Coordinate plane: word problem exercise

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.290
Milena's town is built
on a grid similar

00:00:03.290 --> 00:00:05.150
to the coordinate plane.

00:00:05.150 --> 00:00:08.490
She is riding her bicycle
from her home at point

00:00:08.490 --> 00:00:14.780
negative 3, 4 to the mall at
point negative 3, negative 7.

00:00:14.780 --> 00:00:18.720
Each unit on the graph
denotes one city block.

00:00:18.720 --> 00:00:21.520
Plot the two points,
and find the distance

00:00:21.520 --> 00:00:24.420
between Milena's
home and the mall.

00:00:24.420 --> 00:00:26.700
So let's see, she's
riding her bicycle

00:00:26.700 --> 00:00:30.250
from her home at the
point negative 3, 4.

00:00:30.250 --> 00:00:32.110
So let's plot negative 3, 4.

00:00:32.110 --> 00:00:34.020
So I'll use this
point right over here.

00:00:34.020 --> 00:00:36.960
So negative 3 is
our x-coordinate.

00:00:36.960 --> 00:00:40.940
So we're going to go 3 to the
left of the origin 1, 2, 3.

00:00:40.940 --> 00:00:42.640
That gets us a negative 3.

00:00:42.640 --> 00:00:44.185
And positive 4 is
our y-coordinate.

00:00:44.185 --> 00:00:46.840
So we're going to go
4 above the origin.

00:00:46.840 --> 00:00:50.100
Or I should say, we're
going to go 4 up.

00:00:50.100 --> 00:00:52.280
So we went negative 3,
or we went 3 to the left.

00:00:52.280 --> 00:00:54.950
That's negative 3, positive 4.

00:00:54.950 --> 00:00:57.862
Or you could say we went
positive 4, negative 3.

00:00:57.862 --> 00:01:00.070
This tells us what we do in
the horizontal direction.

00:01:00.070 --> 00:01:02.880
This tells us what we do
in the vertical direction.

00:01:02.880 --> 00:01:03.985
That's where her home is.

00:01:03.985 --> 00:01:05.610
Now let's figure out
where the mall is.

00:01:05.610 --> 00:01:08.920
It's at the point
negative 3, negative 7.

00:01:08.920 --> 00:01:12.480
So negative 3, we
went negative 3

00:01:12.480 --> 00:01:14.880
along the horizontal
direction and then negative 7

00:01:14.880 --> 00:01:16.910
along the vertical direction.

00:01:16.910 --> 00:01:20.991
So we get to negative 3,
negative 7 right over there.

00:01:20.991 --> 00:01:22.740
And now we need to
figure out the distance

00:01:22.740 --> 00:01:24.310
between her home and the mall.

00:01:24.310 --> 00:01:25.470
Now, we could
actually count it out,

00:01:25.470 --> 00:01:26.636
or we could just compute it.

00:01:26.636 --> 00:01:32.711
If we wanted to count it out,
it's 1, 2, 3, 4, 5, 6, 7, 8, 9,

00:01:32.711 --> 00:01:34.350
10, 11 blocks.

00:01:34.350 --> 00:01:35.980
So we could type that in.

00:01:35.980 --> 00:01:38.050
And another way to
think about it is they

00:01:38.050 --> 00:01:42.380
have the exact
same x-coordinate.

00:01:42.380 --> 00:01:44.960
They're both at the
x-coordinate negative 3.

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The only difference
between these two

00:01:46.690 --> 00:01:49.090
is what is happening
in the y-coordinate.

00:01:49.090 --> 00:01:50.580
This is at a positive 4.

00:01:50.580 --> 00:01:51.980
This is at a negative 7.

00:01:51.980 --> 00:01:53.890
Positive 4, negative 7.

00:01:53.890 --> 00:01:56.530
So we're really trying to
find the distance between 4

00:01:56.530 --> 00:01:57.850
and negative 7.

00:01:57.850 --> 00:02:01.310
So if I were to say
4 minus negative 7,

00:02:01.310 --> 00:02:03.950
we would get this
distance right over here.

00:02:03.950 --> 00:02:07.890
So we have 4 minus negative 7,
which is the same thing as 4

00:02:07.890 --> 00:02:12.640
plus 7, which is 11.

00:02:12.640 --> 00:02:15.430
Let's do a couple more.

00:02:15.430 --> 00:02:18.700
Carlos is hanging a
poster in the area

00:02:18.700 --> 00:02:21.050
shown by the red rectangle.

00:02:21.050 --> 00:02:25.400
He is placing a nail in the
center of the blue line.

00:02:25.400 --> 00:02:31.260
In the second graph, plot the
point where he places the nail.

00:02:31.260 --> 00:02:35.520
So he wants to place a nail in
the center of the blue line.

00:02:35.520 --> 00:02:37.660
The blue line is 6 units long.

00:02:37.660 --> 00:02:38.910
The center is right over here.

00:02:38.910 --> 00:02:41.070
That's 3 to the
right, 3 to the left.

00:02:41.070 --> 00:02:46.360
So he wants to put the nail
at the point x equals 0,

00:02:46.360 --> 00:02:48.120
y is equal to 4.

00:02:48.120 --> 00:02:53.150
So he wants to put it at x is
equal to 0, y is equal to 4.

00:02:53.150 --> 00:02:55.300
That's this point
right over here.

00:02:55.300 --> 00:02:57.850
So let's check our answer.

00:02:57.850 --> 00:02:58.780
Let's do one more.

00:03:02.050 --> 00:03:05.060
Town A and Town B are
connected by a train that

00:03:05.060 --> 00:03:08.280
has a station at the
point negative 1, 3.

00:03:08.280 --> 00:03:09.260
I see that.

00:03:09.260 --> 00:03:11.290
The train tracks are in blue.

00:03:11.290 --> 00:03:12.460
Fair enough.

00:03:12.460 --> 00:03:15.955
Which town is closer to
the station along the train

00:03:15.955 --> 00:03:20.130
route, Town A or Town B?

00:03:20.130 --> 00:03:21.860
So they're not just
asking us what's

00:03:21.860 --> 00:03:24.020
the kind of crow's
flight, the distance

00:03:24.020 --> 00:03:25.020
that if you were to fly.

00:03:25.020 --> 00:03:27.670
They're saying,
which town is closer

00:03:27.670 --> 00:03:30.650
to the station along
the train route?

00:03:30.650 --> 00:03:34.500
So if you were to follow the
train route just like that.

00:03:34.500 --> 00:03:40.610
So the A right over here, A if
you were going along the train

00:03:40.610 --> 00:03:45.030
route, you would have
to go 1, 2, 3, 4, 5,

00:03:45.030 --> 00:03:50.090
6 in the x-direction,
and then 1, 2, 3, 4,

00:03:50.090 --> 00:03:52.080
5 along the y-direction.

00:03:52.080 --> 00:03:55.150
So you'd have to
go a total of 11.

00:03:55.150 --> 00:04:00.022
If you're going from B, you're
going 1, 2, 3, 4, 5, 6, 7,

00:04:00.022 --> 00:04:04.438
8 9, 10, 11 along the
x-direction, and then 1,

00:04:04.438 --> 00:04:09.100
2, 3, 4, 5, 6, so 6 along
the vertical direction.

00:04:09.100 --> 00:04:11.040
So you're going
to go a total 17.

00:04:11.040 --> 00:04:15.930
So it's pretty obvious that
A is closer along the tracks.

00:04:15.930 --> 00:04:19.320
Now, you could also think about
it in terms of coordinates

00:04:19.320 --> 00:04:24.170
because A is at the
coordinate of negative 7, 8.

00:04:24.170 --> 00:04:27.390
And if you were to think
about negative 7, 8,

00:04:27.390 --> 00:04:31.220
to get from negative
7 to negative 1

00:04:31.220 --> 00:04:34.070
along the x-coordinate,
you're going to go 6.

00:04:34.070 --> 00:04:37.130
And then to go from 8 to 3,
you're going to go 5 more.

00:04:37.130 --> 00:04:39.097
So you could also not
necessarily count it out,

00:04:39.097 --> 00:04:41.180
you can actually just think
about the coordinates.

00:04:41.180 --> 00:04:45.560
But either way, you see
that town A is closer.


Coordinate plane: graphing points and naming quadrants

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.170
Plot 4 comma negative
1, and select

00:00:04.170 --> 00:00:07.140
the quadrant in
which the point lies.

00:00:07.140 --> 00:00:09.700
So 4, the first
number in our ordered

00:00:09.700 --> 00:00:11.130
pair, that's our x-coordinate.

00:00:11.130 --> 00:00:14.890
That says how far to move in the
horizontal or the x-direction.

00:00:14.890 --> 00:00:16.260
It's a positive 4.

00:00:16.260 --> 00:00:18.440
So I'm going to
go 4 to the right.

00:00:18.440 --> 00:00:20.030
And then the second
coordinate says,

00:00:20.030 --> 00:00:21.654
what do we do in the
vertical direction

00:00:21.654 --> 00:00:23.110
or in the y-direction?

00:00:23.110 --> 00:00:24.600
It's a negative 1.

00:00:24.600 --> 00:00:26.480
Since it's negative,
we're going to go down.

00:00:26.480 --> 00:00:29.280
And it's a negative 1, so
we're going to go down 1.

00:00:29.280 --> 00:00:32.640
So that right over there is
the point 4 comma negative 1.

00:00:32.640 --> 00:00:34.260
So I've plotted
it, but now I have

00:00:34.260 --> 00:00:36.860
to select which quadrant
the point lies in.

00:00:36.860 --> 00:00:38.810
And this is just a
naming convention.

00:00:38.810 --> 00:00:40.410
This is the first quadrant.

00:00:40.410 --> 00:00:41.940
This is the second quadrant.

00:00:41.940 --> 00:00:43.400
This is the third quadrant.

00:00:43.400 --> 00:00:45.330
And this is the fourth quadrant.

00:00:45.330 --> 00:00:49.560
So the point lies in the
fourth quadrant, quadrant IV.

00:00:49.560 --> 00:00:51.560
And I guess you have to
know your Roman numerals

00:00:51.560 --> 00:00:56.280
a little bit to know that's
representing quadrant IV.

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Let's do a couple more of these.

00:00:58.930 --> 00:01:01.200
Plot 8 comma negative
4, and select

00:01:01.200 --> 00:01:03.490
the quadrant in
which the point lies.

00:01:03.490 --> 00:01:07.690
Well, my x-coordinate
is 8 so I go 8

00:01:07.690 --> 00:01:09.690
in the positive x-direction.

00:01:09.690 --> 00:01:14.450
And then my y-coordinate is
negative 4, so I go 4 down.

00:01:14.450 --> 00:01:17.740
And this is sitting,
again, in not the first,

00:01:17.740 --> 00:01:20.740
not the second, not the third,
but the fourth quadrant,

00:01:20.740 --> 00:01:22.626
in quadrant IV.

00:01:22.626 --> 00:01:23.750
Let's do one more of these.

00:01:23.750 --> 00:01:25.470
Hopefully we get a
different quadrant.

00:01:25.470 --> 00:01:28.550
So we want to plot the
point negative 5 comma 5.

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So now my x-coordinate
is negative.

00:01:30.530 --> 00:01:31.650
It's negative 5.

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So I'm going to move to the
left in the x-direction.

00:01:34.660 --> 00:01:36.290
So I go to negative 5.

00:01:36.290 --> 00:01:39.760
And my y-coordinate
is positive so I go up

00:01:39.760 --> 00:01:44.180
5, so negative 5 comma 5.

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And this is sitting not
in the first quadrant,

00:01:46.170 --> 00:01:47.380
but the second quadrant.

00:01:47.380 --> 00:01:49.380
And of course, this is
the third and the fourth.

00:01:49.380 --> 00:01:52.190
So this is sitting in
the second quadrant.

00:01:52.190 --> 00:01:55.460
Check answer, and
we got it right.


Coordinate plane: reflecting points

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:05.690
The point negative 8 comma, 5
is reflected across the y-axis.

00:00:05.690 --> 00:00:11.550
Plot negative 8 comma 5 and its
reflection across the y-axis.

00:00:11.550 --> 00:00:14.230
So first let's plot
negative 8 comma 5.

00:00:14.230 --> 00:00:16.340
So its x-coordinate
is negative 8,

00:00:16.340 --> 00:00:18.090
so I'll just use this
one right over here.

00:00:18.090 --> 00:00:22.340
So the x-coordinate is negative
8, and the y-coordinate is 5,

00:00:22.340 --> 00:00:23.610
so I'll go up 5.

00:00:23.610 --> 00:00:25.830
So the y-coordinate
is 5 right over here.

00:00:25.830 --> 00:00:27.880
You see negative 8 and 5.

00:00:27.880 --> 00:00:31.380
We've gone 8 to the left
because it's negative,

00:00:31.380 --> 00:00:33.790
and then we've gone 5 up,
because it's a positive 5.

00:00:33.790 --> 00:00:36.320
So we've plotted
negative 8 comma 5.

00:00:36.320 --> 00:00:39.147
Now we have to plot its
reflection across the y-axis.

00:00:42.200 --> 00:00:46.979
And so you can imagine if
this was some type of lake

00:00:46.979 --> 00:00:49.520
or something and you were to
see its reflection, and this is,

00:00:49.520 --> 00:00:52.600
say, like the moon, you would
see its reflection roughly

00:00:52.600 --> 00:00:53.260
around here.

00:00:53.260 --> 00:00:57.270
You would see an equal
distance away from the y-axis.

00:00:57.270 --> 00:01:00.880
So you would see it at 8 to
the right of the y-axis, which

00:01:00.880 --> 00:01:08.050
would be at positive 8, and
still 5 above the x-axis.

00:01:08.050 --> 00:01:10.310
So that's its reflection
right over here.

00:01:10.310 --> 00:01:14.330
It's reflection is
the point 8 comma 5.

00:01:14.330 --> 00:01:17.270
Let's do a couple more of these.

00:01:17.270 --> 00:01:20.140
The point negative
6 comma negative 7

00:01:20.140 --> 00:01:25.090
is reflec-- this should say
"reflected" across the x-axis.

00:01:25.090 --> 00:01:28.010
Plot negative 6 comma
negative 7 and its reflection

00:01:28.010 --> 00:01:30.140
across the x-axis.

00:01:30.140 --> 00:01:32.900
So negative 6 comma
negative 7, so we're

00:01:32.900 --> 00:01:35.850
going to go 6 to the
left of the origin,

00:01:35.850 --> 00:01:40.490
and we're going to go down 7.

00:01:40.490 --> 00:01:41.230
So there we go.

00:01:41.230 --> 00:01:44.740
Negative 6 comma negative
7 is right there.

00:01:44.740 --> 00:01:47.940
And we are reflecting
across the x-axis.

00:01:47.940 --> 00:01:50.240
So, once again, if
you imagine that this

00:01:50.240 --> 00:01:52.130
is some type of a lake,
or maybe some type

00:01:52.130 --> 00:01:54.970
of an upside-down
lake, or a mirror,

00:01:54.970 --> 00:01:57.330
where would we think
we see its reflection?

00:01:57.330 --> 00:01:59.882
Well, its reflection would
be the same distance.

00:01:59.882 --> 00:02:01.340
We're reflecting
across the x-axis,

00:02:01.340 --> 00:02:03.230
so it would be the
same distance, but now

00:02:03.230 --> 00:02:04.630
above the x-axis.

00:02:04.630 --> 00:02:06.640
So this was 7 below.

00:02:06.640 --> 00:02:09.580
Now we're going to go
7 above the x-axis,

00:02:09.580 --> 00:02:11.939
and it's going to be at
the same x-coordinate.

00:02:11.939 --> 00:02:13.480
So there you have
it right over here.

00:02:13.480 --> 00:02:15.690
We reflected this
point to right up here,

00:02:15.690 --> 00:02:18.710
because we reflected
across the x-axis.

00:02:18.710 --> 00:02:19.990
Let's check our answer.

00:02:19.990 --> 00:02:21.890
Let's do one more.

00:02:21.890 --> 00:02:29.461
The point B is a reflection
of point A across which axis?

00:02:29.461 --> 00:02:31.210
So let's think about
this right over here.

00:02:31.210 --> 00:02:33.320
This is at the point
negative 6 comma 5.

00:02:33.320 --> 00:02:36.540
This is at the point
negative 5 comma 6.

00:02:36.540 --> 00:02:37.170
Let's see.

00:02:37.170 --> 00:02:39.200
It doesn't look like
it's only one axis.

00:02:39.200 --> 00:02:43.490
If I were to reflect this
point across the y-axis,

00:02:43.490 --> 00:02:47.649
it would go all the
way to positive 6, 5.

00:02:47.649 --> 00:02:49.440
So it would go all the
way right over here.

00:02:49.440 --> 00:02:53.830
So if I reflect A just across
the y-axis, it would go there.

00:02:53.830 --> 00:02:58.400
And then if I reflected that
point across the x-axis,

00:02:58.400 --> 00:03:04.040
then I would end up
at 5 below the x-axis

00:03:04.040 --> 00:03:05.760
at an x-coordinate of 6.

00:03:05.760 --> 00:03:08.830
So to go from A to B, you could
reflect across the y and then

00:03:08.830 --> 00:03:12.680
the x, or you could
reflect across the x,

00:03:12.680 --> 00:03:14.910
and it would get
you right over here.

00:03:14.910 --> 00:03:17.600
It would get you to
negative 6 comma 5,

00:03:17.600 --> 00:03:19.290
and then reflect across the y.

00:03:19.290 --> 00:03:22.230
So it's really reflecting
across both axes.

00:03:22.230 --> 00:03:25.504
So we would reflect across the
x-axis and then the y-axis.

00:03:25.504 --> 00:03:26.920
It would have also
been legitimate

00:03:26.920 --> 00:03:31.510
if we said the y-axis
and then the x-axis.

00:03:31.510 --> 00:03:32.540
Let's check our answer.

00:03:32.540 --> 00:03:34.470
We got it right.


Constructing and solving equations in the real world 1 exercise

WEBVTT
Kind: captions
Language: en

00:00:01.520 --> 00:00:03.740
Anna wants to
celebrate her birthday

00:00:03.740 --> 00:00:06.100
by eating pizza
with her friends.

00:00:06.100 --> 00:00:11.410
p boxes of pizza cost $40 total.

00:00:11.410 --> 00:00:14.710
Each box of pizza costs $8.

00:00:14.710 --> 00:00:18.664
Select all equations that
describe this situation.

00:00:18.664 --> 00:00:20.080
So before I even
look at that, let

00:00:20.080 --> 00:00:22.810
me at least try to construct
one in my own brain.

00:00:22.810 --> 00:00:28.660
So p boxes of pizza cost $40,
each box of pizza cost $8.

00:00:28.660 --> 00:00:30.035
So one way that
I could say, what

00:00:30.035 --> 00:00:33.390
would be the total cost
of pizzas in terms of p?

00:00:33.390 --> 00:00:35.830
Well it's going to be
the number of boxes

00:00:35.830 --> 00:00:38.800
that I get times
the cost per box.

00:00:38.800 --> 00:00:43.440
So it's going to be
p times 8, or 8p.

00:00:43.440 --> 00:00:45.930
And the other way of
saying that, the total cost

00:00:45.930 --> 00:00:50.140
of my box of my pizzas, 8p,
or I could say it's $40.

00:00:50.140 --> 00:00:53.050
So 8p must be equal to $40.

00:00:53.050 --> 00:00:56.650
And we have that captured right
here, in this first equation.

00:00:56.650 --> 00:01:00.030
The number of pizza boxes is
p, you multiply that times 8,

00:01:00.030 --> 00:01:01.890
because it's $8 per box.

00:01:01.890 --> 00:01:05.220
That's my total cost
of boxes, which I also

00:01:05.220 --> 00:01:06.480
know is going to be $40.

00:01:06.480 --> 00:01:09.345
So these two things must
be equal to each other.

00:01:09.345 --> 00:01:11.670
Now anything else
that I select really

00:01:11.670 --> 00:01:14.510
has to be the same thing
as this first equation,

00:01:14.510 --> 00:01:16.350
just manipulated a little bit.

00:01:16.350 --> 00:01:20.380
So over here I have
40 divided by 8 is p.

00:01:20.380 --> 00:01:22.600
So there's a couple of
ways to think about it.

00:01:22.600 --> 00:01:26.980
If I take the total amount
that I spend on pizza, $40,

00:01:26.980 --> 00:01:32.370
and I divide by the cost per
box, so if I divide by 8,

00:01:32.370 --> 00:01:34.160
that should give me
the number of boxes.

00:01:34.160 --> 00:01:35.451
And that's what this is saying.

00:01:35.451 --> 00:01:38.350
Take my total cost,
divide by my cost per box,

00:01:38.350 --> 00:01:40.430
and it should give me
the number of boxes.

00:01:40.430 --> 00:01:42.240
Another way of
thinking about it is,

00:01:42.240 --> 00:01:43.865
if you take both
sides of this equation

00:01:43.865 --> 00:01:47.370
and divide both sides by
8, on the left-hand side

00:01:47.370 --> 00:01:50.280
8p divided by 8 is p.

00:01:50.280 --> 00:01:52.170
And on the right-hand
side, 40 divided by 8

00:01:52.170 --> 00:01:53.830
is, well, 40 divided by 8.

00:01:53.830 --> 00:01:56.960
It doesn't matter if we wrote
p equals 40 divided by 8 or 40

00:01:56.960 --> 00:01:58.521
divided by 8 is equal to p.

00:01:58.521 --> 00:02:01.020
They're just saying that both
of these quantities are equal.

00:02:01.020 --> 00:02:04.890
So this is also another way
to describe the situation.

00:02:04.890 --> 00:02:07.500
Now let's think about
this right over here.

00:02:07.500 --> 00:02:11.670
8 divided by 40 is equal to p.

00:02:11.670 --> 00:02:14.710
Well this is not consistent,
either with this statement,

00:02:14.710 --> 00:02:16.890
or with some of the other
things that we checked.

00:02:16.890 --> 00:02:19.490
Where on the left-hand side
we're taking the reciprocal,

00:02:19.490 --> 00:02:21.290
and on the right-hand
side we aren't.

00:02:21.290 --> 00:02:24.660
So these two things
cannot be equivalent.

00:02:24.660 --> 00:02:27.540
Over here we have 40
divided by p is equal to 8.

00:02:27.540 --> 00:02:31.330
So if I just said, hey
look, I spent $40 on pizza,

00:02:31.330 --> 00:02:34.730
and I bought p boxes, how much
did each of those boxes cost?

00:02:34.730 --> 00:02:38.070
I would say well, I would take
40 divided by p to figure out

00:02:38.070 --> 00:02:39.810
how much each of
those boxes cost.

00:02:39.810 --> 00:02:41.830
And they tell us to
that must be equal to 8,

00:02:41.830 --> 00:02:43.590
that each box equals 8.

00:02:43.590 --> 00:02:46.350
So this seems consistent
with what we just read.

00:02:46.350 --> 00:02:49.175
And you can algebraically
manipulate any of these top two

00:02:49.175 --> 00:02:51.460
to get to this one
right over here.

00:02:51.460 --> 00:02:55.030
Right up here, if you
divide both sides by p,

00:02:55.030 --> 00:02:56.520
on the left-hand
side you have 8p

00:02:56.520 --> 00:02:58.660
divided by p, which would be 8.

00:02:58.660 --> 00:03:01.390
And on the right-hand side
40 divided by p is just 40

00:03:01.390 --> 00:03:02.220
divided by p.

00:03:02.220 --> 00:03:04.450
So it's the same exact thing.

00:03:04.450 --> 00:03:09.500
And the last one, you have
p is equal to 8 times 40.

00:03:09.500 --> 00:03:12.470
Well we already know that p
is equal to 40 divided by 8.

00:03:12.470 --> 00:03:14.820
So it's definitely not
going to be 8 times 40.

00:03:14.820 --> 00:03:17.210
So that one does not apply.

00:03:17.210 --> 00:03:18.520
So let's check our answer.

00:03:18.520 --> 00:03:19.340
We got it right.

00:03:19.340 --> 00:03:22.100
Let's do a couple more.

00:03:22.100 --> 00:03:24.780
Molly and Tory like to
eat ice cream sandwiches.

00:03:24.780 --> 00:03:27.270
Molly ate 5 ice cream
sandwiches and Tory

00:03:27.270 --> 00:03:29.880
ate n ice cream
sandwiches in one week.

00:03:29.880 --> 00:03:32.730
They ate a total of 12 ice
cream sandwiches together.

00:03:32.730 --> 00:03:34.760
Select all of the
equations that accurately

00:03:34.760 --> 00:03:35.890
describe the situation.

00:03:35.890 --> 00:03:38.260
So Molly ate 5, Tory ate n.

00:03:38.260 --> 00:03:41.180
So 5 plus n is the
total that they ate.

00:03:41.180 --> 00:03:43.187
And they tell us that
that must be equal to 12,

00:03:43.187 --> 00:03:45.020
that the total that
they ate together is 12.

00:03:45.020 --> 00:03:47.110
So 5 plus n is 12.

00:03:47.110 --> 00:03:49.962
So we immediately see
that one right over there.

00:03:49.962 --> 00:03:51.420
Now anything else
that we check has

00:03:51.420 --> 00:03:53.930
to be consistent with 5
plus n is equal to 12.

00:03:53.930 --> 00:03:56.480
Or it has to be a way to
algebraically manipulate

00:03:56.480 --> 00:03:58.170
from this to anything
else that is true.

00:03:58.170 --> 00:04:00.040
So let's go one by one.

00:04:00.040 --> 00:04:03.480
So is n minus 5 equal to 12?

00:04:03.480 --> 00:04:05.990
So n minus 5 is going to be
the difference between what

00:04:05.990 --> 00:04:07.300
Tory ate and Molly ate.

00:04:07.300 --> 00:04:08.550
And that's not going to be 12.

00:04:08.550 --> 00:04:11.030
Their sum is 12, so we're
not going to check that.

00:04:11.030 --> 00:04:12.829
Here we have 12 is
equal to n minus 5,

00:04:12.829 --> 00:04:15.120
which is an equivalent
statement to this top one, which

00:04:15.120 --> 00:04:16.286
we already know is not true.

00:04:16.286 --> 00:04:18.610
All they did was swap both
sides of these equalities.

00:04:18.610 --> 00:04:20.350
So that's not true.

00:04:20.350 --> 00:04:22.300
5 minus n is equal to 12.

00:04:22.300 --> 00:04:24.837
Well that's just the difference
between Molly and Tory.

00:04:24.837 --> 00:04:26.920
And that's definitely not
going to be equal to 12.

00:04:26.920 --> 00:04:33.940
The sum of the two is
going to be equal to 12.

00:04:33.940 --> 00:04:36.320
And then finally, well
we know this is right.

00:04:36.320 --> 00:04:40.570
And then this right over
here, n plus 5 and 5--

00:04:40.570 --> 00:04:42.530
We already know that 5
plus n is equal to 12.

00:04:42.530 --> 00:04:44.350
Well it's a completely
equivalent statement

00:04:44.350 --> 00:04:45.850
to say that 12 is
equal to n plus 5.

00:04:45.850 --> 00:04:47.975
You're just swapping the
left- and right-hand sides

00:04:47.975 --> 00:04:49.420
of the equality.

00:04:49.420 --> 00:04:50.590
So let's check our answer.

00:04:50.590 --> 00:04:52.940
Let's do one more.

00:04:52.940 --> 00:04:56.430
Mr. Herman's class is selling
candy for a school fundraiser.

00:04:56.430 --> 00:05:01.920
The class has a
goal of raising $500

00:05:01.920 --> 00:05:07.010
by selling c boxes
of candy for $2 each.

00:05:07.010 --> 00:05:09.050
Write an equation
that the students

00:05:09.050 --> 00:05:11.270
could solve to figure out
how many boxes of candy

00:05:11.270 --> 00:05:13.160
they need to sell.

00:05:13.160 --> 00:05:15.770
So let's think
about in terms of c,

00:05:15.770 --> 00:05:17.950
how much money they would raise.

00:05:17.950 --> 00:05:20.910
So let's say they sell c boxes.

00:05:20.910 --> 00:05:22.640
For each box, they get $2.

00:05:22.640 --> 00:05:25.280
So the total amount they're
going to get is 2 times c,

00:05:25.280 --> 00:05:27.924
or we can even just write
2c right over there.

00:05:27.924 --> 00:05:29.590
So this is the total
amount that they're

00:05:29.590 --> 00:05:31.080
going to raise in terms of c.

00:05:31.080 --> 00:05:34.050
Now we know that the
goal is to raise $500.

00:05:34.050 --> 00:05:38.430
So we want to figure out what c
does the total amount that they

00:05:38.430 --> 00:05:41.830
raise, does that equal 500?

00:05:41.830 --> 00:05:43.420
So this right here
is the equation

00:05:43.420 --> 00:05:45.211
that the students could
solve to figure out

00:05:45.211 --> 00:05:46.980
how many boxes of candy
they need to sell.

00:05:46.980 --> 00:05:48.390
Let me check our answer.

00:05:48.390 --> 00:05:49.920
And of course to
solve this, we're

00:05:49.920 --> 00:05:52.044
essentially saying 2
times c is equal to 500.

00:05:52.044 --> 00:05:53.960
So you might be able to
solve it in your head.

00:05:53.960 --> 00:05:55.940
You might say 2
times what is 500?

00:05:55.940 --> 00:05:57.490
And you would get 250.

00:05:57.490 --> 00:06:00.080
Or if you wanted to solve it a
little bit more systematically,

00:06:00.080 --> 00:06:02.240
you just divide both sides by 2.

00:06:02.240 --> 00:06:04.030
2c divided by 2 is c.

00:06:04.030 --> 00:06:05.800
500 divided by 2 is 250.

00:06:05.800 --> 00:06:08.900
And you would get
c is equal to 250.


How to plot inequalities on a number line

WEBVTT
Kind: captions
Language: en

00:00:02.001 --> 00:00:04.041
the amount of water I intake.

00:00:04.041 --> 00:00:05.801
And so let's define
some variable.

00:00:05.801 --> 00:00:14.721
Let's say W is equal to the
number of ounces of water

00:00:14.721 --> 00:00:16.431
I consume per day.

00:00:21.331 --> 00:00:25.831
And I've read that I
should have at least-- let

00:00:25.831 --> 00:00:30.126
me throw out a number-- 64
ounces of water per day.

00:00:30.126 --> 00:00:32.251
There's one way I could
think about, where I always

00:00:32.251 --> 00:00:34.861
want to drink more
than 64 ounces,

00:00:34.861 --> 00:00:39.971
so that would be W
is greater than 64.

00:00:39.971 --> 00:00:41.901
W here is the thing that
I want to be bigger,

00:00:41.901 --> 00:00:46.501
so the opening is to the W.
W is greater than 64 ounces.

00:00:46.501 --> 00:00:48.471
How would I depict that?

00:00:48.471 --> 00:00:52.301
Well, let me do my number
line right over here.

00:00:52.301 --> 00:00:54.361
Let's say that this is 0.

00:00:54.361 --> 00:00:56.381
This is 64.

00:00:56.381 --> 00:00:58.361
If I wanted to make
strictly greater than, so

00:00:58.361 --> 00:01:01.881
in this situation it's not cool
if I just drink exactly 64.

00:01:01.881 --> 00:01:04.841
That 64 is not greater than 64.

00:01:04.841 --> 00:01:09.181
I have to drink 64.01
ounces or 0.00001 ounces.

00:01:09.181 --> 00:01:12.491
It has to be something
that is greater than 64.

00:01:12.491 --> 00:01:16.411
So I'm not going to include
64, but anything greater

00:01:16.411 --> 00:01:21.141
than that is completely cool.

00:01:21.141 --> 00:01:23.471
Now, what if I want to
loosen things a little bit?

00:01:23.471 --> 00:01:27.381
It's OK if I drink
exactly 64 ounces or more.

00:01:27.381 --> 00:01:35.183
Well, then I could write W is
greater than or equal to 64.

00:01:35.183 --> 00:01:37.391
And the way that I would be
depict that on the number

00:01:37.391 --> 00:01:42.321
line-- and obviously, I'm
not showing all the numbers

00:01:42.321 --> 00:01:44.306
in between-- let's
say this is 0,

00:01:44.306 --> 00:01:47.161
and then we go all
the way up to 64.

00:01:47.161 --> 00:01:51.171
Well, now it's OK if I
drink exactly 64 ounces,

00:01:51.171 --> 00:01:52.881
so I'm going to fill
in the circle now.

00:01:52.881 --> 00:01:55.871
Here I opened it because
64 was not a cool number.

00:01:55.871 --> 00:01:57.921
Now, 64 is completely OK.

00:01:57.921 --> 00:02:02.081
I can drink exactly 64 ounces
of water in the day or more,

00:02:02.081 --> 00:02:06.162
and then I just go up the
number line just like that.


Greater than and less than symbols

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.730
Most of us are familiar with
the equal sign from our earliest

00:00:04.730 --> 00:00:06.330
days of arithmetic.

00:00:06.330 --> 00:00:10.480
You might see something
like 1 plus 1 is equal to 2.

00:00:10.480 --> 00:00:14.180
Now, a lot of people might
think when they see something

00:00:14.180 --> 00:00:17.440
like this that somehow equal
means give me the answer.

00:00:17.440 --> 00:00:19.100
1 plus 1 is the problem.

00:00:19.100 --> 00:00:22.540
Equal means give me the
answer and 1 plus 1 is 2.

00:00:22.540 --> 00:00:24.940
That's not what
equal actually means.

00:00:24.940 --> 00:00:28.860
Equal is actually just trying
to compare two quantities.

00:00:28.860 --> 00:00:32.250
When I write 1 plus 1
equals 2, that literally

00:00:32.250 --> 00:00:34.640
means that what I
have on the left hand

00:00:34.640 --> 00:00:39.370
side of the equal sign is the
exact same quantity as what

00:00:39.370 --> 00:00:42.420
I have on the right hand
side of the equal sign.

00:00:42.420 --> 00:00:48.990
I could have just as easily have
written 2 is equal to 1 plus 1.

00:00:48.990 --> 00:00:50.950
These two things are equal.

00:00:50.950 --> 00:00:54.250
I could have written
2 is equal to 2.

00:00:54.250 --> 00:00:56.040
This is a completely
true statement.

00:00:56.040 --> 00:00:57.500
These two things are equal.

00:00:57.500 --> 00:01:02.970
I could have written 1 plus
1 is equal to 1 plus 1.

00:01:02.970 --> 00:01:12.650
I could have written 1 plus 1
minus 1 is equal to 3 minus 2.

00:01:12.650 --> 00:01:14.860
These are both equal quantities.

00:01:14.860 --> 00:01:18.880
What I have here on
the left hand side,

00:01:18.880 --> 00:01:23.680
this is 1 plus 1 minus 1 is 1
and this right over here is 1.

00:01:23.680 --> 00:01:26.580
These are both equal quantities.

00:01:26.580 --> 00:01:29.600
Now I will introduce
you to other ways

00:01:29.600 --> 00:01:31.520
of comparing numbers.

00:01:31.520 --> 00:01:34.760
The equal sign is when I
have the exact same quantity

00:01:34.760 --> 00:01:35.805
on both sides.

00:01:35.805 --> 00:01:37.180
Now we'll think
about what we can

00:01:37.180 --> 00:01:40.430
do when we have different
quantities on both sides.

00:01:40.430 --> 00:01:47.780
So let's say I have the number
3 and I have the number 1

00:01:47.780 --> 00:01:50.140
and I want to compare them.

00:01:50.140 --> 00:01:53.130
So clearly 3 and
1 are not equal.

00:01:53.130 --> 00:01:54.630
In fact, I could
make that statement

00:01:54.630 --> 00:01:56.080
with a not equal sign.

00:01:56.080 --> 00:01:59.740
So I could say 3
does not equal 1.

00:01:59.740 --> 00:02:02.410
But let's say I want to figure
out which one is a larger

00:02:02.410 --> 00:02:03.860
and which one is smaller.

00:02:03.860 --> 00:02:08.210
So if I want to have some
symbol where I can compare them,

00:02:08.210 --> 00:02:12.900
where I can tell, where I can
state which of these is larger.

00:02:12.900 --> 00:02:16.536
And the symbol for doing that
is the greater than symbol.

00:02:20.230 --> 00:02:26.130
This literally would be
read as 3 is greater than 1.

00:02:26.130 --> 00:02:28.450
3 is a larger quantity.

00:02:28.450 --> 00:02:31.580
And if you have trouble
remembering what this means--

00:02:31.580 --> 00:02:36.105
greater than-- the larger
quantity is on the opening.

00:02:38.740 --> 00:02:41.200
I guess if you could view
this as some type of an arrow,

00:02:41.200 --> 00:02:45.370
or some type of symbol, but
this is the bigger side.

00:02:45.370 --> 00:02:47.540
Here, you have this
little teeny, tiny point

00:02:47.540 --> 00:02:49.895
and here you have the big
side, so the larger quantity

00:02:49.895 --> 00:02:50.850
is on the big side.

00:02:50.850 --> 00:02:52.860
This would literally
be read as 3

00:02:52.860 --> 00:02:55.550
is greater than--
so let me write

00:02:55.550 --> 00:03:02.420
that down-- greater than,
3 is greater than 1.

00:03:02.420 --> 00:03:05.100
And once again, it just doesn't
have to be numbers like this.

00:03:05.100 --> 00:03:06.470
I could write an expression.

00:03:06.470 --> 00:03:18.130
I could write 1 plus 1 plus 1 is
greater than, let's say, well,

00:03:18.130 --> 00:03:20.110
just one 1 right over there.

00:03:20.110 --> 00:03:21.800
This is making a comparison.

00:03:21.800 --> 00:03:23.910
But what if we had things
the other way around.

00:03:23.910 --> 00:03:28.620
What if I wanted to make
a comparison between 5

00:03:28.620 --> 00:03:33.040
and, let's say, 19.

00:03:33.040 --> 00:03:35.500
So now the greater than
symbol wouldn't apply.

00:03:35.500 --> 00:03:38.320
It's not true that 5
is greater than 19.

00:03:38.320 --> 00:03:40.800
I could say that 5
is not equal to 19.

00:03:40.800 --> 00:03:43.647
So I could still
make this statement.

00:03:43.647 --> 00:03:45.980
But what if I wanted to make
a statement about which one

00:03:45.980 --> 00:03:48.124
is larger and which
one is smaller?

00:03:48.124 --> 00:03:49.540
Well, as in plain
English, I would

00:03:49.540 --> 00:03:53.110
want to say 5 is less than 19.

00:03:53.110 --> 00:03:55.720
So I would want to say--
let me write that down-- I

00:03:55.720 --> 00:04:08.890
want to write 5 is less than 19.

00:04:08.890 --> 00:04:10.189
That's what I want to say.

00:04:10.189 --> 00:04:12.480
And so we just have to think
of a mathematical notation

00:04:12.480 --> 00:04:16.769
for writing "is less than."

00:04:16.769 --> 00:04:19.010
Well, if this is
greater than, it

00:04:19.010 --> 00:04:21.329
makes complete sense that
let's just swap it around.

00:04:21.329 --> 00:04:23.292
Let's make, once
again, the point

00:04:23.292 --> 00:04:26.400
point towards the smaller
quantity and the big side

00:04:26.400 --> 00:04:28.710
of the symbol point to
the larger quantity.

00:04:28.710 --> 00:04:30.810
So here 5 is a smaller
quantity so I'll

00:04:30.810 --> 00:04:32.390
make the point point there.

00:04:32.390 --> 00:04:37.050
And 19 is a larger quantity,
so I'll make it open like this.

00:04:37.050 --> 00:04:42.320
And so this would be read
as 5 is less than 19.

00:04:42.320 --> 00:04:46.160
5 is a smaller quantity than 19.

00:04:46.160 --> 00:04:52.700
I could also write this
as 1 plus 1 is less than 1

00:04:52.700 --> 00:04:54.620
plus 1 plus 1.

00:04:54.620 --> 00:04:56.830
It's just saying that this
statement, this quantity,

00:04:56.830 --> 00:05:03.401
1 plus 1 is less
than 1 plus 1 plus 1.


Ratio word problem: centimeters to kilometers

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.150
The scale on a map is 7
centimeters for every 10

00:00:05.150 --> 00:00:09.130
kilometers, or 7 centimeters
for 10 kilometers.

00:00:09.130 --> 00:00:11.730
If the distance
between two cities

00:00:11.730 --> 00:00:15.110
is 60 kilometers-- so
that's the actual distance--

00:00:15.110 --> 00:00:19.360
how far apart in centimeters
are the two cities on the map?

00:00:19.360 --> 00:00:21.290
Well, they give us the scale.

00:00:21.290 --> 00:00:25.300
For every 10 kilometers
in the real world,

00:00:25.300 --> 00:00:29.692
the map is going to
show 7 centimeters.

00:00:29.692 --> 00:00:31.150
Or another way to
think about it is

00:00:31.150 --> 00:00:33.650
if you see 7 centimeters
on the map, that represents

00:00:33.650 --> 00:00:36.640
10 kilometers in the real world.

00:00:36.640 --> 00:00:39.312
Now, they're saying that the
distance between two cities

00:00:39.312 --> 00:00:40.020
is 60 kilometers.

00:00:43.930 --> 00:00:51.380
So it's essentially 6 times
10 kilometers, so times 6.

00:00:51.380 --> 00:00:54.580
So if you have to 6 times
10 kilometers on the map,

00:00:54.580 --> 00:00:59.100
you're going to have 6 times
7 centimeters, so times 6.

00:00:59.100 --> 00:01:05.710
And 6 times 7 centimeters
gets you to 42 centimeters.

00:01:05.710 --> 00:01:08.560
So how far apart in centimeters
are the two cities on the map?

00:01:08.560 --> 00:01:11.202
Well, 42 centimeters.


Specifying planes in three dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.520
We've already been exposed
to points and lines.

00:00:03.520 --> 00:00:05.990
Now let's think about planes.

00:00:05.990 --> 00:00:10.230
And you can view planes as
really a flat surface that

00:00:10.230 --> 00:00:13.590
exists in three dimensions, that
goes off in every direction.

00:00:13.590 --> 00:00:17.440
So for example, if I have
a flat surface like this,

00:00:17.440 --> 00:00:19.260
and it's not curved,
and it just keeps

00:00:19.260 --> 00:00:23.540
going on and on and
on in every direction.

00:00:23.540 --> 00:00:27.579
Now the question is, how
do you specify a plane?

00:00:27.579 --> 00:00:29.120
Well, you might say,
well, let's see.

00:00:29.120 --> 00:00:30.536
Let's think about
it a little bit.

00:00:30.536 --> 00:00:35.120
Could I specify a plane with
a one point, right over here?

00:00:35.120 --> 00:00:37.560
Let's call that point,
A. Would that, alone,

00:00:37.560 --> 00:00:40.170
be able to specify a plane?

00:00:40.170 --> 00:00:41.940
Well, there's an
infinite number of planes

00:00:41.940 --> 00:00:43.630
that could go
through that point.

00:00:43.630 --> 00:00:47.250
I could have a plane that
goes like this, where

00:00:47.250 --> 00:00:50.140
that point, A,
sits on that plane.

00:00:50.140 --> 00:00:52.200
I could have a plane like that.

00:00:52.200 --> 00:00:55.100
Or, I could have
a plane like this.

00:00:55.100 --> 00:00:57.330
I could have a plane
like this where

00:00:57.330 --> 00:00:59.370
point A sits on it, as well.

00:00:59.370 --> 00:01:01.210
So I could have a
plane like that.

00:01:01.210 --> 00:01:03.560
And I could just keep
rotating around A.

00:01:03.560 --> 00:01:06.070
So one point by
itself does not seem

00:01:06.070 --> 00:01:08.600
to be sufficient
to define a plane.

00:01:08.600 --> 00:01:10.460
Well, what about two points?

00:01:10.460 --> 00:01:13.770
Let's say I had a point,
B, right over here.

00:01:13.770 --> 00:01:17.010
Well, notice the way I
drew this, point A and B,

00:01:17.010 --> 00:01:18.330
they would define a line.

00:01:18.330 --> 00:01:21.520
For example, they would define
this line right over here.

00:01:21.520 --> 00:01:24.010
So they would define,
they could define,

00:01:24.010 --> 00:01:26.260
this line right over here.

00:01:26.260 --> 00:01:29.300
But both of these points and
in fact, this entire line,

00:01:29.300 --> 00:01:31.880
exists on both of these
planes that I just drew.

00:01:31.880 --> 00:01:33.650
And I could keep
rotating these planes.

00:01:33.650 --> 00:01:36.120
I could have a plane
that looks like this.

00:01:36.120 --> 00:01:38.010
I could have a plane
that looks like this,

00:01:38.010 --> 00:01:41.760
that both of these
points actually sit on.

00:01:41.760 --> 00:01:44.330
I'm essentially just
rotating around this line

00:01:44.330 --> 00:01:47.560
that is defined by
both of these points.

00:01:47.560 --> 00:01:50.740
So two points does not
seem to be sufficient.

00:01:50.740 --> 00:01:53.170
Let's try three.

00:01:53.170 --> 00:01:55.540
So there's no way
that I could put--

00:01:55.540 --> 00:01:56.920
Well, let's be careful here.

00:01:56.920 --> 00:02:00.150
So I could put a third
point right over here,

00:02:00.150 --> 00:02:02.870
point C. And C
sits on that line,

00:02:02.870 --> 00:02:04.380
and C sits on all
of these planes.

00:02:04.380 --> 00:02:07.500
So it doesn't seem like
just a random third point

00:02:07.500 --> 00:02:12.210
is sufficient to define, to pick
out any one of these planes.

00:02:12.210 --> 00:02:13.830
But what if we
make the constraint

00:02:13.830 --> 00:02:16.840
that the three points are
not all on the same line.

00:02:16.840 --> 00:02:19.820
Obviously, two points
will always define a line.

00:02:19.820 --> 00:02:21.880
But what if the three
points are not collinear.

00:02:21.880 --> 00:02:24.030
So instead of
picking C as a point,

00:02:24.030 --> 00:02:26.560
what if we pick--
Is there any way

00:02:26.560 --> 00:02:29.880
to pick a point, D, that
is not on this line, that

00:02:29.880 --> 00:02:33.501
is on more than one
of these planes?

00:02:33.501 --> 00:02:34.000
We'll, no.

00:02:34.000 --> 00:02:36.083
If I say, well, let's see,
the point D-- Let's say

00:02:36.083 --> 00:02:38.182
point D is right over here.

00:02:38.182 --> 00:02:39.890
So it sits on this
plane right over here,

00:02:39.890 --> 00:02:41.730
one of the first
ones that I drew.

00:02:41.730 --> 00:02:44.040
So point D sits on that plane.

00:02:44.040 --> 00:02:47.190
Between point D, A, and
B, there's only one plane

00:02:47.190 --> 00:02:49.780
that all three of
those points sit on.

00:02:49.780 --> 00:02:54.820
So a plane is defined by
three non-colinear points.

00:02:54.820 --> 00:02:59.050
So D, A, and B, you see, do
not sit on the same line.

00:02:59.050 --> 00:03:00.760
A and B can sit
on the same line.

00:03:00.760 --> 00:03:02.290
D and A can sit
on the same line.

00:03:02.290 --> 00:03:04.520
D and B can sit
on the same line.

00:03:04.520 --> 00:03:09.510
But A, B, and D does not sit
on-- They are non-colinear.

00:03:09.510 --> 00:03:12.720
So for example, right
over here in this diagram,

00:03:12.720 --> 00:03:13.990
we have a plane.

00:03:13.990 --> 00:03:16.190
This plane is labeled, S.

00:03:16.190 --> 00:03:18.990
But another way that we
can specify plane S is we

00:03:18.990 --> 00:03:21.580
could say, plane--
And we just have

00:03:21.580 --> 00:03:25.170
to find three non-collinear
points on that plane.

00:03:25.170 --> 00:03:27.030
So we could call this plane AJB.

00:03:31.570 --> 00:03:33.178
We could call it plane JBW.

00:03:39.820 --> 00:03:43.352
We could call it plane-- and I
could keep going-- plane WJA.

00:03:47.050 --> 00:03:49.330
But I could not specify
this plane, uniquely,

00:03:49.330 --> 00:03:56.864
by saying plane ABW.

00:03:56.864 --> 00:03:58.280
And the reason why
I can't do this

00:03:58.280 --> 00:04:01.080
is because ABW are
all on the same line.

00:04:01.080 --> 00:04:03.420
And this line sits on an
infinite number of planes.

00:04:03.420 --> 00:04:06.280
I could keep rotating around the
line, just as we did over here.

00:04:06.280 --> 00:04:10.669
It does not specify
only one plane.


Regrouping (borrowing) twice example

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:05.660
We've got 9,601 minus 8,023.

00:00:05.660 --> 00:00:08.420
And immediately when we try to
start subtracting in our ones

00:00:08.420 --> 00:00:09.490
place, we have a problem.

00:00:09.490 --> 00:00:11.290
This 3 is larger than this 1.

00:00:11.290 --> 00:00:13.290
And we also have that
problem in the tens place.

00:00:13.290 --> 00:00:16.079
This 2 is larger than this 0.

00:00:16.079 --> 00:00:18.250
So we're going to have to
do some type of borrowing

00:00:18.250 --> 00:00:19.152
or regrouping.

00:00:19.152 --> 00:00:20.860
And so the way I like
to think about it--

00:00:20.860 --> 00:00:22.610
I like to go to the
first place value that

00:00:22.610 --> 00:00:23.640
has something to give.

00:00:23.640 --> 00:00:26.182
Obviously, the tens
place is in no position

00:00:26.182 --> 00:00:27.640
to give anything
to the ones place.

00:00:27.640 --> 00:00:29.112
It needs things itself.

00:00:29.112 --> 00:00:31.070
And so we're going to go
to the hundreds place.

00:00:31.070 --> 00:00:33.690
And the hundreds place
has an abundance of value

00:00:33.690 --> 00:00:36.540
that it can regroup into
the tens and ones place.

00:00:36.540 --> 00:00:40.270
This 6 right over
here represents 600.

00:00:40.270 --> 00:00:43.940
So why don't we take
100 from that 600--

00:00:43.940 --> 00:00:49.220
so then this will become
500-- and then give that 100

00:00:49.220 --> 00:00:51.550
over to the tens place.

00:00:51.550 --> 00:00:55.040
Now, if we give 100 to
the tens place, how would

00:00:55.040 --> 00:00:57.050
I represent that
in the tens place?

00:00:57.050 --> 00:00:59.480
Well, I have zero 10's.

00:00:59.480 --> 00:01:01.380
And now I'm going to give 100.

00:01:01.380 --> 00:01:04.390
100 is the same
thing as 10 10's.

00:01:04.390 --> 00:01:07.390
It's going to be 0 plus 100.

00:01:07.390 --> 00:01:10.980
100 in the tens
place is just 10.

00:01:10.980 --> 00:01:12.720
So let me write it this way.

00:01:12.720 --> 00:01:18.269
So this right over here is now
going to be rewritten as 10.

00:01:18.269 --> 00:01:20.060
Now, you might be
saying, wait, wait, wait.

00:01:20.060 --> 00:01:21.500
What's going on here, Sal?

00:01:21.500 --> 00:01:23.650
You took 100 from
the hundreds place.

00:01:23.650 --> 00:01:25.410
That's why it became 500.

00:01:25.410 --> 00:01:28.520
Now, why did this
become 10 and not 100?

00:01:28.520 --> 00:01:31.050
Now remember, this is 10 10's.

00:01:31.050 --> 00:01:33.740
So this is still
representing 100.

00:01:33.740 --> 00:01:36.430
You have not changed the
value of this top number.

00:01:36.430 --> 00:01:41.730
Before, the value was
9,000 plus 600 plus 1.

00:01:41.730 --> 00:01:49.210
Now it's 9,000 plus 500 plus
100-- 10 10's is 100-- plus 1.

00:01:49.210 --> 00:01:51.380
I have not changed
the value here.

00:01:51.380 --> 00:01:52.812
Now, we're still not done yet.

00:01:52.812 --> 00:01:54.770
We don't want to just
subtract because we still

00:01:54.770 --> 00:01:56.311
have the problem
with the ones place.

00:01:56.311 --> 00:01:59.560
The ones place still
doesn't have enough value.

00:01:59.560 --> 00:02:03.120
Now, the good thing is we've
given some value to the tens

00:02:03.120 --> 00:02:03.670
place.

00:02:03.670 --> 00:02:06.610
So why don't we take
10 from the tens place?

00:02:06.610 --> 00:02:09.490
So if you have 10 10's,
and you take one 10 away,

00:02:09.490 --> 00:02:13.190
you're going to be left
with nine 10's, or 90.

00:02:13.190 --> 00:02:17.320
And then we can take that 10
and give it to the ones place

00:02:17.320 --> 00:02:18.350
So let's do that.

00:02:18.350 --> 00:02:20.190
You take that 10 we
just took from there,

00:02:20.190 --> 00:02:21.640
and you give it
to the ones place.

00:02:21.640 --> 00:02:24.050
You now have 11 here.

00:02:24.050 --> 00:02:26.670
And now we are
ready to subtract.

00:02:26.670 --> 00:02:29.820
11 minus 3 is 8.

00:02:29.820 --> 00:02:34.140
9 minus 2-- and this is
really 90 minus 20-- is 70.

00:02:34.140 --> 00:02:37.810
But in the tens place,
we represent that as a 7.

00:02:37.810 --> 00:02:42.530
500 minus zero hundred is
500, represented as a 5

00:02:42.530 --> 00:02:43.530
in the hundreds place.

00:02:43.530 --> 00:02:47.370
9,000 minus 8,000 is 1,000.

00:02:47.370 --> 00:02:48.540
And we're done.

00:02:48.540 --> 00:02:50.100
And just to make
things really clear,

00:02:50.100 --> 00:02:51.520
I'm going to redo
this problem now

00:02:51.520 --> 00:02:53.370
but with things expanded out.

00:02:53.370 --> 00:03:06.590
So this first number is 9,000
plus 600 plus zero 10's plus 1.

00:03:06.590 --> 00:03:11.220
And this number right here,
we're subtracting 8,000.

00:03:11.220 --> 00:03:13.795
We're subtracting zero 100's.

00:03:13.795 --> 00:03:16.650
We're subtracting two
10's, which is 20.

00:03:16.650 --> 00:03:18.345
Subtracting 20.

00:03:18.345 --> 00:03:20.440
And subtracting three 1's.

00:03:20.440 --> 00:03:23.370
So I have just rewritten
this exact same statement.

00:03:23.370 --> 00:03:25.970
But the regrouping
and the borrowing

00:03:25.970 --> 00:03:27.970
is going to become a
little bit clearer now.

00:03:27.970 --> 00:03:30.660
So the same exact
thing-- we said, hey,

00:03:30.660 --> 00:03:33.900
we can't subtract the 3 from
the 1 or the 20 from the 0.

00:03:33.900 --> 00:03:36.810
But we have a lot of value
right over here in the 600.

00:03:36.810 --> 00:03:39.260
So why don't we
take 100 from that?

00:03:39.260 --> 00:03:41.340
So this becomes 500.

00:03:41.340 --> 00:03:45.310
And we give that 100
to the tens place.

00:03:45.310 --> 00:03:50.830
So this becomes 100.

00:03:50.830 --> 00:03:53.015
Notice, the value
has not changed.

00:03:53.015 --> 00:03:56.310
This is 9,000 plus
500 plus 100 plus 1.

00:03:56.310 --> 00:03:59.300
That's the same thing as
9,000 plus 600 plus 1.

00:03:59.300 --> 00:04:01.620
We've just put the value
in different places.

00:04:01.620 --> 00:04:04.130
And here we have
explicitly written 100.

00:04:04.130 --> 00:04:06.770
But when we represent it
in the tens place, 10 10's

00:04:06.770 --> 00:04:08.750
is the same thing as 100.

00:04:08.750 --> 00:04:11.020
Now, we aren't done
regrouping just yet.

00:04:11.020 --> 00:04:13.440
We want to give some
value to the ones place.

00:04:13.440 --> 00:04:18.620
So we can take 10 from the tens
place-- and this becomes a 90--

00:04:18.620 --> 00:04:21.420
and give that 10
to the ones place.

00:04:21.420 --> 00:04:24.350
10 plus 1 is 11.

00:04:24.350 --> 00:04:26.890
So notice, I did
the exact borrowing,

00:04:26.890 --> 00:04:29.000
the exact regrouping,
that I did here.

00:04:29.000 --> 00:04:30.980
I just represented it
a little bit different.

00:04:30.980 --> 00:04:34.290
This 500 was represented by
a 5 in the hundreds place.

00:04:34.290 --> 00:04:37.510
This 90 was represented
by a 9 in the tens place.

00:04:37.510 --> 00:04:39.570
But either way, we're
ready to subtract now.

00:04:39.570 --> 00:04:42.906
11 minus 3 is 8.

00:04:42.906 --> 00:04:45.168
90 minus 20 is 70.

00:04:48.520 --> 00:04:49.390
Write a plus there.

00:04:49.390 --> 00:04:53.350
500 minus 0 is 500.

00:04:53.350 --> 00:04:57.210
And then 9,000 minus
8,000 is 1,000.

00:04:57.210 --> 00:04:59.550
And we got the same
result because 1,000

00:04:59.550 --> 00:05:03.525
plus 500 plus 70
plus 8 is 1,578.


Borrowing once example 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.630
We've got 6,798 minus 3,359.

00:00:07.630 --> 00:00:09.890
So let's see how far we can
get with the subtraction.

00:00:09.890 --> 00:00:12.026
So immediately, when we
go into the ones place,

00:00:12.026 --> 00:00:13.900
we're going to try to
subtract a 9 from an 8.

00:00:13.900 --> 00:00:17.250
So we immediately reach a
little bit of a stumbling block.

00:00:17.250 --> 00:00:20.000
And to see what our options are
here in terms of regrouping,

00:00:20.000 --> 00:00:22.050
I'm going to rewrite
both of these numbers.

00:00:22.050 --> 00:00:25.290
So I'm going to rewrite
6,798 literally.

00:00:25.290 --> 00:00:27.950
So this is equal to 6,000.

00:00:27.950 --> 00:00:29.800
That's this right over here.

00:00:29.800 --> 00:00:45.990
That's 6,000, plus
700, plus 90, plus 8.

00:00:45.990 --> 00:00:49.110
Minus all of this.

00:00:49.110 --> 00:00:51.630
So I could subtract
each of the places.

00:00:51.630 --> 00:00:59.447
So I could say this is
going to be minus 3,000

00:00:59.447 --> 00:01:10.046
minus 300 minus 50--
a 5 in a tens place

00:01:10.046 --> 00:01:12.190
is just 50-- minus 9.

00:01:16.336 --> 00:01:18.826
So here, we're just
explicitly showing

00:01:18.826 --> 00:01:20.340
what those place
values represent.

00:01:20.340 --> 00:01:22.480
A 6 in the thousands
place is 6,000.

00:01:22.480 --> 00:01:24.992
A 3 in the hundreds
place is 300.

00:01:24.992 --> 00:01:26.200
Let's go back to our problem.

00:01:26.200 --> 00:01:28.350
We wanted to subtract
a 9 from an 8.

00:01:28.350 --> 00:01:30.437
Well, that's a little
bit of a stumbling block.

00:01:30.437 --> 00:01:33.020
But what if we could take some
of the value from some of these

00:01:33.020 --> 00:01:34.330
and give it to the 8?

00:01:34.330 --> 00:01:37.970
In particular, we could go
one place value up to the 90.

00:01:37.970 --> 00:01:39.830
And what if we were to
take 10 from the 90?

00:01:39.830 --> 00:01:40.790
So let's do that.

00:01:40.790 --> 00:01:44.930
If we were to take 10 from
the 90, then 90 becomes 80.

00:01:44.930 --> 00:01:47.607
But we don't want to change
the value of the entire number.

00:01:47.607 --> 00:01:50.350
So we're going to give
that 10 to this 8.

00:01:54.710 --> 00:01:56.740
We're essentially
regrouping right over here.

00:01:56.740 --> 00:01:59.790
And then that 8
can become an 18.

00:01:59.790 --> 00:02:02.030
Notice, I did not change
the value of the number.

00:02:02.030 --> 00:02:04.280
I just essentially changed
how I represented it.

00:02:04.280 --> 00:02:07.360
Instead of saying it's 6,000
plus 700 plus 90 plus 8,

00:02:07.360 --> 00:02:12.170
I'm just saying that it's
6,000 plus 700 plus 80 plus 18.

00:02:12.170 --> 00:02:15.320
Those are both going
to give you 6,798.

00:02:15.320 --> 00:02:17.010
But now it becomes
a little bit easier

00:02:17.010 --> 00:02:19.327
for us to actually subtract.

00:02:19.327 --> 00:02:26.270
Now, if we subtract, I have
18 minus 9, which is 9.

00:02:26.270 --> 00:02:28.160
I have 80-- not 90, now.

00:02:28.160 --> 00:02:33.700
I have 80 minus 50, which is 30.

00:02:33.700 --> 00:02:36.190
And these are all positive,
so this is plus 9.

00:02:36.190 --> 00:02:37.540
This is a positive 30.

00:02:37.540 --> 00:02:39.740
80 minus 50 is 30.

00:02:39.740 --> 00:02:44.030
I have 700 minus
300, which is 400.

00:02:44.030 --> 00:02:49.970
And I have 6,000 minus
3,000, which is 3,000.

00:02:49.970 --> 00:02:53.470
So this is literally going to be
3,000 plus 400 plus 30 plus 9,

00:02:53.470 --> 00:02:56.160
or 3,439.

00:02:56.160 --> 00:02:57.829
Now, how would you
do it if you didn't

00:02:57.829 --> 00:02:59.120
want to write it out like this?

00:02:59.120 --> 00:03:01.536
And this is where you'll see
a kind of shorthand notation.

00:03:01.536 --> 00:03:03.510
This is often called borrowing.

00:03:03.510 --> 00:03:05.620
So you say, look, I've got
an 8 in the ones place.

00:03:05.620 --> 00:03:08.760
I want to take a
10 from this 90.

00:03:08.760 --> 00:03:13.056
So it's going to become an 80.

00:03:13.056 --> 00:03:15.680
But we'll just write it as an 8
because it's in the tens place.

00:03:15.680 --> 00:03:17.890
This 8 represents the 80.

00:03:17.890 --> 00:03:21.220
And I'm going to give
that 10 to the ones place.

00:03:21.220 --> 00:03:26.480
So 10 plus 8 is 18.

00:03:26.480 --> 00:03:27.870
And now you can subtract.

00:03:27.870 --> 00:03:30.030
18 minus 9 is 9.

00:03:30.030 --> 00:03:33.350
8 minus 5 is 3.

00:03:33.350 --> 00:03:36.610
7 minus 3 is 4.

00:03:36.610 --> 00:03:41.390
6 minus 3 is 3-- 3,439.


Borrowing once example 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.730
We have 7,329 minus 6,278.

00:00:07.730 --> 00:00:10.550
So let's go place by place
and see if we can subtract.

00:00:10.550 --> 00:00:15.270
6,278 is clearly
less than 7,329,

00:00:15.270 --> 00:00:17.140
so we should be able to do this.

00:00:17.140 --> 00:00:18.750
So first, we go
to the ones place.

00:00:18.750 --> 00:00:21.200
We're subtracting an 8 from a 9.

00:00:21.200 --> 00:00:22.630
That seems pretty
straightforward.

00:00:22.630 --> 00:00:24.980
That's just going to be a
1, a 1 in the ones place.

00:00:24.980 --> 00:00:26.450
It literally just represents 1.

00:00:26.450 --> 00:00:28.490
Then we go to the tens place.

00:00:28.490 --> 00:00:30.910
And we're trying to
subtract a 7 from a 2.

00:00:30.910 --> 00:00:32.580
And this is really
representing 70.

00:00:32.580 --> 00:00:34.530
And this is really
representing 20.

00:00:34.530 --> 00:00:37.390
Well, now we're hitting a
bit of a stumbling block.

00:00:37.390 --> 00:00:39.800
So we're going to have
to regroup or borrow.

00:00:39.800 --> 00:00:41.610
And to understand
what we're doing,

00:00:41.610 --> 00:00:44.010
let's rewrite both
of these numbers.

00:00:44.010 --> 00:00:52.380
So 7,329-- 7 we can rewrite
as being equal to 7,000.

00:00:52.380 --> 00:00:56.670
Plus 300-- so this 3
in the hundreds place

00:00:56.670 --> 00:00:58.350
is representing 300.

00:00:58.350 --> 00:01:00.580
The 7 in the thousands
place is 7,000.

00:01:00.580 --> 00:01:03.430
3 in the hundreds place is 300.

00:01:03.430 --> 00:01:07.690
The 2 in the tens place,
that's two 10's, or 20.

00:01:07.690 --> 00:01:11.760
And then the 9 in the ones
place is just going to be 9.

00:01:11.760 --> 00:01:16.340
So this is another way
of representing 7,329.

00:01:16.340 --> 00:01:20.310
And then down here, we have
the 6 in the thousands place.

00:01:20.310 --> 00:01:22.200
Well, that's going to be 6,000.

00:01:22.200 --> 00:01:25.800
And we're subtracting,
so minus 6,000.

00:01:25.800 --> 00:01:28.692
And then here, we have a
2 in the hundreds place.

00:01:28.692 --> 00:01:31.150
And once again, we're going to
be subtracting all of these.

00:01:31.150 --> 00:01:34.060
So we're going to
be subtracting 200.

00:01:34.060 --> 00:01:38.099
And then here in the tens
place, we have our 7.

00:01:38.099 --> 00:01:39.140
And we're subtracting it.

00:01:39.140 --> 00:01:41.560
So seven 10's, that's 70.

00:01:41.560 --> 00:01:46.670
And then we are
subtracting that 8.

00:01:46.670 --> 00:01:51.870
And what we've already
done is said, hey, look.

00:01:51.870 --> 00:01:54.260
Subtracting 8 from 9?

00:01:54.260 --> 00:01:56.576
That's just going to be 1.

00:01:56.576 --> 00:01:58.200
But then we got over
here, and we said,

00:01:58.200 --> 00:02:01.400
hey, how are we going
to subtract 70 from 20?

00:02:01.400 --> 00:02:04.380
And the key here is to regroup
some of the value up here

00:02:04.380 --> 00:02:05.940
and give it to
the tens place, so

00:02:05.940 --> 00:02:08.100
that we can subtract 70 from it.

00:02:08.100 --> 00:02:11.920
And the most natural place to
go is one place value above.

00:02:11.920 --> 00:02:18.640
So we could take
100 from the 300.

00:02:18.640 --> 00:02:20.740
So then it will become 200.

00:02:20.740 --> 00:02:24.800
We're going to give 100--
we're going to give that 100

00:02:24.800 --> 00:02:26.010
to the tens place.

00:02:26.010 --> 00:02:31.670
So it is going to become 120.

00:02:31.670 --> 00:02:35.340
Notice, 200 plus 120 is 320.

00:02:35.340 --> 00:02:37.220
300 plus 20 is 320.

00:02:37.220 --> 00:02:39.320
We have not changed the
value of the number.

00:02:39.320 --> 00:02:42.470
We've just changed what place
we're representing it in.

00:02:42.470 --> 00:02:44.830
If we wanted to do it
here, we could say--

00:02:44.830 --> 00:02:48.140
and when you think of it this
way, this is really regrouping,

00:02:48.140 --> 00:02:49.660
and this is really
what's happening.

00:02:49.660 --> 00:02:51.993
But if you want to think of
it in a borrowing framework,

00:02:51.993 --> 00:02:55.420
you could say, hey,
let's take 1 from the 3.

00:02:55.420 --> 00:02:57.670
Although it's a 300, so
you're really taking 100.

00:02:57.670 --> 00:02:59.030
That becomes a 2.

00:02:59.030 --> 00:03:01.320
And you give that 1
to the tens place.

00:03:01.320 --> 00:03:05.030
And so that becomes a 12.

00:03:05.030 --> 00:03:07.670
Now, what was really
happening is you took 100.

00:03:07.670 --> 00:03:08.880
You gave it to the 20.

00:03:08.880 --> 00:03:10.170
It became 120.

00:03:10.170 --> 00:03:11.480
But now you can subtract.

00:03:11.480 --> 00:03:14.240
Here, you'd say, well,
what's 120 minus 70?

00:03:14.240 --> 00:03:17.970
Well, 120 minus 70
is going to be 50.

00:03:17.970 --> 00:03:20.630
Over here you could say,
well, what's 12 minus 7?

00:03:20.630 --> 00:03:22.039
Well, that's 5.

00:03:22.039 --> 00:03:23.830
But it's still representing
the same thing.

00:03:23.830 --> 00:03:25.880
12 10's is 120.

00:03:25.880 --> 00:03:27.590
Seven 10's is 70.

00:03:27.590 --> 00:03:30.720
And they give you five 10's,
which is the same thing as 50.

00:03:30.720 --> 00:03:33.110
This 5 represents that 50.

00:03:33.110 --> 00:03:35.010
And then we can go
to the other places.

00:03:35.010 --> 00:03:36.180
You say 2 minus 2.

00:03:36.180 --> 00:03:37.860
Well, that's zero 100's.

00:03:37.860 --> 00:03:41.660
And then 7,000 minus
6,000 is 1,000.

00:03:41.660 --> 00:03:47.770
And once again, right over here,
200 minus 200 is zero 100's.

00:03:47.770 --> 00:03:52.880
And then 7,000 minus
6,000 is 1,000.

00:03:52.880 --> 00:03:55.010
So this is going to be
1,000 plus 0 plus 50

00:03:55.010 --> 00:03:59.559
plus 1, which is the
exact same thing as 1,051.

00:03:59.559 --> 00:04:01.100
The important thing
is to visualize--

00:04:01.100 --> 00:04:03.135
you don't have to write
this out every time.

00:04:03.135 --> 00:04:04.926
But to make sure you
visualize in your head

00:04:04.926 --> 00:04:08.464
that this 3 is representing
300, that this 2 represents 20,

00:04:08.464 --> 00:04:11.610
that when you're taking
100 from the 300,

00:04:11.610 --> 00:04:14.480
then you would represent that
as a 2 in the hundreds place.

00:04:14.480 --> 00:04:16.740
And then when you give
100 to the tens place,

00:04:16.740 --> 00:04:22.307
it's essentially that the
two 10's will become 12 10's.

00:04:22.307 --> 00:04:23.890
Because you're giving
it 10 more 10's.

00:04:23.890 --> 00:04:25.100
You're giving it 100.

00:04:25.100 --> 00:04:27.810
So hopefully, that
makes some sense.


Using expressions to understand relationships

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:05.820
Let's say that we have
three integers, a, b, and c,

00:00:05.820 --> 00:00:08.880
and we know that all of these
integers are greater than 0.

00:00:08.880 --> 00:00:13.460
So they're integers, and
they are greater than 0.

00:00:13.460 --> 00:00:19.940
And we also know that
the expression a plus b

00:00:19.940 --> 00:00:22.987
over c, that this
is also an integer.

00:00:22.987 --> 00:00:25.070
The entire expression, if
you were to evaluate it,

00:00:25.070 --> 00:00:27.840
is also an integer.

00:00:27.840 --> 00:00:35.080
And then finally, we know
that a is divisible--

00:00:35.080 --> 00:00:38.420
or another way of saying it,
that a is a multiple of c,

00:00:38.420 --> 00:00:42.950
so a is divisible by c, which
is another way of saying

00:00:42.950 --> 00:00:51.020
is a is a multiple of c.

00:00:51.020 --> 00:00:52.510
So this is what we know.

00:00:52.510 --> 00:00:56.300
a, b, and c are integers,
all greater than 0.

00:00:56.300 --> 00:00:58.690
We know that the
expression a plus b over c

00:00:58.690 --> 00:01:01.900
is also an integer, and
that a is a multiple of c,

00:01:01.900 --> 00:01:05.730
or a another way is that c
divides perfectly into a.

00:01:05.730 --> 00:01:09.280
So our question for you or
the question for all of us

00:01:09.280 --> 00:01:16.970
to work out right now
is, is b a multiple?

00:01:16.970 --> 00:01:19.590
Does b have to be
a multiple of c?

00:01:19.590 --> 00:01:21.330
Let me write it that way.

00:01:21.330 --> 00:01:27.400
Does b-- given all of
these constraints--

00:01:27.400 --> 00:01:37.852
does b have to be
a multiple of c?

00:01:37.852 --> 00:01:39.560
So let's see how we
can-- and I encourage

00:01:39.560 --> 00:01:42.710
you to pause the video right now
to come up with your own answer

00:01:42.710 --> 00:01:47.250
about whether b has
to be a multiple of c.

00:01:47.250 --> 00:01:49.649
So now that you've unpaused
things, let's work it out.

00:01:49.649 --> 00:01:51.940
So let's go to our original
expression right over here.

00:01:51.940 --> 00:01:55.300
We have a plus b over
c, and really one way

00:01:55.300 --> 00:01:59.122
to tackle this, is to
really just play around

00:01:59.122 --> 00:02:01.080
with this expression,
and see if we can come up

00:02:01.080 --> 00:02:02.920
with any conclusions here.

00:02:02.920 --> 00:02:06.070
So, one, we could try to
rewrite a plus b over c.

00:02:06.070 --> 00:02:14.130
We could rewrite
that as a/c plus b/c,

00:02:14.130 --> 00:02:16.090
and this expression is
the exact same thing

00:02:16.090 --> 00:02:17.840
as our first expression.

00:02:17.840 --> 00:02:19.300
So we know that
this entire thing

00:02:19.300 --> 00:02:21.130
is going to be an integer.

00:02:21.130 --> 00:02:24.220
That whole thing is
going to be an integer.

00:02:24.220 --> 00:02:26.280
Now, what do we know
about these parts?

00:02:26.280 --> 00:02:30.200
Well a/c, this is
a divided by c.

00:02:30.200 --> 00:02:32.160
We know that a is
divisible by c.

00:02:32.160 --> 00:02:34.240
We know that a is
a multiple of c.

00:02:34.240 --> 00:02:37.330
So divided by c, this is
going to be an integer,

00:02:37.330 --> 00:02:38.980
so let me write that.

00:02:38.980 --> 00:02:44.170
So, this information
right over here

00:02:44.170 --> 00:02:47.720
tells us that this thing right
over here-- that a divided

00:02:47.720 --> 00:02:50.710
by c-- is going
to be an integer.

00:02:50.710 --> 00:02:53.370
This is going to be an integer.

00:02:53.370 --> 00:02:57.200
Now, if I have an integer,
and I add something to it,

00:02:57.200 --> 00:02:59.920
and the whole thing
is an integer,

00:02:59.920 --> 00:03:03.440
well the thing that I'm adding
to it must be an integer.

00:03:03.440 --> 00:03:06.590
The only way that I get
an integer plus something

00:03:06.590 --> 00:03:08.710
to be an integer is if
the thing I'm adding it to

00:03:08.710 --> 00:03:11.670
is also an integer.

00:03:11.670 --> 00:03:13.830
So this-- there's
no way that I could

00:03:13.830 --> 00:03:17.760
add an integer to a non-integer
and then get an integer,

00:03:17.760 --> 00:03:20.735
so this has to be an integer.

00:03:24.780 --> 00:03:29.150
And if b/c is an integer,
that means that, yes,

00:03:29.150 --> 00:03:32.730
b must be a multiple of c.

00:03:32.730 --> 00:03:36.466
So the answer here is yes.


Reasoning through inequality expressions

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.730
You go for a job interview.

00:00:03.730 --> 00:00:06.380
And the first thing that your
interviewer says is, look,

00:00:06.380 --> 00:00:08.080
you have great work experience.

00:00:08.080 --> 00:00:10.370
You seem like a
nice young person.

00:00:10.370 --> 00:00:13.580
But what I really care about
is your logical reasoning

00:00:13.580 --> 00:00:14.162
capabilities.

00:00:14.162 --> 00:00:15.620
So what she says
is, just sit down.

00:00:15.620 --> 00:00:18.890
I'm going to ask you a question
about some math expressions.

00:00:18.890 --> 00:00:21.170
And you say, sure, shoot away.

00:00:21.170 --> 00:00:23.260
Give me some questions
about math expressions.

00:00:23.260 --> 00:00:24.990
And she says, all right.

00:00:24.990 --> 00:00:30.460
So you have two integers,
integer a and integer b.

00:00:30.460 --> 00:00:34.340
And she tells you that
integer a is greater than 0,

00:00:34.340 --> 00:00:38.280
and integer b is less than 0.

00:00:38.280 --> 00:00:43.290
Then she says that
we also know that a/b

00:00:43.290 --> 00:00:48.690
is greater than a times b.

00:00:48.690 --> 00:00:52.020
So then she says, tell me
some interesting things

00:00:52.020 --> 00:00:57.300
about a and b and
a/b and a times b.

00:00:57.300 --> 00:00:59.540
And you say, OK, I'll
give my best shot.

00:00:59.540 --> 00:01:01.950
And I encourage you to
actually pause now and try

00:01:01.950 --> 00:01:04.239
to think about how these
things might relate,

00:01:04.239 --> 00:01:06.030
or how b might be
constrained, or a

00:01:06.030 --> 00:01:09.430
might be constrained,
and then unpause it.

00:01:09.430 --> 00:01:11.840
So I assume you've
unpaused it, so let's think

00:01:11.840 --> 00:01:13.380
about this a little bit.

00:01:13.380 --> 00:01:17.570
So the first thing, we know that
a is positive, b is negative.

00:01:17.570 --> 00:01:19.980
So if I take a positive
divided by a negative,

00:01:19.980 --> 00:01:21.720
what am I going to get?

00:01:21.720 --> 00:01:24.220
Well, this right over here, a
positive divided by a negative

00:01:24.220 --> 00:01:26.220
is going to be a negative.

00:01:26.220 --> 00:01:28.620
What happens if I take a
positive times a negative?

00:01:28.620 --> 00:01:30.360
Well, ab is also going
to be a negative.

00:01:30.360 --> 00:01:32.957
A positive times a negative
is also a negative.

00:01:32.957 --> 00:01:34.540
So really what we're
saying is that we

00:01:34.540 --> 00:01:38.460
have two negative quantities,
and this one right over here

00:01:38.460 --> 00:01:40.510
is greater than that one.

00:01:40.510 --> 00:01:44.460
So let's visualize
that on a number line.

00:01:44.460 --> 00:01:47.840
So let's say that this
right over here is 0.

00:01:47.840 --> 00:01:50.860
This right over here is
the positive direction.

00:01:50.860 --> 00:01:52.230
This is the negative direction.

00:01:52.230 --> 00:01:55.220
We know both of these are
negative, but a/b is larger.

00:01:55.220 --> 00:01:59.030
So a/b is going to be to
the right of a times b.

00:01:59.030 --> 00:02:04.800
So we know that a/b,
it has to be negative.

00:02:04.800 --> 00:02:07.755
It has to be the left of 0.

00:02:07.755 --> 00:02:11.920
a/b is going to be to
the right of a times b.

00:02:16.850 --> 00:02:20.070
So one way to think about it is
they're both negative, but a/b

00:02:20.070 --> 00:02:21.551
is going to be less negative.

00:02:21.551 --> 00:02:23.050
Or another way of
thinking about it,

00:02:23.050 --> 00:02:25.340
it's going to have a
smaller absolute value.

00:02:25.340 --> 00:02:28.320
Its distance from 0, which
is another way of thinking

00:02:28.320 --> 00:02:31.182
about absolute value, its
distance to the left of 0

00:02:31.182 --> 00:02:34.674
is less than a times b's
distance to the left of 0.

00:02:34.674 --> 00:02:36.090
But because they're
both negative,

00:02:36.090 --> 00:02:38.640
and we're talking about
distances from the left of 0,

00:02:38.640 --> 00:02:40.915
the one that has a smaller
distance to the left of 0

00:02:40.915 --> 00:02:44.655
is less negative, and is
therefore a greater number.

00:02:44.655 --> 00:02:46.671
So, so far your interviewer
seems impressed.

00:02:46.671 --> 00:02:47.670
So that was pretty good.

00:02:47.670 --> 00:02:49.657
You were able to get a
lot of clues just out

00:02:49.657 --> 00:02:51.740
of this little bit of
information that I gave you.

00:02:51.740 --> 00:02:53.200
But tell me more.

00:02:53.200 --> 00:02:55.760
Tell me more about
what b has to be

00:02:55.760 --> 00:02:59.190
and whether we can
constrain it in some way.

00:02:59.190 --> 00:03:00.260
And you say, well, OK.

00:03:00.260 --> 00:03:03.470
I've got some clues here, the
fact that this absolute value

00:03:03.470 --> 00:03:06.640
right over here is going to be
less than this absolute value.

00:03:06.640 --> 00:03:09.230
We know that the
absolute value of a/b

00:03:09.230 --> 00:03:12.010
is going to be less
than a times b,

00:03:12.010 --> 00:03:13.930
than the absolute
value of a times b.

00:03:13.930 --> 00:03:18.400
Once again, it's going
to be not as far left

00:03:18.400 --> 00:03:20.145
of 0 as a times b is.

00:03:20.145 --> 00:03:21.930
So that's how we can
make that statement.

00:03:21.930 --> 00:03:25.035
But let's actually manipulate,
algebraically manipulate

00:03:25.035 --> 00:03:26.730
this inequality here.

00:03:26.730 --> 00:03:29.350
We could multiply
both sides times b.

00:03:29.350 --> 00:03:30.620
So let's do that.

00:03:30.620 --> 00:03:33.790
Let's multiply
both sides times b.

00:03:33.790 --> 00:03:37.672
b Is less than the 0.

00:03:37.672 --> 00:03:39.130
So if you're
multiplying both sides

00:03:39.130 --> 00:03:41.760
of an inequality by
something less than 0,

00:03:41.760 --> 00:03:43.535
it swaps the inequality.

00:03:43.535 --> 00:03:44.660
That's why I got rid of it.

00:03:44.660 --> 00:03:45.660
I'm going to rewrite it.

00:03:45.660 --> 00:03:50.564
So b times a/b is
going to negative.

00:03:50.564 --> 00:03:51.480
We know it's negative.

00:03:51.480 --> 00:03:55.150
So it's going to be
less than ab times b.

00:03:55.150 --> 00:03:57.580
If we just multiply it
out, these cancel out.

00:03:57.580 --> 00:04:02.860
We get a is less than
a times b squared.

00:04:02.860 --> 00:04:04.500
Well, now if we
want to simplify it,

00:04:04.500 --> 00:04:06.806
we can divide both sides by a.

00:04:06.806 --> 00:04:08.180
And since a is
greater than 0, it

00:04:08.180 --> 00:04:09.830
does not change the inequality.

00:04:09.830 --> 00:04:12.240
So we divide both sides by a.

00:04:12.240 --> 00:04:15.790
This becomes 1, and this
just becomes b squared.

00:04:15.790 --> 00:04:23.780
So we're left with 1
is less than b squared.

00:04:23.780 --> 00:04:27.700
Or another way we could think
about this, we could say, well,

00:04:27.700 --> 00:04:33.390
if 1 is less than b squared,
that means that b-- let's

00:04:33.390 --> 00:04:37.580
be careful here-- that means
that the absolute value of b

00:04:37.580 --> 00:04:39.430
is going to be greater than 1.

00:04:39.430 --> 00:04:43.870
So this means that the absolute
value of b is greater than 1.

00:04:43.870 --> 00:04:45.720
And you say, wait,
Sal, how did you

00:04:45.720 --> 00:04:50.200
get from that to that
right over there?

00:04:50.200 --> 00:04:51.920
Well, just think about
this a little bit.

00:04:51.920 --> 00:04:55.200
If I square something, and
if it's greater than 1--

00:04:55.200 --> 00:04:59.051
so that means that either
b is less than negative 1,

00:04:59.051 --> 00:05:01.050
because if it was negative
1, if you squared it,

00:05:01.050 --> 00:05:01.940
you would get 1.

00:05:01.940 --> 00:05:05.230
If it was greater than negative
1, or just directly greater

00:05:05.230 --> 00:05:08.700
than negative 1, if it was
negative 0.99, then when

00:05:08.700 --> 00:05:12.707
you square it, it would
be something less than 1.

00:05:12.707 --> 00:05:13.790
So that doesn't work.

00:05:13.790 --> 00:05:17.270
So b has to be less
than negative 1 or b has

00:05:17.270 --> 00:05:23.111
to be greater than 1, because
for the same exact logic.

00:05:23.111 --> 00:05:25.110
If it was exactly equal
to 1, if you squared it,

00:05:25.110 --> 00:05:26.020
this would be equal.

00:05:26.020 --> 00:05:28.070
If it was 0.5, then
squared it would

00:05:28.070 --> 00:05:30.041
be 0.25, which wouldn't
be greater than.

00:05:30.041 --> 00:05:31.790
So you know both of
these things are true.

00:05:31.790 --> 00:05:34.710
And this is another
way of saying

00:05:34.710 --> 00:05:38.310
that the absolute value
of b is greater than 1.

00:05:38.310 --> 00:05:39.780
Now, we have our
other constraint,

00:05:39.780 --> 00:05:42.510
that b is less than 0.

00:05:42.510 --> 00:05:44.320
So since we know
b is less than 0,

00:05:44.320 --> 00:05:46.209
we can take this
out of the picture.

00:05:46.209 --> 00:05:48.250
And if the absolute value
of b is greater than 1,

00:05:48.250 --> 00:05:49.820
and b is less than
0, then we know

00:05:49.820 --> 00:05:55.050
that b has to be
less than negative 1.

00:05:55.050 --> 00:05:58.590
So wherever negative 1 is, b
is going to be less than that.

00:05:58.590 --> 00:06:01.330
And your interviewer is very
impressed and she says, look,

00:06:01.330 --> 00:06:03.300
you've done very
good reasoning here.

00:06:03.300 --> 00:06:05.933
I think you deserve the job.


Dogs cats and bears in a pet store visual argument

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.510
Let's say that someone's running
a fairly unusual pet store that

00:00:03.510 --> 00:00:05.480
only has three types
of animals in it.

00:00:05.480 --> 00:00:07.910
It has cats, and we
represent the number

00:00:07.910 --> 00:00:10.230
of cats with the letter c.

00:00:10.230 --> 00:00:13.590
It has dogs, and we
represent the number of dogs

00:00:13.590 --> 00:00:16.290
by the letter d.

00:00:16.290 --> 00:00:18.200
And it has bears.

00:00:18.200 --> 00:00:19.680
That's what makes it unusual.

00:00:19.680 --> 00:00:21.680
And we're going to represent
the number of bears

00:00:21.680 --> 00:00:24.150
with the letter b.

00:00:24.150 --> 00:00:27.940
Now, this person also tells us
that at this unusual pet store,

00:00:27.940 --> 00:00:32.790
the number of cats is greater
than the number of dogs, which

00:00:32.790 --> 00:00:36.410
is greater than the
number of bears.

00:00:36.410 --> 00:00:38.780
Now based just on
this information,

00:00:38.780 --> 00:00:42.590
they then ask us which
expression is greater.

00:00:42.590 --> 00:00:50.270
b divided by c plus
d plus b or 1/3.

00:00:50.270 --> 00:00:53.250
And I encourage you to pause
the video now and come up

00:00:53.250 --> 00:00:56.120
with your own rationale behind
which of these is greater.

00:00:56.120 --> 00:00:57.787
Or maybe neither is greater.

00:00:57.787 --> 00:00:58.620
Maybe they're equal.

00:00:58.620 --> 00:01:00.578
Or maybe you just don't
have enough information

00:01:00.578 --> 00:01:03.480
from what this pet store owner
told us to actually figure out.

00:01:03.480 --> 00:01:06.067
So pause the video now.

00:01:06.067 --> 00:01:08.400
So let's reason through what
this expression is actually

00:01:08.400 --> 00:01:09.440
representing.

00:01:09.440 --> 00:01:15.140
This is b, the number of
bears, over the number of cats,

00:01:15.140 --> 00:01:21.670
plus the number of dogs,
plus the number of bears.

00:01:21.670 --> 00:01:23.826
So there's the number of
bears over the sum of all

00:01:23.826 --> 00:01:24.325
the animals.

00:01:27.350 --> 00:01:30.390
So this really is the
fraction that are bears.

00:01:37.650 --> 00:01:39.620
Now so this question
really reduces

00:01:39.620 --> 00:01:43.060
to is the fraction that are
bears greater than a third,

00:01:43.060 --> 00:01:47.070
less than a third, equal to
a third, or can we not tell?

00:01:47.070 --> 00:01:49.580
And there's a bunch of different
ways to do this problem.

00:01:49.580 --> 00:01:51.950
And I'll try to expose
you to many ways

00:01:51.950 --> 00:01:54.640
in this video and the next.

00:01:54.640 --> 00:01:57.450
So let's do a visual one
doing different cases.

00:02:00.390 --> 00:02:03.230
Let's do the scenario where
the fraction that are bears

00:02:03.230 --> 00:02:07.170
are greater than one third.

00:02:07.170 --> 00:02:09.070
So let's visualize this.

00:02:09.070 --> 00:02:11.660
So this little diagram
right over here.

00:02:11.660 --> 00:02:14.020
Let's say this is all of
the pets in the pet store.

00:02:14.020 --> 00:02:16.570
And I've divided it into
thirds right over here.

00:02:16.570 --> 00:02:20.140
If the fraction that are bears
is greater than one third,

00:02:20.140 --> 00:02:22.670
then it might look
something like this.

00:02:22.670 --> 00:02:24.720
So this is a third
under the dotted lines.

00:02:24.720 --> 00:02:27.178
If I want to be greater than
1/3, I'll go a little bit more

00:02:27.178 --> 00:02:28.660
than greater than 1/3.

00:02:28.660 --> 00:02:31.830
So we go greater than
1/3 right over there.

00:02:31.830 --> 00:02:34.530
But if the fraction of
bears is greater than 1/3,

00:02:34.530 --> 00:02:38.286
well the fraction of dogs has
to be even greater than that.

00:02:38.286 --> 00:02:40.410
So it has to be at least
as much as this blue area.

00:02:40.410 --> 00:02:42.600
So it's going to be
even greater than that.

00:02:42.600 --> 00:02:45.170
And the fraction
that are cats has

00:02:45.170 --> 00:02:46.780
to be even greater than that.

00:02:46.780 --> 00:02:49.327
And I didn't even make
this one that great.

00:02:49.327 --> 00:02:50.910
It has to be more
than this blue area.

00:02:50.910 --> 00:02:52.090
And the fraction
that are cats has

00:02:52.090 --> 00:02:53.460
to be even greater than that.

00:02:53.460 --> 00:02:56.030
And you see, you can't
have three things

00:02:56.030 --> 00:03:00.620
that are all greater than
1/3 adding up to a whole.

00:03:00.620 --> 00:03:05.140
You can't have all three parts
of it being greater than 1/3.

00:03:05.140 --> 00:03:06.369
So this scenario breaks down.

00:03:06.369 --> 00:03:08.160
Another way you could
have thought about it

00:03:08.160 --> 00:03:12.690
is if this expression right over
here is greater than a third,

00:03:12.690 --> 00:03:17.550
then c plus d over this
has to be less than 2/3.

00:03:17.550 --> 00:03:20.010
But that means one
of them, c or d,

00:03:20.010 --> 00:03:22.450
would have to be definitely
less than a third

00:03:22.450 --> 00:03:23.700
in order for that to work out.

00:03:23.700 --> 00:03:25.400
So at least thinking
about this way,

00:03:25.400 --> 00:03:28.170
you know that this thing
cannot be greater than a third.

00:03:28.170 --> 00:03:30.320
Let's think about whether
it can be equal a third.

00:03:30.320 --> 00:03:33.075
Let's imagine if this
expression, the fraction that

00:03:33.075 --> 00:03:35.080
are bears, were
equal to a third.

00:03:35.080 --> 00:03:38.960
So let's make our diagram again.

00:03:38.960 --> 00:03:43.420
So in this situation, the
bears are exactly 1/3.

00:03:43.420 --> 00:03:48.100
So I'll just color in this
third right over here.

00:03:48.100 --> 00:03:52.010
So exactly 1/3 are bears.

00:03:52.010 --> 00:03:53.469
But we know that
the number of dogs

00:03:53.469 --> 00:03:54.968
is greater than the
number of bears.

00:03:54.968 --> 00:03:56.400
So the fraction
that are dogs has

00:03:56.400 --> 00:03:58.862
to be greater than the
fraction that are bears.

00:03:58.862 --> 00:04:00.320
So the fraction
that are dogs would

00:04:00.320 --> 00:04:01.486
have to be greater than 1/3.

00:04:04.320 --> 00:04:06.840
But if the fraction that are
dogs are greater than 1/3,

00:04:06.840 --> 00:04:09.380
then all we have left is
something less than 1/3

00:04:09.380 --> 00:04:10.260
for the cats.

00:04:10.260 --> 00:04:11.760
That's actually the
largest fraction

00:04:11.760 --> 00:04:13.090
of the animals are cats.

00:04:13.090 --> 00:04:18.575
So we know that this also
is not a possibility.

00:04:18.575 --> 00:04:19.950
So what we're
really left with is

00:04:19.950 --> 00:04:22.040
that this has to
be less than 1/3.

00:04:22.040 --> 00:04:24.130
And let's make sure
that's reasonable.

00:04:24.130 --> 00:04:28.230
So, let's paste another
example right over here.

00:04:28.230 --> 00:04:35.911
So if the bears
are less than 1/3--

00:04:35.911 --> 00:04:37.660
let me make it a little
bit more dramatic,

00:04:37.660 --> 00:04:40.590
so that you can make it clear.

00:04:40.590 --> 00:04:42.040
So this is less than a third.

00:04:46.550 --> 00:04:49.627
let's say that the
purple, the dogs,

00:04:49.627 --> 00:04:51.210
let's say they're
roughly exactly 1/3.

00:04:54.100 --> 00:04:55.600
Then you could make
a scenario where

00:04:55.600 --> 00:04:59.210
you have the largest
proportion of cats.

00:04:59.210 --> 00:05:01.372
So this would be the
cats right over here.

00:05:01.372 --> 00:05:02.955
That's not the same
color as the cats.

00:05:02.955 --> 00:05:05.520
Let me do the same color
as that pink color.

00:05:05.520 --> 00:05:08.060
So this right over
here is our cats.

00:05:08.060 --> 00:05:09.970
So this is complete believable.

00:05:09.970 --> 00:05:17.260
c over c plus d
plus b does indeed

00:05:17.260 --> 00:05:23.480
look bigger than d over c plus
d plus b, which does indeed

00:05:23.480 --> 00:05:30.410
look bigger than b
over c plus d plus b.

00:05:30.410 --> 00:05:33.620
This is completely believable
to be less than 1/3.

00:05:33.620 --> 00:05:36.540
So based on just this visual
argument that we just made,

00:05:36.540 --> 00:05:37.880
you could say hey, look.

00:05:37.880 --> 00:05:41.400
This right over here
is the larger quantity.

00:05:41.400 --> 00:05:42.920
In the next video,
I'll give what

00:05:42.920 --> 00:05:45.120
I would call more of an
analytical argument, where

00:05:45.120 --> 00:05:50.690
I won't draw diagrams
to show that this

00:05:50.690 --> 00:05:53.810
has to be less than 1/3.


Dogs cats and bears in a pet store analytic argument

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.180
In the last video, we made
a visual argument as to why

00:00:04.180 --> 00:00:06.800
this expression has
to be less than 1/3,

00:00:06.800 --> 00:00:08.780
and this expression
we already figured out

00:00:08.780 --> 00:00:10.840
is the fraction that are bears.

00:00:10.840 --> 00:00:12.990
Now we will make an
algebraic argument,

00:00:12.990 --> 00:00:15.221
or I could call it
an analytic argument.

00:00:15.221 --> 00:00:16.720
And to make this
argument, I'm going

00:00:16.720 --> 00:00:19.219
to leave this expression-- we
know this is the fraction that

00:00:19.219 --> 00:00:22.530
are bears-- and I'm going to
write this 1/3 in a form that

00:00:22.530 --> 00:00:24.290
looks a lot like
this, and then based

00:00:24.290 --> 00:00:27.110
on the information we have,
we can directly compare them.

00:00:27.110 --> 00:00:29.160
So how can I write 1/3?

00:00:29.160 --> 00:00:32.060
Maybe with the b as a numerator.

00:00:32.060 --> 00:00:36.950
Well, 1/3 is the same
thing as b over 3b,

00:00:36.950 --> 00:00:44.090
which is the exact same thing
as b over b plus b plus b.

00:00:44.090 --> 00:00:46.030
So now, this is
looking pretty similar.

00:00:46.030 --> 00:00:49.500
The only difference between this
expression right over here, b

00:00:49.500 --> 00:00:52.945
over c plus d plus
b and b over b

00:00:52.945 --> 00:00:56.010
plus b plus b is that our
denominators are different.

00:00:56.010 --> 00:00:58.790
And the only difference in our
denominators, this denominator

00:00:58.790 --> 00:01:03.360
has a c plus d here, while
this has a b plus b over here.

00:01:03.360 --> 00:01:05.770
Now, we have to ask
ourselves a question.

00:01:05.770 --> 00:01:07.000
What is larger?

00:01:07.000 --> 00:01:10.249
Is c plus d larger
than b plus b?

00:01:10.249 --> 00:01:12.040
And I encourage you to
pause that and think

00:01:12.040 --> 00:01:14.480
about that for a second.

00:01:14.480 --> 00:01:16.149
Well, yes.

00:01:16.149 --> 00:01:17.440
We already see right over here.

00:01:17.440 --> 00:01:19.030
It was given to us
that c is greater

00:01:19.030 --> 00:01:20.940
than d that is greater
than b, so both c

00:01:20.940 --> 00:01:22.720
and d are greater than b.

00:01:22.720 --> 00:01:26.920
So c plus d is definitely going
to be greater than b plus b.

00:01:26.920 --> 00:01:29.850
So this denominator right
over here is greater,

00:01:29.850 --> 00:01:31.460
so this has a
larger denominator.

00:01:37.340 --> 00:01:41.550
This right over here has
a smaller denominator.

00:01:45.981 --> 00:01:47.980
And since we know this
has a larger denominator,

00:01:47.980 --> 00:01:48.940
this has a smaller
denominator, they

00:01:48.940 --> 00:01:51.340
have the exact same
numerator-- they both have

00:01:51.340 --> 00:01:56.970
b as a numerator-- we know
that this whole thing must

00:01:56.970 --> 00:01:58.290
be a smaller quantity.

00:02:00.850 --> 00:02:04.710
If you have the same numerator
but one expression has a larger

00:02:04.710 --> 00:02:06.120
denominator, it must be smaller.

00:02:06.120 --> 00:02:07.286
Wait, so how does that work?

00:02:07.286 --> 00:02:08.236
Well, just remember.

00:02:08.236 --> 00:02:09.110
I mean, just imagine.

00:02:09.110 --> 00:02:11.401
You have the same numerator,
what's going to be bigger,

00:02:11.401 --> 00:02:16.800
a over 7 or a over 5?

00:02:16.800 --> 00:02:18.272
Well here, you're
dividing a by 7.

00:02:18.272 --> 00:02:20.480
You're dividing into many
more chunks than over here,

00:02:20.480 --> 00:02:22.000
so this right over
here is smaller.

00:02:22.000 --> 00:02:24.230
This right over here is larger.

00:02:24.230 --> 00:02:25.530
So this is the larger.

00:02:25.530 --> 00:02:29.420
This right over here is smaller.

00:02:29.420 --> 00:02:32.330
So the same numerator, the
larger the denominator,

00:02:32.330 --> 00:02:34.226
the smaller the
quantity is going to be.

00:02:34.226 --> 00:02:35.850
So going back to the
original question,

00:02:35.850 --> 00:02:38.730
this is the smaller quantity,
and this right over here,

00:02:38.730 --> 00:02:43.310
1/3, is the larger quantity.