Adding decimals: example 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.760
Let's see if we can
add 9.087 to 15.31.

00:00:07.760 --> 00:00:09.770
And I encourage you
to pause the video

00:00:09.770 --> 00:00:12.040
and try to do it on your own.

00:00:12.040 --> 00:00:15.050
So I'm assuming you have
tried to do it on your own.

00:00:15.050 --> 00:00:17.880
And now let's see how we
could actually tackle this.

00:00:17.880 --> 00:00:19.890
Now, one thing I want to
point out, some of you

00:00:19.890 --> 00:00:21.431
all might have seen
these numbers all

00:00:21.431 --> 00:00:24.710
lined up and immediately want
to say, hey, 7 plus 1 is 8,

00:00:24.710 --> 00:00:27.790
and 8 plus 3 is 11, carry
the 1, et cetera, et cetera.

00:00:27.790 --> 00:00:31.654
And if you did that, you
would be making a mistake.

00:00:31.654 --> 00:00:33.070
Because, you see,
right over here,

00:00:33.070 --> 00:00:34.950
these decimals aren't lined up.

00:00:34.950 --> 00:00:38.110
Here, if you did that, you would
be adding the 7 thousandths

00:00:38.110 --> 00:00:39.310
to 1 hundredths.

00:00:39.310 --> 00:00:42.800
You would be adding
0 tenths to 5 ones.

00:00:42.800 --> 00:00:46.630
You would be adding 9 to 1
tens, or essentially, this

00:00:46.630 --> 00:00:47.780
is a 10 right over here.

00:00:47.780 --> 00:00:50.840
So the places would
be all mixed up.

00:00:50.840 --> 00:00:53.920
So what you need to do is to
actually align the decimals

00:00:53.920 --> 00:00:56.660
so that your place
values are aligned.

00:00:56.660 --> 00:01:00.050
So what you want to do is
you want to align things up.

00:01:00.050 --> 00:01:04.610
So we could write 9.087.

00:01:04.610 --> 00:01:06.380
And then we want to
align the decimal.

00:01:06.380 --> 00:01:07.570
So let's align the decimal.

00:01:07.570 --> 00:01:09.500
This is what has to match up.

00:01:09.500 --> 00:01:13.750
And this is going to be 15.31.

00:01:13.750 --> 00:01:16.000
And this should hopefully
make sense to you as well.

00:01:16.000 --> 00:01:20.145
This is 9 point something
plus 15 point something

00:01:20.145 --> 00:01:22.450
so it's going to be--
If you add 9 to 15,

00:01:22.450 --> 00:01:26.410
it will be 24 point something,
give or take a little bit.

00:01:26.410 --> 00:01:27.340
And you see that here.

00:01:27.340 --> 00:01:29.220
Here you have a 9 plus the 15.

00:01:29.220 --> 00:01:31.690
So you have lined up the
appropriate place values.

00:01:31.690 --> 00:01:34.530
And now we are ready to add.

00:01:34.530 --> 00:01:37.190
It's a good idea to start
with the smallest place value,

00:01:37.190 --> 00:01:39.270
so if you have any extra
at a certain place,

00:01:39.270 --> 00:01:43.080
you can bring something
into the next place value.

00:01:43.080 --> 00:01:46.590
So here you say 7 plus--
well, this is 7 thousandths.

00:01:46.590 --> 00:01:48.350
It's in the thousandths place.

00:01:48.350 --> 00:01:50.260
And you might want to-- you
say, well, what do I add it to?

00:01:50.260 --> 00:01:51.870
There are no thousandths
right over here.

00:01:51.870 --> 00:01:52.300
And you're right.

00:01:52.300 --> 00:01:53.341
There are no thousandths.

00:01:53.341 --> 00:01:55.600
So we could literally
write 0 thousandths.

00:01:55.600 --> 00:02:00.000
So 7 thousandths plus 0
thousandths is 7 thousandths.

00:02:00.000 --> 00:02:04.570
8 hundredths plus 1
hundredth is 9 hundredths.

00:02:04.570 --> 00:02:10.759
0 plus 0 tenths plus
3 tenths is 3 tenths.

00:02:10.759 --> 00:02:12.400
We got our decimal.

00:02:12.400 --> 00:02:17.660
Then you have 9 ones
plus 5 ones is 14 ones.

00:02:17.660 --> 00:02:23.080
Well, 14 ones is the same
thing as 4 ones and 1 ten.

00:02:23.080 --> 00:02:25.540
So we'll carry that
1 right over there.

00:02:25.540 --> 00:02:29.740
This is just 1 ten plus
4 ones, which is 14.

00:02:29.740 --> 00:02:33.960
And so then finally, you have
1 ten plus another ten is 2.

00:02:33.960 --> 00:02:36.220
So we get 24.397.


Adding decimals: example 2

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:06.820
We're asked to
add 0.822 to 5.65.

00:00:06.820 --> 00:00:08.430
So let me rewrite this.

00:00:08.430 --> 00:00:11.150
And when I rewrite it, I
want to line up the decimals

00:00:11.150 --> 00:00:14.030
so that we add the right
place to the right place.

00:00:14.030 --> 00:00:16.260
And so we could write
either number first,

00:00:16.260 --> 00:00:18.410
although I like to write
the larger number first.

00:00:18.410 --> 00:00:22.395
So let's write 5.65.

00:00:22.395 --> 00:00:23.770
And remember, the
important thing

00:00:23.770 --> 00:00:26.120
is that we line up
the decimal points.

00:00:26.120 --> 00:00:30.235
So if we write 0.822-- so
we line up the decimal.

00:00:30.235 --> 00:00:31.805
Let me line up
the decimal first.

00:00:31.805 --> 00:00:34.710
So I'll write the decimal
right below the other decimal.

00:00:34.710 --> 00:00:40.070
And it is 0.822.

00:00:40.070 --> 00:00:44.320
And now we are ready to add.

00:00:44.320 --> 00:00:46.290
So let's see what's
going on here.

00:00:46.290 --> 00:00:49.040
So I like to start in
the smallest place.

00:00:49.040 --> 00:00:51.460
That way, the carrying
works out well.

00:00:51.460 --> 00:00:53.640
So you might say, wait,
I need to add this 2

00:00:53.640 --> 00:00:55.000
thousandths to something.

00:00:55.000 --> 00:00:56.640
I don't see anything up here.

00:00:56.640 --> 00:01:00.254
Well, you could say there's just
a 0 thousandths right up here.

00:01:00.254 --> 00:01:01.420
Then it makes it very clear.

00:01:01.420 --> 00:01:04.060
Well, 0 thousandths
plus 2 thousandths

00:01:04.060 --> 00:01:06.330
is going to be 2 thousandths.

00:01:06.330 --> 00:01:10.992
5 hundredths plus 2
hundredths is 7 hundredths.

00:01:10.992 --> 00:01:16.530
6 tenths plus 8
tenths is 14 tenths.

00:01:16.530 --> 00:01:22.524
Well, 14 tenths is the same
thing as 4 tenths and 1 one.

00:01:22.524 --> 00:01:24.940
Another way of thinking about
it is you're carrying the 1.

00:01:24.940 --> 00:01:27.450
But really, what you're saying
is, look, this is 14 tenths.

00:01:27.450 --> 00:01:32.690
I could write it as 4 tenths
and a 1, or a ones place, a 1

00:01:32.690 --> 00:01:33.930
in a ones place.

00:01:33.930 --> 00:01:36.400
Then you have 1 plus 5 is 6.

00:01:36.400 --> 00:01:39.210
And of course, you cannot
forget the decimal.

00:01:39.210 --> 00:01:41.090
The decimal goes right there.

00:01:41.090 --> 00:01:43.110
And this is 6.472.


LCM and GCF word problems

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.130
William and Luis are in
different physics classes

00:00:03.130 --> 00:00:04.370
at Santa Rita.

00:00:04.370 --> 00:00:07.750
Luis's teacher always gives
exams with 30 questions

00:00:07.750 --> 00:00:10.870
on them, while William's
teacher gives more

00:00:10.870 --> 00:00:14.150
frequent exams with
only 24 questions.

00:00:14.150 --> 00:00:17.802
Luis's teacher also assigns
three projects per year.

00:00:17.802 --> 00:00:20.260
Even though the two classes
have to take a different number

00:00:20.260 --> 00:00:22.270
of exams, their
teachers have told them

00:00:22.270 --> 00:00:25.250
that both classes-- let me
underline-- both classes will

00:00:25.250 --> 00:00:29.040
get the same total number
of exam questions each year.

00:00:29.040 --> 00:00:32.850
What is the minimum
number of exam questions

00:00:32.850 --> 00:00:36.807
William's or Luis's class can
expect to get in a given year?

00:00:36.807 --> 00:00:38.390
So let's think about
what's happening.

00:00:38.390 --> 00:00:40.014
So if we think about
Luis's teacher who

00:00:40.014 --> 00:00:44.590
gives 30 questions per test,
so after the first test,

00:00:44.590 --> 00:00:46.850
he would have done 30 questions.

00:00:46.850 --> 00:00:48.750
So this is 0 right over here.

00:00:48.750 --> 00:00:52.240
Then after the second test,
he would have done 60.

00:00:52.240 --> 00:00:56.150
Then after the third test,
he would have done 90.

00:00:56.150 --> 00:01:00.070
And after the fourth test,
he would have done 120.

00:01:00.070 --> 00:01:03.480
And after the fifth test,
if there is a fifth test,

00:01:03.480 --> 00:01:06.700
he would do-- so this is if
they have that many tests-- he

00:01:06.700 --> 00:01:08.912
would get to 150
total questions.

00:01:08.912 --> 00:01:10.620
And we could keep
going on and on looking

00:01:10.620 --> 00:01:12.467
at all the multiples of 30.

00:01:12.467 --> 00:01:14.800
So this is probably a hint
of what we're thinking about.

00:01:14.800 --> 00:01:16.549
We're looking at
multiples of the numbers.

00:01:16.549 --> 00:01:19.710
We want the minimum multiples
or the least multiple.

00:01:19.710 --> 00:01:20.950
So that's with Luis.

00:01:20.950 --> 00:01:22.710
Well what's going
on with William?

00:01:22.710 --> 00:01:25.650
Will William's teacher,
after the first test,

00:01:25.650 --> 00:01:29.220
they're going to
get to 24 questions.

00:01:29.220 --> 00:01:32.770
Then they're going to get
to 48 after the second test.

00:01:32.770 --> 00:01:37.420
Then they're going to get
to 72 after the third test.

00:01:37.420 --> 00:01:39.250
Then they're going to get to 96.

00:01:39.250 --> 00:01:41.820
I'm just taking multiples of 24.

00:01:41.820 --> 00:01:45.030
They're going to get to
96 after the fourth test.

00:01:45.030 --> 00:01:49.610
And then after the fifth test,
they're going to get to 120.

00:01:49.610 --> 00:01:55.160
And if there's a sixth test,
then they would get to 144.

00:01:55.160 --> 00:01:57.430
And we could keep going
on and on in there.

00:01:57.430 --> 00:01:58.300
But let's see what
they're asking us.

00:01:58.300 --> 00:02:00.180
What is the minimum
number of exam questions

00:02:00.180 --> 00:02:03.200
William's or Luis's class
can expect to get in a year?

00:02:03.200 --> 00:02:04.710
Well the minimum
number is the point

00:02:04.710 --> 00:02:07.380
at which they've gotten the
same number of exam questions,

00:02:07.380 --> 00:02:09.190
despite the fact
that the tests had

00:02:09.190 --> 00:02:10.617
a different number of items.

00:02:10.617 --> 00:02:12.950
And you see the point at which
they have the same number

00:02:12.950 --> 00:02:14.880
is at 120.

00:02:14.880 --> 00:02:16.770
This happens at 120.

00:02:16.770 --> 00:02:19.300
They both could have
exactly 120 questions

00:02:19.300 --> 00:02:21.840
even though Luis's teacher
is giving 30 at a time

00:02:21.840 --> 00:02:25.240
and even though William's
teacher is giving 24 at a time.

00:02:25.240 --> 00:02:28.469
And so the answer is 120.

00:02:28.469 --> 00:02:30.510
And notice, they had a
different number of exams.

00:02:30.510 --> 00:02:33.650
Luis had one, two,
three, four exams

00:02:33.650 --> 00:02:36.300
while William would have to
have one, two, three, four,

00:02:36.300 --> 00:02:37.570
five exams.

00:02:37.570 --> 00:02:41.270
But that gets them both
to 120 total questions.

00:02:41.270 --> 00:02:44.100
Now thinking of it in terms
of some of the math notation

00:02:44.100 --> 00:02:47.370
or the least common multiple
notation we've seen before,

00:02:47.370 --> 00:02:55.650
this is really asking us what is
the least common multiple of 30

00:02:55.650 --> 00:02:56.980
and 24.

00:02:56.980 --> 00:03:02.692
And that least common
multiple is equal to 120.

00:03:02.692 --> 00:03:04.150
Now there's other
ways that you can

00:03:04.150 --> 00:03:06.399
find the least common multiple
other than just looking

00:03:06.399 --> 00:03:07.870
at the multiples like this.

00:03:07.870 --> 00:03:10.440
You could look at it
through prime factorization.

00:03:10.440 --> 00:03:15.290
30 is 2 times 15,
which is 3 times 5.

00:03:15.290 --> 00:03:20.420
So we could say that 30 is
equal to 2 times 3 times 5.

00:03:20.420 --> 00:03:28.580
And 24-- that's a different
color than that blue-- 24

00:03:28.580 --> 00:03:31.570
is equal to 2 times 12.

00:03:31.570 --> 00:03:33.846
12 is equal to 2 times 6.

00:03:33.846 --> 00:03:36.080
6 is equal to 2 times 3.

00:03:36.080 --> 00:03:44.660
So 24 is equal to 2
times 2 times 2 times 3.

00:03:44.660 --> 00:03:47.250
So another way to come up with
the least common multiple,

00:03:47.250 --> 00:03:49.720
if we didn't even do this
exercise up here, says, look,

00:03:49.720 --> 00:03:52.820
the number has to be
divisible by both 30 and 24.

00:03:52.820 --> 00:03:54.810
If it's going to
be divisible by 30,

00:03:54.810 --> 00:04:00.060
it's going to have to
have 2 times 3 times 5

00:04:00.060 --> 00:04:01.430
in its prime factorization.

00:04:01.430 --> 00:04:03.420
That is essentially 30.

00:04:03.420 --> 00:04:05.830
So this makes it
divisible by 30.

00:04:05.830 --> 00:04:10.050
And say, well in order
to be divisible by 24,

00:04:10.050 --> 00:04:13.750
its prime factorization is
going to need 3 twos and a 3.

00:04:13.750 --> 00:04:15.230
Well we already have 1 three.

00:04:15.230 --> 00:04:18.040
And we already have 1 two,
so we just need 2 more twos.

00:04:18.040 --> 00:04:20.740
So 2 times 2.

00:04:20.740 --> 00:04:24.340
So this makes it--
let me scroll up

00:04:24.340 --> 00:04:29.080
a little bit-- this right over
here makes it divisible by 24.

00:04:29.080 --> 00:04:32.030
And so this is essentially
the prime factorization

00:04:32.030 --> 00:04:34.920
of the least common
multiple of 30 and 24.

00:04:34.920 --> 00:04:37.300
You take any one of
these numbers away,

00:04:37.300 --> 00:04:40.251
you are no longer going to be
divisible by one of these two

00:04:40.251 --> 00:04:40.750
numbers.

00:04:40.750 --> 00:04:43.333
If you take a two away, you're
not going to be divisible by 24

00:04:43.333 --> 00:04:43.950
anymore.

00:04:43.950 --> 00:04:45.830
If you take a two
or a three away.

00:04:45.830 --> 00:04:50.520
If you take a three
or a five away,

00:04:50.520 --> 00:04:53.145
you're not going to be
divisible by 30 anymore.

00:04:53.145 --> 00:04:55.020
And so if you were to
multiply all these out,

00:04:55.020 --> 00:05:04.170
this is 2 times 2 times 2 is 8
times 3 is 24 times 5 is 120.

00:05:04.170 --> 00:05:06.740
Now let's do one more of these.

00:05:06.740 --> 00:05:09.971
Umama just bought one
package of 21 binders.

00:05:09.971 --> 00:05:11.220
Let me write that number down.

00:05:11.220 --> 00:05:12.660
21 binders.

00:05:12.660 --> 00:05:14.800
She also bought a
package of 30 pencils.

00:05:17.860 --> 00:05:20.240
She wants to use all of
the binders and pencils

00:05:20.240 --> 00:05:23.060
to create identical
sets of office supplies

00:05:23.060 --> 00:05:24.650
for her classmates.

00:05:24.650 --> 00:05:27.540
What is the greatest
number of identical sets

00:05:27.540 --> 00:05:29.456
Umama can make using
all the supplies?

00:05:29.456 --> 00:05:31.330
So the fact that we're
talking about greatest

00:05:31.330 --> 00:05:33.246
is clue that it's probably
going to be dealing

00:05:33.246 --> 00:05:34.620
with greatest common divisors.

00:05:34.620 --> 00:05:36.710
And it's also dealing with
dividing these things.

00:05:36.710 --> 00:05:39.660
We want to divide these
both into the greatest

00:05:39.660 --> 00:05:44.764
number of identical sets.

00:05:44.764 --> 00:05:46.930
So there's a couple of ways
we could think about it.

00:05:46.930 --> 00:05:49.060
Let's think about what the
greatest common divisor

00:05:49.060 --> 00:05:51.100
of both these numbers are.

00:05:51.100 --> 00:05:53.450
Or I could even say the
greatest common factor.

00:05:53.450 --> 00:06:00.500
The greatest common
divisor of 21 and 30.

00:06:00.500 --> 00:06:04.280
So what's the largest number
that divides into both of them?

00:06:04.280 --> 00:06:05.902
So we could go with
the prime factor.

00:06:05.902 --> 00:06:07.610
We could list all of
their normal factors

00:06:07.610 --> 00:06:09.570
and see what is the
greatest common one.

00:06:09.570 --> 00:06:16.700
Or we could look at the
prime factorization.

00:06:16.700 --> 00:06:18.820
So let's just do the prime
factorization method.

00:06:18.820 --> 00:06:21.760
So 21 is the same
thing as 3 times 7.

00:06:21.760 --> 00:06:23.690
These are both prime numbers.

00:06:23.690 --> 00:06:27.140
30 is, let's see,
it's 3-- actually,

00:06:27.140 --> 00:06:30.210
I could write it this
way-- it is 2 times 15.

00:06:30.210 --> 00:06:32.110
We already did it
actually just now.

00:06:32.110 --> 00:06:34.620
And 15 is 3 times 5.

00:06:34.620 --> 00:06:37.680
So what's the largest
number of prime numbers that

00:06:37.680 --> 00:06:39.780
are common to both
factorizations?

00:06:39.780 --> 00:06:42.820
Well you only have a
three right over here.

00:06:42.820 --> 00:06:44.820
Then you don't have a
three times anything else.

00:06:44.820 --> 00:06:47.420
So this is just going
to be equal to 3.

00:06:47.420 --> 00:06:48.900
So this is essentially
telling us,

00:06:48.900 --> 00:06:54.760
look, we can divide both
of these numbers into 3

00:06:54.760 --> 00:06:56.740
and that will give
us the largest

00:06:56.740 --> 00:06:58.504
number of identical sets.

00:06:58.504 --> 00:07:00.170
So just to be clear
of what we're doing.

00:07:00.170 --> 00:07:02.260
So we've answered
the question is 3,

00:07:02.260 --> 00:07:04.360
but just to visualize
it for this question,

00:07:04.360 --> 00:07:07.070
let's actually draw 21 binders.

00:07:07.070 --> 00:07:13.728
So let's say the 21 binders so
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

00:07:13.728 --> 00:07:19.320
11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21.

00:07:19.320 --> 00:07:22.760
And then 30 pencils, so
I'll just do those in green.

00:07:22.760 --> 00:07:27.700
So 1, 2, 3, 4, 5,
6, 7, 8, 9, 10.

00:07:27.700 --> 00:07:29.480
Let me just copy and paste that.

00:07:29.480 --> 00:07:31.660
This is getting tedious.

00:07:31.660 --> 00:07:35.510
So copy and paste.

00:07:35.510 --> 00:07:41.630
So that's 20 and then
paste that is 30.

00:07:41.630 --> 00:07:45.030
Now, we figured out that 3
is the largest number that

00:07:45.030 --> 00:07:46.750
divides into both
of these evenly.

00:07:46.750 --> 00:07:50.670
So I can divide both of
these into groups of 3.

00:07:50.670 --> 00:07:55.390
So for the binders, I could
do it into three groups of 7.

00:07:55.390 --> 00:07:58.400
And then for the
pencils, I could do it

00:07:58.400 --> 00:08:01.320
into three groups of 10.

00:08:01.320 --> 00:08:03.050
So if there are
three people that

00:08:03.050 --> 00:08:05.710
are coming into
this classroom, I

00:08:05.710 --> 00:08:11.640
could give them each seven
binders and 10 pencils.

00:08:11.640 --> 00:08:14.290
But that's the greatest
number of identical sets

00:08:14.290 --> 00:08:15.270
Umama can make.

00:08:15.270 --> 00:08:16.450
I would have three sets.

00:08:16.450 --> 00:08:22.000
Each set would have seven
binders and 10 pencils.

00:08:22.000 --> 00:08:23.500
And we essentially
are just thinking

00:08:23.500 --> 00:08:27.960
about what's the number that we
can divide both of these sets

00:08:27.960 --> 00:08:30.050
into evenly, the
largest number that we

00:08:30.050 --> 00:08:33.263
can divide both of
these sets into evenly.


Greatest common factor exercise

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.840
We're asked, what is the
greatest common divisor

00:00:03.840 --> 00:00:04.679
of 20 and 40?

00:00:04.679 --> 00:00:06.220
And they just say,
another way to say

00:00:06.220 --> 00:00:09.840
this is the GCD, or greatest
common divisor, of 20 of 40

00:00:09.840 --> 00:00:11.750
is equal to question mark.

00:00:11.750 --> 00:00:14.870
And greatest common divisor
sounds like a very fancy term,

00:00:14.870 --> 00:00:16.260
but it's really
just saying, what

00:00:16.260 --> 00:00:22.900
is the largest number that is
divisible into both 20 and 40?

00:00:22.900 --> 00:00:25.680
Well, this seems like a pretty
straightforward situation,

00:00:25.680 --> 00:00:28.610
because 20 is actually
divisible into 40.

00:00:28.610 --> 00:00:30.630
Or another way to
say it is 40 can

00:00:30.630 --> 00:00:34.090
be divided by 20
without a remainder.

00:00:34.090 --> 00:00:36.450
So the largest
number that is a-- I

00:00:36.450 --> 00:00:41.250
guess you could say-- factor of
both 20 and 40 is actually 20.

00:00:41.250 --> 00:00:45.902
20 is 20 times 1,
and 40 is 20 times 2.

00:00:45.902 --> 00:00:47.360
So in this situation,
we don't even

00:00:47.360 --> 00:00:48.730
have to break out our paper.

00:00:48.730 --> 00:00:50.550
We can just write 20.

00:00:50.550 --> 00:00:53.370
Let's do a couple more of these.

00:00:53.370 --> 00:00:56.590
So we're asked, what is
the greatest common divisor

00:00:56.590 --> 00:00:58.460
of 10 and 7?

00:00:58.460 --> 00:01:00.780
So let's now break out
our paper for this.

00:01:00.780 --> 00:01:04.080
So our greatest common
divisor of 10 and 7.

00:01:04.080 --> 00:01:05.209
So let me write that down.

00:01:05.209 --> 00:01:06.460
So we have 10.

00:01:06.460 --> 00:01:15.120
We want to think about what
is our GCD of 10 and 7?

00:01:15.120 --> 00:01:18.130
And there's two ways that
you can approach this.

00:01:18.130 --> 00:01:20.990
One way, you could literally
list all of the factors--

00:01:20.990 --> 00:01:23.710
not prime factors, just
regular factors-- of each

00:01:23.710 --> 00:01:26.190
of these numbers and figure
out which one is greater

00:01:26.190 --> 00:01:28.410
or what is the largest
factor of both.

00:01:28.410 --> 00:01:34.050
So, for example, you could
say, well, I got a 10,

00:01:34.050 --> 00:01:46.960
and 10 can be expressed
1 times 10 or 2 times 5.

00:01:46.960 --> 00:01:48.615
1, 2, 5, and 10.

00:01:48.615 --> 00:01:51.450
These are all factors of 10.

00:01:51.450 --> 00:01:54.650
These are all, we could
say, divisors of 10.

00:01:54.650 --> 00:01:57.800
And sometimes this is called
greatest common factor.

00:01:57.800 --> 00:02:00.190
Seven-- what are
all of its factors?

00:02:00.190 --> 00:02:01.180
Well, 7 is prime.

00:02:01.180 --> 00:02:04.860
It only has two
factors-- 1 and itself.

00:02:04.860 --> 00:02:07.550
So what is the
greatest common factor?

00:02:07.550 --> 00:02:11.360
Well, there's only one
common factor here, 1.

00:02:11.360 --> 00:02:13.570
1 is the only common factor.

00:02:13.570 --> 00:02:18.050
So the greatest common
factor of 10 and 7,

00:02:18.050 --> 00:02:24.470
or the greatest common divisor,
is going to be equal to 1.

00:02:24.470 --> 00:02:26.360
So let's write that down.

00:02:26.360 --> 00:02:27.290
1.

00:02:27.290 --> 00:02:29.490
Let's do one more.

00:02:29.490 --> 00:02:32.960
What is the greatest common
divisor of 21 and 30?

00:02:32.960 --> 00:02:35.430
And this is just another
way of saying that.

00:02:35.430 --> 00:02:39.270
So 21 and 30 are the two
numbers that we care about.

00:02:39.270 --> 00:02:43.370
So we want to figure out
the greatest common divisor,

00:02:43.370 --> 00:02:45.610
and I could have written
greatest common factor,

00:02:45.610 --> 00:02:53.070
of 21 and 30.

00:02:53.070 --> 00:02:56.050
So once again, there's
two ways of doing this.

00:02:56.050 --> 00:02:58.550
And so there's the way I did
the last time where I literally

00:02:58.550 --> 00:02:59.800
list all the factors.

00:02:59.800 --> 00:03:01.850
Let me do it that
way really fast.

00:03:01.850 --> 00:03:04.110
So if I say 21, what
are all the factors?

00:03:04.110 --> 00:03:11.220
Well, it's 1 and
21, and 3, and 7.

00:03:11.220 --> 00:03:13.030
I think I've got all of them.

00:03:13.030 --> 00:03:25.512
And 30 can be written as 1 and
30, 2 and 15, and 3-- actually,

00:03:25.512 --> 00:03:26.720
I'm going to run out of them.

00:03:26.720 --> 00:03:29.170
Let me write it this way so
I get a little more space.

00:03:29.170 --> 00:03:31.880
So 1 and 30.

00:03:31.880 --> 00:03:34.160
2 and 15.

00:03:34.160 --> 00:03:37.100
3 and 10.

00:03:37.100 --> 00:03:40.710
And 5 and 6.

00:03:40.710 --> 00:03:46.210
So here are all of
the factors of 30.

00:03:46.210 --> 00:03:48.610
And now what are
the common factors?

00:03:48.610 --> 00:03:51.130
Well, 1 is a common factor.

00:03:51.130 --> 00:03:54.490
3 is also a common factor.

00:03:54.490 --> 00:03:56.435
But what is the
greatest common factor

00:03:56.435 --> 00:03:58.500
or the greatest common divisor?

00:03:58.500 --> 00:04:03.020
Well, it is going to be 3.

00:04:03.020 --> 00:04:04.780
So we could write 3 here.

00:04:04.780 --> 00:04:06.800
Now, I keep talking
about another technique.

00:04:06.800 --> 00:04:08.300
Let me show you the
other technique,

00:04:08.300 --> 00:04:10.690
and that involves the
prime factorization.

00:04:10.690 --> 00:04:13.320
So if you say the prime
factorization of 21-- well,

00:04:13.320 --> 00:04:14.700
let's see, it's divisible by 3.

00:04:14.700 --> 00:04:17.399
It is 3 times 7.

00:04:17.399 --> 00:04:24.610
And the prime factorization
of 30 is equal to 3

00:04:24.610 --> 00:04:29.550
times 10, and 10 is 2 times 5.

00:04:29.550 --> 00:04:32.310
So what are the
most factors that we

00:04:32.310 --> 00:04:35.620
can take from both 21 and
30 to make the largest

00:04:35.620 --> 00:04:36.920
possible numbers?

00:04:36.920 --> 00:04:40.190
So when you look at the
prime factorization,

00:04:40.190 --> 00:04:45.540
the only thing that's common
right over here is a 3.

00:04:45.540 --> 00:04:50.440
And so we would say that
the greatest common factor

00:04:50.440 --> 00:04:54.090
or the greatest common
divisor of 21 and 30 is 3.

00:04:54.090 --> 00:04:56.140
If you saw nothing in
common right over here,

00:04:56.140 --> 00:04:59.310
then you say the greatest
common divisor is one.

00:04:59.310 --> 00:05:01.630
Let me give you another
interesting example, just so

00:05:01.630 --> 00:05:03.980
that we can get a
sense of things.

00:05:03.980 --> 00:05:08.470
So let's say these two
numbers were not 21 and 30,

00:05:08.470 --> 00:05:13.630
but let's say we care about
the greatest common divisor not

00:05:13.630 --> 00:05:19.430
of 21, but let's
say of 105 and 30.

00:05:22.257 --> 00:05:24.090
So if we did the prime
factorization method,

00:05:24.090 --> 00:05:25.822
it might become a
little clearer now.

00:05:25.822 --> 00:05:28.280
Actually figuring out, hey,
what are all the factors of 105

00:05:28.280 --> 00:05:30.080
might be a little bit
of a pain, but if you

00:05:30.080 --> 00:05:32.610
do a prime factorization,
you'd say, well, let's see,

00:05:32.610 --> 00:05:36.460
105-- it's divisible
by 5, definitely.

00:05:36.460 --> 00:05:42.880
So it's 5 times 21,
and 21 is 3 times 7.

00:05:42.880 --> 00:05:46.360
So the prime
factorization of 105

00:05:46.360 --> 00:05:49.620
is equal to-- if I write them
in increasing order-- 3 times

00:05:49.620 --> 00:05:51.790
5 times 7.

00:05:51.790 --> 00:05:55.890
The prime factorization of
30, we already figured out

00:05:55.890 --> 00:06:00.520
is 30 is equal to
2 times 3 times 5.

00:06:00.520 --> 00:06:02.650
So what's the most
number of factors

00:06:02.650 --> 00:06:05.360
or prime factors that
they have in common?

00:06:05.360 --> 00:06:11.180
Well, these two both have a
3, and they both have a 5.

00:06:11.180 --> 00:06:14.040
So the greatest common factor
or greatest common divisor

00:06:14.040 --> 00:06:15.770
is going to be a
product of these two.

00:06:15.770 --> 00:06:19.090
In this situation,
the GCD of 105 and 30

00:06:19.090 --> 00:06:23.230
is 3 times 5, is equal to 15.

00:06:23.230 --> 00:06:24.480
So you could do it either way.

00:06:24.480 --> 00:06:27.120
You could just list out the
traditional divisors or factors

00:06:27.120 --> 00:06:28.600
and, say, figure
out which of those

00:06:28.600 --> 00:06:31.110
is common and is the greatest.

00:06:31.110 --> 00:06:34.470
Or you can break it down
into its core constituencies,

00:06:34.470 --> 00:06:36.700
its prime factors,
and then figure out

00:06:36.700 --> 00:06:39.946
what is the largest set
of common prime factors,

00:06:39.946 --> 00:06:41.320
and the product
of those is going

00:06:41.320 --> 00:06:43.135
to be your greatest
common factor.

00:06:43.135 --> 00:06:48.541
It's the largest number that
is divisible into both numbers.


Least common multiple exercise 2

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:05.550
We need to figure out the least
common multiple of 30 and 25.

00:00:05.550 --> 00:00:07.850
So let's get our little
scratch pad out here.

00:00:07.850 --> 00:00:13.160
And we care about 30
and we care about 25.

00:00:13.160 --> 00:00:15.820
And I'm going to do this
using the prime factorization

00:00:15.820 --> 00:00:18.560
method which I just like more.

00:00:18.560 --> 00:00:20.080
Let's find the
prime factorization

00:00:20.080 --> 00:00:21.160
of both of these numbers.

00:00:21.160 --> 00:00:23.470
So 30, it's divisible by 2.

00:00:23.470 --> 00:00:25.520
It's 2 times 15.

00:00:25.520 --> 00:00:28.160
15 is 3 times 5.

00:00:28.160 --> 00:00:30.770
And now we've expressed
30 as the product

00:00:30.770 --> 00:00:35.030
of only prime numbers,
2 times 3 times 5.

00:00:35.030 --> 00:00:37.360
Now let's do the
same thing for 25.

00:00:37.360 --> 00:00:41.357
25 is-- well that's
just 5 times 5.

00:00:41.357 --> 00:00:42.440
So let me write that down.

00:00:42.440 --> 00:00:47.060
25 is equal to 5 times 5.

00:00:47.060 --> 00:00:49.000
Now to find the least
common multiple,

00:00:49.000 --> 00:00:51.870
let me write this down,
the least common multiple

00:00:51.870 --> 00:01:00.320
of 30 and 25 is going to
have a number whose prime

00:01:00.320 --> 00:01:03.510
factorization is a super
set of both of these

00:01:03.510 --> 00:01:06.370
or has all of these numbers
in them as many times

00:01:06.370 --> 00:01:08.734
as we have in any one of these.

00:01:08.734 --> 00:01:10.150
So it's the least
common multiple.

00:01:10.150 --> 00:01:12.050
Well it has to be
divisible by 30.

00:01:12.050 --> 00:01:18.360
So it's going to need a
2 times a 3 times a 5.

00:01:18.360 --> 00:01:22.220
This is what makes
it divisible by 30.

00:01:22.220 --> 00:01:24.740
But it needs to also
be divisible by 25.

00:01:24.740 --> 00:01:27.010
And in order to be
divisible by 25,

00:01:27.010 --> 00:01:30.430
you need to have two 5s in
your prime factorization.

00:01:30.430 --> 00:01:34.210
Right now our prime
factorization only has one 5.

00:01:34.210 --> 00:01:34.885
So let's throw.

00:01:34.885 --> 00:01:36.260
So we have one 5
right over here.

00:01:36.260 --> 00:01:37.110
We need another 5.

00:01:37.110 --> 00:01:39.670
So let's throw another
5 right over here.

00:01:39.670 --> 00:01:42.550
So now this thing
clearly has a 25 in it.

00:01:42.550 --> 00:01:45.110
It's clearly divisible by 25.

00:01:45.110 --> 00:01:47.040
And this is the least
common multiple.

00:01:47.040 --> 00:01:50.797
I could have, if we just
wanted a common multiple,

00:01:50.797 --> 00:01:52.380
we could have thrown
more factors here

00:01:52.380 --> 00:01:55.000
and it would have definitely
been divisible by 30 or 25,

00:01:55.000 --> 00:01:57.340
but this has the bare
minimum of prime factors

00:01:57.340 --> 00:02:01.060
necessary to be
divisible by 30 and 25.

00:02:01.060 --> 00:02:03.210
If I got rid of
any one of these,

00:02:03.210 --> 00:02:04.950
I wouldn't be divisible
by both anymore.

00:02:04.950 --> 00:02:07.170
If I got rid of this 2, I
wouldn't be divisible by 30

00:02:07.170 --> 00:02:07.670
anymore.

00:02:07.670 --> 00:02:10.860
If I got rid of one of the 5s,
I wouldn't be divisible by 25

00:02:10.860 --> 00:02:11.610
anymore.

00:02:11.610 --> 00:02:12.860
So let's just multiply it out.

00:02:12.860 --> 00:02:14.680
This is essentially
the prime factorization

00:02:14.680 --> 00:02:16.540
of our least common multiple.

00:02:16.540 --> 00:02:23.230
And this is equal to 2 times 3
is 6, 6 times 5 is 30, 30 times

00:02:23.230 --> 00:02:26.810
5 is equal to 150.

00:02:26.810 --> 00:02:31.630
And of course, we can
check our answer, 150.

00:02:31.630 --> 00:02:34.340
Check it, and we got it right.


Least common multiple exercise

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:05.090
What is the least common
multiple of 36 and 12?

00:00:05.090 --> 00:00:08.492
So another way to say this is
LCM, in parentheses, 36 to 12.

00:00:08.492 --> 00:00:09.950
And this is literally
saying what's

00:00:09.950 --> 00:00:13.250
the least common
multiple of 36 and 12?

00:00:13.250 --> 00:00:14.840
Well, this one might
pop out at you,

00:00:14.840 --> 00:00:18.740
because 36 itself
is a multiple of 12.

00:00:18.740 --> 00:00:20.600
And 36 is also a multiple of 36.

00:00:20.600 --> 00:00:22.900
It's 1 times 36.

00:00:22.900 --> 00:00:26.870
So the smallest number that is
both a multiple of 36 and 12--

00:00:26.870 --> 00:00:32.920
because 36 is a multiple
of 12-- is actually 36.

00:00:32.920 --> 00:00:33.637
There we go.

00:00:33.637 --> 00:00:34.970
Let's do a couple more of these.

00:00:34.970 --> 00:00:37.570
That one was too easy.

00:00:37.570 --> 00:00:40.910
What is the least common
multiple of 18 and 12?

00:00:40.910 --> 00:00:43.780
And they just state this
with a different notation.

00:00:43.780 --> 00:00:45.580
The least common
multiple of 18 and 12

00:00:45.580 --> 00:00:47.090
is equal to question mark.

00:00:47.090 --> 00:00:48.984
So let's think about
this a little bit.

00:00:48.984 --> 00:00:51.400
So there's a couple of ways
you can think about-- so let's

00:00:51.400 --> 00:00:53.358
just write down our
numbers that we care about.

00:00:53.358 --> 00:00:57.530
We care about 18,
and we care about 12.

00:00:57.530 --> 00:00:59.840
So there's two ways that
we could approach this.

00:00:59.840 --> 00:01:01.850
One is the prime
factorization approach.

00:01:01.850 --> 00:01:04.030
We can take the prime
factorization of both

00:01:04.030 --> 00:01:06.340
of these numbers
and then construct

00:01:06.340 --> 00:01:08.960
the smallest number
whose prime factorization

00:01:08.960 --> 00:01:11.190
has all of the ingredients
of both of these numbers,

00:01:11.190 --> 00:01:13.070
and that will be the
least common multiple.

00:01:13.070 --> 00:01:14.310
So let's do that.

00:01:14.310 --> 00:01:20.770
18 is 2 times 9, which is
the same thing as 2 times 3

00:01:20.770 --> 00:01:23.570
times 3, or 18 is 2 times 9.

00:01:23.570 --> 00:01:24.860
9 is 3 times 3.

00:01:24.860 --> 00:01:30.030
So we could write 18 is
equal to 2 times 3 times 3.

00:01:30.030 --> 00:01:32.230
That's its prime factorization.

00:01:32.230 --> 00:01:35.430
12 is 2 times 6.

00:01:35.430 --> 00:01:38.020
6 is 2 times 3.

00:01:38.020 --> 00:01:43.980
So 12 is equal to
2 times 2 times 3.

00:01:43.980 --> 00:01:47.570
Now, the least common
multiple of 18 and 12--

00:01:47.570 --> 00:01:52.280
let me write this down-- so
the least common multiple of 18

00:01:52.280 --> 00:02:01.220
and 12 is going to have to have
enough prime factors to cover

00:02:01.220 --> 00:02:03.090
both of these
numbers and no more,

00:02:03.090 --> 00:02:06.130
because we want the least
common multiple or the smallest

00:02:06.130 --> 00:02:07.990
common multiple.

00:02:07.990 --> 00:02:08.990
So let's think about it.

00:02:08.990 --> 00:02:12.020
Well, it needs to have at
least 1, 2, a 3 and a 3

00:02:12.020 --> 00:02:14.670
in order to be divisible by 18.

00:02:14.670 --> 00:02:15.810
So let's write that down.

00:02:15.810 --> 00:02:19.870
So we have to have
a 2 times 3 times 3.

00:02:19.870 --> 00:02:21.590
This makes it divisible by 18.

00:02:21.590 --> 00:02:24.142
If you multiply this
out, you actually get 18.

00:02:24.142 --> 00:02:25.350
And now let's look at the 12.

00:02:25.350 --> 00:02:27.516
So this part right over
here-- let me make it clear.

00:02:27.516 --> 00:02:29.260
This part right over
here is the part

00:02:29.260 --> 00:02:32.439
that makes up 18, makes
it divisible by 18.

00:02:32.439 --> 00:02:33.230
And then let's see.

00:02:33.230 --> 00:02:37.290
12, we need two 2's and a 3.

00:02:37.290 --> 00:02:42.220
Well, we already have one 3,
so our 3 is taken care of.

00:02:42.220 --> 00:02:45.750
We have one 2, so this
2 is taken care of.

00:02:45.750 --> 00:02:47.170
But we don't have two 2s's.

00:02:47.170 --> 00:02:49.180
So we need another 2 here.

00:02:49.180 --> 00:02:52.850
So, notice, now this number
right over here has a 2 times

00:02:52.850 --> 00:02:56.990
2 times 3 in it, or it has a
12 in it, and it has a 2 times

00:02:56.990 --> 00:02:59.620
3 times 3, or an 18 in it.

00:02:59.620 --> 00:03:02.110
So this right over here is
the least common multiple

00:03:02.110 --> 00:03:03.500
of 18 and 12.

00:03:03.500 --> 00:03:07.580
If we multiply it out,
so 2 times 2 is 4.

00:03:07.580 --> 00:03:10.000
4 times 3 is 12.

00:03:10.000 --> 00:03:15.730
12 times 3 is equal to 36.

00:03:15.730 --> 00:03:17.226
And we are done.

00:03:17.226 --> 00:03:18.850
Now, the other way
you could've done it

00:03:18.850 --> 00:03:21.136
is what I would say
just the brute force

00:03:21.136 --> 00:03:23.510
method of just looking at the
multiples of these numbers.

00:03:23.510 --> 00:03:24.960
You would say, well, let's see.

00:03:24.960 --> 00:03:29.470
The multiples of 18
are 18, 36, and I

00:03:29.470 --> 00:03:32.030
could keep going
higher and higher, 54.

00:03:32.030 --> 00:03:33.420
And I could keep going.

00:03:33.420 --> 00:03:38.960
And the multiples of
12 are 12, 24, 36.

00:03:38.960 --> 00:03:41.920
And immediately I say, well, I
don't have to go any further.

00:03:41.920 --> 00:03:44.230
I already found a
multiple of both,

00:03:44.230 --> 00:03:46.260
and this is the smallest
multiple of both.

00:03:46.260 --> 00:03:48.145
It is 36.

00:03:48.145 --> 00:03:49.770
You might say, hey,
why would I ever do

00:03:49.770 --> 00:03:52.310
this one right over here
as opposed to this one?

00:03:52.310 --> 00:03:53.530
A couple of reasons.

00:03:53.530 --> 00:03:55.220
This one, you're
kind of-- it's fun,

00:03:55.220 --> 00:03:57.280
because you're actually
decomposing the number

00:03:57.280 --> 00:03:58.624
and then building it back up.

00:03:58.624 --> 00:04:00.790
And also, this is a better
way, especially if you're

00:04:00.790 --> 00:04:03.020
doing it with really, really
large and hairy numbers.

00:04:03.020 --> 00:04:05.170
Really, really, really
large and hairy numbers

00:04:05.170 --> 00:04:07.190
where you keep trying to
find all the multiples,

00:04:07.190 --> 00:04:09.100
you might have to go pretty
far to actually figure out

00:04:09.100 --> 00:04:10.840
what their least
common multiple is.

00:04:10.840 --> 00:04:13.464
Here, you'll be able to do it a
little bit more systematically,

00:04:13.464 --> 00:04:15.740
and you'll know
what you're doing.


Prime factorization exercise

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.560
We're asked, what is the
prime factorization of 36?

00:00:04.560 --> 00:00:07.190
Let me get my little
scratch paper out.

00:00:07.190 --> 00:00:11.230
So the prime
factorization of 36.

00:00:11.230 --> 00:00:14.060
So let's start with the
smallest prime number we know,

00:00:14.060 --> 00:00:15.130
and that is 2.

00:00:15.130 --> 00:00:17.730
And think about,
does 2 go into 36?

00:00:17.730 --> 00:00:18.770
Well, sure, it does.

00:00:18.770 --> 00:00:21.447
36 is 2 times 18.

00:00:21.447 --> 00:00:22.530
So we can write that down.

00:00:22.530 --> 00:00:27.150
36 is 2 times 18.

00:00:27.150 --> 00:00:30.270
So now we have 36 as a
product of a prime number,

00:00:30.270 --> 00:00:32.890
and 18 is clearly
a composite number.

00:00:32.890 --> 00:00:35.640
It has factors
other than 1 and 18.

00:00:35.640 --> 00:00:37.610
So let's try to
factor this further.

00:00:37.610 --> 00:00:39.720
So is this divisible by 2?

00:00:39.720 --> 00:00:40.510
Sure.

00:00:40.510 --> 00:00:42.370
18 is 2 times 9.

00:00:46.120 --> 00:00:49.570
So now 9 is a composite number
that we haven't fully factored.

00:00:49.570 --> 00:00:51.790
Obviously, the 2's
are both prime.

00:00:51.790 --> 00:00:55.710
9 is not divisible by 2,
but it is divisible by 3.

00:00:55.710 --> 00:00:59.800
9 is 3 times 3.

00:00:59.800 --> 00:01:08.450
So we can say that 36 is equal
to 2 times 2 times 3 times 3.

00:01:08.450 --> 00:01:10.460
This is its prime factorization.

00:01:10.460 --> 00:01:12.710
All of these numbers are prime.

00:01:12.710 --> 00:01:17.790
So now let's input that to
make sure we got it right.

00:01:17.790 --> 00:01:22.050
2 times 2 times 3 times 3.

00:01:22.050 --> 00:01:23.410
And you can check yourself.

00:01:23.410 --> 00:01:26.740
If you have the product of
numbers that are all prime

00:01:26.740 --> 00:01:29.800
and the product actually is
36, you have successfully

00:01:29.800 --> 00:01:32.350
prime factorized the number.

00:01:32.350 --> 00:01:35.080
Let's do a couple more of these.

00:01:35.080 --> 00:01:37.170
What is the prime
factorization of 30?

00:01:37.170 --> 00:01:39.300
So I'll get my scratch
paper out again.

00:01:39.300 --> 00:01:41.230
So we'll do the same process.

00:01:41.230 --> 00:01:44.820
So 30-- well, it's
divisible by 2.

00:01:44.820 --> 00:01:48.911
So we can write
that as 2 times 15.

00:01:48.911 --> 00:01:50.950
15 isn't divisible by 2.

00:01:50.950 --> 00:01:52.970
But it is divisible by 3.

00:01:52.970 --> 00:01:56.160
It's the same
thing as 3 times 5.

00:01:56.160 --> 00:01:58.560
And both 3 and 5
are prime numbers.

00:01:58.560 --> 00:02:01.810
They are only divisible
by 1 and themselves.

00:02:01.810 --> 00:02:07.850
So the prime factorization
of 30 is 2 times 3 times 5.

00:02:07.850 --> 00:02:09.419
Let's enter that in.

00:02:09.419 --> 00:02:15.960
So it is 2 times 3 times 5.

00:02:15.960 --> 00:02:19.230
Let's do one more of these.

00:02:19.230 --> 00:02:22.490
What is the prime
factorization of 73?

00:02:22.490 --> 00:02:23.925
Now, 73 is interesting.

00:02:26.510 --> 00:02:28.370
I'll get my scratch
paper out for this.

00:02:28.370 --> 00:02:31.050
We could try to factor 73.

00:02:31.050 --> 00:02:32.150
So you might try 2.

00:02:32.150 --> 00:02:33.650
Well, this is clearly
an odd number.

00:02:33.650 --> 00:02:35.380
So 2 isn't going to
be divisible into 73.

00:02:35.380 --> 00:02:37.120
You might try 3.

00:02:37.120 --> 00:02:39.550
You would immediately see,
well, 3 is divisible into 72.

00:02:39.550 --> 00:02:42.515
If you divide into 73,
you have a remainder of 1.

00:02:42.515 --> 00:02:45.070
Well, 4 isn't a prime number,
so we wouldn't even try.

00:02:45.070 --> 00:02:46.790
5 isn't divisible into 73.

00:02:46.790 --> 00:02:49.050
It doesn't end in a 5 or 0.

00:02:49.050 --> 00:02:52.310
7 is not divisible into 73.

00:02:52.310 --> 00:02:55.522
7 goes into 70, so you'd
have a remainder of 3.

00:02:55.522 --> 00:02:57.890
11 isn't divisible into 73.

00:02:57.890 --> 00:03:02.480
It's divisible into
66 or 77, so not 73.

00:03:02.480 --> 00:03:04.420
As I test more and
more numbers, it

00:03:04.420 --> 00:03:07.080
doesn't look like there's
any easy thing that

00:03:07.080 --> 00:03:10.020
divides into 73.

00:03:10.020 --> 00:03:15.330
So I'm willing to go with
73 itself is a prime number.

00:03:15.330 --> 00:03:18.010
So this is its
prime factorization.

00:03:18.010 --> 00:03:19.350
It's just 73.

00:03:19.350 --> 00:03:21.390
So let's write that down.

00:03:21.390 --> 00:03:24.190
So the answer here,
let's just write 73.

00:03:24.190 --> 00:03:25.910
And you don't want
to write 1 times 73,

00:03:25.910 --> 00:03:28.540
because 1 is not a prime number.

00:03:28.540 --> 00:03:34.070
Remember, 1 only has
one factor, itself.

00:03:34.070 --> 00:03:37.380
A prime number has two
factors, 1 and itself.

00:03:37.380 --> 00:03:39.690
Two different prime
factors-- 1 and itself.

00:03:39.690 --> 00:03:41.200
And itself is not one.

00:03:41.200 --> 00:03:43.250
So we just want to write
prime numbers here.

00:03:43.250 --> 00:03:44.660
73 is a prime number.

00:03:44.660 --> 00:03:46.240
Let's check our answer.

00:03:46.240 --> 00:03:47.867
And we got it right.


Rational number word problem with proportions

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.620
Shares of stock represent
how much of a company

00:00:03.620 --> 00:00:05.110
a person owns?

00:00:05.110 --> 00:00:09.120
Puff Incorporated is owned
by Peter, Paul, and Mary.

00:00:09.120 --> 00:00:15.570
Peter owns 4,050 shares,
Paul owns 2,510 shares,

00:00:15.570 --> 00:00:19.930
And Mary owns 42
hundred or 4,200 shares.

00:00:19.930 --> 00:00:26.250
Suppose the company made a
profit this year of $1,500,000.

00:00:26.250 --> 00:00:30.370
If each shareholder gets a
proportion of the total profit

00:00:30.370 --> 00:00:33.920
that is equal to the proportion
of the share they own,

00:00:33.920 --> 00:00:38.220
how much money
does Mary receive?

00:00:38.220 --> 00:00:42.060
So first, let's think about
what fraction or what percentage

00:00:42.060 --> 00:00:46.260
or what proportion of the total
profit should Mary receive?

00:00:46.260 --> 00:00:54.220
So Mary owns 4,200 shares.

00:00:54.220 --> 00:00:57.390
And how many total shares
are there in the company?

00:00:57.390 --> 00:01:00.120
Well, the only owners
of this company

00:01:00.120 --> 00:01:01.236
are Peter, Paul, and Mary.

00:01:01.236 --> 00:01:02.610
So the total number
of shares are

00:01:02.610 --> 00:01:05.900
the sum of Peter, Paul,
and Mary's shares.

00:01:05.900 --> 00:01:14.530
So we know that Peter
owns 4,050 shares.

00:01:14.530 --> 00:01:21.100
We know that Paul owns 2,510.

00:01:21.100 --> 00:01:27.339
And of course, we know
that Mary owns 4,200.

00:01:27.339 --> 00:01:28.130
So let's figure it.

00:01:28.130 --> 00:01:31.990
This is the fraction
of the profit

00:01:31.990 --> 00:01:34.390
that Mary should be getting.

00:01:34.390 --> 00:01:36.780
So let's figure out
this with a calculator.

00:01:36.780 --> 00:01:42.860
So this is going
to be 4,200 divided

00:01:42.860 --> 00:01:56.080
by 4,050 plus 2,510 plus 4,200.

00:01:56.080 --> 00:02:00.794
And so Mary should get
0.39 of the profit.

00:02:00.794 --> 00:02:02.710
Or if we talk about it
in terms of percentage,

00:02:02.710 --> 00:02:05.890
it would be 39% of the profit.

00:02:05.890 --> 00:02:09.602
So this right over
here is 0.390.

00:02:09.602 --> 00:02:11.310
Actually, let me just
leave it like that.

00:02:11.310 --> 00:02:20.070
So this is approximately
equal to 0.39.

00:02:20.070 --> 00:02:23.920
And so if we want to figure
out how much profit she gets,

00:02:23.920 --> 00:02:26.650
she gets this much
times the total profit.

00:02:26.650 --> 00:02:35.954
So Mary gets 0.39, and
this is the exact quantity

00:02:35.954 --> 00:02:36.620
right over here.

00:02:36.620 --> 00:02:41.360
0.39 is the approximation,
so approximately 0.39.

00:02:41.360 --> 00:02:45.280
This little squiggly equal sign
is how I say approximately.

00:02:45.280 --> 00:02:48.850
She's going to get 0.39
times the total profit, which

00:02:48.850 --> 00:02:54.655
is $1,500,000.

00:02:54.655 --> 00:02:57.220
And so let's go back
to our calculator.

00:02:57.220 --> 00:02:59.940
And so we said
approximately 0.39.

00:02:59.940 --> 00:03:01.570
This is the exact value.

00:03:01.570 --> 00:03:03.270
I can get that
exact value again,

00:03:03.270 --> 00:03:06.210
at least on this calculator,
by saying Second Answer.

00:03:06.210 --> 00:03:08.620
This literally means
the previous answer.

00:03:08.620 --> 00:03:13.170
So this is literally this number
right over here-- 0.390334

00:03:13.170 --> 00:03:13.720
whatever.

00:03:13.720 --> 00:03:18.465
So this is the exact
quantity times $1,500,000.

00:03:18.465 --> 00:03:20.280
One, two, three, there we go.

00:03:20.280 --> 00:03:22.530
There's $1,500,000.

00:03:22.530 --> 00:03:25.160
I've got to make sure I have
the right number zeroes there.

00:03:25.160 --> 00:03:30.980
And that gets us
$585,50- and we need

00:03:30.980 --> 00:03:34.990
to round to the nearest
penny-- and $0.86.

00:03:34.990 --> 00:03:39.980
So let me put that someplace
where I can see it.

00:03:39.980 --> 00:03:55.730
So this is going to be equal
to $585,501 and, if we round,

00:03:55.730 --> 00:03:58.040
$0.86.

00:03:58.040 --> 00:04:00.460
So not bad for
one year for Mary.

00:04:00.460 --> 00:04:05.260
She should be able to live
quite well just on her dividends

00:04:05.260 --> 00:04:07.139
from Puff Incorporated.


Rational number word problem with ratios

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:06.970
In the year 1944, computers
weighed as much as 4,500

00:00:06.970 --> 00:00:08.520
kilograms.

00:00:08.520 --> 00:00:14.530
A modern laptop weighs
around 2.7 kilograms.

00:00:14.530 --> 00:00:18.830
What is the ratio of how much
computers weighed in 1944

00:00:18.830 --> 00:00:21.490
to how much a modern
laptop weighs?

00:00:21.490 --> 00:00:25.430
Express your answer as
a ratio of two integers.

00:00:25.430 --> 00:00:31.330
So the ratio of how much
computers weighed in 1944--

00:00:31.330 --> 00:00:36.010
so we know that's 4,500
kilograms-- we want

00:00:36.010 --> 00:00:41.520
the ratio of that to how
much a modern laptop weighs,

00:00:41.520 --> 00:00:44.550
and that's 2.7 kilograms.

00:00:44.550 --> 00:00:46.690
So this right over
here is a ratio.

00:00:46.690 --> 00:00:50.620
But we haven't expressed it
as a ratio of two integers.

00:00:50.620 --> 00:00:52.790
In particular,
4,500 is an integer.

00:00:52.790 --> 00:00:55.710
But 2.7 is not an integer.

00:00:55.710 --> 00:00:58.900
So the easy way to
convert 2.7 to an integer

00:00:58.900 --> 00:01:00.950
is to move the decimal
place one to the right.

00:01:00.950 --> 00:01:04.099
Or another way of thinking about
it is to multiply it by 10.

00:01:04.099 --> 00:01:06.160
So we can multiply this by 10.

00:01:06.160 --> 00:01:08.820
But if we just multiplied
the denominator by 10,

00:01:08.820 --> 00:01:10.696
that would change the
value of the ratio.

00:01:10.696 --> 00:01:12.070
In order to not
change the value,

00:01:12.070 --> 00:01:14.860
we have to multiply the
numerator and the denominator

00:01:14.860 --> 00:01:15.650
by 10.

00:01:15.650 --> 00:01:17.970
This is equivalent to just
multiplying this fraction

00:01:17.970 --> 00:01:20.240
by 10/10, which is
the same thing as one.

00:01:20.240 --> 00:01:21.850
It does not change the value.

00:01:21.850 --> 00:01:23.150
So what do we get?

00:01:23.150 --> 00:01:28.754
Well, in the numerator,
4,500 times 10 is 45,000.

00:01:28.754 --> 00:01:29.670
I'll put a comma here.

00:01:29.670 --> 00:01:31.416
It makes it a little
bit easier to read.

00:01:31.416 --> 00:01:32.790
And in the
denominator-- and this

00:01:32.790 --> 00:01:36.800
is the whole point of why
we multiplied by 10-- 2.7

00:01:36.800 --> 00:01:40.470
multiplied by 10 is 27.

00:01:40.470 --> 00:01:44.940
So we now have
expressed our answer

00:01:44.940 --> 00:01:46.350
as a ratio of two integers.

00:01:46.350 --> 00:01:48.090
So this is completely
legitimate.

00:01:48.090 --> 00:01:50.170
But we could also simplify this.

00:01:50.170 --> 00:01:53.700
Just looking at this,
it looks like 45,000

00:01:53.700 --> 00:01:58.340
is divisible by 45,
which is divisible by 9.

00:01:58.340 --> 00:02:00.057
And 27 is also divisible by 9.

00:02:00.057 --> 00:02:02.390
So why don't we divide the
numerator and the denominator

00:02:02.390 --> 00:02:04.490
both by 9?

00:02:04.490 --> 00:02:06.860
So we're going to divide
by 9 in the numerator,

00:02:06.860 --> 00:02:10.460
and we're going to divide
by 9 in the denominator.

00:02:10.460 --> 00:02:15.050
And we are going to get
45 divided by 9 is 5.

00:02:15.050 --> 00:02:18.400
So 45,000 divided by 9 is 5,000.

00:02:18.400 --> 00:02:26.210
So we're going to get 5,000
over-- 27 divided by 9 is 3.

00:02:26.210 --> 00:02:30.580
And I think we have now
simplified this about as much

00:02:30.580 --> 00:02:32.510
as we can.


Rational number word problem with fractions

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.460
Most liquids, when cooled,
will simply shrink.

00:00:04.460 --> 00:00:06.340
Water, on the other
hand, actually

00:00:06.340 --> 00:00:08.620
expands when it is frozen.

00:00:08.620 --> 00:00:12.910
Its volume will
increase by about 9%.

00:00:12.910 --> 00:00:18.350
Suppose you have 1/3 of a gallon
of water that gets frozen.

00:00:18.350 --> 00:00:22.670
What is the volume of the
ice that you now have?

00:00:22.670 --> 00:00:26.380
So you're starting with
1/3 of a gallon of water.

00:00:26.380 --> 00:00:29.740
They tell us that when it gets
frozen, when it turns into ice,

00:00:29.740 --> 00:00:33.230
its volume is going
to expand by 9%.

00:00:33.230 --> 00:00:36.010
So the new volume is going
to be your existing volume.

00:00:36.010 --> 00:00:41.800
So this is the original
volume, 1/3 of a gallon,

00:00:41.800 --> 00:00:44.010
and it's going to expand by 9%.

00:00:44.010 --> 00:00:47.420
So your frozen volume is going
to be your original volume

00:00:47.420 --> 00:00:50.290
plus 9% of your original volume.

00:00:55.710 --> 00:00:59.050
So you could say
it's 9% times 1/3.

00:00:59.050 --> 00:01:01.897
So this right over here is
going to be the expanded volume.

00:01:01.897 --> 00:01:03.980
Now, there's a bunch of
ways we can figure it out.

00:01:03.980 --> 00:01:06.470
We could turn things to
decimals or whatever else,

00:01:06.470 --> 00:01:11.590
but they tell us to express
your answer as a fraction.

00:01:11.590 --> 00:01:13.840
So let's make sure that
everything here is a fraction,

00:01:13.840 --> 00:01:15.567
and then we'll just
try to simplify.

00:01:15.567 --> 00:01:18.025
So the one thing that's sitting
here that is not a fraction

00:01:18.025 --> 00:01:20.320
is our 9%.

00:01:20.320 --> 00:01:23.020
Well, what does 9%
actually represent?

00:01:23.020 --> 00:01:27.200
Well, 9% literally
means 9 per 100.

00:01:27.200 --> 00:01:30.140
So we could rewrite
this as-- so this

00:01:30.140 --> 00:01:34.980
is going to be equal to 1/3
plus, instead of writing 9%,

00:01:34.980 --> 00:01:40.470
I'll write that as 9 per 100,
and then once again times 1/3.

00:01:44.620 --> 00:01:47.720
And we can simplify this
expression right over here.

00:01:47.720 --> 00:01:50.270
We have a 9 in the numerator,
a 3 in the denominator.

00:01:50.270 --> 00:01:54.810
If we divide both of them
by 3, we get a 3 and a 1.

00:01:54.810 --> 00:01:59.741
And so we're left with
1/3 plus 300 times 1/1.

00:01:59.741 --> 00:02:01.240
Well, that's just
going to be 3/100.

00:02:01.240 --> 00:02:07.440
So this is just going to
be equal to 1/3 plus-- I'll

00:02:07.440 --> 00:02:09.250
write this in orange
still, or maybe I'll

00:02:09.250 --> 00:02:14.690
do it in a new
color-- plus 3/100.

00:02:14.690 --> 00:02:18.554
And now we have to add
something, or two numbers that

00:02:18.554 --> 00:02:19.720
have different denominators.

00:02:19.720 --> 00:02:22.110
So let's find a
common denominator.

00:02:22.110 --> 00:02:25.840
So this is going to be equal to,
well, the least common multiple

00:02:25.840 --> 00:02:28.594
of 3 and 100.

00:02:28.594 --> 00:02:31.010
And they share no common factor,
so it's really just going

00:02:31.010 --> 00:02:34.160
to be the product of 3 and 100--
the least common multiple is

00:02:34.160 --> 00:02:34.820
300.

00:02:34.820 --> 00:02:38.574
So it's going to be
something over 300

00:02:38.574 --> 00:02:42.000
plus something over 300.

00:02:42.000 --> 00:02:45.040
Now to go from 3 to 300, in
the denominator you multiply

00:02:45.040 --> 00:02:48.710
by 100, so you have to multiply
the numerator by 100 as well.

00:02:48.710 --> 00:02:52.700
So 1/3 is the same
thing as 100/300.

00:02:52.700 --> 00:02:54.990
And to go from 100
to 300, we have

00:02:54.990 --> 00:02:57.800
to multiply by 3
in the denominator,

00:02:57.800 --> 00:03:00.350
so we have to multiply by
3 in the numerator as well.

00:03:00.350 --> 00:03:06.276
So 3/100 is the
same thing as 9/300.

00:03:06.276 --> 00:03:07.400
And now we're ready to add.

00:03:07.400 --> 00:03:14.480
This is going to be 100 plus
9/300, which is 109/300.

00:03:14.480 --> 00:03:16.680
So this is the volume
of ice that I now

00:03:16.680 --> 00:03:20.398
have expressed as a fraction.


Constructing and interpreting absolute value

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.040
A riparian buffer is
an area around a stream

00:00:04.040 --> 00:00:06.230
that's covered in vegetation.

00:00:06.230 --> 00:00:08.720
The buffer protects
the water in the stream

00:00:08.720 --> 00:00:10.932
from activity happening
near its location.

00:00:10.932 --> 00:00:12.390
So let's imagine
this a little bit.

00:00:12.390 --> 00:00:15.460
So let's say that I have
a stream right over here.

00:00:15.460 --> 00:00:18.840
So this right over
here is my stream,

00:00:18.840 --> 00:00:20.720
and the riparian
buffer is an area

00:00:20.720 --> 00:00:23.070
around the stream that's
covered in vegetation.

00:00:23.070 --> 00:00:25.950
So this could be the riparian
buffer right over here

00:00:25.950 --> 00:00:28.160
on this side of the
stream, and maybe there's

00:00:28.160 --> 00:00:32.259
another one on this side of
the stream right over here.

00:00:32.259 --> 00:00:35.750
Suppose you run a farm
near a riparian buffer.

00:00:35.750 --> 00:00:39.530
The buffer begins S feet
from the edge of your farm

00:00:39.530 --> 00:00:44.020
and ends B feet from
the edge of your farm.

00:00:44.020 --> 00:00:46.870
How wide is the riparian buffer?

00:00:46.870 --> 00:00:49.860
So this right over here
could be maybe my farm.

00:00:49.860 --> 00:00:57.360
So I've got my farm here, those
are my crops, just like that.

00:00:57.360 --> 00:00:59.500
And they're saying
the buffer begins

00:00:59.500 --> 00:01:01.170
S feet from the
edge of your farm.

00:01:01.170 --> 00:01:03.720
So, actually, let's say
that this right over here

00:01:03.720 --> 00:01:06.810
is the edge of my farm.

00:01:06.810 --> 00:01:09.890
And it says it begins S feet
from the edge of your farm.

00:01:09.890 --> 00:01:11.640
So it depends which
side you consider

00:01:11.640 --> 00:01:14.210
to be the beginning of
the riparian buffer.

00:01:14.210 --> 00:01:16.590
Maybe we'll take this
side to be the beginning.

00:01:16.590 --> 00:01:19.510
And so they're saying this
distance right over here

00:01:19.510 --> 00:01:24.580
is S feet, and it ends B feet
from the edge of your farm.

00:01:24.580 --> 00:01:29.150
So they're saying that this
distance right over here all

00:01:29.150 --> 00:01:33.210
the way to the other side of
the riparian buffer is B feet.

00:01:33.210 --> 00:01:35.690
How wide is the riparian buffer?

00:01:35.690 --> 00:01:39.380
So the way I made my assumptions
about where the riparian buffer

00:01:39.380 --> 00:01:42.350
starts and ends,
it looks like it's

00:01:42.350 --> 00:01:46.130
going to be B minus
S. It looks like it's

00:01:46.130 --> 00:01:48.800
going to be B minus s,
which would be this distance

00:01:48.800 --> 00:01:49.530
right over there.

00:01:49.530 --> 00:01:54.390
So that is B minus
S. But I just assumed

00:01:54.390 --> 00:01:57.810
the buffer starts
here and ends here.

00:01:57.810 --> 00:01:59.690
I could have viewed it
the other way around.

00:01:59.690 --> 00:02:03.260
I could have said, hey,
look, the buffer starts here.

00:02:03.260 --> 00:02:06.750
The buffer starts on
this side, and this

00:02:06.750 --> 00:02:09.300
would be my S. And
I could have said,

00:02:09.300 --> 00:02:11.210
maybe it ends on
this side, and then

00:02:11.210 --> 00:02:14.840
this would be the B. We
really don't know which one

00:02:14.840 --> 00:02:17.640
is the start or the
end of the buffer.

00:02:17.640 --> 00:02:19.720
And in this world
right over here,

00:02:19.720 --> 00:02:22.800
we would say that
the difference,

00:02:22.800 --> 00:02:31.620
or how wide the riparian
buffer is, would be S minus B.

00:02:31.620 --> 00:02:33.580
But one way that we
make sure that we're

00:02:33.580 --> 00:02:36.038
going to get a positive answer
either way-- because we want

00:02:36.038 --> 00:02:37.584
our distance to
be positive-- is,

00:02:37.584 --> 00:02:39.000
well, we could
say, look, we don't

00:02:39.000 --> 00:02:41.700
know if it's S minus
B or B minus S,

00:02:41.700 --> 00:02:43.785
so let's just take the
absolute value of these.

00:02:43.785 --> 00:02:48.760
So absolute value of B minus
S, even in this situation right

00:02:48.760 --> 00:02:51.180
over here, is going to
give the same exact answer

00:02:51.180 --> 00:02:53.470
as the absolute value
of S minus B. It's just

00:02:53.470 --> 00:02:57.200
going to find the absolute
difference between these two

00:02:57.200 --> 00:03:00.050
values regardless of
which one is larger.

00:03:00.050 --> 00:03:01.150
So let me make that clear.

00:03:01.150 --> 00:03:03.640
The absolute value of B
minus S is the same thing

00:03:03.640 --> 00:03:07.710
as the absolute
value of S minus B.

00:03:07.710 --> 00:03:11.090
And so I would go with
either one of these.

00:03:11.090 --> 00:03:13.900
Let's do another one.

00:03:13.900 --> 00:03:18.490
Doctor When is a time-traveling
alien with a PhD.

00:03:18.490 --> 00:03:23.340
In the year 1999, Doctor When
is sitting in Central Park,

00:03:23.340 --> 00:03:26.130
New York City and
eating a hot dog.

00:03:26.130 --> 00:03:29.340
He then realizes that
he's forgotten his hat.

00:03:29.340 --> 00:03:32.540
To retrieve his hat,
he travels T years,

00:03:32.540 --> 00:03:36.320
where T is a
negative to represent

00:03:36.320 --> 00:03:38.580
going backwards in time.

00:03:38.580 --> 00:03:42.750
What do we know about
the sum 1999 plus T?

00:03:42.750 --> 00:03:44.730
So we know that T is
a negative number.

00:03:44.730 --> 00:03:45.990
He's going backwards in time.

00:03:45.990 --> 00:03:47.610
So let's look at the choices.

00:03:47.610 --> 00:03:53.330
1999 plus T is the absolute
value of T greater than 1999.

00:03:53.330 --> 00:03:56.110
No, that would've been true
if T is a positive number.

00:03:56.110 --> 00:03:58.080
Then we'd say, hey,
this right over here

00:03:58.080 --> 00:04:02.540
is going to be the absolute
value of T greater than 1999.

00:04:02.540 --> 00:04:04.570
This would have been
only in a positive,

00:04:04.570 --> 00:04:08.650
so that would have been in a
situation if T is positive.

00:04:08.650 --> 00:04:09.617
So let's read this one.

00:04:09.617 --> 00:04:15.070
1999 plus T is the absolute
value of T less than 1999.

00:04:15.070 --> 00:04:15.777
That's right.

00:04:15.777 --> 00:04:17.860
Imagine a situation-- we
know that T is negative--

00:04:17.860 --> 00:04:21.920
imagine T as being
equal to negative 3.

00:04:21.920 --> 00:04:29.710
Then 1999 plus T would
be equal to 1996,

00:04:29.710 --> 00:04:41.180
which is 3 years less than, so
this is 3 years less than 1999.

00:04:41.180 --> 00:04:43.260
And this 3 right
over here-- or we

00:04:43.260 --> 00:04:47.594
could write that
1996 is-- instead

00:04:47.594 --> 00:04:49.010
of saying 3 years
less than-- it's

00:04:49.010 --> 00:04:57.560
the absolute value of negative
3 years less than 1999.

00:04:57.560 --> 00:04:59.480
So this one works out.

00:04:59.480 --> 00:05:03.100
1999 plus T is going to
be the absolute value,

00:05:03.100 --> 00:05:06.090
the absolute magnitude
of that negative number

00:05:06.090 --> 00:05:08.590
T less than 1999.

00:05:08.590 --> 00:05:11.590
So we could go with
this, right over here.

00:05:11.590 --> 00:05:13.430
Let's do one more.

00:05:13.430 --> 00:05:17.130
You're using an auger to
drill a hole for a fence post.

00:05:17.130 --> 00:05:20.950
The bottom of the hole is at
a position of negative 2.3

00:05:20.950 --> 00:05:22.210
relative to ground level.

00:05:22.210 --> 00:05:23.680
So let's draw the ground.

00:05:23.680 --> 00:05:28.840
So here is the ground right over
here, that's the ground level.

00:05:28.840 --> 00:05:31.255
And the bottom of my
hole-- so I drill a hole.

00:05:33.930 --> 00:05:37.020
And the bottom of the
hole is at a position

00:05:37.020 --> 00:05:39.970
of negative 2.3 feet.

00:05:39.970 --> 00:05:47.090
So this position right over here
is negative 2.3 feet relative

00:05:47.090 --> 00:05:48.550
to ground level.

00:05:48.550 --> 00:05:51.880
The fence post has a
length of 4.7 feet.

00:05:51.880 --> 00:05:53.770
How much higher
is the fence post

00:05:53.770 --> 00:05:55.930
when it stands atop
the ground than when

00:05:55.930 --> 00:05:58.050
it is sitting in
the hole you dug?

00:05:58.050 --> 00:06:00.070
So when it is sitting
atop the ground-- so let

00:06:00.070 --> 00:06:02.390
me draw the fence post.

00:06:02.390 --> 00:06:05.480
So when it's sitting
atop the ground,

00:06:05.480 --> 00:06:08.360
the fence post looks
something like this.

00:06:08.360 --> 00:06:12.180
And so when it's sitting atop
the ground, it's 4.7 feet high.

00:06:12.180 --> 00:06:15.490
4.7, so this distance
right over here.

00:06:15.490 --> 00:06:18.960
So if I were to draw-- so
let me make this clear.

00:06:18.960 --> 00:06:22.990
So this right over
here is negative 2.3.

00:06:22.990 --> 00:06:26.190
This right over here is 4.7.

00:06:26.190 --> 00:06:28.905
That's how tall it is when it's
sitting on top of the ground.

00:06:28.905 --> 00:06:30.280
How much higher
is the fence post

00:06:30.280 --> 00:06:31.946
when it stands atop
the ground than when

00:06:31.946 --> 00:06:33.690
it's sitting in
the hole you dug?

00:06:33.690 --> 00:06:36.920
So when you put this fence
post in the hole you dug,

00:06:36.920 --> 00:06:39.500
it's going to go down 2.3 feet.

00:06:39.500 --> 00:06:43.015
So it's going to
go down 2.3 feet.

00:06:45.890 --> 00:06:47.967
You could figure out
what this height is,

00:06:47.967 --> 00:06:50.550
but they're not even asking us
this height above ground level.

00:06:50.550 --> 00:06:53.890
They're just saying, how
much does the top go down

00:06:53.890 --> 00:06:55.630
when it's sitting
in the hole you dug?

00:06:55.630 --> 00:06:59.700
So it's pretty clear that
this distance right over here

00:06:59.700 --> 00:07:02.440
is going to be 2.3 feet.

00:07:02.440 --> 00:07:05.910
It's going to go
2.3 feet lower, when

00:07:05.910 --> 00:07:08.080
you stick it into the hole.

00:07:08.080 --> 00:07:09.610
So how much higher
is the fence post

00:07:09.610 --> 00:07:11.076
when it stands atop
the ground than when

00:07:11.076 --> 00:07:12.540
it's sitting in
the hole you dug?

00:07:12.540 --> 00:07:18.589
It's 2.3 feet higher when
it's stands atop the ground.


Distance between a point and a line

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.110
Find the distance between the
point negative 2, negative 4.

00:00:04.110 --> 00:00:05.380
This point right here.

00:00:05.380 --> 00:00:08.600
And the line y is equal
to negative 1/3 x plus 2.

00:00:08.600 --> 00:00:10.740
That's this line
right over here.

00:00:10.740 --> 00:00:15.660
Now to do it, we just need
to figure out a perpendicular

00:00:15.660 --> 00:00:18.770
line to this blue line, to
y is equal to negative 1/3

00:00:18.770 --> 00:00:24.420
x plus 2, that contains
this point right over here.

00:00:24.420 --> 00:00:29.829
So we need to figure out
the equation of this line.

00:00:29.829 --> 00:00:31.870
And then we need to figure
out where do these two

00:00:31.870 --> 00:00:34.190
lines intersect,
and then we need

00:00:34.190 --> 00:00:36.190
to find the distance
between these two

00:00:36.190 --> 00:00:39.270
points of intersection, and
we have the shortest distance

00:00:39.270 --> 00:00:43.120
between this point and
this line right over here.

00:00:43.120 --> 00:00:44.720
So the first step
is to figure out

00:00:44.720 --> 00:00:47.930
what is the slope of
this perpendicular line.

00:00:47.930 --> 00:00:50.310
Well, the slope of
a perpendicular line

00:00:50.310 --> 00:00:52.490
is going to be the
negative inverse

00:00:52.490 --> 00:00:55.460
of the slope of this blue line.

00:00:55.460 --> 00:00:58.170
So the negative
inverse of negative 1/3

00:00:58.170 --> 00:00:59.840
is going to be positive 3.

00:00:59.840 --> 00:01:02.980
So this line right over here
is going to have a slope of 3.

00:01:02.980 --> 00:01:04.560
So it's going to
have the form y is

00:01:04.560 --> 00:01:10.234
equal to 3x plus b, where
b is its y-intercept.

00:01:10.234 --> 00:01:11.650
It looks, just
eyeballing it here,

00:01:11.650 --> 00:01:13.370
that it's going to
be pretty close to 2.

00:01:13.370 --> 00:01:15.020
But let's verify that.

00:01:15.020 --> 00:01:18.724
So to figure out what b actually
is, let's substitute this point

00:01:18.724 --> 00:01:19.390
right over here.

00:01:19.390 --> 00:01:21.970
We know that not only
is this line slope 3,

00:01:21.970 --> 00:01:23.970
but this point has to sit on it.

00:01:23.970 --> 00:01:26.890
So this point has to
satisfy this equation.

00:01:26.890 --> 00:01:30.200
So when x is negative
2, y is negative 4.

00:01:30.200 --> 00:01:37.160
Or we have negative 4 is equal
to 3 times negative 2 plus b.

00:01:37.160 --> 00:01:39.030
Let me write the
negative 2 in there.

00:01:39.030 --> 00:01:41.390
3 times negative 2 plus b.

00:01:41.390 --> 00:01:42.750
And now we can solve for b.

00:01:42.750 --> 00:01:47.400
We get negative 4 is equal
to negative 6 plus b.

00:01:47.400 --> 00:01:52.270
Add 6 to both sides, you
get 2 is equal to b or b

00:01:52.270 --> 00:01:53.590
is equal to 2.

00:01:53.590 --> 00:01:54.680
So we were right.

00:01:54.680 --> 00:01:58.370
The y-intercept for the
second line is at 2.

00:01:58.370 --> 00:02:00.560
So we immediately
can eyeball, or we

00:02:00.560 --> 00:02:02.440
can verify, where
they both intersect.

00:02:02.440 --> 00:02:08.210
They both intersect the
y-axis at y equals 2.

00:02:08.210 --> 00:02:11.536
For both of these, when x is
equal to 0, y is equal to 2.

00:02:11.536 --> 00:02:12.910
If it wasn't so
obvious, we could

00:02:12.910 --> 00:02:15.020
set these two equations
equal to each other.

00:02:15.020 --> 00:02:19.510
We could say look,
we have 3x plus 2.

00:02:19.510 --> 00:02:23.300
We know that this is now
3x plus 2, because b is 2.

00:02:23.300 --> 00:02:30.390
When does 3x plus 2 equal
negative 1/3 x plus 2?

00:02:30.390 --> 00:02:32.320
Well, let's see.

00:02:32.320 --> 00:02:34.660
If we subtract 2
from both sides,

00:02:34.660 --> 00:02:38.904
when does 3x equal
negative 1/3 x?

00:02:38.904 --> 00:02:40.570
Well, there's a couple
of things that we

00:02:40.570 --> 00:02:42.470
could do right over here.

00:02:42.470 --> 00:02:47.740
We could add 1/3
x to both sides.

00:02:47.740 --> 00:02:52.580
And then we will
get 3 and 1/3, which

00:02:52.580 --> 00:02:58.320
is the same thing as
(10/3)x is equal to 0.

00:02:58.320 --> 00:03:04.060
And if you multiply
both sides by 3/10,

00:03:04.060 --> 00:03:08.850
you get x is equal to 0.

00:03:08.850 --> 00:03:11.110
So these two lines intersect
when x is equal to 0.

00:03:11.110 --> 00:03:14.790
For both of them, when x is
equal to 0, y is equal to 2.

00:03:17.600 --> 00:03:19.654
But you could have
eyeballed it here.

00:03:19.654 --> 00:03:21.820
You could have seen that
both of their y-intercepts,

00:03:21.820 --> 00:03:25.010
which happens when x is
equal to 0, y is equal to 2.

00:03:25.010 --> 00:03:27.792
So this point right over
here is the point 0, 2.

00:03:27.792 --> 00:03:29.750
We already know that this
point right over here

00:03:29.750 --> 00:03:33.742
is the point negative
2, negative 4.

00:03:33.742 --> 00:03:36.200
And now we just need to find
the distance between these two

00:03:36.200 --> 00:03:37.140
points.

00:03:37.140 --> 00:03:39.290
And the distance
formula really is just

00:03:39.290 --> 00:03:41.900
an application of the
Pythagorean theorem.

00:03:41.900 --> 00:03:43.610
We just need to
find the distance

00:03:43.610 --> 00:03:46.660
in the change in the y
direction and the change

00:03:46.660 --> 00:03:47.620
in the x direction.

00:03:47.620 --> 00:03:49.380
So let's do that separately.

00:03:49.380 --> 00:03:57.110
So in the y direction, what is
this distance right over here?

00:03:57.110 --> 00:04:01.310
So we went from y is equal to
negative 4 to y is equal to 2.

00:04:01.310 --> 00:04:04.630
This distance right
over here is 6.

00:04:04.630 --> 00:04:07.620
And what is this
distance right over here?

00:04:07.620 --> 00:04:11.860
Well, we go from x equals
negative 2 to x equals 0.

00:04:11.860 --> 00:04:14.560
So this distance
right over here is 2.

00:04:14.560 --> 00:04:16.290
So the distance between
these two points

00:04:16.290 --> 00:04:19.079
is really just the hypotenuse
of a right triangle

00:04:19.079 --> 00:04:22.420
that has sides 6 and 2.

00:04:22.420 --> 00:04:24.890
If we call this
distance d, we could

00:04:24.890 --> 00:04:27.804
say that the distance
squared is equal to.

00:04:27.804 --> 00:04:30.470
And all I'm really doing here is
restating the distance formula.

00:04:30.470 --> 00:04:35.190
The distance formula
tells you all this Y2

00:04:35.190 --> 00:04:39.034
minus Y1, which is 6, squared.

00:04:39.034 --> 00:04:40.700
But that's just the
Pythagorean theorem.

00:04:40.700 --> 00:04:47.040
That's just saying 6 squared
plus x2 minus x1, which

00:04:47.040 --> 00:04:52.000
is 0 minus negative 2, which
is positive 2 squared is going

00:04:52.000 --> 00:04:53.500
to be equal to the
distance squared.

00:04:53.500 --> 00:04:55.940
But we see that's just
the Pythagorean theorem.

00:04:55.940 --> 00:04:57.947
But anyway, let's
solve for the distance.

00:04:57.947 --> 00:04:59.530
So the distance
squared is going to be

00:04:59.530 --> 00:05:04.900
equal to 36 plus 4, which is 40.

00:05:04.900 --> 00:05:06.540
And now, let's see.

00:05:11.030 --> 00:05:14.092
The distance is equal to
the square root of 40.

00:05:14.092 --> 00:05:15.550
Square root of 40
is the same thing

00:05:15.550 --> 00:05:18.840
as the square root
of 4 times 10.

00:05:18.840 --> 00:05:20.150
And so that's the same.

00:05:20.150 --> 00:05:23.857
So the distance is equal to
2 if we factor out the 4.

00:05:23.857 --> 00:05:25.940
It's the square root of 4
times square root of 10.

00:05:25.940 --> 00:05:27.050
2 is the square root of 4.

00:05:27.050 --> 00:05:30.120
2 square roots of 10.

00:05:30.120 --> 00:05:32.060
And we're done.


Dividing by two digits example 2

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.037
Let's divide 7,182 by 42.

00:00:04.037 --> 00:00:05.620
And what's different
here is we're now

00:00:05.620 --> 00:00:08.039
dividing by a two-digit
number, not a one-digit number,

00:00:08.039 --> 00:00:09.810
but the same idea holds.

00:00:09.810 --> 00:00:12.310
So we say, hey, how many
times does 42 go into 7?

00:00:12.310 --> 00:00:14.210
Well, it doesn't really
go into 7 at all,

00:00:14.210 --> 00:00:16.219
so let's add one
more place value.

00:00:16.219 --> 00:00:19.190
How many times
does 42 go into 71?

00:00:19.190 --> 00:00:22.324
Well, it goes into 71 one time.

00:00:22.324 --> 00:00:24.990
Just a reminder, whoever's doing
the process where you say, hey,

00:00:24.990 --> 00:00:26.820
42 goes into 71 one time.

00:00:26.820 --> 00:00:30.722
But what we're really saying,
42 goes into 7,100 100 times

00:00:30.722 --> 00:00:32.930
because we're putting this
one in the hundreds place.

00:00:32.930 --> 00:00:34.888
But let's put that on
the side for a little bit

00:00:34.888 --> 00:00:36.420
and focus on the process.

00:00:36.420 --> 00:00:41.950
So 1 times 42 is 42,
and now we subtract.

00:00:41.950 --> 00:00:44.860
Now, you might be able to do 71
minus 42 in your head, knowing,

00:00:44.860 --> 00:00:46.810
hey, 72 minus 42 would be 30.

00:00:46.810 --> 00:00:49.900
So 71 minus 42 would be
29, but we could also

00:00:49.900 --> 00:00:51.670
do it by regrouping.

00:00:51.670 --> 00:00:55.210
To regroup, you want to
subtract a 2 from a 1.

00:00:55.210 --> 00:00:58.130
You can't really do that
in any traditional way.

00:00:58.130 --> 00:01:02.040
So let's take a 10 from the
70, so that it becomes a 60,

00:01:02.040 --> 00:01:03.580
and give that 10
to the ones place,

00:01:03.580 --> 00:01:05.870
and then that becomes an 11.

00:01:05.870 --> 00:01:10.860
And so 11 minus 2 is
9, and 6 minus 4 is 2.

00:01:10.860 --> 00:01:12.140
So you get 29.

00:01:12.140 --> 00:01:15.440
And we can bring down
the next place value.

00:01:15.440 --> 00:01:16.800
Bring down an 8.

00:01:16.800 --> 00:01:18.540
And now, this is
where the art happens

00:01:18.540 --> 00:01:20.424
when we're dividing by
a multi-digit number

00:01:20.424 --> 00:01:21.090
right over here.

00:01:21.090 --> 00:01:24.610
We have to estimate how many
times does 42 go into 298.

00:01:24.610 --> 00:01:27.235
And sometimes it might involve
a little bit of trial and error.

00:01:27.235 --> 00:01:29.151
So you really just kind
of have to eyeball it.

00:01:29.151 --> 00:01:30.660
If you make a
mistake, try again.

00:01:30.660 --> 00:01:32.118
The way you know
you make a mistake

00:01:32.118 --> 00:01:35.210
is, if say it goes into it 9
times, and you do 9 times 42

00:01:35.210 --> 00:01:38.600
and you get a number larger than
298, then you overestimated.

00:01:38.600 --> 00:01:41.459
If you say it goes into it
three times, you do 3 times 42,

00:01:41.459 --> 00:01:42.500
you get some number here.

00:01:42.500 --> 00:01:45.449
When you subtract, you get
something larger than 42,

00:01:45.449 --> 00:01:47.990
then you also made a mistake,
and you have to adjust upwards.

00:01:47.990 --> 00:01:49.531
Well, let's see if
we can eyeball it.

00:01:49.531 --> 00:01:51.490
So this is roughly 40.

00:01:51.490 --> 00:01:53.790
This is roughly 300.

00:01:53.790 --> 00:01:57.480
40 goes into 300 the same
times as 4 goes into the 30,

00:01:57.480 --> 00:02:00.570
so it's going to be roughly 7.

00:02:00.570 --> 00:02:01.960
Let's see if that's right.

00:02:01.960 --> 00:02:05.970
7 times 2 is 14.

00:02:05.970 --> 00:02:09.479
7 times 1 is 28, plus 1 is 29.

00:02:09.479 --> 00:02:12.060
So I got pretty close.

00:02:12.060 --> 00:02:15.310
My remainder here-- notice
294 is less than 298.

00:02:15.310 --> 00:02:16.740
So I'm cool there.

00:02:16.740 --> 00:02:21.600
And my remainder is less
than 42, so I'm cool as well.

00:02:21.600 --> 00:02:25.740
So now let's add
another place value.

00:02:25.740 --> 00:02:27.779
Let's bring this 2 down.

00:02:27.779 --> 00:02:29.320
And here we're just
asking ourselves,

00:02:29.320 --> 00:02:31.125
how many times
does 42 go into 42?

00:02:31.125 --> 00:02:34.360
Well, 42 goes into
42 exactly one time.

00:02:34.360 --> 00:02:39.600
1 times 42 is 42, and
we have no remainder.

00:02:39.600 --> 00:02:42.110
So this one luckily
divided exactly.

00:02:42.110 --> 00:02:48.240
42 goes into 7,182
exactly 171 times.


Dividing numbers: example with remainders

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:06.036
Let's divide 3,771 by 8.

00:00:06.036 --> 00:00:08.119
So we're going to do a
little bit of long division

00:00:08.119 --> 00:00:09.480
is the way we set it up.

00:00:09.480 --> 00:00:11.350
So the first thing we
want to do and think

00:00:11.350 --> 00:00:13.340
about is, well,
does 8 go into 3?

00:00:13.340 --> 00:00:15.200
Well, no, 8 does not go into 3.

00:00:15.200 --> 00:00:16.210
So we'll keep moving.

00:00:16.210 --> 00:00:18.580
Does 8 go into 37?

00:00:18.580 --> 00:00:19.100
Sure.

00:00:19.100 --> 00:00:20.900
8 times 4 is 37.

00:00:20.900 --> 00:00:21.950
8 times 5 is 40.

00:00:21.950 --> 00:00:22.930
So that's too big.

00:00:22.930 --> 00:00:27.840
So 8 goes into 37 four times.

00:00:27.840 --> 00:00:31.170
4 times 8 is 32.

00:00:31.170 --> 00:00:37.267
And we subtract, and we
get 37 minus 32 is 5.

00:00:37.267 --> 00:00:39.600
And just so you have a little
glimpse of what's actually

00:00:39.600 --> 00:00:41.150
going on here,
we're really saying

00:00:41.150 --> 00:00:45.080
how many times does
8 go into 3,700?

00:00:45.080 --> 00:00:48.964
And we're saying that 8
goes into 3,700 400 times.

00:00:48.964 --> 00:00:50.630
There's 4 sitting in
the hundreds place,

00:00:50.630 --> 00:00:53.640
and that 400 times 8 is 3,200.

00:00:53.640 --> 00:00:57.210
And then when you subtract 3,200
from 3,700, you're getting 500.

00:00:57.210 --> 00:01:00.010
But we'll revert back to just
the traditional long division

00:01:00.010 --> 00:01:00.510
mechanics.

00:01:00.510 --> 00:01:02.780
But that's what's
really going on here.

00:01:02.780 --> 00:01:04.680
When we say 8 goes
into 37 four times,

00:01:04.680 --> 00:01:08.050
we're really saying 8
goes into 3,700 400 times.

00:01:08.050 --> 00:01:09.310
But let's keep going.

00:01:09.310 --> 00:01:10.989
So now, we have
this 5 to deal with,

00:01:10.989 --> 00:01:12.530
which is, as we
already talked about,

00:01:12.530 --> 00:01:13.654
it's in the hundreds place.

00:01:13.654 --> 00:01:15.370
It's really a 500.

00:01:15.370 --> 00:01:18.020
But now, let's bring
down another place value.

00:01:18.020 --> 00:01:20.690
So let's bring down the 7.

00:01:20.690 --> 00:01:23.780
And let's think about how
many times does 8 go into 57.

00:01:23.780 --> 00:01:26.580
And we're really thinking
about how much 8 goes into 570.

00:01:26.580 --> 00:01:33.110
But 8 goes into 57-- let's
see, 8 times 5 is 40.

00:01:33.110 --> 00:01:35.350
8 times 6 is 48.

00:01:35.350 --> 00:01:38.350
8 times 7 is 54.

00:01:38.350 --> 00:01:40.750
So let's go 7 times.

00:01:40.750 --> 00:01:42.869
7 times 8.

00:01:42.869 --> 00:01:44.160
Oh, actually, I did that right.

00:01:44.160 --> 00:01:48.110
8 times 7-- my multiplication
tables are weak.

00:01:48.110 --> 00:01:52.000
It is 8 times-- so it
does go in seven times.

00:01:52.000 --> 00:01:55.120
But 7 times 8 isn't 54; it's 56.

00:01:55.120 --> 00:01:57.280
I always have trouble
with that one.

00:01:57.280 --> 00:01:59.120
It is 56.

00:01:59.120 --> 00:02:01.340
And so now, we subtract again.

00:02:01.340 --> 00:02:04.970
And 57 minus 56 is 1.

00:02:04.970 --> 00:02:07.140
And now we have one more
place value to bring down.

00:02:07.140 --> 00:02:09.949
We have-- let me do it in in
a color I haven't used yet.

00:02:09.949 --> 00:02:10.930
I'll do blue.

00:02:10.930 --> 00:02:13.404
We have a 1 to bring down.

00:02:13.404 --> 00:02:15.320
And now we're literally
saying, how many times

00:02:15.320 --> 00:02:16.510
does 8 go into 11?

00:02:16.510 --> 00:02:19.600
Well, 8 goes into 11 one time.

00:02:19.600 --> 00:02:22.060
1 times 8 is 8.

00:02:22.060 --> 00:02:23.730
And we are left
with-- because we

00:02:23.730 --> 00:02:28.080
have no more places to bring
down-- 11 minus 8 is 3.

00:02:28.080 --> 00:02:39.740
So 8 goes into 3,771 471 times,
and the remainder here is 3.


Negative number word problem

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:05.000
One of the coldest temperatures
ever recorded outside

00:00:05.000 --> 00:00:12.710
was negative 128 degrees
Fahrenheit in Antarctica.

00:00:12.710 --> 00:00:14.620
One of the warmest
temperatures ever

00:00:14.620 --> 00:00:19.530
recorded outside was
134 degrees Fahrenheit

00:00:19.530 --> 00:00:22.130
in Death Valley, California.

00:00:22.130 --> 00:00:24.640
How many degrees
difference are there

00:00:24.640 --> 00:00:28.850
between the coldest and warmest
recorded outside temperatures?

00:00:28.850 --> 00:00:30.490
So let's think about
this a little bit.

00:00:30.490 --> 00:00:32.604
Now, what I'll do is I'll
plot them on a number line.

00:00:32.604 --> 00:00:34.970
But I'm going to plot it on
a vertical number line that

00:00:34.970 --> 00:00:36.527
has a resemblance
to a thermometer,

00:00:36.527 --> 00:00:38.110
since we're talking
about temperature.

00:00:38.110 --> 00:00:41.390
So I'm going to make my number
line vertical right over here.

00:00:41.390 --> 00:00:45.180
So there's my little
vertical number line.

00:00:45.180 --> 00:00:51.450
And this right over here
is 0 degrees Fahrenheit,

00:00:51.450 --> 00:00:53.070
which really is of
no significance.

00:00:53.070 --> 00:00:54.509
If it was Celsius,
we'd be talking

00:00:54.509 --> 00:00:55.550
about the freezing point.

00:00:55.550 --> 00:00:58.240
But for Fahrenheit, that
happens at 32 degrees.

00:00:58.240 --> 00:01:00.330
But let's say this is
0 degrees Fahrenheit.

00:01:00.330 --> 00:01:02.690
And let's plot these two points.

00:01:02.690 --> 00:01:04.769
So one of the
coldest ever recorded

00:01:04.769 --> 00:01:08.880
temperatures was negative
128 degrees Fahrenheit.

00:01:08.880 --> 00:01:11.890
So let's say that's
right over here.

00:01:11.890 --> 00:01:18.340
This is negative 128
degrees Fahrenheit.

00:01:18.340 --> 00:01:20.640
And one of the warmest
temperatures ever recorded

00:01:20.640 --> 00:01:22.160
was 134 degrees.

00:01:22.160 --> 00:01:24.130
This is a positive 134.

00:01:24.130 --> 00:01:26.360
So it's about that far
and a little bit further.

00:01:26.360 --> 00:01:32.060
So it's a positive 134
degrees Fahrenheit.

00:01:32.060 --> 00:01:35.790
So when they're asking us
how many degrees difference

00:01:35.790 --> 00:01:37.786
are there between the
coldest and the warmest,

00:01:37.786 --> 00:01:39.160
they're essentially
saying, well,

00:01:39.160 --> 00:01:41.580
what is this distance
between the coldest

00:01:41.580 --> 00:01:45.510
and the warmest right over here?

00:01:45.510 --> 00:01:47.201
What is this distance?

00:01:47.201 --> 00:01:49.450
And there's a couple of ways
you could think about it.

00:01:49.450 --> 00:01:51.908
You could say, hey, if I started
at the coldest temperature

00:01:51.908 --> 00:01:54.380
and I wanted to go all
the way up to the warmest,

00:01:54.380 --> 00:01:56.230
how much would I have to add?

00:01:56.230 --> 00:01:58.430
Or you could say, well,
what's the difference

00:01:58.430 --> 00:02:00.220
between the coldest
and the warmest?

00:02:00.220 --> 00:02:02.220
So you could take
the larger number.

00:02:02.220 --> 00:02:04.660
So it's, say, 134.

00:02:04.660 --> 00:02:10.919
And from that, you could
subtract the smaller number,

00:02:10.919 --> 00:02:18.296
which is negative 128.

00:02:18.296 --> 00:02:19.670
So this essentially
saying what's

00:02:19.670 --> 00:02:21.040
the difference between
these two numbers?

00:02:21.040 --> 00:02:23.206
It's going to be positive,
because we're subtracting

00:02:23.206 --> 00:02:24.720
the smaller one
from the larger one.

00:02:24.720 --> 00:02:27.800
This is going to give you
the exact same thing as this.

00:02:27.800 --> 00:02:30.000
Now, there's several
ways to think about it.

00:02:30.000 --> 00:02:32.700
One is we know that if you
subtract a negative number,

00:02:32.700 --> 00:02:36.660
that's the same thing as adding
the positive of that number,

00:02:36.660 --> 00:02:38.790
or adding the absolute value.

00:02:38.790 --> 00:02:42.370
So this is the same thing.

00:02:42.370 --> 00:02:52.810
This is going to be equal to
134 plus positive 128 degrees.

00:02:52.810 --> 00:02:54.870
And what's the
intuition behind that?

00:02:54.870 --> 00:02:56.432
Why does this happen?

00:02:56.432 --> 00:02:57.890
Well, look at this
right over here.

00:02:57.890 --> 00:02:59.598
We're trying to figure
out this distance.

00:02:59.598 --> 00:03:03.610
This distance is 134
minus negative 128.

00:03:03.610 --> 00:03:05.240
And if you look at
that, it's going

00:03:05.240 --> 00:03:09.360
to be the absolute value of 134.

00:03:09.360 --> 00:03:11.810
It's going to be this distance
right over here, which

00:03:11.810 --> 00:03:15.010
is just 134-- which is just
that right over there--

00:03:15.010 --> 00:03:20.170
plus this distance
right over here.

00:03:20.170 --> 00:03:21.550
Now, what is this distance?

00:03:21.550 --> 00:03:23.920
Well, it's the absolute
value of negative 128.

00:03:23.920 --> 00:03:25.780
It's just 128.

00:03:25.780 --> 00:03:29.350
So it's going to be that
distance, 134, plus 128.

00:03:29.350 --> 00:03:30.790
And that's why it made sense.

00:03:30.790 --> 00:03:31.910
This way, you're
thinking of what's

00:03:31.910 --> 00:03:34.390
the difference between a larger
number and a smaller number.

00:03:34.390 --> 00:03:36.140
But since it's a smaller
number and you're

00:03:36.140 --> 00:03:39.030
subtracting a negative,
it's the same thing

00:03:39.030 --> 00:03:40.050
as adding a positive.

00:03:40.050 --> 00:03:42.550
And hopefully this gives you a
little bit of that intuition.

00:03:42.550 --> 00:03:45.330
But needless to say, we can now
figure out what's going to be.

00:03:45.330 --> 00:03:48.440
And this is going
to be equal to-- let

00:03:48.440 --> 00:03:50.630
me figure this out
separately over here.

00:03:50.630 --> 00:04:01.710
So if I were to add 134
plus 128, I get 4 plus 8

00:04:01.710 --> 00:04:06.000
is 12, 1 plus 3 plus 2 is 6.

00:04:06.000 --> 00:04:08.740
It's 262.

00:04:08.740 --> 00:04:11.099
This right over here
is equal to 262.

00:04:11.099 --> 00:04:12.640
How many degrees
difference are there

00:04:12.640 --> 00:04:15.660
between the coldest and warmest
recorded outside temperature?

00:04:15.660 --> 00:04:20.932
262 degrees
Fahrenheit difference.


Coordinate plane: graphing points

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.550
We are asked to plot 8 comma 10.

00:00:03.550 --> 00:00:06.910
So the first number in
this ordered pair, this

00:00:06.910 --> 00:00:07.960
is our x-coordinate.

00:00:07.960 --> 00:00:10.700
This tells us how far do
we move in the x direction.

00:00:10.700 --> 00:00:14.180
It's a positive 8, so we
move 8 in the x direction.

00:00:16.710 --> 00:00:20.760
And then the second number
in our ordered pair is 10.

00:00:20.760 --> 00:00:22.060
That is our y-coordinate.

00:00:22.060 --> 00:00:24.390
That tells us how far we
move in the y direction.

00:00:24.390 --> 00:00:26.830
Since it's positive,
we move up 10.

00:00:26.830 --> 00:00:30.112
So we move up 10, all
the way over here.

00:00:30.112 --> 00:00:32.070
And you could have thought
about it either way.

00:00:32.070 --> 00:00:34.403
You could have said, hey,
look this is our y-coordinate.

00:00:34.403 --> 00:00:35.510
This is 10.

00:00:35.510 --> 00:00:37.940
So I could move up 10.

00:00:37.940 --> 00:00:39.920
And then my x-coordinate
is positive 8.

00:00:39.920 --> 00:00:42.760
So I'll move 8 along
the positive x-axis,

00:00:42.760 --> 00:00:44.820
or I'll move 8 to the right.

00:00:44.820 --> 00:00:46.970
You see right over
here I have moved 8

00:00:46.970 --> 00:00:51.210
to the right-- 1,
2, 3, 4, 5, 6, 7, 8.

00:00:51.210 --> 00:00:57.620
And I have moved 10 up-- 1,
2, 3, 4, 5, 6, 7, 8, 9, 10.

00:00:57.620 --> 00:01:01.300
You might be tempted to
move 8 up and then 10

00:01:01.300 --> 00:01:03.110
to the right, which
would put you there.

00:01:03.110 --> 00:01:05.430
But then you would have had
the two numbers mixed up.

00:01:05.430 --> 00:01:08.410
You would have had the x and
the y-coordinates mixed up.

00:01:08.410 --> 00:01:12.070
The 8 tells you how far to move
in the horizontal direction.

00:01:12.070 --> 00:01:16.350
The 10 tells you how far to
move in the vertical direction.

00:01:16.350 --> 00:01:19.810
Let's do a couple more of these.

00:01:19.810 --> 00:01:22.240
Plot 6 comma 10.

00:01:22.240 --> 00:01:26.230
Well once again, the first
number in the ordered pair

00:01:26.230 --> 00:01:29.800
is the x-coordinate, how far
we move in the x direction.

00:01:29.800 --> 00:01:31.270
So we move 6 to the right.

00:01:31.270 --> 00:01:32.750
And then the second
number, the 10,

00:01:32.750 --> 00:01:35.550
tells us our vertical
coordinate, our y-coordinate.

00:01:35.550 --> 00:01:36.970
So it's positive 10.

00:01:36.970 --> 00:01:40.880
So we move 10 up.

00:01:40.880 --> 00:01:45.070
Let's do one more-- 5 comma 7.

00:01:45.070 --> 00:01:50.020
So my horizontal coordinate is
5, so I move 5 to the right.

00:01:50.020 --> 00:01:52.930
And then my vertical one is
7, or my y-coordinate is 7.

00:01:52.930 --> 00:01:58.045
So I move 7 up-- 1,
2, 3, 4, 5, 6, 7.

00:01:58.045 --> 00:02:00.170
And as long as you remember
which one is horizontal

00:02:00.170 --> 00:02:02.586
and which one is vertical or
which one is the x-coordinate

00:02:02.586 --> 00:02:04.650
and which one is the
y, you should be fine.

00:02:04.650 --> 00:02:07.310
You could say, hey look,
this is my y-coordinate, 7.

00:02:07.310 --> 00:02:09.139
So I'm going to move 7 up.

00:02:09.139 --> 00:02:12.270
And my x-coordinate is 5, so I'm
going to move 5 to the right.

00:02:12.270 --> 00:02:14.320
And it will get you to
that exact same point.

00:02:14.320 --> 00:02:18.940
This specifies exactly one
point in the coordinate plane.