:00:59.750 --> 00:01:02.920
So let's add 7 to both
sides of this inequality.

00:01:02.920 --> 00:01:04.959
Negative 7 plus 7 cancel out.

00:01:04.959 --> 00:01:07.720
All we're left with
is negative 4p.

00:01:07.720 --> 00:01:11.710
On the right-hand side, we
have 9 plus 7 equals 16.

00:01:11.710 --> 00:01:13.690
And it's still less than.

00:01:13.690 --> 00:01:15.770
Now, the last step
to isolate the p

00:01:15.770 --> 00:01:17.870
is to get rid of this
negative 4 coefficient.

00:01:17.870 --> 00:01:19.900
And the easiest way I
can think of to get rid

00:01:19.900 --> 00:01:22.750
of this negative
4 coefficient is

00:01:22.750 --> 00:01:25.810
to divide both
sides by negative 4.

00:01:25.810 --> 00:01:27.792
So if we divide this
side by negative 4,

00:01:27.792 --> 00:01:29.250
these guys are
going to cancel out.

00:01:29.250 --> 00:01:30.708
We're just going
to be left with p.

00:01:30.708 --> 00:01:32.680
We also have to do it
to the right-hand side.

00:01:32.680 --> 00:01:35.130
Now, there's one thing that
you really have to remember,

00:01:35.130 --> 00:01:37.800
since this is an
inequality, not an equation.

00:01:37.800 --> 00:01:39.660
If you're dealing
with an inequality

00:01:39.660 --> 00:01:43.830
and you multiply or divide
both sides of an equation

00:01:43.830 --> 00:01:47.640
by a negative number, you
have to swap the inequality.

00:01:47.640 --> 00:01:50.250
So in this case, the less
than becomes greater than,

00:01:50.250 --> 00:01:52.400
since we're dividing
by a negative number.

00:01:52.400 --> 00:01:55.710
And so negative 4 divided by
negative 4-- those cancel out.

00:01:55.710 --> 00:01:59.870
We have p is greater than
16 divided by negative 4,

00:01:59.870 --> 00:02:01.290
which is negative 4.

00:02:01.290 --> 00:02:04.480
And we can plot this
solution set right over here.

00:02:04.480 --> 00:02:06.080
And then we can
try out some values

00:02:06.080 --> 00:02:09.289
to help us feel good about
the idea of it working.

00:02:09.289 --> 00:02:13.680
So let's say this is negative
5, negative 4, negative 3,

00:02:13.680 --> 00:02:17.637
negative 2, negative 1, 0.

00:02:17.637 --> 00:02:19.220
Let me write that a
little bit neater.

00:02:22.290 --> 00:02:24.220
And then we can keep
going to the right.

00:02:24.220 --> 00:02:27.100
And so our solution is p is
not greater than or equal,

00:02:27.100 --> 00:02:29.060
so we have to
exclude negative 4.

00:02:29.060 --> 00:02:33.240
p is greater than negative 4,
so all the values above that.

00:02:33.240 --> 00:02:38.110
So negative 3.9999999 will work.

00:02:38.110 --> 00:02:40.490
Negative 4 will not work.

00:02:40.490 --> 00:02:42.370
And let's just try
some values out

00:02:42.370 --> 00:02:44.740
to feel good that this is
really the solution set.

00:02:44.740 --> 00:02:48.790
So first let's try out when
p is equal to negative 3.

00:02:48.790 --> 00:02:50.184
This should work.

00:02:50.184 --> 00:02:52.100
The way I've drawn it,
this is in our solution

00:02:52.100 --> 00:02:55.120
set. p equals negative 3
is greater than negative 4.

00:02:55.120 --> 00:02:56.300
So let's try that out.

00:02:56.300 --> 00:03:00.020
We have negative 3
times negative 3.

00:03:00.020 --> 00:03:01.900
The first negative
3 is this one,

00:03:01.900 --> 00:03:03.650
and then we're saying
p is negative 3.

00:03:03.650 --> 00:03:06.111
Minus 7 should be less
than-- instead of a p,

00:03:06.111 --> 00:03:07.610
we're going to
putting a negative 3.

00:03:07.610 --> 00:03:10.000
Should be less than
negative 3 plus 9.

00:03:10.000 --> 00:03:12.030
Negative 3 times
negative 3 is 9,

00:03:12.030 --> 00:03:17.070
minus 7 should be less than
negative 3 plus 9 is 6.

00:03:17.070 --> 00:03:18.180
9 minus 7 is 2.

00:03:18.180 --> 00:03:21.430
2 should be less than 6,
which, of course, it is.

00:03:21.430 --> 00:03:24.400
Now let's try a value that
definitely should not work.

00:03:24.400 --> 00:03:26.690
So let's try negative 5.

00:03:26.690 --> 00:03:28.510
Negative 5 is not
in our solution set,

00:03:28.510 --> 00:03:30.080
so it should not work.

00:03:30.080 --> 00:03:34.170
So we have negative 3
times negative 5 minus 7.

00:03:34.170 --> 00:03:38.860
Let's see whether it is
less than negative 5 plus 9.

00:03:38.860 --> 00:03:42.940
Negative 3 times negative
5 is 15, minus 7.

00:03:42.940 --> 00:03:47.380
It really should not be
less than negative 5 plus 9.

00:03:47.380 --> 00:03:50.720
So we're just seeing if p
equals negative 5 works.

00:03:50.720 --> 00:03:55.230
15 minus 7 is 8.

00:03:55.230 --> 00:03:57.520
And so we get 8 is
less than 4, which

00:03:57.520 --> 00:04:00.730
is definitely not the case.

00:04:00.730 --> 00:04:02.910
So p equals negative
5 doesn't work.

00:04:02.910 --> 00:04:05.494
And it shouldn't work, because
that's not in our solution set.

00:04:05.494 --> 00:04:07.493
And now if we really want
to feel good about it,

00:04:07.493 --> 00:04:09.170
we can actually try
this boundary point.

00:04:09.170 --> 00:04:11.550
Negative 4 should
not work, but it

00:04:11.550 --> 00:04:13.536
should satisfy the
related equation.

00:04:13.536 --> 00:04:15.160
When I talk about
the related equation,

00:04:15.160 --> 00:04:19.750
negative 4 should satisfy
negative 3 minus 7

00:04:19.750 --> 00:04:22.060
is equal to p plus 9.

00:04:22.060 --> 00:04:25.240
It'll satisfy this, but
it won't satisfy this.

00:04:25.240 --> 00:04:27.630
Because when we get the
same value on both sides,

00:04:27.630 --> 00:04:30.160
the same value is not
less than the same value.

00:04:30.160 --> 00:04:31.250
So let's try it out.

00:04:31.250 --> 00:04:33.690
Let's see whether
negative 4 at least

00:04:33.690 --> 00:04:36.870
satisfies the related equation.

00:04:36.870 --> 00:04:41.190
So if we get negative 3
times negative 4 minus 7,

00:04:41.190 --> 00:04:43.870
this should be equal
to negative 4 plus 9.

00:04:43.870 --> 00:04:48.370
So this is 12 minus 7 should
be equal to negative 4 plus 9.

00:04:48.370 --> 00:04:49.480
It should be equal to 5.

00:04:49.480 --> 00:04:50.688
And this, of course, is true.

00:04:50.688 --> 00:04:52.560
5 is equal to 5.

00:04:52.560 --> 00:04:54.360
So it satisfies the
related equation,

00:04:54.360 --> 00:04:55.760
but it should not satisfy this.

00:04:55.760 --> 00:04:57.910
If you put negative
4 for p here--

00:04:57.910 --> 00:04:59.390
and I encourage you to do so.

00:04:59.390 --> 00:05:00.848
Actually, we could
do it over here.

00:05:00.848 --> 00:05:03.010
Instead of an equals
sign, if you put it

00:05:03.010 --> 00:05:07.090
into the original inequality--
let me delete all of that--

00:05:07.090 --> 00:05:08.960
it really just becomes this.

00:05:08.960 --> 00:05:11.650
The original inequality
is this right over here.

00:05:11.650 --> 00:05:14.730
If you put negative
4, you have less than.

00:05:14.730 --> 00:05:18.470
And then you get 5 is less
than 5, which is not the case.

00:05:18.470 --> 00:05:19.950
And that's good,
because we did not

00:05:19.950 --> 00:05:21.460
include that in
the solution set.

00:05:21.460 --> 00:05:22.780
We put an open circle.

00:05:22.780 --> 00:05:25.020
If negative 4 was included,
we would fill that in.

00:05:25.020 --> 00:05:26.978
But the only reason why
we'd include negative 4

00:05:26.978 --> 00:05:28.650
is if this was
greater than or equal.

00:05:28.650 --> 00:05:32.650
So it's good that this does
not work, because negative 4

00:05:32.650 --> 00:05:34.540
is not part of our solution set.

00:05:34.540 --> 00:05:37.640
You can kind of view
it as a boundary point.


Adding negative numbers (example)

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:06.380
Find the sum of negative 15 plus
negative 46 plus negative 29.

00:00:06.380 --> 00:00:08.180
To do this, let's
just first visualize

00:00:08.180 --> 00:00:09.830
what each of these
numbers look like.

00:00:09.830 --> 00:00:12.300
So I'm going to draw a
number line for each of them.

00:00:12.300 --> 00:00:15.140
Negative 15 might look
something like this.

00:00:15.140 --> 00:00:20.790
So if this is 0 and let's
say that this is negative 15,

00:00:20.790 --> 00:00:23.270
I could represent
negative 15 as--

00:00:23.270 --> 00:00:27.960
if I'd like to-- an arrow that
points from 0 to negative 15.

00:00:27.960 --> 00:00:31.340
The length of the arrow is
the absolute value of this.

00:00:31.340 --> 00:00:32.920
It's the distance from 0.

00:00:32.920 --> 00:00:35.680
So the length here is 15.

00:00:35.680 --> 00:00:39.130
And the negative says that
we are pointing to the left.

00:00:39.130 --> 00:00:41.600
So the absolute value is 15.

00:00:41.600 --> 00:00:43.320
That's the length of that arrow.

00:00:43.320 --> 00:00:45.760
Let's do the same
thing for negative 46.

00:00:45.760 --> 00:00:47.820
So once again, let me
draw my number line.

00:00:53.070 --> 00:00:55.270
0 is going to be
right over there.

00:00:55.270 --> 00:00:58.420
And negative 46 is going
to be someplace over here.

00:00:58.420 --> 00:00:59.610
Negative 46.

00:00:59.610 --> 00:01:01.730
Notice, same exact idea.

00:01:01.730 --> 00:01:04.720
The distance between negative
46 and 0-- or another way

00:01:04.720 --> 00:01:08.130
to think about that is the
absolute value of negative 46--

00:01:08.130 --> 00:01:11.215
this distance right over
here is going to be 46.

00:01:14.700 --> 00:01:17.980
And its direction
is to the left.

00:01:17.980 --> 00:01:21.167
That is why we get to
the number negative 46.

00:01:21.167 --> 00:01:23.500
The negative really tells you
whether you're to the left

00:01:23.500 --> 00:01:26.390
or to the right of 0,
but the absolute value

00:01:26.390 --> 00:01:29.850
says how far are you to
the left or the right of 0.

00:01:29.850 --> 00:01:32.345
And then, finally, let's do
the same thing for negative 29.

00:01:35.280 --> 00:01:37.390
So once again, let me
draw my number line.

00:01:40.280 --> 00:01:42.740
I want to use the yellow again.

00:01:42.740 --> 00:01:44.640
So draw my number line.

00:01:44.640 --> 00:01:46.190
Let's say this is 0.

00:01:46.190 --> 00:01:49.180
Then negative 29 is going
to be roughly over here.

00:01:49.180 --> 00:01:50.600
I'm estimating it.

00:01:50.600 --> 00:01:54.510
And once again, negative 29
is exactly 29 away from 0.

00:01:54.510 --> 00:02:00.120
So this length right over here
is 29, and it is to the left.

00:02:00.120 --> 00:02:01.630
That is why it is negative 29.

00:02:01.630 --> 00:02:04.930
If it was positive 29, it
would be 29 to the right of 0.

00:02:04.930 --> 00:02:07.090
So we've represented
all of these numbers.

00:02:07.090 --> 00:02:09.389
And you can see what their
absolute values are like.

00:02:09.389 --> 00:02:11.680
And now let's think about
what happens if you add them.

00:02:11.680 --> 00:02:13.910
So one way to think about
adding these numbers

00:02:13.910 --> 00:02:16.090
is if you added
these arrows, if you

00:02:16.090 --> 00:02:19.120
put this arrow on
top of this arrow

00:02:19.120 --> 00:02:21.660
or to the left of this
arrow, if you started where

00:02:21.660 --> 00:02:24.940
this arrow leaves off-- and
then you put this green arrow,

00:02:24.940 --> 00:02:26.390
and then you put
the orange arrow.

00:02:26.390 --> 00:02:27.160
So let's do that.

00:02:31.170 --> 00:02:32.100
Let's draw that.

00:02:32.100 --> 00:02:34.410
It's going to be a
longer arrow now.

00:02:34.410 --> 00:02:35.720
So we're going to start at 0.

00:02:35.720 --> 00:02:38.180
First, we have the negative 15.

00:02:38.180 --> 00:02:41.520
So we're going to move
15 spaces to the left

00:02:41.520 --> 00:02:42.765
to get us to negative 15.

00:02:46.580 --> 00:02:48.740
Then we're going to go
46 spaces to the left

00:02:48.740 --> 00:02:53.350
to get us to negative
15 plus negative 46.

00:02:53.350 --> 00:02:54.770
So let me draw that.

00:02:54.770 --> 00:02:58.740
So then we're going
to go-- and we're

00:02:58.740 --> 00:03:01.160
going to figure out what
that number is in a second.

00:03:01.160 --> 00:03:03.370
So then we're going
to go 46 to the left.

00:03:03.370 --> 00:03:04.710
That's about that far.

00:03:04.710 --> 00:03:08.120
Really just this arrow,
I'm now placing it.

00:03:08.120 --> 00:03:13.260
I'm starting off where
the negative 15 left off,

00:03:13.260 --> 00:03:15.410
and then I'm putting
that arrow after that.

00:03:15.410 --> 00:03:17.555
And we don't know yet what
number this gets us to.

00:03:17.555 --> 00:03:19.430
We're going to have to
do a little math here.

00:03:19.430 --> 00:03:21.030
And that's actually the
point of this problem.

00:03:21.030 --> 00:03:22.960
But we know what the
length of this arrow is.

00:03:22.960 --> 00:03:24.645
The length of this arrow is 46.

00:03:24.645 --> 00:03:27.260
It is 46 to the
left, and we know

00:03:27.260 --> 00:03:30.500
the length of this
magenta arrow is 15.

00:03:30.500 --> 00:03:32.910
And then, finally, we
have this orange arrow.

00:03:32.910 --> 00:03:35.810
The orange arrow, which we
know has a length of 29.

00:03:35.810 --> 00:03:37.670
Although, it's 29 to the left.

00:03:37.670 --> 00:03:39.080
It looks like that.

00:03:39.080 --> 00:03:40.930
It has a length of 29.

00:03:40.930 --> 00:03:43.790
So when all is said
and done, where are we?

00:03:43.790 --> 00:03:46.590
Where do we end up
on our number line?

00:03:46.590 --> 00:03:49.460
Well, the total length
that we are to the left

00:03:49.460 --> 00:03:52.220
is going to be 15
plus 46 plus 29.

00:03:52.220 --> 00:03:53.620
But it's going to
be to the left,

00:03:53.620 --> 00:03:55.160
so it's going to be negative.

00:03:55.160 --> 00:03:57.490
So what we can really view
this as, since all of these

00:03:57.490 --> 00:03:59.020
have the same sign,
is we could say,

00:03:59.020 --> 00:04:01.620
look, this is the same
thing as the absolute value

00:04:01.620 --> 00:04:04.680
of negative 15, which is
15, plus the absolute value

00:04:04.680 --> 00:04:07.210
of negative 46, which is
46, plus the absolute value

00:04:07.210 --> 00:04:09.510
of negative 29,
which is 29, but then

00:04:09.510 --> 00:04:11.260
take the negative value of it.

00:04:11.260 --> 00:04:12.300
So let's just add that.

00:04:12.300 --> 00:04:24.580
15 plus 46 plus 29 is equal
to, let's see-- 5 plus 6 is 11.

00:04:24.580 --> 00:04:26.480
11 plus 9 is 20.

00:04:26.480 --> 00:04:27.330
Carry the 2.

00:04:27.330 --> 00:04:28.900
2 plus 1 is 3.

00:04:28.900 --> 00:04:30.186
3 plus 4 is 7.

00:04:30.186 --> 00:04:32.330
7 plus 2 is 9.

00:04:32.330 --> 00:04:33.330
So you get 90.

00:04:33.330 --> 00:04:36.120
So this entire
length here is 90.

00:04:36.120 --> 00:04:40.300
The entire length, if you add
up the arrows, you get a 90.

00:04:40.300 --> 00:04:42.220
But it is not 90 to the right.

00:04:42.220 --> 00:04:44.870
So if it was 90 to the right,
these would all be positives.

00:04:44.870 --> 00:04:46.578
And then we would just
get a positive 90.

00:04:46.578 --> 00:04:48.020
But this is 90 to the left.

00:04:48.020 --> 00:04:51.560
So when you add these guys up,
you don't get to positive 90.

00:04:51.560 --> 00:04:53.860
You get to negative 90.

00:04:53.860 --> 00:04:56.760
So one way to think about it
is these are all the same sign.

00:04:56.760 --> 00:04:59.000
So this is going to be
equal to the negative

00:04:59.000 --> 00:05:04.560
of the absolute
value of negative 15

00:05:04.560 --> 00:05:07.710
plus the absolute
value of negative 46

00:05:07.710 --> 00:05:12.410
plus the absolute
value of negative 29.

00:05:12.410 --> 00:05:14.590
And the reason why we're
writing this here-- this

00:05:14.590 --> 00:05:16.089
might look fancy,
but this is really

00:05:16.089 --> 00:05:21.110
just the length of
this purple arrow.

00:05:21.110 --> 00:05:23.440
This absolute value
of negative 46,

00:05:23.440 --> 00:05:25.750
that's really just the
length of the green arrow.

00:05:25.750 --> 00:05:26.890
46.

00:05:26.890 --> 00:05:31.590
And this is just the
length of the orange arrow.

00:05:31.590 --> 00:05:35.380
So this is 15 plus 46
plus 29, gave us 90,

00:05:35.380 --> 00:05:36.600
but it is to the left.

00:05:36.600 --> 00:05:40.391
So that is why it
is negative 90.


Number sets

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.235
Consider the numbers
1/3, 7, and negative 15.

00:00:06.790 --> 00:00:09.620
To which sets of
numbers, listed below,

00:00:09.620 --> 00:00:11.890
does each of these
numbers belong?

00:00:11.890 --> 00:00:14.230
And they give us a bunch of
different sets of numbers.

00:00:14.230 --> 00:00:15.521
And we could just review these.

00:00:15.521 --> 00:00:16.990
The natural numbers,
and you'll see

00:00:16.990 --> 00:00:19.870
this used in different
contexts in different places,

00:00:19.870 --> 00:00:21.760
but for the sake of
this video, we'll

00:00:21.760 --> 00:00:25.020
assume this is counting numbers,
and it does not equal 0.

00:00:25.020 --> 00:00:31.345
So the natural numbers are 1,
2, 3, 4, and so on and so forth.

00:00:31.345 --> 00:00:33.470
We'll assume that the whole
numbers are essentially

00:00:33.470 --> 00:00:35.340
the natural numbers plus 0.

00:00:35.340 --> 00:00:41.700
So the whole numbers are 0, 1,
2, 3, 4, so on and so forth.

00:00:41.700 --> 00:00:44.610
The integers we'll assume
it's the whole numbers

00:00:44.610 --> 00:00:47.010
plus the opposite of
the natural numbers,

00:00:47.010 --> 00:00:49.340
or the negative version
of the natural numbers.

00:00:49.340 --> 00:00:52.630
So we would go from
very negative values

00:00:52.630 --> 00:00:55.700
to maybe negative 4,
negative 3, negative 2,

00:00:55.700 --> 00:01:02.410
negative 1, 0, 1, 2, 3,
4, so on and so forth.

00:01:02.410 --> 00:01:03.890
Rational numbers.

00:01:03.890 --> 00:01:06.060
These are numbers
that can be expressed

00:01:06.060 --> 00:01:08.900
as the ratio of two integers.

00:01:08.900 --> 00:01:11.680
So a rational number
can be expressed as m

00:01:11.680 --> 00:01:15.380
over n, where both m
and n are integers.

00:01:15.380 --> 00:01:21.535
And of course n cannot be 0,
because anything divided by 0

00:01:21.535 --> 00:01:23.170
is not defined.

00:01:23.170 --> 00:01:26.740
M could be 0, and in fact,
0 is a rational number,

00:01:26.740 --> 00:01:29.750
because 0 can be
represented as 0 over 1.

00:01:29.750 --> 00:01:33.220
Or really 0 over 2, or 0 over 3.

00:01:33.220 --> 00:01:34.450
I think you get the idea.

00:01:34.450 --> 00:01:39.350
Or 0 over negative 537.

00:01:39.350 --> 00:01:41.459
These are all
representations of 0,

00:01:41.459 --> 00:01:43.500
and you have an integer
in the numerator, integer

00:01:43.500 --> 00:01:44.520
in the denominator.

00:01:44.520 --> 00:01:46.340
So 0 is definitely
a rational number.

00:01:46.340 --> 00:01:49.830
You just can't have n, you can't
have the denominator, be 0.

00:01:49.830 --> 00:01:52.750
Irrational numbers are numbers
that cannot be expressed

00:01:52.750 --> 00:01:53.830
in this way.

00:01:53.830 --> 00:01:56.610
And so common versions
of irrational numbers

00:01:56.610 --> 00:02:00.690
are numbers that you will never
see digits that repeat on, go

00:02:00.690 --> 00:02:03.550
on and on forever, or have
series of sequences of numbers

00:02:03.550 --> 00:02:06.570
that keep repeating
on and on and on.

00:02:06.570 --> 00:02:10.520
And examples of irrational
numbers are square root of 2,

00:02:10.520 --> 00:02:12.400
or pi.

00:02:12.400 --> 00:02:14.865
And the ratio of the
circumference, the diameter

00:02:14.865 --> 00:02:17.600
of a circle, or you might learn
about the irrational number

00:02:17.600 --> 00:02:19.200
e, which is 2.71.

00:02:19.200 --> 00:02:21.580
And it keeps going,
and it doesn't repeat.

00:02:21.580 --> 00:02:26.420
And then real numbers are
really all of these combined.

00:02:26.420 --> 00:02:28.080
If you combine the
irrational numbers

00:02:28.080 --> 00:02:31.730
and the rational
numbers, then you

00:02:31.730 --> 00:02:33.520
are talking about
the real numbers.

00:02:33.520 --> 00:02:37.310
Every real number is either
rational or irrational.

00:02:37.310 --> 00:02:39.770
So let's classify
what we have up here.

00:02:39.770 --> 00:02:48.250
1/3 over here-- let me do that
in a different color-- so 1/3,

00:02:48.250 --> 00:02:49.520
is it a natural number?

00:02:49.520 --> 00:02:50.314
Well no, it's not.

00:02:50.314 --> 00:02:51.980
It's not just a natural
counting number.

00:02:51.980 --> 00:02:52.880
Is it a whole number?

00:02:52.880 --> 00:02:55.620
No, it's not a whole number,
it's in fact a fraction.

00:02:55.620 --> 00:02:56.550
Is it an integer?

00:02:56.550 --> 00:02:58.450
No once again, it
is not an integer.

00:02:58.450 --> 00:03:00.310
Is it a rational number?

00:03:00.310 --> 00:03:01.110
Well sure it is.

00:03:01.110 --> 00:03:03.720
It can be represented
as 1 over 3.

00:03:03.720 --> 00:03:07.020
Both 1 and 3 are integers.

00:03:07.020 --> 00:03:10.050
So 1/3 is rational.

00:03:12.950 --> 00:03:16.260
And if something is rational,
it cannot be irrational.

00:03:16.260 --> 00:03:19.000
There's nothing that's both
rational and irrational.

00:03:19.000 --> 00:03:20.750
This literally
means not rational.

00:03:20.750 --> 00:03:22.155
And it is also a real number.

00:03:24.670 --> 00:03:27.210
Pretty much all of these are
going to be real numbers.

00:03:27.210 --> 00:03:28.280
Now, let's try 7 out.

00:03:31.830 --> 00:03:35.360
So the number 7, it
is a natural number.

00:03:35.360 --> 00:03:37.480
It is just a regular
counting number.

00:03:37.480 --> 00:03:38.500
It is a whole number.

00:03:41.500 --> 00:03:43.930
Any natural number is also
going to be a whole number.

00:03:43.930 --> 00:03:45.920
And anything that is a
natural number that's

00:03:45.920 --> 00:03:48.544
also going to be a whole number
is also going to be an integer.

00:03:48.544 --> 00:03:51.350
So this is also an integer.

00:03:51.350 --> 00:03:53.600
And anything that is a natural
number, a whole number,

00:03:53.600 --> 00:03:56.325
and integer, is also going
to be a rational number.

00:03:59.220 --> 00:04:02.540
And to see that, you
can write 7 as 7 over 1.

00:04:02.540 --> 00:04:04.970
In fact, you can
write any integer

00:04:04.970 --> 00:04:09.010
as that number over 1, which
hopefully makes it obvious

00:04:09.010 --> 00:04:11.320
that that is also
a rational number.

00:04:11.320 --> 00:04:13.450
And if it's rational,
you can't be irrational.

00:04:13.450 --> 00:04:15.180
That literally
means not rational.

00:04:15.180 --> 00:04:17.490
And then of course,
you're going to be real.

00:04:17.490 --> 00:04:19.890
It's going to be a real number.

00:04:19.890 --> 00:04:21.390
And you're going
to appreciate later

00:04:21.390 --> 00:04:23.194
on in your mathematical
careers why we even

00:04:23.194 --> 00:04:25.360
go through the trouble of
saying it's a real number.

00:04:25.360 --> 00:04:27.180
Because as far as
it seems right now,

00:04:27.180 --> 00:04:29.161
this kind of includes
all the numbers.

00:04:29.161 --> 00:04:30.910
What you're going to
see in the future is,

00:04:30.910 --> 00:04:32.660
we're going to start
defining numbers that

00:04:32.660 --> 00:04:33.940
don't fall into this category.

00:04:33.940 --> 00:04:35.850
And they're a little bit
philosophically interesting

00:04:35.850 --> 00:04:36.530
as well.

00:04:36.530 --> 00:04:38.350
But anyway, back to the problem.

00:04:38.350 --> 00:04:41.580
We have a negative 15.

00:04:41.580 --> 00:04:45.842
Negative 15 is not a natural
number, it's a negative number.

00:04:45.842 --> 00:04:47.675
So it's not a natural
number, natural number

00:04:47.675 --> 00:04:50.857
is only the positive
integers, we could say.

00:04:50.857 --> 00:04:52.440
Whole number, it's
not a whole number,

00:04:52.440 --> 00:04:54.400
because that's just the
natural numbers and 0.

00:04:54.400 --> 00:04:57.240
It is an integer.

00:04:57.240 --> 00:05:00.030
It is the opposite
of positive 15.

00:05:00.030 --> 00:05:03.360
It is an integer.

00:05:03.360 --> 00:05:07.240
It is rational, it can be
represented as 15 over 1.

00:05:07.240 --> 00:05:08.710
So it is a rational.

00:05:08.710 --> 00:05:11.570
And because it is rational,
it cannot be rational.

00:05:11.570 --> 00:05:14.280
And of course, it is also real.

00:05:14.280 --> 00:05:16.430
And we are done.


Compound inequalities 3

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.100
We're asked to solve for x.

00:00:02.100 --> 00:00:03.850
And we have this
compound inequality here,

00:00:03.850 --> 00:00:07.530
negative 16 is less than or
equal to 3x plus 5, which

00:00:07.530 --> 00:00:09.137
is less than or equal to 20.

00:00:09.137 --> 00:00:10.970
And really, there's two
ways to approach it,

00:00:10.970 --> 00:00:12.150
which are really the same way.

00:00:12.150 --> 00:00:13.191
And I'll do both of them.

00:00:13.191 --> 00:00:15.500
And I'll actually do both
of them simultaneously.

00:00:15.500 --> 00:00:18.480
So one is to just solve
this compound inequality

00:00:18.480 --> 00:00:18.980
all at once.

00:00:18.980 --> 00:00:20.021
And I'll just rewrite it.

00:00:20.021 --> 00:00:23.930
Negative 16 is less than or
equal to 3x plus 5, which

00:00:23.930 --> 00:00:25.540
is less than or equal to 20.

00:00:25.540 --> 00:00:27.050
And the other way
is to think of it

00:00:27.050 --> 00:00:29.510
as two separate inequalities,
but both of them

00:00:29.510 --> 00:00:30.870
need to be true.

00:00:30.870 --> 00:00:34.770
So you could also view it as
negative 16 has to be less than

00:00:34.770 --> 00:00:37.488
or equal to 3x plus 5.

00:00:37.488 --> 00:00:45.790
And 3x plus 5 needs to be
less than or equal to 20.

00:00:45.790 --> 00:00:48.849
This statement and this
statement are equivalent.

00:00:48.849 --> 00:00:51.140
This one may seem a little
bit more familiar because we

00:00:51.140 --> 00:00:53.600
can independently solve
each of these inequalities

00:00:53.600 --> 00:00:54.767
and just remember the "and."

00:00:54.767 --> 00:00:57.266
This one might seem a little
less traditional because now we

00:00:57.266 --> 00:00:58.870
have three sides
to the statement.

00:00:58.870 --> 00:01:00.870
We have three parts of
this compound inequality.

00:01:00.870 --> 00:01:02.619
But what we can see
is that we're actually

00:01:02.619 --> 00:01:04.190
going to solve it
the exact same way.

00:01:04.190 --> 00:01:07.650
In any situation, we really just
want to isolate the x on one

00:01:07.650 --> 00:01:09.370
side of the inequality,
or in this case,

00:01:09.370 --> 00:01:11.860
one part of the
compound inequality.

00:01:11.860 --> 00:01:14.450
Well, the best way to
isolate this x right here

00:01:14.450 --> 00:01:16.540
is to first get rid of
this positive 5 that's

00:01:16.540 --> 00:01:17.456
sitting in the middle.

00:01:17.456 --> 00:01:19.260
So let's subtract
5 from every part

00:01:19.260 --> 00:01:21.030
of this compound inequality.

00:01:21.030 --> 00:01:23.590
So I'm going to subtract
5 there, subtract 5

00:01:23.590 --> 00:01:26.230
there, and subtract
5 over there.

00:01:26.230 --> 00:01:32.150
And so we get negative 16
minus 5 is negative 21,

00:01:32.150 --> 00:01:36.780
is less than or equal to 3x
plus 5 minus 5 is 3x, which

00:01:36.780 --> 00:01:39.891
is less than or equal to
20 minus 5, which is 15.

00:01:39.891 --> 00:01:41.890
And we could essentially
do the same thing here.

00:01:41.890 --> 00:01:47.410
If we want to isolate the 3x, we
can subtract 5 from both sides.

00:01:47.410 --> 00:01:49.190
We get negative 21.

00:01:49.190 --> 00:01:54.400
Negative 21 is less
than or equal to 3x.

00:01:54.400 --> 00:01:58.950
And we get, subtracting
5 from both sides.

00:01:58.950 --> 00:02:00.450
And notice, we're
just subtracting 5

00:02:00.450 --> 00:02:02.283
from every part of this
compound inequality.

00:02:02.283 --> 00:02:06.578
We get 3x is less
than or equal to 15.

00:02:06.578 --> 00:02:08.619
So this statement and this
statement, once again,

00:02:08.619 --> 00:02:10.139
are the exact same thing.

00:02:10.139 --> 00:02:13.470
Now, going back here, if
we want to isolate the x,

00:02:13.470 --> 00:02:14.882
we can divide by 3.

00:02:14.882 --> 00:02:17.090
And we have to do it to
every part of the inequality.

00:02:17.090 --> 00:02:20.410
And since 3 is positive, we
don't have to change the sign.

00:02:20.410 --> 00:02:24.583
So let's divide every part of
this compound inequality by 3.

00:02:24.583 --> 00:02:26.420
You divide every part by 3.

00:02:26.420 --> 00:02:28.671
This is equivalent to
dividing every part of each

00:02:28.671 --> 00:02:32.820
of these inequalities by 3.

00:02:32.820 --> 00:02:36.810
And then we get negative 21
divided by 3 is negative 7,

00:02:36.810 --> 00:02:40.620
is less than or equal
to x, which is less than

00:02:40.620 --> 00:02:43.140
or equal to 15
divided by 3 is 5.

00:02:43.140 --> 00:02:43.920
You do it here.

00:02:43.920 --> 00:02:47.990
You get negative 7 is
less than or equal to x,

00:02:47.990 --> 00:02:54.940
and x is less than or
equal to 15/3, which is 5.

00:02:54.940 --> 00:02:57.952
This statement and this
statement are completely equal.

00:02:57.952 --> 00:02:58.910
And we've solved for x.

00:02:58.910 --> 00:03:00.590
We've given you
the solution set.

00:03:00.590 --> 00:03:04.630
And if we want to graph
it on a number line,

00:03:04.630 --> 00:03:08.170
it would look like this.

00:03:08.170 --> 00:03:09.270
This is 0.

00:03:09.270 --> 00:03:10.470
This is 5.

00:03:10.470 --> 00:03:12.080
This is negative 7.

00:03:12.080 --> 00:03:15.730
Our solution set includes
everything between negative 7

00:03:15.730 --> 00:03:17.650
and 5, including
negative 7 and 5.

00:03:17.650 --> 00:03:22.530
So we have to fill in
the circles on negative 7

00:03:22.530 --> 00:03:23.940
and positive 5.

00:03:23.940 --> 00:03:26.420
And it is everything in between.

00:03:26.420 --> 00:03:28.060
That's our solution set.

00:03:28.060 --> 00:03:29.760
And so we can verify
that these work.

00:03:29.760 --> 00:03:31.134
You could try out
a number that's

00:03:31.134 --> 00:03:33.380
well inside of our
solution set, like 0.

00:03:33.380 --> 00:03:35.120
3 times 0 is 0.

00:03:35.120 --> 00:03:37.720
So you're just left
with 5 is greater

00:03:37.720 --> 00:03:39.690
than or equal to negative
16, which is true.

00:03:39.690 --> 00:03:41.730
And 5 is less than
or equal to 20.

00:03:41.730 --> 00:03:44.300
Or negative 16 is less than or
equal to 5, which is less than

00:03:44.300 --> 00:03:44.930
or equal to 20.

00:03:44.930 --> 00:03:47.130
So that works, and
that makes sense.

00:03:47.130 --> 00:03:48.480
You could try 5.

00:03:48.480 --> 00:03:50.320
If you put 5 here,
you get 3 times

00:03:50.320 --> 00:03:52.400
5 plus 5-- well, that's just 20.

00:03:52.400 --> 00:03:55.146
Negative 16 is less than or
equal to 20, which is less than

00:03:55.146 --> 00:03:55.770
or equal to 20.

00:03:55.770 --> 00:03:56.610
That works.

00:03:56.610 --> 00:03:58.200
Negative 7 should also work.

00:03:58.200 --> 00:04:02.060
3 times negative 7 is negative
21, plus 5 is negative 16.

00:04:02.060 --> 00:04:03.850
So you get negative
16, which is less than

00:04:03.850 --> 00:04:07.739
or equal to negative 16, which
is less than or equal to 20.

00:04:07.739 --> 00:04:09.030
And you could try other values.

00:04:09.030 --> 00:04:10.738
You could go outside
of our solution set.

00:04:10.738 --> 00:04:11.660
Try something like 10.

00:04:11.660 --> 00:04:12.850
10 should not work.

00:04:12.850 --> 00:04:15.986
And you see here, if you put 10
here, you get 3 times 10 plus 5

00:04:15.986 --> 00:04:18.310
is 35.

00:04:18.310 --> 00:04:20.880
Negative 16 is less
than or equal to 35,

00:04:20.880 --> 00:04:23.620
but 35 is not less
than or equal to 20.

00:04:23.620 --> 00:04:27.640
And that's why 10 is not
part of our solution set.


Proportion word problem (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:05.790
A recipe for oatmeal cookies
calls for 2 cups of flour

00:00:05.790 --> 00:00:08.160
for every 3 cups of oatmeal.

00:00:08.160 --> 00:00:10.630
How much flour is
needed for a big batch

00:00:10.630 --> 00:00:12.592
of cookies that uses
9 cups of oatmeal?

00:00:12.592 --> 00:00:14.300
So let's think about
what they're saying.

00:00:14.300 --> 00:00:16.239
They're saying 2 cups of flour.

00:00:16.239 --> 00:00:25.345
So 2 cups of flour for
every 3 cups of oatmeal.

00:00:39.349 --> 00:00:40.890
And so they're
saying, how much flour

00:00:40.890 --> 00:00:42.570
is needed for a big
batch of cookies

00:00:42.570 --> 00:00:46.700
that uses 9 cups of oatmeal?

00:00:46.700 --> 00:00:48.200
Now we're going to
go to a situation

00:00:48.200 --> 00:00:50.580
where we are using
9 cups of oatmeal.

00:00:50.580 --> 00:00:56.187
Let me write it this
way-- 9 cups of oatmeal.

00:00:56.187 --> 00:00:58.020
And I'll show you a
couple of different ways

00:00:58.020 --> 00:00:58.769
to think about it.

00:00:58.769 --> 00:01:01.900
And whatever works
for you, that works.

00:01:01.900 --> 00:01:04.695
So one way to think about
it, so we're wondering.

00:01:04.695 --> 00:01:06.070
We're going to
say, look, we know

00:01:06.070 --> 00:01:09.130
if we have 3 cups of oatmeal,
we should use 2 cups of flour.

00:01:09.130 --> 00:01:12.170
But what we don't know is if
we have 9 cups of oatmeal,

00:01:12.170 --> 00:01:14.220
how many cups of
flour do we use?

00:01:14.220 --> 00:01:15.812
That's what they're asking us.

00:01:15.812 --> 00:01:17.520
But if we're going
from 3 cups of oatmeal

00:01:17.520 --> 00:01:21.330
to 9 cups of oatmeal, how much
more oatmeal are we using?

00:01:21.330 --> 00:01:25.530
Well, we're using three
times more oatmeal, Right?

00:01:25.530 --> 00:01:26.934
We're multiplying by 3.

00:01:26.934 --> 00:01:28.600
3 cups of oatmeal and
9 cups of oatmeal,

00:01:28.600 --> 00:01:30.700
we're using 3 times the oatmeal.

00:01:30.700 --> 00:01:33.070
Well, if we want to use
flour in the same proportion,

00:01:33.070 --> 00:01:35.650
we have to use 3
times the flour.

00:01:35.650 --> 00:01:39.190
So then we're also going to
multiply the flour times 3.

00:01:39.190 --> 00:01:40.940
We're going to multiply
the flour times 3,

00:01:40.940 --> 00:01:43.220
so we're going to have
to use 6 cups of flour.

00:01:48.920 --> 00:01:50.050
Ignore that question mark.

00:01:53.160 --> 00:01:54.550
And that answers the question.

00:01:54.550 --> 00:01:57.110
That's how much flour we need
for a big batch of cookies

00:01:57.110 --> 00:01:58.840
that uses 9 cups of oatmeal.

00:01:58.840 --> 00:02:02.040
The other thing is you
could set up a proportion.

00:02:02.040 --> 00:02:15.300
You could say 2 cups of
flour over 3 cups of oatmeal

00:02:15.300 --> 00:02:18.060
is equal to question mark.

00:02:18.060 --> 00:02:19.610
And instead of
writing question mark,

00:02:19.610 --> 00:02:21.030
I'll put a variable in there.

00:02:23.161 --> 00:02:24.910
Actually, let me put
a question mark there

00:02:24.910 --> 00:02:26.560
just so you really
understand it is

00:02:26.560 --> 00:02:29.530
equal to a question
mark in a box number

00:02:29.530 --> 00:02:37.615
cups of flour over
9 cups of oatmeal.

00:02:42.200 --> 00:02:44.047
And so I like this
first way we did it

00:02:44.047 --> 00:02:45.630
because it's really
just common sense.

00:02:45.630 --> 00:02:48.370
If we're tripling the
oatmeal, then we're

00:02:48.370 --> 00:02:50.440
going to have to
triple the flour

00:02:50.440 --> 00:02:52.820
to make the recipe in
the same proportion.

00:02:52.820 --> 00:02:55.030
Another way, once you set
up an equation like this,

00:02:55.030 --> 00:02:57.160
is actually to do a
little bit of algebra.

00:02:57.160 --> 00:02:59.140
Some people might call
it cross-multiplying,

00:02:59.140 --> 00:03:00.770
but that
cross-multiplying is still

00:03:00.770 --> 00:03:02.420
using a little bit of algebra.

00:03:02.420 --> 00:03:04.780
And I'll show you why they're
really the same thing.

00:03:04.780 --> 00:03:06.300
In cross-multiplication,
whenever

00:03:06.300 --> 00:03:08.160
you have a proportion
set up like this,

00:03:08.160 --> 00:03:10.920
people will multiply
the diagonals.

00:03:10.920 --> 00:03:14.030
So when you use
cross-multiplication,

00:03:14.030 --> 00:03:21.020
you'll say that 2 times 9
must be equal to question mark

00:03:21.020 --> 00:03:27.620
times 3, must be equal to
whatever is in this question

00:03:27.620 --> 00:03:31.420
mark, the number of
cups of flour times 3.

00:03:31.420 --> 00:03:36.750
Or we get 18 is
equal to whatever

00:03:36.750 --> 00:03:41.380
our question mark was times 3.

00:03:41.380 --> 00:03:44.640
So the number of cups of
flour we need to use times 3

00:03:44.640 --> 00:03:46.170
needs to be equal to 18.

00:03:46.170 --> 00:03:47.430
What times 3 is equal 18?

00:03:47.430 --> 00:03:49.180
You might be able to
do that in your head.

00:03:49.180 --> 00:03:50.030
That is 6.

00:03:50.030 --> 00:03:53.400
Or you could divide both sides
by 3, and you will get 6.

00:03:53.400 --> 00:03:56.770
So we get question
mark in a box needs

00:03:56.770 --> 00:03:59.070
to be equal to 6 cups of flour.

00:03:59.070 --> 00:04:01.575
Same answer we got through
kind of common sense.

00:04:01.575 --> 00:04:02.950
Now, you might be
wondering, hey,

00:04:02.950 --> 00:04:05.530
this cross-multiplying doesn't
make any intuitive sense.

00:04:05.530 --> 00:04:07.090
Why does that work?

00:04:07.090 --> 00:04:09.510
If I have something set up
like this proportion set up,

00:04:09.510 --> 00:04:12.490
why does it work that if I
take the denominator here

00:04:12.490 --> 00:04:14.490
and multiply it by the
numerator there that that

00:04:14.490 --> 00:04:15.906
needs to be equal
to the numerator

00:04:15.906 --> 00:04:17.560
here times the
denominator there?

00:04:17.560 --> 00:04:19.560
And that comes from
straight up algebra.

00:04:19.560 --> 00:04:22.760
And to do that, I'm just going
to rewrite this part as x just

00:04:22.760 --> 00:04:24.580
to simplify the
writing a little bit.

00:04:24.580 --> 00:04:28.350
So we have 2/3 is equal to--
instead of that question mark,

00:04:28.350 --> 00:04:31.110
I'll write x over 9.

00:04:31.110 --> 00:04:32.830
And in algebra,
all you're saying

00:04:32.830 --> 00:04:34.500
is that this
quantity over here is

00:04:34.500 --> 00:04:36.850
equal to this
quantity over here.

00:04:36.850 --> 00:04:38.840
So if you do anything
to what's on the left,

00:04:38.840 --> 00:04:40.310
if you want it to
still be equal,

00:04:40.310 --> 00:04:42.400
if the thing on the right
still needs to be equal,

00:04:42.400 --> 00:04:44.470
you have to do the
same thing to it.

00:04:44.470 --> 00:04:47.220
Now, what we want to do is we
want to simplify this so all

00:04:47.220 --> 00:04:49.530
we have on the
right-hand side is an x.

00:04:49.530 --> 00:04:52.309
So what can we
multiply this by so

00:04:52.309 --> 00:04:53.600
that we're just left with an x?

00:04:53.600 --> 00:04:55.520
So that we've solved for x?

00:04:55.520 --> 00:04:57.550
Well, if we multiply
this times 9,

00:04:57.550 --> 00:04:59.390
the 9's are going to cancel out.

00:04:59.390 --> 00:05:00.970
So let's multiply
the right by 9.

00:05:00.970 --> 00:05:02.844
But of course, if we
multiply the right by 9,

00:05:02.844 --> 00:05:04.850
we have to still
multiply the left by 9.

00:05:04.850 --> 00:05:06.740
Otherwise they still
wouldn't be equal.

00:05:06.740 --> 00:05:09.470
If they were equal before
being multiplied by 9, for them

00:05:09.470 --> 00:05:12.720
to still be equal, you have to
multiply 9 times both sides.

00:05:12.720 --> 00:05:14.760
On the right-hand side,
the 9's cancel out,

00:05:14.760 --> 00:05:16.330
so you're just left with an x.

00:05:16.330 --> 00:05:21.490
On the left-hand side, you have
9 times 2/3, or 9/1 times 2/3.

00:05:21.490 --> 00:05:24.770
Or this is equal to 18/3.

00:05:24.770 --> 00:05:28.300
And we know that 18/3
is the same thing as 6.

00:05:28.300 --> 00:05:30.440
So these are all
legitimate ways to do it.

00:05:30.440 --> 00:05:32.340
I wanted you to understand
that what I'm doing right here

00:05:32.340 --> 00:05:32.840
is algebra.

00:05:32.840 --> 00:05:35.840
That's actually the reasoning
why cross-multiplication works.

00:05:35.840 --> 00:05:37.950
But for a really simple
problem like this,

00:05:37.950 --> 00:05:39.740
you could really just
use common sense.

00:05:39.740 --> 00:05:43.520
If you're increasing the cups
of oatmeal by a factor of 3,

00:05:43.520 --> 00:05:46.799
then increase the cups of
flour by a factor of 3.


Converting Fahrenheit to Celsius

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.950
A thermometer in a science
lab displays the temperature

00:00:02.950 --> 00:00:04.730
in both Celsius and Fahrenheit.

00:00:04.730 --> 00:00:06.410
If the mercury in
the thermometer

00:00:06.410 --> 00:00:08.880
rises to 56 degrees
Fahrenheit-- they're

00:00:08.880 --> 00:00:11.110
giving us the
Fahrenheit temperature--

00:00:11.110 --> 00:00:13.540
what is the corresponding
Celsius temperature?

00:00:13.540 --> 00:00:15.704
And then they give
us the two formulas,

00:00:15.704 --> 00:00:17.370
that if we know the
Celsius temperature,

00:00:17.370 --> 00:00:19.382
how do we figure out the
Fahrenheit temperature,

00:00:19.382 --> 00:00:21.090
or if we know the
Fahrenheit temperature,

00:00:21.090 --> 00:00:23.120
how do we figure out
the Celsius temperature.

00:00:23.120 --> 00:00:25.130
And these are actually
derived from each other,

00:00:25.130 --> 00:00:27.338
and you'll learn more about
that when you do algebra.

00:00:27.338 --> 00:00:29.060
And we also-- maybe
in another video--

00:00:29.060 --> 00:00:30.843
will explain how
to derive these.

00:00:30.843 --> 00:00:32.259
It's actually kind
of interesting,

00:00:32.259 --> 00:00:33.530
involves a little
bit of algebra.

00:00:33.530 --> 00:00:34.750
But they gave us the formula.

00:00:34.750 --> 00:00:37.810
So they really just want us to
apply it, and maybe make sure

00:00:37.810 --> 00:00:39.877
we understand which
one we should apply.

00:00:39.877 --> 00:00:41.960
Well, they're giving us
the Fahrenheit temperature

00:00:41.960 --> 00:00:45.530
right here, so F is
going to be equal to 56.

00:00:45.530 --> 00:00:47.990
And they're asking us for
the Celsius temperature,

00:00:47.990 --> 00:00:50.570
so we need to figure out what
the Celsius temperature is.

00:00:50.570 --> 00:00:53.570
Well, in this one over here,
if you know the Fahrenheit

00:00:53.570 --> 00:00:56.260
temperature, then you can solve
for the Celsius temperature.

00:00:56.260 --> 00:00:58.360
So let's use this
right over here.

00:00:58.360 --> 00:00:59.880
So our Celsius
temperature is going

00:00:59.880 --> 00:01:05.780
to be 5/9 times the Fahrenheit
temperature-- the Fahrenheit

00:01:05.780 --> 00:01:10.578
temperature is 56 degrees
Fahrenheit-- minus 32.

00:01:13.290 --> 00:01:16.090
Well, 56 minus 32 is 24.

00:01:16.090 --> 00:01:23.490
So this is going to be
equal to 5/9 times 24.

00:01:23.490 --> 00:01:28.920
And this is the same thing
as 5 times 24, over 9.

00:01:28.920 --> 00:01:32.420
And before I even
multiply out 5 times 24,

00:01:32.420 --> 00:01:35.260
we can divide the numerator
and the denominator by 3.

00:01:35.260 --> 00:01:36.270
So let's do that.

00:01:36.270 --> 00:01:38.760
If we divide the numerator
by 3 and the denominator,

00:01:38.760 --> 00:01:40.190
we're not changing the value.

00:01:40.190 --> 00:01:42.850
24 divided by 3 is 8.

00:01:42.850 --> 00:01:45.440
9 divided by 3 is 3.

00:01:45.440 --> 00:01:50.880
So it becomes 5 times 8,
which is 40, over 3 degrees.

00:01:50.880 --> 00:01:52.921
And if we want to write
this as a number that

00:01:52.921 --> 00:01:55.170
makes a little bit more sense
in terms of temperature,

00:01:55.170 --> 00:02:01.460
let's divide 3 into 40 to get
the number of degrees we have.

00:02:01.460 --> 00:02:03.060
3 goes into 4 one time.

00:02:03.060 --> 00:02:04.610
1 times 3 is 3.

00:02:04.610 --> 00:02:05.550
Subtract.

00:02:05.550 --> 00:02:08.120
4 minus 3 is 1.

00:02:08.120 --> 00:02:09.280
Bring down the 0.

00:02:09.280 --> 00:02:11.960
3 goes into 10 three times.

00:02:11.960 --> 00:02:13.810
3 times 3 is 9.

00:02:13.810 --> 00:02:14.880
Subtract.

00:02:14.880 --> 00:02:16.020
Get a 1.

00:02:16.020 --> 00:02:17.860
And then you could
bring down another 0.

00:02:17.860 --> 00:02:19.810
We now have a decimal
point over here.

00:02:19.810 --> 00:02:21.050
You're going to get a 0 here.

00:02:21.050 --> 00:02:23.150
3 goes into, once
again, 10 three times.

00:02:23.150 --> 00:02:24.980
And this 3 is going
to repeat forever.

00:02:24.980 --> 00:02:29.960
So you could view this--
this is equal to 13.333--

00:02:29.960 --> 00:02:31.290
it'll just keep repeating.

00:02:31.290 --> 00:02:34.700
This little line on top means
repeating-- degrees Celsius.

00:02:34.700 --> 00:02:40.096
Or you could say that, look,
3 goes into 40 13 times

00:02:40.096 --> 00:02:42.910
with a remainder of 1.

00:02:42.910 --> 00:02:47.040
So you could say that this is
also equal to 13, remainder 1.

00:02:47.040 --> 00:02:50.600
So 13 and 1/3 degrees Celsius.

00:02:50.600 --> 00:02:51.740
Either way, it works.

00:02:51.740 --> 00:02:53.470
But that's our
Celsius temperature

00:02:53.470 --> 00:02:57.520
when our Fahrenheit
temperature is 56 degrees.


Solving a proportion with an unknown variable (example)

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.140
We're asked to solve
the proportion.

00:00:02.140 --> 00:00:07.290
We have 8 36ths is
equal to 10 over what.

00:00:07.290 --> 00:00:11.584
Or the ratio of 8/36 is equal
to the ratio of 10 to what.

00:00:11.584 --> 00:00:13.750
And there's a bunch of
different ways to solve this.

00:00:13.750 --> 00:00:16.540
And I'll explore
really all of them,

00:00:16.540 --> 00:00:17.810
or a good selection of them.

00:00:17.810 --> 00:00:19.900
So one way to think about
it is, these two need

00:00:19.900 --> 00:00:23.460
to be equivalent ratios, or
really, equivalent fractions.

00:00:23.460 --> 00:00:25.240
So whatever happened
to the numerator also

00:00:25.240 --> 00:00:27.420
has to happen to
the denominator.

00:00:27.420 --> 00:00:30.400
So what do we have to
multiply 8 by to get 10?

00:00:30.400 --> 00:00:34.560
Well you could
multiply 8 times 10/8.

00:00:34.560 --> 00:00:36.660
It will definitely give you 10.

00:00:36.660 --> 00:00:39.340
So we're multiplying
by 10/8 over here.

00:00:39.340 --> 00:00:44.340
Or another way to write 10/8,
10/8 is the same thing as 5/4.

00:00:44.340 --> 00:00:49.417
So we're multiplying by 5/4
to get to 10, from 8 to 10.

00:00:49.417 --> 00:00:51.000
Well, if we did that
to the numerator,

00:00:51.000 --> 00:00:52.270
in order to have an
equivalent fraction,

00:00:52.270 --> 00:00:54.440
you have to do the same
thing to the denominator.

00:00:54.440 --> 00:00:56.100
You have to multiply it.

00:00:56.100 --> 00:00:59.030
You have to multiply
it times 5/4.

00:00:59.030 --> 00:01:02.110
And so we could say this n, this
thing that we just solved for,

00:01:02.110 --> 00:01:08.202
this n is going to be equal
to 36 times 5 divided by 4.

00:01:08.202 --> 00:01:09.660
Or you could say
that this is going

00:01:09.660 --> 00:01:14.080
to be equal to 36
times 5 divided by 4.

00:01:14.080 --> 00:01:17.340
And now, 36 divided by
4, we know what that is.

00:01:17.340 --> 00:01:19.640
We could divide both the
numerator and the denominator

00:01:19.640 --> 00:01:20.410
by 4.

00:01:20.410 --> 00:01:22.660
You divide the numerator
by 4, you get 9.

00:01:22.660 --> 00:01:24.950
Divide the denominator
by 4 you get 1.

00:01:24.950 --> 00:01:26.991
You get 45.

00:01:26.991 --> 00:01:28.490
So that's one way
to think about it.

00:01:28.490 --> 00:01:32.575
8/36 is equal to 10/45.

00:01:35.190 --> 00:01:37.910
Another way to
think about it is,

00:01:37.910 --> 00:01:41.750
what do we have to multiply
8 by to get its denominator.

00:01:41.750 --> 00:01:45.300
How much larger is the
denominator 36 than 8?

00:01:45.300 --> 00:01:47.540
Well let's just divide 36/8.

00:01:47.540 --> 00:01:53.170
So 36/8 is the same
thing as-- so we

00:01:53.170 --> 00:01:55.580
can simplify, dividing the
numerator and the denominator

00:01:55.580 --> 00:01:57.320
by 4.

00:01:57.320 --> 00:01:59.170
That's the greatest
common divisor.

00:01:59.170 --> 00:02:01.480
That's the same thing as 9/2.

00:02:01.480 --> 00:02:04.080
So if you multiply
the numerator by 9/2,

00:02:04.080 --> 00:02:05.640
you get the denominator.

00:02:05.640 --> 00:02:09.527
So we're multiplying by 9/2 to
get the denominator over here.

00:02:09.527 --> 00:02:11.610
Well, then we have to do
the same thing over here.

00:02:11.610 --> 00:02:17.510
If 36 is 9/2 times
8, let me write this.

00:02:17.510 --> 00:02:24.480
8 times 9/2 is equal to 36.

00:02:24.480 --> 00:02:24.980
Right?

00:02:24.980 --> 00:02:26.980
That's how we go from the
numerator to the denominator.

00:02:26.980 --> 00:02:28.790
Then to figure out what
the denominator here

00:02:28.790 --> 00:02:30.165
is, if we want
the same fraction,

00:02:30.165 --> 00:02:33.320
we have to multiply
by 9/2 again.

00:02:33.320 --> 00:02:39.880
So then we'll get 10 times
9/2 is going to be equal to n,

00:02:39.880 --> 00:02:42.290
is going to be equal
to this denominator.

00:02:42.290 --> 00:02:46.474
And so this is the same
thing as saying 10 times 9/2.

00:02:46.474 --> 00:02:48.390
Divide the numerator and
the denominator by 2,

00:02:48.390 --> 00:02:52.170
you get 5/1, which is 45.

00:02:52.170 --> 00:02:54.480
So 45 is equal to n.

00:02:54.480 --> 00:02:57.880
Once again, we got the same
way, completely legitimate way,

00:02:57.880 --> 00:02:59.180
to solve it.

00:02:59.180 --> 00:03:02.990
Now sometimes when you
see proportion like this,

00:03:02.990 --> 00:03:05.045
sometimes people say, oh
you can cross-multiply.

00:03:05.045 --> 00:03:06.170
And you can cross-multiply.

00:03:06.170 --> 00:03:07.690
And I'll teach you
how to do that.

00:03:07.690 --> 00:03:09.570
And that's sometimes
a quick way to do it.

00:03:09.570 --> 00:03:12.389
But I don't like teaching
it the first time you

00:03:12.389 --> 00:03:14.680
look at proportions, because
it's really just something

00:03:14.680 --> 00:03:15.260
mechanical.

00:03:15.260 --> 00:03:16.610
You really don't understand
what you're doing.

00:03:16.610 --> 00:03:18.590
And it really comes out of
a little bit of algebra.

00:03:18.590 --> 00:03:20.190
And I'll show you
the algebra as well.

00:03:20.190 --> 00:03:21.330
But if you don't
understand it, or if it

00:03:21.330 --> 00:03:23.329
doesn't make as much sense
to you at this point,

00:03:23.329 --> 00:03:25.060
don't worry too much about it.

00:03:25.060 --> 00:03:29.860
So we have 8/36
is equal to 10/n.

00:03:29.860 --> 00:03:31.780
When you cross-multiply,
you're saying

00:03:31.780 --> 00:03:34.990
that the numerator here, times
the denominator over here,

00:03:34.990 --> 00:03:38.600
is going to be equal
to, so 8 times n,

00:03:38.600 --> 00:03:44.060
is going to be equal to
the denominator over here,

00:03:44.060 --> 00:03:47.170
let me just different color,
the denominator over here,

00:03:47.170 --> 00:03:48.420
times the numerator over here.

00:03:48.420 --> 00:03:50.190
This is what it means
to cross-multiply.

00:03:50.190 --> 00:03:52.760
So this is going to be
equal to 36 times 10.

00:03:57.350 --> 00:03:59.690
Let me do this in a
neutral color now.

00:03:59.690 --> 00:04:03.840
You could say that
8n is equal to 360.

00:04:03.840 --> 00:04:07.390
And so you're saying 8
times what is equal to 360.

00:04:07.390 --> 00:04:09.090
Or to figure out
what that times what

00:04:09.090 --> 00:04:11.339
is, you divide 360 divided by 8.

00:04:11.339 --> 00:04:13.880
So we could divide, and this is
a little bit of algebra here,

00:04:13.880 --> 00:04:16.529
we're dividing both sides
of the equation by 8.

00:04:16.529 --> 00:04:20.574
And we're getting n is
equal to 360 divided by 8.

00:04:20.574 --> 00:04:21.990
You could do that
without thinking

00:04:21.990 --> 00:04:23.730
in strict algebraic terms.

00:04:23.730 --> 00:04:26.090
You could say 8
times what is 360.

00:04:26.090 --> 00:04:28.980
Well 8 times 360/8.

00:04:28.980 --> 00:04:33.620
If I write 8 times question
mark is equal to 360,

00:04:33.620 --> 00:04:38.230
well, question mark could
definitely be 360/8.

00:04:38.230 --> 00:04:40.890
If I multiply these out, this
guy and that guy cancel out,

00:04:40.890 --> 00:04:43.060
and it's definitely 360.

00:04:43.060 --> 00:04:44.884
And that's why it's 360/8.

00:04:44.884 --> 00:04:46.300
But now we want
to actually divide

00:04:46.300 --> 00:04:49.690
this to actually get our right
answer, or a simplified answer.

00:04:49.690 --> 00:04:55.390
8 goes into 360, 8 goes into
36 4 times, 4 times 8 is 32.

00:04:55.390 --> 00:04:56.920
You have a remainder of 4.

00:04:56.920 --> 00:04:58.210
Bring down the 0.

00:04:58.210 --> 00:05:00.300
8 goes into 40 5 times.

00:05:00.300 --> 00:05:02.430
5 times 8 is 40.

00:05:02.430 --> 00:05:05.444
And then you have no remainder.

00:05:05.444 --> 00:05:06.110
And you're done.

00:05:06.110 --> 00:05:10.404
Once again, we got
n is equal to 45.

00:05:10.404 --> 00:05:12.070
Now the last thing
I'm going to show you

00:05:12.070 --> 00:05:13.740
involves a little
bit of algebra.

00:05:13.740 --> 00:05:15.990
If any of the ways before
this worked, that's fine.

00:05:15.990 --> 00:05:17.590
And where this is
sitting in the playlist,

00:05:17.590 --> 00:05:19.280
you're not expected
to know the algebra.

00:05:19.280 --> 00:05:21.070
But I want to show
you the algebra just

00:05:21.070 --> 00:05:23.528
because I wanted to show you
that this cross-multiplication

00:05:23.528 --> 00:05:25.380
isn't some magic,
that using algebra,

00:05:25.380 --> 00:05:27.450
we will get this
exact same thing.

00:05:27.450 --> 00:05:28.930
But you could stop
watching this,

00:05:28.930 --> 00:05:31.280
if you'll find this
part confusing.

00:05:31.280 --> 00:05:36.340
So let's rewrite our proportion,
8/36 is equal to 10/n.

00:05:38.860 --> 00:05:43.390
And we want to solve for n.

00:05:43.390 --> 00:05:46.912
Well the easiest way to solve
for n is maybe multiply both--

00:05:46.912 --> 00:05:49.370
this thing on the left is equal
to this thing on the right.

00:05:49.370 --> 00:05:51.910
So we can multiply them
both by the same thing.

00:05:51.910 --> 00:05:53.500
And the equality
will still hold.

00:05:53.500 --> 00:05:57.950
So we could multiply
both of them by n.

00:05:57.950 --> 00:06:00.080
On the right-hand side,
the n's cancel out.

00:06:00.080 --> 00:06:07.220
On the left-hand side, we have
8/36 times n is equal to 10.

00:06:07.220 --> 00:06:11.150
Now if we want to solve for n,
we could literally multiply.

00:06:11.150 --> 00:06:12.920
If we want just an
n here, we would

00:06:12.920 --> 00:06:15.775
want to multiply
this side times 36--

00:06:15.775 --> 00:06:17.350
I'll do that in a
different color--

00:06:17.350 --> 00:06:20.960
we'd want to multiply this
side times 36 times 8,

00:06:20.960 --> 00:06:23.510
because if you multiply
these guys out, you get 1.

00:06:23.510 --> 00:06:24.820
And you just have an n.

00:06:24.820 --> 00:06:26.778
But since we're doing it
to the left-hand side,

00:06:26.778 --> 00:06:32.170
we also have to do it to the
right-hand side, so times 36/8.

00:06:32.170 --> 00:06:35.330
These guys cancel out
and we're left with n

00:06:35.330 --> 00:06:40.480
is equal to 10
times 36 is 360/8.

00:06:40.480 --> 00:06:43.410
And notice, we're getting
the exact same value

00:06:43.410 --> 00:06:45.220
that we got with
cross-multiplying.

00:06:45.220 --> 00:06:47.160
And with
cross-multiplying, you're

00:06:47.160 --> 00:06:48.500
actually doing two steps.

00:06:48.500 --> 00:06:50.250
Actually, you're doing
an extra step here.

00:06:50.250 --> 00:06:53.540
You're multiplying both sides
by n, so that you had your 8n.

00:06:53.540 --> 00:06:56.110
And then you're multiplying
both sides by 36,

00:06:56.110 --> 00:06:59.510
so that you get your
36 on both sides.

00:06:59.510 --> 00:07:00.740
And you get this value here.

00:07:00.740 --> 00:07:02.281
But at the end, when
you simplify it,

00:07:02.281 --> 00:07:04.330
you'll get the
exact same answer.

00:07:04.330 --> 00:07:06.930
So those are all different
ways to solve this proportion.

00:07:06.930 --> 00:07:09.340
Probably the most obvious
way, or the easiest way

00:07:09.340 --> 00:07:12.345
to do it in your head,
was either just looking

00:07:12.345 --> 00:07:14.220
at what you have to
multiply the numerator by

00:07:14.220 --> 00:07:16.830
and then doing the same
thing to the denominator,

00:07:16.830 --> 00:07:19.332
or maybe by
cross-multiplication.


Apsidal precession (perihelion precession) and Milankovitch cycles

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.260
We've learned that
axial precession,

00:00:03.260 --> 00:00:06.180
it's not a change in the
tilt or the obliquity

00:00:06.180 --> 00:00:08.820
of our rotational axis, it's
a change in the direction.

00:00:08.820 --> 00:00:12.420
And over a long period
of time, 26,000 years,

00:00:12.420 --> 00:00:14.780
it kind of traces out a circle.

00:00:14.780 --> 00:00:16.900
And the main affect
of that is that if we

00:00:16.900 --> 00:00:20.230
wait long enough that our
rotational axis, or you could

00:00:20.230 --> 00:00:22.020
say almost the
North Pole, will be

00:00:22.020 --> 00:00:24.290
pointed in a
different direction.

00:00:24.290 --> 00:00:26.330
And so if our rotational
axis is pointed

00:00:26.330 --> 00:00:30.020
in a different direction
after a long enough time,

00:00:30.020 --> 00:00:32.299
then the absolute
point in our orbit,

00:00:32.299 --> 00:00:34.480
if we use the sun as
our frame of reference,

00:00:34.480 --> 00:00:37.400
the point in our orbit when
we are most pointed away

00:00:37.400 --> 00:00:39.880
from the sun, or when the
Northern Hemisphere is most

00:00:39.880 --> 00:00:44.310
pointed away from the sun,
will be earlier in the orbit.

00:00:44.310 --> 00:00:47.290
Now I emphasize that
that won't necessarily

00:00:47.290 --> 00:00:49.830
mean earlier in our calendar,
because our calendar,

00:00:49.830 --> 00:00:52.880
by definition, takes
into consideration,

00:00:52.880 --> 00:00:55.470
I guess, or it's
more based on when

00:00:55.470 --> 00:01:00.220
we are furthest tilted away
from the sun or furthest tilted

00:01:00.220 --> 00:01:01.410
towards the sun.

00:01:01.410 --> 00:01:06.490
So even though if we wait 1800
years, like the example I gave,

00:01:06.490 --> 00:01:09.980
we will be most tilted
away from the sun.

00:01:09.980 --> 00:01:14.280
The Northern Hemisphere
will have its winter equinox

00:01:14.280 --> 00:01:16.060
at an earlier
point in the orbit.

00:01:16.060 --> 00:01:20.630
According to our calendar, it
will still be December 22nd.

00:01:20.630 --> 00:01:22.830
If our calendar
instead was based--

00:01:22.830 --> 00:01:25.280
and it's not based on
this-- but if our calendar

00:01:25.280 --> 00:01:31.570
was based on the
exact point in orbit,

00:01:31.570 --> 00:01:37.140
then our year would be about
20, 25 minutes longer every year

00:01:37.140 --> 00:01:41.800
and then the date for
the start of winter

00:01:41.800 --> 00:01:43.729
actually would go back.

00:01:43.729 --> 00:01:45.770
1800 years later, the date
of the start of winter

00:01:45.770 --> 00:01:47.299
would be November 22nd.

00:01:47.299 --> 00:01:49.090
But that's not how we
measure our calendar.

00:01:49.090 --> 00:01:51.790
Our calendar is actually
measured from equinox

00:01:51.790 --> 00:01:52.940
to equinox.

00:01:52.940 --> 00:01:57.550
From December 22nd or 21st,
there's slight fluctuations

00:01:57.550 --> 00:02:00.200
depending on the calendar,
but that will always

00:02:00.200 --> 00:02:03.280
be the date that we are most
pointed away from the sun.

00:02:03.280 --> 00:02:09.539
That will not be necessarily
the date at this exact position

00:02:09.539 --> 00:02:11.300
relative to the sun itself.

00:02:11.300 --> 00:02:15.900
And that's why the actual
perihelion does change.

00:02:15.900 --> 00:02:18.220
Because if this is
always December 22nd,

00:02:18.220 --> 00:02:21.280
and if we, at first, assume
that the perihelion is always

00:02:21.280 --> 00:02:23.929
at the same fixed point in
space relative to the sun,

00:02:23.929 --> 00:02:25.470
although that's not
exactly the case,

00:02:25.470 --> 00:02:27.110
but if we make that
assumption, then it

00:02:27.110 --> 00:02:30.890
will be further and further
after that December 22nd,

00:02:30.890 --> 00:02:32.890
further and further
after that time,

00:02:32.890 --> 00:02:35.670
that we are most pointed
away from the sun.

00:02:35.670 --> 00:02:40.480
And that's why you have
this kind of pushing back

00:02:40.480 --> 00:02:41.720
of the perihelion.

00:02:41.720 --> 00:02:43.310
Now, what I want to
add to this video

00:02:43.310 --> 00:02:45.840
is that the perihelion
itself is also changing.

00:02:45.840 --> 00:02:50.900
So if I draw the sun
again, and right now

00:02:50.900 --> 00:02:54.460
our orbit looks
something like this,

00:02:54.460 --> 00:02:59.890
and I'm going to exaggerate
the eccentricity of it

00:02:59.890 --> 00:03:01.920
so that the perihelion
and the aphelion

00:03:01.920 --> 00:03:03.090
are a little bit clearer.

00:03:03.090 --> 00:03:04.620
So right now this
is the perihelion.

00:03:04.620 --> 00:03:06.161
This is the aphelion
based on the way

00:03:06.161 --> 00:03:08.670
I drew it right over
there in different colors.

00:03:08.670 --> 00:03:11.090
I don't want to show that's
necessarily where Earth is.

00:03:11.090 --> 00:03:13.690
Perihelion and
aphelion, there is also

00:03:13.690 --> 00:03:17.050
a rotation of this,
of the perihelion.

00:03:17.050 --> 00:03:21.650
And sometimes this is called the
precession of the perihelion,

00:03:21.650 --> 00:03:26.720
or perihelion precession,
or apsidal precession.

00:03:26.720 --> 00:03:28.380
These are all very hard to say.

00:03:28.380 --> 00:03:31.280
And so if we wait several
thousands of years

00:03:31.280 --> 00:03:35.520
our orbit might look a
little bit like this.

00:03:35.520 --> 00:03:39.210
The actual perihelion
will have rotated.

00:03:39.210 --> 00:03:41.050
So our orbit will
look like this.

00:03:41.050 --> 00:03:44.600
The actual ellipse would
have rotated a little bit.

00:03:44.600 --> 00:03:48.869
You wait a little bit longer,
it might look like this.

00:03:48.869 --> 00:03:50.410
And obviously, I'm
once again talking

00:03:50.410 --> 00:03:52.650
about over thousands
and thousands of years.

00:03:52.650 --> 00:03:55.210
From a year-to-year
basis, you really

00:03:55.210 --> 00:03:56.870
wouldn't notice the difference.

00:03:56.870 --> 00:03:58.900
But what that does
is, is we talked

00:03:58.900 --> 00:04:02.040
about the axial precession,
that this change in direction

00:04:02.040 --> 00:04:05.420
of our rotational axis,
it takes 26,000 years

00:04:05.420 --> 00:04:06.630
to complete one period.

00:04:10.850 --> 00:04:15.580
So 26,000 years from
today, our polar axis,

00:04:15.580 --> 00:04:19.180
if we don't think about
our rotational axis,

00:04:19.180 --> 00:04:22.630
if we aren't too concerned
about the actual change in tilt,

00:04:22.630 --> 00:04:25.220
which there will be some
small change in tilt,

00:04:25.220 --> 00:04:27.480
but 26,000 years
from now, our pole

00:04:27.480 --> 00:04:30.390
will roughly point in
the same direction again.

00:04:30.390 --> 00:04:32.780
We would have completed
one whole period

00:04:32.780 --> 00:04:34.780
of axial precession.

00:04:34.780 --> 00:04:39.090
However, it does not take
26,000 years for whatever

00:04:39.090 --> 00:04:40.930
our date of perihelion is today.

00:04:40.930 --> 00:04:42.049
So it's in January.

00:04:42.049 --> 00:04:43.590
I actually don't
know the exact date.

00:04:43.590 --> 00:04:44.590
You can look that up.

00:04:44.590 --> 00:04:46.275
But whatever that
data is in January,

00:04:46.275 --> 00:04:50.190
it will not take 26,000 years
for it to be that date again.

00:04:50.190 --> 00:04:55.909
And it would have taken 26,000
years if the perihelion itself

00:04:55.909 --> 00:04:58.200
were not changing, if it
always stayed fixed over here,

00:04:58.200 --> 00:05:02.690
if we did not have this
apsidal precession.

00:05:02.690 --> 00:05:04.740
But since it is
also changing, you

00:05:04.740 --> 00:05:06.750
could kind of say it
is over 1,000 years

00:05:06.750 --> 00:05:13.780
moving in that direction while
our January date is moving

00:05:13.780 --> 00:05:16.270
in that direction, they
will actually meet sooner,

00:05:16.270 --> 00:05:18.820
so that the precession will
be back on whatever date

00:05:18.820 --> 00:05:21.924
it is on January, less
than 26,000 years from now.

00:05:21.924 --> 00:05:24.590
And actually the exact time, and
I haven't done the calculation,

00:05:24.590 --> 00:05:26.260
but this is what
I've read, is that it

00:05:26.260 --> 00:05:30.910
will be 21,000 years from now.

00:05:30.910 --> 00:05:33.200
And then on top of that,
if that's enough for you

00:05:33.200 --> 00:05:36.620
that not only is the direction
of Earth's rotational axis

00:05:36.620 --> 00:05:38.480
changing and the
tilt is changing

00:05:38.480 --> 00:05:40.440
and that the perihelion
and the aphelion

00:05:40.440 --> 00:05:43.050
are also rotating around,
it's also the case

00:05:43.050 --> 00:05:46.510
that the eccentricity of the
orbit itself is changing.

00:05:46.510 --> 00:05:48.530
So over long periods
of time, Earth's orbit

00:05:48.530 --> 00:05:50.110
becomes more or less eccentric.

00:05:50.110 --> 00:05:53.140
And we've learned that
almost circular has--

00:05:53.140 --> 00:05:55.140
well if you're circular
you have no eccentricity

00:05:55.140 --> 00:05:57.140
and then you can become
more and more eccentric,

00:05:57.140 --> 00:05:59.240
which means you're
more and more of kind

00:05:59.240 --> 00:06:01.140
of this flattened out ellipse.

00:06:01.140 --> 00:06:04.700
And these cycles occur,
these eccentricities cycles

00:06:04.700 --> 00:06:09.410
occur over approximately
100,000 years.

00:06:09.410 --> 00:06:12.262
And so to revisit the
Milankovitch cycles-- and once

00:06:12.262 --> 00:06:13.220
again this is a theory.

00:06:13.220 --> 00:06:15.950
We're not sure whether this
is necessarily causing our ice

00:06:15.950 --> 00:06:17.700
ages or whether
this is necessarily

00:06:17.700 --> 00:06:20.930
a major influence over
long-term climate change,

00:06:20.930 --> 00:06:25.040
but the theory of
Milankovitch cycles

00:06:25.040 --> 00:06:26.560
is that over long
periods of time,

00:06:26.560 --> 00:06:29.230
if the eccentricity
changes enough

00:06:29.230 --> 00:06:32.540
and if it coincides with when
the perihelion and the seasons

00:06:32.540 --> 00:06:36.520
also coincide, maybe that's
enough to start an ice age.

00:06:36.520 --> 00:06:39.390
Or maybe that's enough to
take us out of an ice age.

00:06:39.390 --> 00:06:41.640
And actually, if you want
to throw something even more

00:06:41.640 --> 00:06:43.630
on that, the actual
plane of our orbit

00:06:43.630 --> 00:06:46.910
also changes over time mainly
because of interactions

00:06:46.910 --> 00:06:48.152
with the outer planets.

00:06:48.152 --> 00:06:49.360
Anyway, I'll leave you there.

00:06:49.360 --> 00:06:51.841
As you could imagine, this
is a very complex topic,

00:06:51.841 --> 00:06:53.590
but hopefully you now
have an appreciation

00:06:53.590 --> 00:06:56.509
of all the different ways our
orbit can change and maybe

00:06:56.509 --> 00:06:58.800
start to think about how that
might affect our weather.

00:06:58.800 --> 00:07:00.880
Although, we don't
necessarily know how it does

00:07:00.880 --> 00:07:03.600
or whether it even
really does affect

00:07:03.600 --> 00:07:06.670
going into or out of ice ages.