Absolute value as distance between numbers

WEBVTT
Kind: captions
Language: en

00:00:00.289 --> 00:00:02.112
- [Voiceover] Let's say
that I have two numbers

00:00:02.112 --> 00:00:03.013
on a number line.

00:00:03.013 --> 00:00:04.812
So let me draw a little quick number line

00:00:04.812 --> 00:00:06.037
right over here.

00:00:06.037 --> 00:00:08.104
The two numbers on my number line

00:00:08.104 --> 00:00:12.596
that I care about, the
number a and the number

00:00:12.596 --> 00:00:13.996
represented by b here.

00:00:13.996 --> 00:00:15.865
The way I've drawn it, b is to the right

00:00:15.865 --> 00:00:16.864
of a on our number line,

00:00:16.864 --> 00:00:18.669
and by our convention, b is going

00:00:18.669 --> 00:00:20.352
to be greater than a.

00:00:20.352 --> 00:00:21.757
So if I were to figure out the distance

00:00:21.757 --> 00:00:24.555
between a and b, what
is this distance going

00:00:24.555 --> 00:00:27.875
from a, I want to draw
a straight line here,

00:00:27.875 --> 00:00:31.857
this distance going from a to b,

00:00:31.857 --> 00:00:34.040
so this distance right over there,

00:00:34.040 --> 00:00:35.730
how would I figure it out?

00:00:35.730 --> 00:00:36.949
Well I could just take the larger

00:00:36.949 --> 00:00:38.934
of these two numbers,
which is going to be b,

00:00:38.934 --> 00:00:41.117
and then subtract out the smaller.

00:00:41.117 --> 00:00:42.730
So I subtract out a, and I'll be left

00:00:42.730 --> 00:00:44.029
with this distance.

00:00:44.029 --> 00:00:45.848
This will give me a positive value.

00:00:45.848 --> 00:00:48.019
When I want a distance, I just think

00:00:48.019 --> 00:00:49.539
in terms of a positive value.

00:00:49.539 --> 00:00:52.059
How far apart are these two things?

00:00:52.059 --> 00:00:54.647
But I was only able to
know to do b minus a

00:00:54.647 --> 00:00:56.946
because I knew that b was greater than a.

00:00:56.946 --> 00:00:58.885
This was going to give
me a positive value.

00:00:58.885 --> 00:01:01.218
What if I knew that a was greater than b?

00:01:01.218 --> 00:01:02.896
Well then I would do it
the other way around.

00:01:02.896 --> 00:01:04.280
So let me draw that again.

00:01:04.280 --> 00:01:07.135
Let me draw another number
line right over here.

00:01:07.135 --> 00:01:09.469
In this world, in this world,

00:01:09.469 --> 00:01:12.047
I'm going to make a greater than b.

00:01:14.065 --> 00:01:18.306
This is b, that is a,

00:01:18.306 --> 00:01:19.745
and if I wanted to calculate the distance

00:01:19.745 --> 00:01:22.880
between b and a here,

00:01:22.880 --> 00:01:26.165
well now I would take
the larger of the two,

00:01:26.165 --> 00:01:28.557
a, remember I want the
positive distance here,

00:01:28.557 --> 00:01:30.216
and then I would subtract out the smaller.

00:01:30.216 --> 00:01:32.324
I would do a minus b.

00:01:32.324 --> 00:01:34.484
Well so here I did b minus a,

00:01:34.484 --> 00:01:35.947
here I did a minus b,

00:01:35.947 --> 00:01:37.943
but what if I didn't know
which one was greater?

00:01:37.943 --> 00:01:39.883
If I didn't know whether
b or a was greater,

00:01:39.883 --> 00:01:41.918
what could I do?

00:01:41.918 --> 00:01:44.797
Well what you could do is just take

00:01:44.797 --> 00:01:48.000
either a minus b or b minus a

00:01:48.000 --> 00:01:50.438
and take the absolute value.

00:01:50.438 --> 00:01:51.680
If you do that, it doesn't matter

00:01:51.680 --> 00:01:54.803
if you take b minus a or a minus b.

00:01:54.803 --> 00:01:57.706
It turns out that regardless of whether a

00:01:57.706 --> 00:01:59.539
is greater than b, or b is greater than a,

00:01:59.539 --> 00:02:02.336
or they're equivalent,
that the absolute value

00:02:02.336 --> 00:02:06.523
of a minus b is equivalent
to the absolute value

00:02:06.523 --> 00:02:09.249
of b minus a, and this is equivalent,

00:02:09.249 --> 00:02:11.243
either of these
expressions is the distance

00:02:11.243 --> 00:02:13.611
between these numbers.

00:02:13.611 --> 00:02:15.191
I encourage you to play
around with the negatives

00:02:15.191 --> 00:02:16.641
to see if you can factor
out some negatives

00:02:16.641 --> 00:02:18.075
and think about the absolute value.

00:02:18.075 --> 00:02:19.921
It will actually make a lot
of sense why this is true.

00:02:19.921 --> 00:02:21.402
In another video, I might
do a little bit more

00:02:21.402 --> 00:02:24.049
of a rigorous justification for it.

00:02:24.049 --> 00:02:24.943
But I think the important thing

00:02:24.943 --> 00:02:26.765
for this video is to see that this

00:02:26.765 --> 00:02:30.085
is actually true.

00:02:30.085 --> 00:02:32.071
So let's say we're in a world,

00:02:32.071 --> 00:02:34.231
let's get a number line out,

00:02:34.231 --> 00:02:36.645
and let's look at some examples.

00:02:36.645 --> 00:02:38.850
So let's say that we want to figure out

00:02:38.850 --> 00:02:41.893
the distance between,

00:02:42.240 --> 00:02:45.830
between, let's say negative two,

00:02:45.830 --> 00:02:48.245
the distance between negative two

00:02:48.245 --> 00:02:51.102
and positive three.

00:02:51.971 --> 00:02:53.820
So we can look at the number line

00:02:53.820 --> 00:02:55.050
and figure out what that distance is.

00:02:55.050 --> 00:02:57.762
To go from negative two to positive three,

00:02:57.762 --> 00:02:58.784
or the distance between them,

00:02:58.784 --> 00:03:02.040
we see is one, two, three, four, five.

00:03:02.040 --> 00:03:04.072
Actually, let me draw
a straighter line here.

00:03:04.072 --> 00:03:06.475
This distance right over here,

00:03:06.475 --> 00:03:08.192
this distance right over here

00:03:08.192 --> 00:03:11.037
is equal to five.

00:03:11.037 --> 00:03:11.659
You see it right over here.

00:03:11.659 --> 00:03:15.212
One, two, three, four, five.

00:03:15.212 --> 00:03:17.173
Or you'd have to go five backwards

00:03:17.173 --> 00:03:19.391
to go from three to negative two.

00:03:19.391 --> 00:03:20.691
But let's see that what I just wrote

00:03:20.691 --> 00:03:23.049
actually applies right over here.

00:03:23.049 --> 00:03:26.114
So if we took negative two to be our a

00:03:26.114 --> 00:03:27.796
and three to be our b,

00:03:27.796 --> 00:03:29.537
then we could write this as

00:03:29.537 --> 00:03:33.684
the absolute value of negative two

00:03:33.684 --> 00:03:37.648
minus three,

00:03:40.381 --> 00:03:42.540
what is this going to be equal to?

00:03:42.540 --> 00:03:43.666
Well this is going to be equal

00:03:43.666 --> 00:03:47.594
to negative two minus
three is negative five,

00:03:47.594 --> 00:03:50.451
so it's the absolute
value of negative five.

00:03:50.451 --> 00:03:53.097
So this indeed equals five.

00:03:53.097 --> 00:03:55.118
So notice I subtracted the larger number

00:03:55.118 --> 00:03:56.928
from the smaller number.

00:03:56.928 --> 00:03:58.077
I got a negative value, but then I took

00:03:58.077 --> 00:03:59.552
the absolute value of it.

00:03:59.552 --> 00:04:01.258
That gave me the actual distance

00:04:01.258 --> 00:04:03.313
between these two numbers.

00:04:03.313 --> 00:04:05.333
Now what if I did it the other way around?

00:04:05.333 --> 00:04:09.652
What if I took three minus negative two?

00:04:09.652 --> 00:04:12.310
So it's going to be the absolute value

00:04:12.310 --> 00:04:15.897
of three, let me do it in the blue color,

00:04:15.897 --> 00:04:20.388
the absolute value of three minus,

00:04:21.969 --> 00:04:24.722
and in parentheses I'll
write the negative two.

00:04:25.026 --> 00:04:26.187
Negative two.

00:04:26.187 --> 00:04:28.103
Now if you subtract a smaller number

00:04:28.103 --> 00:04:29.379
from a larger number, you should get

00:04:29.379 --> 00:04:30.425
a positive value.

00:04:30.425 --> 00:04:33.478
So the absolute value sign
here is just kind of extra.

00:04:33.478 --> 00:04:34.019
You don't really need it,

00:04:34.019 --> 00:04:35.855
unless to verify that that's true.

00:04:35.855 --> 00:04:38.756
This is going to be
three minus negative two.

00:04:38.756 --> 00:04:40.277
That's the same thing as three plus

00:04:40.277 --> 00:04:42.799
positive two, or five.

00:04:42.799 --> 00:04:44.008
So this is just going
to be the absolute value

00:04:44.008 --> 00:04:47.478
of five, which of
course, is equal to five.

00:04:47.478 --> 00:04:48.708
So hopefully this makes you feel good

00:04:48.708 --> 00:04:49.719
that if you want the distance

00:04:49.719 --> 00:04:51.728
between two numbers, you subtract one

00:04:51.728 --> 00:04:53.364
from the other, and it doesn't matter

00:04:53.364 --> 00:04:54.029
which order you do it.

00:04:54.029 --> 00:04:55.746
You could subtract
three from negative two,

00:04:55.746 --> 00:04:58.306
or negative two from three,

00:04:58.306 --> 00:05:00.105
be careful with the negative symbols here,

00:05:00.105 --> 00:05:01.684
and then take the absolute value,

00:05:01.684 --> 00:05:02.787
and then that is going to give you

00:05:02.787 --> 00:05:05.050
the distance between these two numbers.

00:05:05.050 --> 00:05:06.862
This is super important because later

00:05:06.862 --> 00:05:08.128
in your mathematical careers,

00:05:08.128 --> 00:05:09.183
you're going to see a math professor

00:05:09.183 --> 00:05:11.402
just say, oh, you know,
I care about the distance

00:05:11.402 --> 00:05:13.480
between two variables, you know,

00:05:13.480 --> 00:05:16.602
a and b, so the distance is a minus b,

00:05:16.602 --> 00:05:18.483
and then later they
might write it like this.

00:05:18.483 --> 00:05:19.505
And then to realize that these

00:05:19.505 --> 00:05:20.839
are actually the same thing,

00:05:20.839 --> 00:05:23.721
and these are gonna give the same value,

00:05:23.721 --> 00:05:25.358
and they represent the distance

00:05:25.358 --> 00:05:28.142
between these numbers.


Interpreting negative number statements

WEBVTT
Kind: captions
Language: en

00:00:00.616 --> 00:00:02.972
- [Voiceover] Lina was
watching a football game.

00:00:02.972 --> 00:00:07.554
The team she was cheering for
gained six yards on one play.

00:00:07.554 --> 00:00:10.433
It lost eight yards on the next play.

00:00:10.433 --> 00:00:13.857
It lost another two yards
on the following play.

00:00:13.857 --> 00:00:17.015
Lina wants to know what
her team's net change

00:00:17.015 --> 00:00:21.055
in field position was
after these three plays.

00:00:21.055 --> 00:00:22.529
Which of the following equations

00:00:22.529 --> 00:00:24.550
matches the situation above?

00:00:24.550 --> 00:00:25.931
So, let's think about it.

00:00:25.931 --> 00:00:27.035
The team she was cheering for,

00:00:27.035 --> 00:00:28.768
first they gained six yards.

00:00:28.768 --> 00:00:31.182
So that's positive six yards.

00:00:31.182 --> 00:00:33.028
Then they lost eight.

00:00:33.028 --> 00:00:35.965
So it'd be six minus eight.

00:00:35.965 --> 00:00:37.892
And then it lost another two.

00:00:37.892 --> 00:00:40.468
So, six minus eight minus two.

00:00:40.468 --> 00:00:43.241
This is six minus eight but then plus two.

00:00:43.241 --> 00:00:44.788
So, this isn't right.

00:00:44.788 --> 00:00:47.330
This is six plus negative eight

00:00:47.330 --> 00:00:49.281
which is equivalent to six minus eight,

00:00:49.281 --> 00:00:50.465
plus negative two.

00:00:50.465 --> 00:00:53.227
So, this would be equivalent
to six minus eight minus two.

00:00:53.227 --> 00:00:54.230
Or another way to think about it,

00:00:54.230 --> 00:00:56.188
they gained six yards.

00:00:56.188 --> 00:00:58.521
Then they lost eight yards.

00:00:58.521 --> 00:01:00.425
And then they lost another two yards.

00:01:00.425 --> 00:01:01.865
So, definitely go with that one.

00:01:02.541 --> 00:01:03.397
Let's keep going.

00:01:04.290 --> 00:01:07.471
Eddie and Fran are scuba diving.

00:01:07.471 --> 00:01:11.569
Eddie is 35 meters below
the surface of the water,

00:01:11.569 --> 00:01:15.853
and Fran is eight meters
directly above him.

00:01:15.853 --> 00:01:18.561
The following situation describes this,

00:01:18.561 --> 00:01:22.470
or the following equation
describes this situation.

00:01:22.470 --> 00:01:24.990
So let's see, Eddie is 35 meters

00:01:24.990 --> 00:01:26.672
below the surface of the water.

00:01:26.672 --> 00:01:28.123
So, when we're below the surface,

00:01:28.723 --> 00:01:31.107
or the more below the surface we are,

00:01:31.107 --> 00:01:33.870
the more negative, I guess
is what we're doing here.

00:01:33.870 --> 00:01:35.382
So, if we're at the surface of the water,

00:01:36.335 --> 00:01:38.972
you would be at zero meters.

00:01:39.849 --> 00:01:42.681
And they tell us Fran is
eight meters above him.

00:01:42.681 --> 00:01:46.826
So we're taking Eddie's depth at 35 meters

00:01:46.826 --> 00:01:48.822
and then Fran is eight meters above.

00:01:48.822 --> 00:01:50.181
So, we're adding eight,

00:01:50.181 --> 00:01:54.325
to figure out Fran's depth
of negative 27 meters.

00:01:54.325 --> 00:01:57.761
Notice, this is eight
meters above Eddie's depth.

00:01:57.761 --> 00:02:00.919
So, negative 27 is Fran's depth.

00:02:00.919 --> 00:02:04.866
She is 27 meters below
the surface of the water.

00:02:05.342 --> 00:02:07.002
So, select all that apply.

00:02:07.002 --> 00:02:10.183
Fran is 27 meters below
the surface of the water.

00:02:10.183 --> 00:02:11.785
Yeah, that's what we just talked about.

00:02:13.445 --> 00:02:15.465
If your depth is, or
I guess you could say,

00:02:17.264 --> 00:02:18.286
if you're at negative 27

00:02:18.286 --> 00:02:20.283
that means 27 meters below the water.

00:02:20.283 --> 00:02:21.722
If you were at positive 27,

00:02:21.722 --> 00:02:24.427
that means you're at 27
meters above the water.

00:02:24.427 --> 00:02:26.018
So that's what they're telling us.

00:02:26.018 --> 00:02:27.678
Fran's position relative to the surface

00:02:27.678 --> 00:02:31.103
of the water is negative 27 meters.

00:02:31.826 --> 00:02:35.325
Yeah, I think that's a
reasonable way to say it.

00:02:35.325 --> 00:02:38.080
It's a little bit more confusing.

00:02:38.080 --> 00:02:40.064
But, yeah, she is negative 27 meters

00:02:40.064 --> 00:02:42.444
below the surface of the water

00:02:42.444 --> 00:02:43.826
which we just said up here.

00:02:43.826 --> 00:02:46.078
So I think that's fair
enough to say that as well.

00:02:46.924 --> 00:02:48.655
Let's do one more.

00:02:48.655 --> 00:02:51.000
Fred's basement is flooded.

00:02:51.000 --> 00:02:54.100
The basement floor is 3.4 meters

00:02:54.100 --> 00:02:55.579
below the ground floor.

00:02:55.579 --> 00:02:56.561
Alright, that makes sense

00:02:56.561 --> 00:02:59.974
that the basement is below
the ground, 3.4 meters.

00:02:59.974 --> 00:03:02.585
The water is 1.2 meters high.

00:03:03.201 --> 00:03:06.012
The following equation
describes this situation.

00:03:06.597 --> 00:03:10.875
So negative 3.4, that's where the floor

00:03:10.875 --> 00:03:12.279
of the basement is.

00:03:12.279 --> 00:03:15.091
It is 3.4 meters below the ground floor.

00:03:15.091 --> 00:03:17.562
So the ground floor would be zero.

00:03:17.562 --> 00:03:19.732
So, it's 3.4 meters
below the ground floor.

00:03:19.732 --> 00:03:21.520
So, that's the negative 3.4.

00:03:21.520 --> 00:03:23.331
But then the water is 1.2 meters.

00:03:23.331 --> 00:03:25.579
The top of, the surface of the water,

00:03:25.579 --> 00:03:28.737
is 1.2 meters above the
floor of the basement.

00:03:28.737 --> 00:03:31.778
So this negative 2.2 is how high

00:03:32.362 --> 00:03:34.133
the top of the water is.

00:03:34.657 --> 00:03:36.777
So, what does negative 2.2 tell us?

00:03:37.239 --> 00:03:41.135
Well, the top of the water is 2.2 meters

00:03:41.135 --> 00:03:42.328
below the ground floor.

00:03:42.328 --> 00:03:43.654
Yep, that's right.

00:03:43.654 --> 00:03:46.197
The water's 1.2 meters above the floor

00:03:46.197 --> 00:03:50.074
of the basement which
is negative 2.2 meters.

00:03:50.074 --> 00:03:52.976
And this is all relative
to the ground floor.

00:03:52.976 --> 00:03:53.928
So if the ground floor was zero,

00:03:53.928 --> 00:03:56.239
the top of the water is negative 2.2,

00:03:56.239 --> 00:03:58.328
which means the top of
the water is 2.2 meters

00:03:58.328 --> 00:04:00.429
below, that's what the negative tells us,

00:04:00.429 --> 00:04:02.193
below the ground floor.

00:04:02.809 --> 00:04:04.493
The water is 2.2 meters deep.

00:04:04.493 --> 00:04:05.294
No, that's not right.

00:04:05.294 --> 00:04:07.917
They tell us that the
water is 1.2 meters high

00:04:07.917 --> 00:04:09.531
or 1.2 meters deep.

00:04:09.531 --> 00:04:10.552
So, that's not right.

00:04:10.552 --> 00:04:11.887
And we're not going to
click None of the above

00:04:11.887 --> 00:04:14.753
because we definitely
found, we found an answer.


Finding initial temperature from temperature changes

WEBVTT
Kind: captions
Language: en

00:00:00.757 --> 00:00:03.825
- Amanda lives in Fairbanks, Alaska.

00:00:03.825 --> 00:00:06.629
The table below shows
the temperature changes

00:00:06.629 --> 00:00:10.387
in Fairbanks over a two-hour period.

00:00:10.387 --> 00:00:12.702
So when we went from 3pm to 4pm

00:00:12.702 --> 00:00:15.690
the temperature increased
by two degrees Celsius,

00:00:15.690 --> 00:00:17.741
and then from four to five the temperature

00:00:17.741 --> 00:00:20.850
decreased by five degrees Celsius.

00:00:20.850 --> 00:00:23.857
Amanda notices the temperature at 5pm

00:00:23.857 --> 00:00:26.925
is negative nine degrees Celsius.

00:00:26.925 --> 00:00:30.440
What was the temperature
in Fairbanks at 3pm?

00:00:30.440 --> 00:00:32.045
So they're essentially
telling us the temperature

00:00:32.045 --> 00:00:33.752
at the end of these changes,

00:00:33.752 --> 00:00:35.215
and they want us to
figure out the temperature

00:00:35.215 --> 00:00:37.961
at the beginning of these changes.

00:00:37.961 --> 00:00:39.647
So let's see if we can do that.

00:00:39.647 --> 00:00:40.927
Let me draw.

00:00:40.927 --> 00:00:43.690
Since, in thermometers we're
used to thinking vertically.

00:00:43.690 --> 00:00:45.519
So I'll draw a vertical number line here

00:00:45.519 --> 00:00:47.717
to think about this a little bit.

00:00:47.717 --> 00:00:51.536
So here we go.

00:00:51.845 --> 00:00:54.181
Alright, so at 5pm we end up at

00:00:54.181 --> 00:00:56.152
negative nine degrees Celsius,

00:00:56.152 --> 00:00:59.103
so let's just say this
is zero, negative one,

00:00:59.103 --> 00:01:04.103
two, three, four, five, six,

00:01:04.141 --> 00:01:06.600
seven, eight, nine.

00:01:06.600 --> 00:01:10.013
So we end up at negative
nine degrees Celsius.

00:01:10.013 --> 00:01:13.975
This is zero degrees
Celsius right over here.

00:01:13.975 --> 00:01:15.113
So from zero...

00:01:15.113 --> 00:01:20.113
Sorry, from 4pm to 5pm
the temperature dropped

00:01:20.354 --> 00:01:22.650
by five degrees Celsius

00:01:22.650 --> 00:01:25.393
and that got us to negative nine.

00:01:25.393 --> 00:01:28.806
So we drop by five to
get to negative nine,

00:01:28.806 --> 00:01:31.041
so we must've started
five above negative nine.

00:01:31.041 --> 00:01:33.520
So one, two, three, four, five...

00:01:33.520 --> 00:01:35.084
Right over here.

00:01:35.084 --> 00:01:36.912
So this is that drop.

00:01:36.912 --> 00:01:39.412
This is that five degree drop.

00:01:39.412 --> 00:01:43.052
Like we said, one, two, three, four, five.

00:01:43.052 --> 00:01:45.795
So that's that five degree
drop right over here.

00:01:45.795 --> 00:01:49.207
This is a negative five degrees Celsius.

00:01:49.207 --> 00:01:54.207
So at 4pm, right at 4pm, we
must've had a temperature of...

00:01:55.852 --> 00:01:59.793
Well, five above negative nine
is going to be negative four.

00:01:59.793 --> 00:02:01.500
Negative four degrees Celsius.

00:02:01.500 --> 00:02:04.852
This is what the temperature was at 4pm.

00:02:04.852 --> 00:02:09.626
So this is at 4pm.

00:02:09.626 --> 00:02:11.882
And then, to figure out 3pm,

00:02:11.882 --> 00:02:13.974
remember when we go from 3pm to 4pm

00:02:13.974 --> 00:02:16.859
our temperature increased
two degrees Celsius.

00:02:16.859 --> 00:02:21.593
So we increased by two
degrees Celsius from that hour

00:02:21.593 --> 00:02:24.802
to get to negative four degrees Celsius.

00:02:24.802 --> 00:02:27.973
So we increased by two degrees Celsius.

00:02:27.973 --> 00:02:30.709
So if we increase by two degrees Celsius

00:02:30.709 --> 00:02:33.117
to get to negative four, that means at 3pm

00:02:33.117 --> 00:02:35.027
we must've been two below that.

00:02:35.027 --> 00:02:37.546
So this is what we did from three to four.

00:02:37.546 --> 00:02:40.369
>From three to four we increased by two

00:02:40.369 --> 00:02:43.885
to get to negative four.

00:02:43.885 --> 00:02:47.380
So this right over here is
the plus two degrees Celsius

00:02:47.380 --> 00:02:48.477
right over there.

00:02:48.477 --> 00:02:52.276
So we must've started right here at 3pm.

00:02:52.276 --> 00:02:54.897
We must've started right there at 3pm.

00:02:54.897 --> 00:02:56.055
All I did is underline this.

00:02:56.055 --> 00:02:58.615
If you know we're at
negative nine to end with,

00:02:58.615 --> 00:03:01.013
and from four to five
we dropped negative five

00:03:01.013 --> 00:03:03.065
to get there, that means
at four we must've been

00:03:03.065 --> 00:03:05.259
five above negative nine.

00:03:05.259 --> 00:03:06.803
So we're at negative four,

00:03:06.803 --> 00:03:08.408
and then from three to four,

00:03:08.408 --> 00:03:11.232
if we increased two degrees to get there,

00:03:11.232 --> 00:03:12.979
that means at three we must've been

00:03:12.979 --> 00:03:15.539
two degrees below negative four.

00:03:15.539 --> 00:03:19.399
So this right over here is
negative six degrees Celsius.

00:03:19.399 --> 00:03:22.894
So what was the temperature
in Fairbanks at 3pm?

00:03:22.894 --> 00:03:25.474
It was negative six degrees Celsius.

00:03:25.474 --> 00:03:26.998
And you can verify that.

00:03:26.998 --> 00:03:30.066
If you start at negative six, you add two,

00:03:30.066 --> 00:03:31.630
you get to negative four,

00:03:31.630 --> 00:03:34.332
and then you subtract five,
you get to negative nine,

00:03:34.332 --> 00:03:37.083
which is exactly what we had here.


Interpreting integer expressions

WEBVTT
Kind: captions
Language: en

00:00:00.533 --> 00:00:04.433
- Which of the following
is a correct interpretation

00:00:04.433 --> 00:00:09.066
of the expression one plus six?

00:00:09.066 --> 00:00:10.656
Well before I even read
this, let's just think about

00:00:10.656 --> 00:00:13.666
what's happening in one
plus six on the number line.

00:00:13.666 --> 00:00:15.474
I'm going to start at one,
which is going to be one

00:00:15.474 --> 00:00:17.200
to the right of zero.

00:00:17.200 --> 00:00:18.967
And then I'm going to add another six.

00:00:18.967 --> 00:00:21.200
I'm going to go six more
to the right of zero

00:00:21.200 --> 00:00:22.766
to end up at seven,

00:00:22.766 --> 00:00:25.733
which is going to be seven
to the right of zero.

00:00:25.733 --> 00:00:29.266
The number that is six to the left of one

00:00:29.266 --> 00:00:30.733
on the number line.

00:00:30.733 --> 00:00:34.200
Now six to the left of one,
in this type of a number line,

00:00:34.200 --> 00:00:35.700
which is the standard number line,

00:00:35.700 --> 00:00:38.533
where to the right is positive
and to the left is negative.

00:00:38.533 --> 00:00:39.986
If you move six to the
left that means you're

00:00:39.986 --> 00:00:42.033
subtracting six from
one so that's not right.

00:00:42.033 --> 00:00:43.733
We're adding six.

00:00:43.733 --> 00:00:45.571
The number that is six to the right of one

00:00:45.571 --> 00:00:46.366
on the number line.

00:00:46.366 --> 00:00:47.866
Yeah, that's exactly right.

00:00:47.866 --> 00:00:50.700
We're going six to the right.

00:00:50.700 --> 00:00:52.966
If you're adding six
you're moving to the right.

00:00:52.966 --> 00:00:54.700
You're going to increase.

00:00:54.700 --> 00:00:57.333
So that's definitely
going to be the choice.

00:00:57.333 --> 00:00:59.000
Let's keep going.

00:00:59.000 --> 00:01:01.233
Which of the following is
a correct interpretation

00:01:01.233 --> 00:01:04.833
of the expression three
plus negative seven?

00:01:04.833 --> 00:01:07.067
So before I even look at these choices,

00:01:07.067 --> 00:01:09.633
one way I would think about
it is I would start at three

00:01:09.633 --> 00:01:12.300
and then I would move
seven to the left of three.

00:01:12.300 --> 00:01:14.066
I'm adding negative sevens.

00:01:14.066 --> 00:01:17.000
So this is the same thing
as three minus seven.

00:01:17.000 --> 00:01:19.433
So I'd move seven to the left of three.

00:01:19.433 --> 00:01:21.033
So let's see they would say.

00:01:21.033 --> 00:01:22.700
Start at seven on the number line

00:01:22.700 --> 00:01:24.133
and move three to the left.

00:01:24.133 --> 00:01:24.933
Well that doesn't make sense.

00:01:24.933 --> 00:01:26.000
I want to start at three.

00:01:26.000 --> 00:01:27.533
And let's skip over to here.

00:01:27.533 --> 00:01:28.933
Start at three on the number line

00:01:28.933 --> 00:01:30.533
and move seven to the left.

00:01:30.533 --> 00:01:31.966
Yeah, that's exactly right.

00:01:31.966 --> 00:01:33.366
We're going to start at three,

00:01:33.366 --> 00:01:34.566
and then we're going to move seven

00:01:34.566 --> 00:01:36.842
in the negative direction into the left,

00:01:36.842 --> 00:01:39.629
if our convention, which
is our standard convention,

00:01:39.629 --> 00:01:41.129
that to the left is negative

00:01:41.129 --> 00:01:42.912
and to the right is more positive.

00:01:42.912 --> 00:01:45.133
So that's going to be exactly right.

00:01:45.133 --> 00:01:46.433
Let's do one more.

00:01:46.433 --> 00:01:50.000
So now they have three
minus negative four.

00:01:50.000 --> 00:01:52.904
Well one way to think about it

00:01:52.904 --> 00:01:55.200
is three minus negative
four is the same thing

00:01:55.200 --> 00:01:57.100
if you subtract a negative,
it's the same thing as

00:01:57.100 --> 00:01:58.833
adding the positive version.

00:01:58.833 --> 00:02:01.300
So it's the same thing as three plus four.

00:02:01.300 --> 00:02:03.773
And so I would want to start at three

00:02:03.773 --> 00:02:05.900
and then move four to the right.

00:02:05.900 --> 00:02:07.639
Because once again this is the same thing

00:02:07.639 --> 00:02:09.509
as three plus four.

00:02:09.509 --> 00:02:11.666
Three minus negative
four is the same thing

00:02:11.666 --> 00:02:13.733
as three plus four.

00:02:13.733 --> 00:02:17.200
So I want to start at three
and move four to the right.

00:02:17.200 --> 00:02:19.700
Start at three and move four to the right.

00:02:19.700 --> 00:02:21.567
This would be the same
thing as three plus four

00:02:21.567 --> 00:02:25.078
which is the same thing
as that expression.


Integer equations to describe diagram

WEBVTT
Kind: captions
Language: en

00:00:00.558 --> 00:00:02.485
- [Voiceover] We're told
to fill in the blanks

00:00:02.485 --> 00:00:07.292
to complete the equation
that describes the diagram.

00:00:07.292 --> 00:00:09.544
So let's think about
what's going on over here.

00:00:09.544 --> 00:00:14.536
If we start at zero, and we
move one, two, three, four

00:00:14.536 --> 00:00:17.810
spaces to the right of zero,
this arrow right over here

00:00:17.810 --> 00:00:20.062
represents positive four.

00:00:20.062 --> 00:00:22.430
We already see that right
over here in the equation.

00:00:22.430 --> 00:00:25.913
Then from positive four,
from the tip of this arrow,

00:00:25.913 --> 00:00:30.913
we then go one, two,
three, four, five, six

00:00:31.532 --> 00:00:34.597
spaces to the left.

00:00:34.597 --> 00:00:36.640
So what we just did here is we just added

00:00:36.640 --> 00:00:39.914
a negative six to the positive four.

00:00:39.914 --> 00:00:41.841
Positive four plus negative six.

00:00:41.841 --> 00:00:43.374
Where does that put us?

00:00:43.374 --> 00:00:46.137
Well we see it puts us one, two spaces

00:00:46.137 --> 00:00:48.505
to the left of zero and
each of these spaces

00:00:48.505 --> 00:00:49.898
in this diagram are one.

00:00:49.898 --> 00:00:52.429
So two spaces to the left of zero

00:00:52.429 --> 00:00:56.400
is going to be negative two.

00:00:56.400 --> 00:01:01.260
This is fun. Let's keep
doing more examples.

00:01:01.260 --> 00:01:04.339
Write an addition equation
or a subtraction equation,

00:01:04.339 --> 00:01:06.823
your choice, so they're
giving us some choice,

00:01:06.823 --> 00:01:08.263
to describe the diagram.

00:01:08.263 --> 00:01:09.935
Alright, let's see what's going on here.

00:01:09.935 --> 00:01:12.814
We're starting at zero
and we're going one, two,

00:01:12.814 --> 00:01:16.065
three, four to the left.

00:01:16.065 --> 00:01:17.876
So if we're going four to the left

00:01:17.876 --> 00:01:22.876
or so we can say negative four, -4.

00:01:22.984 --> 00:01:26.583
And then we're going
to go another one, two,

00:01:26.583 --> 00:01:31.583
three, four, five, six, seven,
eight, nine to the left.

00:01:31.784 --> 00:01:35.731
So we could write this as
negative four minus nine

00:01:35.731 --> 00:01:37.937
is equal to.

00:01:37.937 --> 00:01:39.818
And when you go four to the left

00:01:39.818 --> 00:01:41.629
and then you go another nine to the left,

00:01:41.629 --> 00:01:46.629
you end up 13 to the left of
zero which is negative 13.

00:01:47.088 --> 00:01:49.145
Equals negative 13.

00:01:49.145 --> 00:01:51.746
So this way I've written it
as a subtraction equation

00:01:51.746 --> 00:01:52.582
I guess you could say.

00:01:52.582 --> 00:01:56.761
Negative four minus nine,
is equal to negative 13.

00:01:56.761 --> 00:01:57.992
Now another way I could have done it,

00:02:00.992 --> 00:02:05.283
I could have said negative
four plus negative nine

00:02:05.283 --> 00:02:08.568
is equal to negative thirteen as well.

00:02:08.568 --> 00:02:10.008
Either of those would
have been legitimate.

00:02:10.008 --> 00:02:13.746
Now I've written it as
an addition equation.

00:02:13.746 --> 00:02:15.279
Let's keep going.

00:02:15.279 --> 00:02:17.345
Fill in the blanks to
complete the equation

00:02:17.345 --> 00:02:19.621
that describes the diagram.

00:02:19.621 --> 00:02:21.514
So we're starting at zero, we go three

00:02:21.514 --> 00:02:24.360
to the left of zero,
that's negative three.

00:02:24.360 --> 00:02:27.768
Then we go another three
to the left of that.

00:02:27.768 --> 00:02:30.485
So we're going to add
another negative three.

00:02:30.485 --> 00:02:32.621
We're going to add another negative three

00:02:32.621 --> 00:02:35.268
and that puts us six to the left of zero.

00:02:35.268 --> 00:02:40.005
Well six to the left of
zero is negative six.

00:02:40.005 --> 00:02:42.075
And we're done.


Adding negative numbers on number line examples

WEBVTT
Kind: captions
Language: en

00:00:00.293 --> 00:00:01.837
- [Voiceover] So we've
already spent some time

00:00:01.837 --> 00:00:04.002
introducing ourselves to the idea

00:00:04.002 --> 00:00:05.904
of adding or subtracting

00:00:05.904 --> 00:00:08.045
positive and negative numbers.

00:00:08.045 --> 00:00:10.176
What I now what do in this video

00:00:10.176 --> 00:00:12.039
is do a bunch of examples using the

00:00:12.039 --> 00:00:15.008
adding negative numbers on
the number line exercise

00:00:15.008 --> 00:00:16.410
on Khan Academy,

00:00:16.410 --> 00:00:19.638
so that we can think about
different ways to model,

00:00:19.638 --> 00:00:20.480
or think about,

00:00:20.480 --> 00:00:22.913
or visualize adding and subtracting

00:00:22.913 --> 00:00:25.296
positive and negative numbers.

00:00:25.296 --> 00:00:27.112
So this question asks us,

00:00:27.112 --> 00:00:30.493
"Which number line model represents

00:00:30.493 --> 00:00:31.977
"the expression negative two

00:00:31.977 --> 00:00:33.646
"plus negative seven?"

00:00:33.646 --> 00:00:34.943
And a number line model seems

00:00:34.943 --> 00:00:36.644
like this really fancy thing,

00:00:36.644 --> 00:00:37.580
but they're just saying,

00:00:37.580 --> 00:00:39.671
which of these diagrams,

00:00:39.671 --> 00:00:42.695
which of these drawings
are a way to think about

00:00:42.695 --> 00:00:45.612
what negative two plus negative seven is.

00:00:45.612 --> 00:00:46.876
So let's think about it.

00:00:46.876 --> 00:00:49.269
I want to start at negative two

00:00:49.269 --> 00:00:52.599
and then to that I want
to add negative seven.

00:00:52.599 --> 00:00:54.178
So let's see what's going on here.

00:00:54.178 --> 00:00:57.240
This number line model

00:00:57.240 --> 00:00:58.780
I guess we could call it,

00:00:58.780 --> 00:01:01.753
it's saying this is positive
two right over here.

00:01:01.753 --> 00:01:03.509
It's to the right of zero.

00:01:03.509 --> 00:01:06.826
Then it subtracts seven.

00:01:06.826 --> 00:01:08.153
This is the model that would be

00:01:08.153 --> 00:01:11.103
for positive two plus negative seven,

00:01:11.103 --> 00:01:14.677
or two, or positive two minus seven.

00:01:14.677 --> 00:01:16.346
But that's not what we have over here,

00:01:16.346 --> 00:01:18.617
we have negative two minus seven.

00:01:18.617 --> 00:01:19.813
This one right over here,

00:01:19.813 --> 00:01:21.477
let's see, this first...

00:01:21.477 --> 00:01:22.773
I guess you could say this thing,

00:01:22.773 --> 00:01:25.471
this arrow,

00:01:25.471 --> 00:01:27.078
it's going

00:01:27.078 --> 00:01:29.278
one, two, three, four, five

00:01:29.278 --> 00:01:30.374
to the left of zero.

00:01:30.374 --> 00:01:32.211
So starting at negative five,

00:01:32.211 --> 00:01:33.596
and then it goes another

00:01:33.596 --> 00:01:35.706
one, two, three, four

00:01:35.706 --> 00:01:37.110
to the left of that.

00:01:37.110 --> 00:01:38.170
So you could view this as

00:01:38.170 --> 00:01:40.634
negative five minus four.

00:01:40.634 --> 00:01:41.867
Or you could view this as

00:01:41.867 --> 00:01:45.533
negative five plus negative four,

00:01:45.533 --> 00:01:47.812
but that's not what we have up here.

00:01:47.812 --> 00:01:48.594
And then this last one,

00:01:48.594 --> 00:01:49.577
hopefully this is the answer,

00:01:49.577 --> 00:01:50.543
otherwise there would be a mistake

00:01:50.543 --> 00:01:51.804
in the exercise.

00:01:51.804 --> 00:01:53.910
Let's see, we have this first arrow

00:01:53.910 --> 00:01:56.500
that takes us two steps
to the left of zero.

00:01:56.500 --> 00:01:58.262
So, one, two.

00:01:58.262 --> 00:01:59.935
So this is negative two,

00:01:59.935 --> 00:02:01.746
and then we take another seven steps

00:02:01.746 --> 00:02:02.931
to the left of that.

00:02:02.931 --> 00:02:06.344
So one, two, three,
four, five, six, seven.

00:02:06.344 --> 00:02:07.997
So you can think of this arrow

00:02:07.997 --> 00:02:10.309
as representing negative two,

00:02:10.309 --> 00:02:11.975
and then we're going to that,

00:02:11.975 --> 00:02:13.426
so we're starting from
this tip of this arrow.

00:02:13.426 --> 00:02:14.914
We're going to add a negative seven

00:02:14.914 --> 00:02:16.141
which makes us move seven

00:02:16.141 --> 00:02:19.049
to the left again to get to negative nine.

00:02:19.049 --> 00:02:20.210
So that's definitely the model

00:02:20.210 --> 00:02:22.361
that represents that expression.

00:02:22.361 --> 00:02:23.644
Let's keep going.

00:02:23.644 --> 00:02:25.165
All right.

00:02:26.873 --> 00:02:28.776
Which number line model represents

00:02:28.776 --> 00:02:31.606
the expression six plus negative two.

00:02:31.606 --> 00:02:33.145
So here, they've taken our number lines

00:02:33.145 --> 00:02:35.375
but they've made them go up and down,

00:02:35.375 --> 00:02:36.701
they've made them vertical.

00:02:36.701 --> 00:02:38.340
So we wanna really think about it.

00:02:38.340 --> 00:02:39.369
I want to start at six

00:02:39.369 --> 00:02:41.164
and I want to add negative two.

00:02:41.164 --> 00:02:43.068
So on a vertical number line like this,

00:02:43.068 --> 00:02:44.980
it seems like we're
increasing as we go up.

00:02:44.980 --> 00:02:48.565
I want to start at six above zero.

00:02:48.565 --> 00:02:50.893
So, I want to start at six above zero,

00:02:50.893 --> 00:02:52.366
but then I want to add negative two,

00:02:52.366 --> 00:02:54.566
which should take me two steps back down.

00:02:54.566 --> 00:02:56.472
This one takes me two steps even higher,

00:02:56.472 --> 00:02:57.967
so this one right over here

00:02:57.967 --> 00:03:00.168
is positive six plus two,

00:03:00.168 --> 00:03:03.290
or positive six plus positive two.

00:03:03.290 --> 00:03:04.488
This one over here we're starting

00:03:04.488 --> 00:03:06.500
at positive four and
then we're adding two,

00:03:06.500 --> 00:03:08.235
so that's not what we're talking about.

00:03:08.235 --> 00:03:09.201
This one over here,

00:03:09.201 --> 00:03:11.348
this big arrow, this tall arrow.

00:03:11.348 --> 00:03:13.128
This is going, it's pointing up.

00:03:13.128 --> 00:03:15.346
It gets us to positive six.

00:03:15.346 --> 00:03:17.346
The tip of the arrows are at positive six,

00:03:17.346 --> 00:03:19.012
But then we're going back down too.

00:03:19.012 --> 00:03:22.038
So this would be positive
six plus negative two.

00:03:22.038 --> 00:03:23.145
You could think of this arrow

00:03:23.145 --> 00:03:25.033
as representing positive six,

00:03:25.033 --> 00:03:27.733
and then this arrow is
representing negative two.

00:03:27.733 --> 00:03:29.666
It takes us two down.

00:03:29.666 --> 00:03:32.168
So if you take positive
six plus negative two,

00:03:32.168 --> 00:03:33.333
you do the positive six.

00:03:33.333 --> 00:03:35.206
And from the tip of your arrow,

00:03:35.206 --> 00:03:36.935
from there you can start the negative two,

00:03:36.935 --> 00:03:39.744
and you go two back down
and you end up at four.

00:03:39.744 --> 00:03:43.104
So it's definitely that
one right over there.

00:03:43.104 --> 00:03:45.449
This is surprisingly fun.

00:03:45.449 --> 00:03:46.545
Let's keep going.

00:03:46.545 --> 00:03:48.167
Which number line model represents

00:03:48.167 --> 00:03:51.933
the expression negative 2/5 plus 4/5?

00:03:51.933 --> 00:03:53.442
So we're going to start at negative 2/5

00:03:53.442 --> 00:03:55.174
which is going to be, it's negative,

00:03:55.174 --> 00:03:57.084
it's going to be to the left of zero.

00:03:57.084 --> 00:03:57.684
So let's see,

00:03:57.684 --> 00:03:59.278
this arrow right over here,

00:03:59.278 --> 00:04:01.146
this is negative 4/5,

00:04:01.146 --> 00:04:04.496
so that's not going to be quite right.

00:04:04.496 --> 00:04:05.873
So the second choice,

00:04:05.873 --> 00:04:07.106
what's going on here?

00:04:07.106 --> 00:04:09.288
So if I start at zero,

00:04:09.288 --> 00:04:11.708
and then I'm going 2/5

00:04:11.708 --> 00:04:12.911
to the left of zero.

00:04:12.911 --> 00:04:15.813
So that's negative 2/5 and from that,

00:04:15.813 --> 00:04:17.811
I'm going to the right

00:04:17.811 --> 00:04:21.770
1/5, 2/5, 3/5, 4/5.

00:04:21.770 --> 00:04:22.769
So this looks right.

00:04:22.769 --> 00:04:23.928
This looks like negative 2/5,

00:04:23.928 --> 00:04:25.610
so I'm starting at zero,

00:04:25.610 --> 00:04:27.200
I'm going to the left 2/5.

00:04:27.200 --> 00:04:29.013
So that's negative 2/5,

00:04:29.013 --> 00:04:30.133
and then I go to the right

00:04:30.133 --> 00:04:31.458
4/5 from that tip.

00:04:31.458 --> 00:04:33.003
>From negative 2/5,

00:04:33.003 --> 00:04:34.938
I'm adding 4/5.

00:04:34.938 --> 00:04:36.078
So that looks right.

00:04:36.078 --> 00:04:37.015
Let's do one more.

00:04:37.015 --> 00:04:39.481
This is exciting.

00:04:39.481 --> 00:04:41.308
Even if you don't find it fully exciting,

00:04:41.308 --> 00:04:42.549
anything you do you should

00:04:42.549 --> 00:04:43.970
convince yourself it's exciting.

00:04:43.970 --> 00:04:45.380
You'll do a better job.

00:04:45.380 --> 00:04:46.898
Which number line model represents

00:04:46.898 --> 00:04:50.650
the expression five 1/2
plus negative three?

00:04:50.650 --> 00:04:52.942
This is positive five 1/2,

00:04:52.942 --> 00:04:54.619
so I want to go five 1/2

00:04:54.619 --> 00:04:56.106
to the right of zero.

00:04:56.106 --> 00:04:57.979
This is five 1/2 to the left of zero,

00:04:57.979 --> 00:04:59.400
so that's negative five 1/2.

00:04:59.400 --> 00:05:00.804
That's not what I want to deal with.

00:05:00.804 --> 00:05:02.379
So this one looks interesting,

00:05:02.379 --> 00:05:03.909
and this is actually five 1/2

00:05:03.909 --> 00:05:05.607
to the left of zero as well,

00:05:05.607 --> 00:05:06.747
so it's probably going to be this

00:05:06.747 --> 00:05:08.162
but let's just look at it.

00:05:08.162 --> 00:05:11.185
So let's see, we go five
1/2 to the right of zero.

00:05:11.185 --> 00:05:12.845
So this is five 1/2,

00:05:12.845 --> 00:05:16.347
and then from that I go three left.

00:05:16.347 --> 00:05:18.975
So that in the end gets me to two 1/2.

00:05:18.975 --> 00:05:22.760
So it's this choice, right over there.

00:05:22.760 --> 00:05:24.113
Exciting.


Rewriting a fraction to decimal example 2

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:07.576
Let's see if we can
rewrite 6/12 as a decimal.

00:00:07.576 --> 00:00:09.700
And once again, I encourage
you to pause this video

00:00:09.700 --> 00:00:10.870
and try this on your own.

00:00:10.870 --> 00:00:12.203
And I'll give you a little hint.

00:00:12.203 --> 00:00:14.090
See if you can rewrite
this fraction, not

00:00:14.090 --> 00:00:18.490
in terms of 12ths,
but in terms of 10ths.

00:00:18.490 --> 00:00:20.210
Well, let's just
visualize this thing.

00:00:20.210 --> 00:00:25.800
So let's say that this
right over here is a 12th.

00:00:25.800 --> 00:00:27.970
So that is a 12th.

00:00:27.970 --> 00:00:31.100
And so let me try
to make 12 12ths.

00:00:31.100 --> 00:00:34.806
So copy and paste that,
so that's two 12ths.

00:00:37.630 --> 00:00:40.720
And that is three 12ths.

00:00:40.720 --> 00:00:43.530
And that is-- actually
let me just copy and paste

00:00:43.530 --> 00:00:45.240
the three 12ths.

00:00:45.240 --> 00:00:49.710
So copy and paste.

00:00:49.710 --> 00:00:53.420
So now we have six 12ths, which
I'll color it in a second.

00:00:53.420 --> 00:01:00.700
And now we have 9/12, and now we
have 12 12ths right over there.

00:01:00.700 --> 00:01:03.210
And we care about
six of these 12ths,

00:01:03.210 --> 00:01:06.630
so this is representing
six of these 12ths.

00:01:06.630 --> 00:01:09.630
So let me see if I can
color in six of them.

00:01:09.630 --> 00:01:15.230
So let me go to-- let me
see if I can pull this off.

00:01:15.230 --> 00:01:17.130
So this is going to be 1.

00:01:17.130 --> 00:01:18.880
Let me do a bigger paint brush.

00:01:18.880 --> 00:01:21.339
So 1, 2, 3, 4, 5, 6.

00:01:21.339 --> 00:01:21.880
That's right.

00:01:21.880 --> 00:01:22.870
That's 6/12.

00:01:22.870 --> 00:01:26.090
I'm just going to color
all of this business in.

00:01:26.090 --> 00:01:26.910
So there you have.

00:01:26.910 --> 00:01:30.380
We have represented
our fraction.

00:01:30.380 --> 00:01:34.290
Now, how do we represent
this thing right over here

00:01:34.290 --> 00:01:37.460
in terms of 10ths?

00:01:37.460 --> 00:01:39.140
So the one thing that
I might want to do

00:01:39.140 --> 00:01:45.250
is well look, 6 and 12
are both divisible by 6.

00:01:45.250 --> 00:01:48.240
So what if we divide this, not
if we rewrite this whole, not

00:01:48.240 --> 00:01:52.210
in terms of 12ths, but we
write it in terms of halves?

00:01:52.210 --> 00:01:53.500
Well, how would we do that?

00:01:53.500 --> 00:01:56.560
Well, we could
say-- we could call

00:01:56.560 --> 00:02:03.490
all of this stuff-- this is
1/2, and this is another 1/2.

00:02:03.490 --> 00:02:05.810
Or another way of
thinking about it is we're

00:02:05.810 --> 00:02:07.679
taking six of our
12ths together,

00:02:07.679 --> 00:02:09.220
and then we're
putting them together,

00:02:09.220 --> 00:02:11.910
and we're turning
them into a 1/2.

00:02:11.910 --> 00:02:14.240
So in terms of the
number of sections

00:02:14.240 --> 00:02:15.880
we have, we're
dividing the number

00:02:15.880 --> 00:02:18.290
of sections we have by 6.

00:02:18.290 --> 00:02:19.720
So let's do that.

00:02:19.720 --> 00:02:22.450
So let's divide the
number of sections by 6.

00:02:22.450 --> 00:02:27.040
So now we're only going
to have two sections.

00:02:27.040 --> 00:02:29.170
We're only going
to have two halves.

00:02:29.170 --> 00:02:33.980
That right over there is 1/2,
and that right over there

00:02:33.980 --> 00:02:34.860
is the other 1/2.

00:02:34.860 --> 00:02:37.190
Notice we had 12/12.

00:02:37.190 --> 00:02:40.300
Now we're taking 6 of those at
a time putting them together,

00:02:40.300 --> 00:02:43.220
so now we only have two
sections instead of 12 sections.

00:02:43.220 --> 00:02:47.630
But those 6/12, what are
they going to be in halves?

00:02:47.630 --> 00:02:50.522
Well, once again, where we're
taking six of the things

00:02:50.522 --> 00:02:51.980
and putting them
together into one.

00:02:51.980 --> 00:02:53.950
So we're going to divide
that by 6 as well.

00:02:56.510 --> 00:02:59.100
And so this is going to
be the same thing as 1/2.

00:02:59.100 --> 00:03:05.420
And you see this
1/2 right over here.

00:03:05.420 --> 00:03:07.540
And notice, all we
did in the past,

00:03:07.540 --> 00:03:10.650
we had multiplied
and divided-- sorry,

00:03:10.650 --> 00:03:13.040
in the past we had
multiplied the numerator

00:03:13.040 --> 00:03:14.860
and the denominator
by the same quantity

00:03:14.860 --> 00:03:16.270
to get an equivalent fraction.

00:03:16.270 --> 00:03:19.270
Now, we're dividing the
numerator and the denominator

00:03:19.270 --> 00:03:21.710
by the same quantity to
get the same fraction.

00:03:21.710 --> 00:03:26.012
Hopefully, this makes sense,
that 1/2 is equivalent to 6/12.

00:03:26.012 --> 00:03:26.970
Now, why did I do that?

00:03:26.970 --> 00:03:29.980
Didn't I say that I wanted to
get this in terms of 10ths?

00:03:29.980 --> 00:03:32.300
And yes, I do want to
get it in terms of 10ths,

00:03:32.300 --> 00:03:35.650
but 1/2 is fairly easy to
rewrite in terms of 10ths.

00:03:35.650 --> 00:03:39.060
So let me re-express 1/2.

00:03:39.060 --> 00:03:41.720
So 1/2, let me
write it like this.

00:03:41.720 --> 00:03:47.290
So 1/2, I could
represent 1/2 again.

00:03:47.290 --> 00:03:49.550
I'm going to re-represent 1/2.

00:03:49.550 --> 00:03:58.960
So let's say this right
over here is clearly 1/2,

00:03:58.960 --> 00:04:01.690
so I'm just representing it.

00:04:01.690 --> 00:04:05.170
So I'm taking a whole,
dividing it into 2,

00:04:05.170 --> 00:04:08.895
and 1/2-- let me
just color it in--

00:04:08.895 --> 00:04:11.270
would be that right over there,
which as we already seen,

00:04:11.270 --> 00:04:14.790
it's equivalent to the 6/12.

00:04:14.790 --> 00:04:16.680
Now, I want to
write this as 10ths.

00:04:16.680 --> 00:04:19.430
So what if I take
each of those halves,

00:04:19.430 --> 00:04:22.219
and I turn it into five
times as many pieces?

00:04:25.860 --> 00:04:27.980
So let me do that.

00:04:27.980 --> 00:04:42.240
So I'm going to do it into
1, 1, 2, 2, 3, 4, and 5.

00:04:42.240 --> 00:04:43.620
So what did I just do?

00:04:43.620 --> 00:04:44.850
I multiplied.

00:04:44.850 --> 00:04:47.280
I now have five
times as many pieces

00:04:47.280 --> 00:04:48.950
and the 1/2 is
going to represent

00:04:48.950 --> 00:04:51.710
five times as many 10ths.

00:04:51.710 --> 00:04:53.855
So this is going to
be equal to 5/10.

00:05:00.810 --> 00:05:05.130
Let me do it in a
green color, 5/10.

00:05:05.130 --> 00:05:08.560
Notice, 6/12 is the
same thing as 1/2,

00:05:08.560 --> 00:05:10.810
which is the same thing as 5/10.

00:05:10.810 --> 00:05:12.660
All I did here is I
multiplied the numerator

00:05:12.660 --> 00:05:14.600
and the denominator by
the same quantity 5,

00:05:14.600 --> 00:05:16.620
so I could get 10
in the denominator.

00:05:16.620 --> 00:05:22.090
And 5/10, well that's the
same thing as 5 times 1/10.

00:05:22.090 --> 00:05:27.230
It's literally 5/10, and we can
represent this as a decimal,

00:05:27.230 --> 00:05:29.730
or you could say in
decimal notation.

00:05:29.730 --> 00:05:34.030
5/10, we know that-- let's say
that that is the ones place,

00:05:34.030 --> 00:05:37.060
and if we go one place to the
right of the decimal, that

00:05:37.060 --> 00:05:38.730
is the tenths place.

00:05:38.730 --> 00:05:42.010
That right over there
is the tenths place.

00:05:42.010 --> 00:05:46.720
And we have 5/10, so we would
put a 5 right over there.

00:05:46.720 --> 00:05:53.487
So this could be rewritten
as the decimal, 0.5.

00:05:53.487 --> 00:05:55.820
Now, there's other ways you
could have thought about it.

00:05:55.820 --> 00:05:56.820
You could say hey, look.

00:05:56.820 --> 00:06:00.610
6 is half of 12, and
so what is half of 10?

00:06:00.610 --> 00:06:04.410
Well, 5 is half
of 10, and so 5/10

00:06:04.410 --> 00:06:09.270
is the same thing as 5/10,
which is the same thing as 5/10

00:06:09.270 --> 00:06:11.389
written as a decimal.


How to solve one-step multiplication and division equations with fractions and decimals

WEBVTT
Kind: captions
Language: en

00:00:00.325 --> 00:00:02.600
Let's get some practice
solving some equations,

00:00:02.600 --> 00:00:04.342
and we're gonna set up some equations

00:00:04.342 --> 00:00:05.967
that are a little bit hairier than normal,

00:00:05.967 --> 00:00:08.127
they're gonna have some
decimals and fractions in them.

00:00:08.127 --> 00:00:12.585
So let's say I had the
equation 1.2 times c

00:00:12.585 --> 00:00:15.998
is equal to 0.6.

00:00:15.998 --> 00:00:17.298
So what do I have to multiply

00:00:17.298 --> 00:00:20.851
times 1.2 to get 0.6?

00:00:20.851 --> 00:00:23.405
And it might not jump out
immediately in your brain but

00:00:23.405 --> 00:00:24.960
lucky for us we can think about this

00:00:24.960 --> 00:00:26.818
a little bit methodically.

00:00:26.818 --> 00:00:28.281
So one thing I like to do is say okay,

00:00:28.281 --> 00:00:30.277
I have the c on the left hand side,

00:00:30.277 --> 00:00:32.483
and I'm just multiplying it by 1.2,

00:00:32.483 --> 00:00:34.875
it would be great if this just said c.

00:00:34.875 --> 00:00:37.638
If this just said c instead of 1.2c.

00:00:37.638 --> 00:00:39.217
So what can I do there?

00:00:39.217 --> 00:00:41.237
Well I could just divide by 1.2

00:00:41.237 --> 00:00:43.118
but as we've seen multiple times,

00:00:43.118 --> 00:00:45.114
you can't just do that
to the left hand side,

00:00:45.114 --> 00:00:47.808
that would change, you no longer could say

00:00:47.808 --> 00:00:51.639
that this is equal to that if
you only operate on one side.

00:00:51.639 --> 00:00:54.797
So you have to divide
by 1.2 on both sides.

00:00:54.797 --> 00:00:57.931
So on your left hand
side, 1.2c divided by 1.2,

00:00:57.931 --> 00:01:00.091
well that's just going to be c.

00:01:00.091 --> 00:01:02.250
You're just going to be left with c,

00:01:02.250 --> 00:01:07.250
and you're going to have
c is equal to 0.6 over 1.2

00:01:07.637 --> 00:01:08.844
Now what is that equal to?

00:01:08.844 --> 00:01:10.911
There's a bunch of ways
you could approach it.

00:01:10.911 --> 00:01:12.908
The way I like to do
it is, well let's just,

00:01:12.908 --> 00:01:14.301
let's just get rid of the decimals.

00:01:14.301 --> 00:01:16.832
Let's just multiply the
numerator and denominator

00:01:16.832 --> 00:01:19.641
by a large enough number so
that the decimals go away.

00:01:19.641 --> 00:01:20.848
So what happens if we multiply

00:01:20.848 --> 00:01:23.217
the numerator and the denominator by...

00:01:23.217 --> 00:01:25.817
Let's see if we multiply them by 10,

00:01:25.817 --> 00:01:29.486
you're gonna have a 6 in the numerator

00:01:29.486 --> 00:01:31.668
and 12 in the denominator,
actually let's do that.

00:01:31.668 --> 00:01:35.592
Let's multiply the numerator
and denominator by 10.

00:01:35.592 --> 00:01:37.845
So once again, this is the
same thing as multiplying by

00:01:37.845 --> 00:01:41.049
10 over 10, it's not changing
the value of the fraction.

00:01:41.049 --> 00:01:46.049
So 0.6 times 10 is 6,
and 1.2 times 10 is 12.

00:01:50.931 --> 00:01:53.134
So it's equal to six
twelfths, and if we want

00:01:53.134 --> 00:01:55.952
we can write that in a
little bit of a simpler way.

00:01:55.952 --> 00:01:57.902
We could rewrite that
as, divide the numerator

00:01:57.902 --> 00:02:01.803
and denominator by 6, you get 1 over 2,

00:02:01.803 --> 00:02:04.427
so this is equal to one half.

00:02:04.427 --> 00:02:08.443
And if you look back at
the original equation,

00:02:08.443 --> 00:02:12.948
1.2 times one half, you could
view this as twelve tenths.

00:02:12.948 --> 00:02:14.504
Twelve tenths times one half

00:02:14.504 --> 00:02:17.545
is going to be equal to six tenths,

00:02:17.545 --> 00:02:18.706
so we can feel pretty good

00:02:18.706 --> 00:02:20.866
that c is equal to one half.

00:02:20.866 --> 00:02:22.561
Let's do another one.

00:02:22.561 --> 00:02:27.561
Let's say that we have 1 over 4

00:02:28.812 --> 00:02:33.085
is equal to y over 12.

00:02:33.085 --> 00:02:35.337
So how do we solve for y here?

00:02:35.337 --> 00:02:37.845
So we have a y on the right hand side,

00:02:37.845 --> 00:02:40.492
and it's being divided by 12.

00:02:40.492 --> 00:02:41.653
Well the best way I can think of

00:02:41.653 --> 00:02:44.044
of getting rid of this
12 and just having a y

00:02:44.044 --> 00:02:49.044
on the right hand side is
multiplying both sides by 12.

00:02:49.176 --> 00:02:51.683
We do that in yellow.

00:02:51.683 --> 00:02:54.632
So if I multiply the
right hand side by 12,

00:02:54.632 --> 00:02:57.000
I have to multiply the
left hand side by 12.

00:02:57.000 --> 00:02:58.835
And once again, why did I pick 12?

00:02:58.835 --> 00:03:00.808
Well I wanted to multiply by some number,

00:03:00.808 --> 00:03:03.177
that when I multiply it by y over 12

00:03:03.177 --> 00:03:04.218
I'm just left with y.

00:03:04.218 --> 00:03:06.169
And so y times 12 divided by 12,

00:03:06.169 --> 00:03:07.910
well that's just going to be 1.

00:03:07.910 --> 00:03:10.720
And then on the left hand
side you're going to have

00:03:10.720 --> 00:03:13.204
12 times one fourth,
which is twelve fourths.

00:03:13.204 --> 00:03:18.204
So you get 12 over 4, is equal to y.

00:03:19.868 --> 00:03:24.868
Or you could say y is equal
to 12 over 4, y is equal to,

00:03:25.165 --> 00:03:27.798
let me do that just so you
can see what I'm doing,

00:03:27.798 --> 00:03:31.446
just flopping the sides, doesn't
change what's being said,

00:03:31.446 --> 00:03:33.233
y is equal to 12 over 4.

00:03:33.233 --> 00:03:35.671
Now what is twelve fourths?

00:03:35.671 --> 00:03:37.599
Well, you can view this
as 12 divided by 4,

00:03:37.599 --> 00:03:40.896
which is 3, or you could
view this as twelve fourths

00:03:40.896 --> 00:03:43.287
which would be literally, 3 wholes.

00:03:43.287 --> 00:03:45.284
So you could say this would be equal to 3.

00:03:45.284 --> 00:03:47.908
Y is equal to 3, and you can check that.

00:03:47.908 --> 00:03:52.389
One fourth is equal to 3 over 12,

00:03:52.389 --> 00:03:53.480
so it all works out.

00:03:53.480 --> 00:03:54.688
That's the neat thing about equations,

00:03:54.688 --> 00:03:58.612
you can always check to see
if you got the right answer.

00:03:58.612 --> 00:04:02.071
Let's do another one, can't stop.

00:04:02.071 --> 00:04:07.071
4.5 is equal to 0.5n

00:04:08.039 --> 00:04:11.104
So like always, I have my n
already on the right hand side.

00:04:11.104 --> 00:04:13.124
But it's being multiplied by 0.5,

00:04:13.124 --> 00:04:15.097
it would be great if it just said n.

00:04:15.097 --> 00:04:16.003
So what can I do?

00:04:16.003 --> 00:04:19.857
Well I can divide both sides,
I can divide both sides

00:04:19.857 --> 00:04:22.852
by 0.5, once again, if I do
it to the right hand side

00:04:22.852 --> 00:04:24.733
I have to do it to the left hand side.

00:04:24.733 --> 00:04:26.312
And why am I dividing by 0.5?

00:04:26.312 --> 00:04:29.098
So I'm just left with an
n on the right hand side.

00:04:29.098 --> 00:04:31.699
So this is going to be,
so on the left hand side,

00:04:31.699 --> 00:04:36.180
I have 4.5 over 0.5, let me just,

00:04:36.180 --> 00:04:37.388
I don't want to skip too many steps.

00:04:37.388 --> 00:04:42.388
4.5 over 0.5, is equal to n,

00:04:43.819 --> 00:04:46.170
because you have 0.5 divided by 0.5,

00:04:46.170 --> 00:04:48.144
you're just left with an n over here.

00:04:48.144 --> 00:04:50.210
So what does that equal to?

00:04:50.210 --> 00:04:53.368
Well 4.5 divided by 0.5,

00:04:53.368 --> 00:04:54.575
there's a couple ways to view this.

00:04:54.575 --> 00:04:56.247
You could view this as forty-five tenths

00:04:56.247 --> 00:04:58.151
divided by five tenths,
which would tell you

00:04:58.151 --> 00:04:59.985
okay, this is going to be 9.

00:04:59.985 --> 00:05:01.680
Or if that seems a little bit confusing

00:05:01.680 --> 00:05:04.141
or a little bit daunting, you
can do what we did over here.

00:05:04.141 --> 00:05:05.883
You could multiply the
numerator and the denominator

00:05:05.883 --> 00:05:08.367
by the same number, so that
we get rid of the decimals.

00:05:08.367 --> 00:05:10.643
And in this case, if you multiply by 10

00:05:10.643 --> 00:05:12.965
you can move the decimal one to the right.

00:05:12.965 --> 00:05:15.102
So once again, it has to be multiplying

00:05:15.102 --> 00:05:16.755
the numerator and the
denominator by the same thing.

00:05:16.755 --> 00:05:19.101
We're multiplying by 10 over
10, which is equivalent to 1,

00:05:19.101 --> 00:05:20.703
which tells us that we're not

00:05:20.703 --> 00:05:23.234
changing the value of this fraction.

00:05:23.234 --> 00:05:27.738
So let's see, this is going to be 45

00:05:27.738 --> 00:05:32.738
over 5, is equal to n.

00:05:33.427 --> 00:05:35.191
And some of you might say wait wait wait,

00:05:35.191 --> 00:05:37.327
hold on a second, you just
told us whatever we do

00:05:37.327 --> 00:05:41.019
to one side of the equation,
we have to do to the other side

00:05:41.019 --> 00:05:42.552
of the equation and here you are,

00:05:42.552 --> 00:05:44.758
you're just multiplying the
left hand side of this equation

00:05:44.758 --> 00:05:46.336
by 10 over 10.

00:05:46.336 --> 00:05:48.217
Now remember, what is 10 over 10?

00:05:48.217 --> 00:05:49.773
10 over 10 is just 1.

00:05:49.773 --> 00:05:52.281
Yes, if I wanted to, I could
multiply the left hand side

00:05:52.281 --> 00:05:55.392
by 10 over 10, and I could
multiply the right hand side

00:05:55.392 --> 00:05:58.225
by 10 over 10, but that's
not going to change the value

00:05:58.225 --> 00:05:59.548
of the right hand side.

00:05:59.548 --> 00:06:02.009
I'm not actually changing
the values of the two sides.

00:06:02.009 --> 00:06:04.401
I'm just trying to
rewrite the left hand side

00:06:04.401 --> 00:06:07.303
by multiplying it by 1 in
kind of a creative way.

00:06:07.303 --> 00:06:10.415
But notice, n times 10 over
10, well that's still going

00:06:10.415 --> 00:06:11.599
to just be n.

00:06:11.599 --> 00:06:13.805
So I'm not violating this principle

00:06:13.805 --> 00:06:15.198
of whatever I do to the left hand side

00:06:15.198 --> 00:06:16.196
I do to the right hand side.

00:06:16.196 --> 00:06:18.750
You can always multiply one side by 1

00:06:18.750 --> 00:06:20.584
and you can do that as
many times as you want.

00:06:20.584 --> 00:06:22.651
Like the same way you can add 0

00:06:22.651 --> 00:06:24.926
or subtract 0 from one side,

00:06:24.926 --> 00:06:26.993
without necessarily having to show

00:06:26.993 --> 00:06:28.177
you're doing it to the other side,

00:06:28.177 --> 00:06:30.151
because it doesn't change the value.

00:06:30.151 --> 00:06:33.076
But anyways, you have n
is equal to 45 over 5,

00:06:33.076 --> 00:06:34.284
well what's 45 over 5?

00:06:34.284 --> 00:06:35.932
Well that's going to be 9.

00:06:35.932 --> 00:06:40.414
So we have 9 is equal to,
why did I switch to green?

00:06:40.414 --> 00:06:45.414
We have 9 is equal to n, or
we could say n is equal to 9.

00:06:46.149 --> 00:06:47.658
And you could check that:

00:06:47.658 --> 00:06:52.658
4.5 is equal to 0.5 times
9, yup half of 9 is 4.5

00:06:54.484 --> 00:06:57.201
Let's do one more, because
once again I can't stop.

00:06:57.201 --> 00:07:00.126
Alright, let me get some space here,

00:07:00.126 --> 00:07:04.747
so we can keep the different
problems apart that we had.

00:07:04.747 --> 00:07:09.747
So let's do, let's have
a different variable now.

00:07:09.832 --> 00:07:14.832
Let's say we have g
over 4 is equal to 3.2.

00:07:14.940 --> 00:07:16.914
Well I wanna get rid
of this dividing by 4,

00:07:16.914 --> 00:07:18.562
so the easiest way I
can think of doing that

00:07:18.562 --> 00:07:21.743
is multiplying both sides by 4.

00:07:21.743 --> 00:07:23.438
So I'm multiplying both sides by 4,

00:07:23.438 --> 00:07:26.573
and the whole reason is 4
divided by 4 gives me 1,

00:07:26.573 --> 00:07:31.573
so I'm gonna have g is equal
to, what's 3.2 times 4?

00:07:32.879 --> 00:07:36.743
Let's see 3 times 4 is
12, and two tenths times 4

00:07:36.743 --> 00:07:41.665
is eight tenths, so it's
gonna be 12 and eight tenths.

00:07:41.665 --> 00:07:45.241
G is going to be 12.8, and
you can verify this is right.

00:07:45.241 --> 00:07:49.601
12.8 divided by 4 is 3.2.


How to find equivalent expressions by combining like terms and using the distributive property

WEBVTT
Kind: captions
Language: en

00:00:00.419 --> 00:00:02.573
- Which expressions are equivalent

00:00:02.573 --> 00:00:06.860
to x plus two-y plus x plus two?

00:00:06.860 --> 00:00:09.867
Select all that apply. Alright.

00:00:09.867 --> 00:00:13.281
Let's see if I can manipulate
this thing a little bit.

00:00:13.281 --> 00:00:16.657
Let me just rewrite it so I have x plus...

00:00:16.657 --> 00:00:21.611
x plus two-y plus x plus two.

00:00:21.611 --> 00:00:23.379
So, the first thing that jumps out at me,

00:00:23.379 --> 00:00:24.637
before I even look at these choices,

00:00:24.637 --> 00:00:27.299
here I have an x over here,
I have an x over there.

00:00:27.299 --> 00:00:29.351
Well, if I have one x, and
then I can add it to another x

00:00:29.351 --> 00:00:32.846
that would be two-x so
I could rewrite this...

00:00:32.846 --> 00:00:34.350
let me do this in a different color.

00:00:34.350 --> 00:00:37.376
I could rewrite this x and this x,

00:00:37.376 --> 00:00:40.078
if I add them together
that's going to be two-x.

00:00:40.078 --> 00:00:43.471
So, this is...actually, I
don't want to skip any steps.

00:00:43.471 --> 00:00:46.945
So, that's x plus x plus two-y...

00:00:46.945 --> 00:00:49.242
now I'm just switching
the order...plus two.

00:00:49.242 --> 00:00:51.517
And then these two x's right over here,

00:00:51.517 --> 00:00:54.220
I can just rewrite that as two-x.

00:00:54.220 --> 00:00:57.575
So, I have two-x plus two-y...

00:00:57.575 --> 00:01:00.955
plus two-y plus two.

00:01:00.955 --> 00:01:03.240
Now, let's see, out of all my choices...

00:01:03.240 --> 00:01:05.767
so, this one,

00:01:05.767 --> 00:01:10.247
this is two-x plus four-y plus four.

00:01:10.247 --> 00:01:13.581
So, that's not right, I have
two-x plus two-y plus 2,

00:01:13.581 --> 00:01:15.247
so I can rule this one out.

00:01:15.247 --> 00:01:16.871
Now, this one's interesting, it looks like

00:01:16.871 --> 00:01:17.950
they have factored out a two.

00:01:17.950 --> 00:01:20.509
Let's see if we factor out
a two here, what happens?

00:01:20.509 --> 00:01:22.642
So, we do see that two
is a factor of that term,

00:01:22.642 --> 00:01:25.507
it's a factor of that term,
and it's a factor of that term.

00:01:25.507 --> 00:01:28.717
So, let's see if we can factor it out.

00:01:28.717 --> 00:01:32.516
So, this is going to be two times x,

00:01:32.516 --> 00:01:37.373
I'll do the x in that same
magenta color, two times x plus,

00:01:37.373 --> 00:01:40.847
we have just a y left when
you factor out the two.

00:01:40.847 --> 00:01:42.350
And then if you factor out a two here,

00:01:42.350 --> 00:01:43.895
you're just going to have a one left.

00:01:43.895 --> 00:01:46.983
So, two times x plus y plus one.

00:01:46.983 --> 00:01:49.705
Which is exactly what they have over here.

00:01:49.705 --> 00:01:51.513
And since I was able to find a choice,

00:01:51.513 --> 00:01:56.355
I will not pick "None of the above".


Basic linear equation word problem

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.810
Marcia has just opened
her new computer store.

00:00:03.810 --> 00:00:07.480
She makes $27 on every
computer she sells

00:00:07.480 --> 00:00:10.910
and her monthly
expenses are $10,000.

00:00:10.910 --> 00:00:12.850
What is the minimum
number of computer

00:00:12.850 --> 00:00:15.410
she needs to sell in a
month to make a profit?

00:00:15.410 --> 00:00:17.975
So I'll let you think
about that for a second.

00:00:17.975 --> 00:00:20.100
Well, let's think about
what we have to figure out.

00:00:20.100 --> 00:00:21.766
We have to figure out
the minimum number

00:00:21.766 --> 00:00:23.240
of computers she needs to sell.

00:00:23.240 --> 00:00:25.770
So let's set that to a
variable or set a variable

00:00:25.770 --> 00:00:26.660
to represent that.

00:00:26.660 --> 00:00:30.770
So let's let x equal the
number of computers she sells.

00:00:30.770 --> 00:00:36.220
Number of computers sold.

00:00:36.220 --> 00:00:38.250
Now, let's think about
how much net profit

00:00:38.250 --> 00:00:39.770
she will make in a month.

00:00:39.770 --> 00:00:41.311
And that's what
we're thinking about.

00:00:41.311 --> 00:00:43.660
How many computers,
the minimum number she

00:00:43.660 --> 00:00:45.570
needs to sell in order
to make a net profit?

00:00:45.570 --> 00:00:48.770
So I'll write her
profit is going

00:00:48.770 --> 00:00:52.830
to be how much money she brings
in from selling the computers.

00:00:52.830 --> 00:00:56.370
And she makes $27 on
every computer she sells.

00:00:56.370 --> 00:01:00.870
So her profit is going to be
$27 times the number of computer

00:01:00.870 --> 00:01:01.410
she sells.

00:01:01.410 --> 00:01:05.550
She gets $27 per computer
times the number she sells.

00:01:05.550 --> 00:01:07.310
But we're not done yet.

00:01:07.310 --> 00:01:12.250
She still has expenses
of $10,000 per month.

00:01:12.250 --> 00:01:17.860
So we're going to have to
subtract out the $10,000.

00:01:17.860 --> 00:01:19.800
What we care about
is making a profit.

00:01:19.800 --> 00:01:25.530
We want this number right over
here to be greater than 0.

00:01:25.530 --> 00:01:28.000
So let's just think about
what number of computers

00:01:28.000 --> 00:01:29.460
would get us to 0.

00:01:29.460 --> 00:01:32.300
And then, maybe she needs
to sell a little bit more

00:01:32.300 --> 00:01:33.880
than that.

00:01:33.880 --> 00:01:36.180
So let's see what gets
her to break even.

00:01:36.180 --> 00:01:38.150
So break even-- that's 0 profit.

00:01:38.150 --> 00:01:41.411
Neither positive or negative--
is equal to 27 times--

00:01:41.411 --> 00:01:42.910
and I'll do it all
in one color now.

00:01:42.910 --> 00:01:46.300
27x minus 10,000.

00:01:46.300 --> 00:01:48.530
Well, we've seen equations
like this before.

00:01:48.530 --> 00:01:51.710
We can add 10,000 to both sides.

00:01:51.710 --> 00:01:56.890
Add 10,000 to both
sides, so it's no longer

00:01:56.890 --> 00:01:58.610
on the right-hand side.

00:01:58.610 --> 00:02:03.630
And we are left with 10,000.

00:02:03.630 --> 00:02:08.579
10,000 is equal to 27x.

00:02:08.579 --> 00:02:10.120
And then to solve
for x, we just have

00:02:10.120 --> 00:02:13.290
to divide both sides by 27.

00:02:13.290 --> 00:02:13.910
Let's do that.

00:02:13.910 --> 00:02:16.440
Divide both sides by 27.

00:02:16.440 --> 00:02:18.580
On our right-hand
side, we have x.

00:02:18.580 --> 00:02:20.070
So let me just write this down.

00:02:20.070 --> 00:02:24.110
So we have x on our right-hand
side is going to be equal

00:02:24.110 --> 00:02:29.190
to 10,000/27.

00:02:29.190 --> 00:02:31.890
I switched the right and
the left-hand sides here.

00:02:31.890 --> 00:02:33.140
Now, what is this going to be?

00:02:33.140 --> 00:02:35.800
Well, we can do a little bit of
long division to handle that.

00:02:35.800 --> 00:02:39.950
So 27 goes into 10,000.

00:02:39.950 --> 00:02:41.490
So 27 doesn't go into 1.

00:02:41.490 --> 00:02:42.340
Doesn't go into 10.

00:02:42.340 --> 00:02:45.080
It goes into 100 three times.

00:02:45.080 --> 00:02:48.754
3 times 27 is what?

00:02:48.754 --> 00:02:51.556
81.

00:02:51.556 --> 00:02:55.410
100 minus 81 is 19.

00:02:55.410 --> 00:02:58.230
Then we can bring down a 0.

00:02:58.230 --> 00:03:00.332
27 goes into 190?

00:03:00.332 --> 00:03:02.877
It looks like it will go
into it about six times.

00:03:02.877 --> 00:03:03.960
Let's see if that's right.

00:03:03.960 --> 00:03:06.330
6 times 7 is 42.

00:03:06.330 --> 00:03:08.220
6 times 2 is 12.

00:03:08.220 --> 00:03:10.558
Plus 4.

00:03:10.558 --> 00:03:11.420
16.

00:03:11.420 --> 00:03:16.620
Let's see, 90 minus
62 is actually 28.

00:03:16.620 --> 00:03:18.110
Let me turn this back.

00:03:18.110 --> 00:03:19.870
So it goes 7 times.

00:03:19.870 --> 00:03:22.470
7 times 7 is 49.

00:03:22.470 --> 00:03:25.020
7 times 2 is 14.

00:03:25.020 --> 00:03:26.260
Plus 4 is 18.

00:03:26.260 --> 00:03:27.030
There you go.

00:03:27.030 --> 00:03:27.950
Look at that.

00:03:27.950 --> 00:03:30.090
So 190 minus 189.

00:03:30.090 --> 00:03:31.820
We get 1.

00:03:31.820 --> 00:03:33.820
Let's bring down another 0.

00:03:33.820 --> 00:03:36.705
We bring down another 0.

00:03:36.705 --> 00:03:38.900
We have a 0 right there.

00:03:38.900 --> 00:03:42.182
27 goes into 10 how many times?

00:03:42.182 --> 00:03:43.640
Well, it doesn't
go into 10 at all.

00:03:43.640 --> 00:03:45.250
So we'll put a 0 right there.

00:03:45.250 --> 00:03:48.450
0 times 27 is 0.

00:03:48.450 --> 00:03:50.020
Then we subtract.

00:03:50.020 --> 00:03:52.210
And then, we get 10 again.

00:03:52.210 --> 00:03:54.220
And now we're in
the decimal range.

00:03:54.220 --> 00:03:56.400
Or we're going to start
getting decimal values.

00:03:56.400 --> 00:03:57.695
We can bring down another 0.

00:03:57.695 --> 00:04:01.560
We could get 27 goes
into 100 three times.

00:04:01.560 --> 00:04:07.288
So our x-value is going
to be approximately 370.3.

00:04:07.288 --> 00:04:10.340
And then we're going to keep
going on and on and on and on.

00:04:10.340 --> 00:04:13.760
But this is enough information
for us to answer our question.

00:04:13.760 --> 00:04:15.630
What is the minimum
number of computers

00:04:15.630 --> 00:04:19.019
she needs to sell in a
month to make a profit?

00:04:19.019 --> 00:04:21.570
Well, she can't sell
a decimal number

00:04:21.570 --> 00:04:23.750
of computers, a
third of a computer.

00:04:23.750 --> 00:04:26.470
She could either
sell 370 computers.

00:04:26.470 --> 00:04:29.580
If she sells 370
computers, then she's

00:04:29.580 --> 00:04:31.520
not going to get to
break even because that's

00:04:31.520 --> 00:04:34.530
less than the quantity
she needs for break even.

00:04:34.530 --> 00:04:38.500
So she needs to sell 371.

00:04:38.500 --> 00:04:44.477
She needs to sell 371 computers
in a month to make a profit.


More missing value addition and subtraction problems with negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:02.879
- Let's give ourselves a
little bit more practice

00:00:02.879 --> 00:00:05.933
adding and subtracting
positive and negative numbers.

00:00:05.933 --> 00:00:08.092
So, let's say that we have the statement,

00:00:08.092 --> 00:00:11.943
four is equal to negative eight

00:00:11.943 --> 00:00:13.828
minus blank.

00:00:13.828 --> 00:00:15.930
Like always, I encourage
you to pause the video

00:00:15.930 --> 00:00:18.309
and see if you can figure
out what blank would be

00:00:18.309 --> 00:00:19.551
to make the statement true.

00:00:19.551 --> 00:00:21.211
That negative eight minus blank

00:00:21.211 --> 00:00:23.011
is going to be equivalent to four.

00:00:23.011 --> 00:00:26.108
That four is equal to
negative eight minus blank.

00:00:26.749 --> 00:00:28.444
So now, let's work it through together.

00:00:28.444 --> 00:00:31.509
To help us I will get a number line.

00:00:31.509 --> 00:00:33.827
So there's a number line right over there.

00:00:34.561 --> 00:00:37.051
And, one way to think about it is,

00:00:37.051 --> 00:00:39.335
I wanna start at negative eight,

00:00:39.335 --> 00:00:41.542
so this is negative five, negative six,

00:00:41.542 --> 00:00:44.346
negative seven, negative eight.

00:00:44.346 --> 00:00:46.221
I wanna start at negative eight

00:00:46.221 --> 00:00:48.287
and I wanna do something to negative eight

00:00:48.287 --> 00:00:49.895
to end up at four.

00:00:50.512 --> 00:00:52.807
To end up at four.

00:00:52.807 --> 00:00:56.544
So four is positive one, two, three, four.

00:00:56.932 --> 00:00:58.646
So that's four right over there.

00:00:59.113 --> 00:01:02.167
So, what do I need to do
to get from negative eight

00:01:02.167 --> 00:01:03.870
to positive four?

00:01:04.371 --> 00:01:05.556
Well let's see.

00:01:05.556 --> 00:01:08.876
I have to move to the
right one, two, three,

00:01:08.876 --> 00:01:12.869
four, five, six, seven, eight,

00:01:12.869 --> 00:01:16.134
nine, 10, 11, 12.

00:01:16.134 --> 00:01:18.781
I have to the right 12 spaces.

00:01:18.781 --> 00:01:20.232
Or another way to think about it,

00:01:20.232 --> 00:01:22.218
to get from negative
eight to positive four

00:01:22.218 --> 00:01:23.790
I have to, I'm just gonna make sure

00:01:23.790 --> 00:01:25.427
I have some space up here,

00:01:25.427 --> 00:01:28.293
I have to add 12.

00:01:28.712 --> 00:01:29.814
Another way to think about it,

00:01:29.814 --> 00:01:31.394
to get from negative eight to zero,

00:01:31.394 --> 00:01:32.949
you have to add eight.

00:01:32.949 --> 00:01:34.853
Then to get from zero to four,

00:01:34.853 --> 00:01:36.374
you have to add another four.

00:01:36.374 --> 00:01:39.668
So you have to add eight plus four, or 12.

00:01:40.195 --> 00:01:41.196
But how do you do that?

00:01:41.196 --> 00:01:43.216
'Cause they're saying
negative eight minus something

00:01:43.216 --> 00:01:44.732
should be equal to four.

00:01:45.155 --> 00:01:47.373
Whatever you have over here,

00:01:47.373 --> 00:01:51.320
you want this to end up
the equivalent to plus 12,

00:01:51.320 --> 00:01:52.980
so what can you do there?

00:01:52.980 --> 00:01:54.860
Well you could subtract a negative.

00:01:54.860 --> 00:01:58.063
If you subtracted negative 12,

00:01:58.063 --> 00:02:01.083
this would be equivalent to adding 12.

00:02:01.083 --> 00:02:03.592
So notice, let me rewrite it,

00:02:03.592 --> 00:02:05.374
four is equal to,

00:02:05.374 --> 00:02:07.089
let me just write it
without my boxes there,

00:02:07.089 --> 00:02:11.181
is equal to negative
eight minus negative 12.

00:02:11.181 --> 00:02:13.978
Well what happens when
you subtract a negative?

00:02:13.978 --> 00:02:16.904
Well that ends up being
adding the positive.

00:02:16.904 --> 00:02:19.305
So this is gonna be the same thing as

00:02:19.305 --> 00:02:22.116
negative eight plus positive 12,

00:02:22.116 --> 00:02:26.145
or four is equal to negative 8 plus 12,

00:02:26.145 --> 00:02:27.379
which we know is true.

00:02:27.379 --> 00:02:29.258
You add eight you get to zero,

00:02:29.258 --> 00:02:31.023
negative eight plus eight is zero,

00:02:31.023 --> 00:02:34.542
then you add another four,
you get to positive four.

00:02:34.542 --> 00:02:36.933
So that one was pretty interesting.

00:02:36.933 --> 00:02:39.232
Let's do another one, we
can't get enough practice,

00:02:39.232 --> 00:02:40.614
this is too much fun.

00:02:40.614 --> 00:02:41.693
All right.

00:02:41.693 --> 00:02:43.434
So let's say we had...

00:02:43.434 --> 00:02:44.828
let's say we had

00:02:44.828 --> 00:02:47.468
something blank

00:02:47.468 --> 00:02:50.765
minus negative two

00:02:50.765 --> 00:02:53.864
is equal to negative seven.

00:02:54.352 --> 00:02:56.545
How do we think about this?

00:02:57.150 --> 00:02:58.787
Well the first thing my brain wants to do

00:02:58.787 --> 00:03:02.571
is to simplify subtracting a negative.

00:03:02.571 --> 00:03:04.105
So if you subtract negative two

00:03:04.105 --> 00:03:06.147
that's the same thing
as adding positive two.

00:03:06.147 --> 00:03:08.341
So this is an equivalent statement to say,

00:03:08.341 --> 00:03:11.104
something subtracting a negative two

00:03:11.104 --> 00:03:13.646
is the same thing as
adding a positive two.

00:03:13.646 --> 00:03:17.430
Something plus two is
equal to negative seven.

00:03:17.780 --> 00:03:20.843
And now we can get the number
line out to think about it.

00:03:21.541 --> 00:03:23.942
So, get my number line out.

00:03:24.559 --> 00:03:27.682
So, my goal, my goal is

00:03:27.682 --> 00:03:29.841
I wanna get to negative seven.

00:03:29.841 --> 00:03:31.867
So there's negative five, negative six,

00:03:32.375 --> 00:03:34.844
negative seven right there.

00:03:34.844 --> 00:03:37.642
And I wanna add two to something
to get to negative seven.

00:03:37.642 --> 00:03:40.521
So if I'm adding two that
means I'm jumping two,

00:03:40.521 --> 00:03:41.938
I'm jumping two to the right,

00:03:41.938 --> 00:03:43.760
that's what adding two looks like.

00:03:43.760 --> 00:03:45.966
So what number do I need to start at

00:03:45.966 --> 00:03:48.241
that if I jump two to
the right, if I add two

00:03:48.241 --> 00:03:50.061
I end up at negative seven?

00:03:50.061 --> 00:03:51.689
Well I would have to start two to the left

00:03:51.689 --> 00:03:53.334
of negative seven.

00:03:53.334 --> 00:03:56.511
If I start two to the left, one, two,

00:03:56.511 --> 00:03:57.792
if I started...

00:03:57.792 --> 00:03:59.394
Let me do this in a different color.

00:03:59.394 --> 00:04:01.692
If I start at this number right here

00:04:01.692 --> 00:04:04.999
and if I add two, plus one, plus two,

00:04:06.231 --> 00:04:08.019
I end up at negative seven.

00:04:08.019 --> 00:04:09.297
Well what number is this?

00:04:09.297 --> 00:04:13.116
This number that is two to
the left of negative seven.

00:04:13.116 --> 00:04:15.089
Well it's negative nine.

00:04:15.089 --> 00:04:17.598
So this right over here is negative nine.

00:04:17.598 --> 00:04:18.865
Negative nine plus two,

00:04:18.865 --> 00:04:20.885
so if you start at negative nine

00:04:20.885 --> 00:04:22.337
and you move two to the right

00:04:22.337 --> 00:04:24.051
you end up at negative seven.

00:04:24.051 --> 00:04:25.933
Or, you could say, negative nine

00:04:25.933 --> 00:04:29.531
minus negative two, 'cause
if you subtract negative two

00:04:29.531 --> 00:04:32.178
you're going to move to the right.

00:04:32.178 --> 00:04:34.894
If you subtract positive
two, you move to the left.

00:04:34.894 --> 00:04:36.171
But then if you subtract negative two,

00:04:36.171 --> 00:04:39.339
you move two to the
right, to negative seven.

00:04:39.793 --> 00:04:41.244
Another fascinating one.

00:04:41.244 --> 00:04:42.707
Let's do one more of these,

00:04:42.707 --> 00:04:44.866
I'm having maybe a
little bit too much fun.

00:04:44.866 --> 00:04:46.166
All right.

00:04:46.166 --> 00:04:50.218
So let's say we have negative four

00:04:50.218 --> 00:04:54.111
is equal to blank minus nine.

00:04:55.008 --> 00:04:56.388
So let's get the number line out

00:04:56.388 --> 00:04:58.327
and always pause the video, see if you can

00:04:58.327 --> 00:04:59.717
try it on your own.

00:05:00.474 --> 00:05:03.196
But let's get my number line out

00:05:03.917 --> 00:05:05.554
and so let's think about it.

00:05:05.554 --> 00:05:08.622
Our goal, we wanna take some number

00:05:08.622 --> 00:05:11.837
subtract nine from it
and get negative four.

00:05:11.837 --> 00:05:13.737
So we wanna get negative four

00:05:14.621 --> 00:05:15.840
and then there's some number

00:05:15.840 --> 00:05:17.500
that if we take nine away from it.

00:05:17.500 --> 00:05:19.682
So if you take nine away from something

00:05:19.682 --> 00:05:22.004
that's moving nine to the left.

00:05:22.004 --> 00:05:25.475
So, another way to think
about subtracting nine,

00:05:25.475 --> 00:05:27.569
let me do this in a different color,

00:05:27.569 --> 00:05:31.085
subtracting nine.

00:05:31.644 --> 00:05:33.594
That's literally moving nine to the left.

00:05:33.594 --> 00:05:35.405
One, two, three, four, five,

00:05:35.405 --> 00:05:38.087
six, seven, eight, nine.

00:05:38.087 --> 00:05:40.804
So there's some number that we start with,

00:05:40.804 --> 00:05:42.812
we're gonna subtract nine

00:05:42.812 --> 00:05:45.013
and then we end up with negative four.

00:05:45.013 --> 00:05:47.451
We're gonna end up with negative four.

00:05:47.451 --> 00:05:50.144
Some number, that's the
blank right over here,

00:05:50.144 --> 00:05:52.373
you subtract nine, you
move nine to the left,

00:05:52.373 --> 00:05:54.811
that's gonna be equal to negative four.

00:05:54.811 --> 00:05:56.455
So another way to think about it is,

00:05:56.455 --> 00:06:00.576
this number is gonna be nine
to the right of negative four.

00:06:00.576 --> 00:06:02.892
So let's think about what
that number's gonna be.

00:06:03.370 --> 00:06:04.844
So this number is gonna be nine

00:06:04.844 --> 00:06:07.131
to the right of negative four.

00:06:07.131 --> 00:06:09.882
One, two, three, four, five,

00:06:09.882 --> 00:06:12.902
six, seven, eight, nine.

00:06:12.902 --> 00:06:15.327
It looks like positive five.

00:06:15.327 --> 00:06:17.742
So let's see if we're right about that.

00:06:17.742 --> 00:06:20.400
If we start as positive
five and we subtract nine,

00:06:20.400 --> 00:06:24.080
so let's do that, one, two, three, four,

00:06:24.080 --> 00:06:28.401
five, six, seven, eight, nine.

00:06:28.401 --> 00:06:30.672
Yup, you subtract nine,

00:06:30.672 --> 00:06:33.629
you're gonna end up at negative four.

00:06:33.629 --> 00:06:37.748
So negative four is
equal to five minus nine.