Equivalent fraction word problem example 4

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.910
Sarah has $48.

00:00:02.910 --> 00:00:07.050
She wants to save 1/3 of
her money for a trip.

00:00:07.050 --> 00:00:09.990
How many dollars should
she set aside?

00:00:09.990 --> 00:00:13.530
So we essentially want to think
about what 1/3 of 48 is.

00:00:13.530 --> 00:00:16.170
Use 48 as the denominator
and find an

00:00:16.170 --> 00:00:19.480
equivalent fraction to 1/3.

00:00:19.480 --> 00:00:22.070
So what they want us to do in
this problem is they want us

00:00:22.070 --> 00:00:26.500
to say, OK, we want 1/3 of her
money, but we want to write

00:00:26.500 --> 00:00:30.530
this as an equivalent fraction
where we have 48 in the

00:00:30.530 --> 00:00:31.100
denominator.

00:00:31.100 --> 00:00:36.590
So this is equal to something,
some blank up here.

00:00:36.590 --> 00:00:39.425
This is equal to something
over 48.

00:00:39.425 --> 00:00:44.120
So how can we get it to that
something over 48?

00:00:44.120 --> 00:00:46.160
So let's think about what
this means for a second.

00:00:46.160 --> 00:00:50.430
So 1/3, if we were to draw
1/3, it looks like this.

00:00:50.430 --> 00:00:55.980
You could imagine a box
or a pie, I guess.

00:00:55.980 --> 00:01:00.690
So let's say that this is my
pie, and I have it split into

00:01:00.690 --> 00:01:01.840
three pieces.

00:01:01.840 --> 00:01:04.860
So let me split it into
three even pieces.

00:01:04.860 --> 00:01:09.020
And 1/3 is one of those
three pieces.

00:01:09.020 --> 00:01:11.550
That is what 1/3 means.

00:01:11.550 --> 00:01:15.140
Now, if we want express this as
a fraction over 48, how can

00:01:15.140 --> 00:01:15.740
we do that?

00:01:15.740 --> 00:01:17.220
Well, we're going to
have to split this

00:01:17.220 --> 00:01:19.280
thing into 48 pieces.

00:01:19.280 --> 00:01:22.350
How can we split something
into 48?

00:01:22.350 --> 00:01:28.540
Well, 3 times 16 is 48, so if we
split each of these into 16

00:01:28.540 --> 00:01:30.810
pieces-- and it's going
to be hard to draw

00:01:30.810 --> 00:01:32.300
here, but you can imagine.

00:01:32.300 --> 00:01:38.130
Let's see, you split it into
two, now we've split it into

00:01:38.130 --> 00:01:42.620
four, now you split
it into eight.

00:01:42.620 --> 00:01:44.280
You're just going to end up with
a bunch of lines here,

00:01:44.280 --> 00:01:47.360
but you can imagine, you can
just split each of these.

00:01:47.360 --> 00:01:51.120
If you split each of these into
enough, you would have 16

00:01:51.120 --> 00:01:54.140
pieces, so those would
be 16 right there.

00:01:54.140 --> 00:01:58.050
You would have 16 right there
and you have 16 right there.

00:01:58.050 --> 00:01:59.740
And I can just keep doing it.

00:01:59.740 --> 00:02:01.280
Let me do it in the
green over here.

00:02:01.280 --> 00:02:05.780
So if we just kept splitting
it up, we would get 48,

00:02:05.780 --> 00:02:09.050
because you have this first
third would be 16 pieces, the

00:02:09.050 --> 00:02:11.700
second third would be 16,
and then this third

00:02:11.700 --> 00:02:13.820
third would be 16 pieces.

00:02:13.820 --> 00:02:16.470
Altogether, you would
have 48 pieces.

00:02:16.470 --> 00:02:19.110
Now, that 1/3, what does
that represent?

00:02:19.110 --> 00:02:23.230
Well, that represents
16 of the 48.

00:02:23.230 --> 00:02:26.450
It represents these
16 right here.

00:02:26.450 --> 00:02:31.370
It represents these 16 right
there, so 1 over 3 is the

00:02:31.370 --> 00:02:33.090
exact same thing.

00:02:33.090 --> 00:02:41.120
So 1 over 3 is the exact same
thing as 16 over 48.

00:02:41.120 --> 00:02:43.420
Now, we did it just by thinking
about it kind of

00:02:43.420 --> 00:02:48.890
intuitively what 1/3 of 48 is,
but one way to do it more-- I

00:02:48.890 --> 00:02:53.600
guess a process for doing it--
we would say, well, look, to

00:02:53.600 --> 00:02:58.840
get the denominator, the bottom
number, from 3 to 48,

00:02:58.840 --> 00:03:02.020
we multiply by 16.

00:03:02.020 --> 00:03:04.790
3 times 16 is 48.

00:03:04.790 --> 00:03:07.760
And that's literally the process
of going from 3 pieces

00:03:07.760 --> 00:03:08.960
to 48 pieces.

00:03:08.960 --> 00:03:11.460
We have to multiply by 16.

00:03:11.460 --> 00:03:14.500
We have to turn each of our
pieces into 16 pieces.

00:03:14.500 --> 00:03:15.990
That's what we did.

00:03:15.990 --> 00:03:20.030
Now, you can't just multiply
only the denominator by 16.

00:03:20.030 --> 00:03:24.650
You have to multiply the
numerator by the same number.

00:03:24.650 --> 00:03:29.750
And so if each of my pieces
now become 16 pieces, then

00:03:29.750 --> 00:03:32.730
that one piece will
now become 16.

00:03:32.730 --> 00:03:36.370
So one way to think about it,
you just say, well, 3 times 16

00:03:36.370 --> 00:03:41.920
is 48, so 1 times 16 will be my
numerator, so it'll be 16.

00:03:41.920 --> 00:03:45.180
So 1/3 is equal to 16/48.

00:03:45.180 --> 00:03:47.040
And another way you could think
about it, which you'll

00:03:47.040 --> 00:03:53.930
learn in more detail later on,
is we want 1/3 of 48, right?

00:03:53.930 --> 00:04:00.060
That's how much she wants to
save. 1/3 of 48 is equal to

00:04:00.060 --> 00:04:03.950
1/3 times 48.

00:04:03.950 --> 00:04:08.130
And when you multiply-- let me
write it like this-- 1/3 times

00:04:08.130 --> 00:04:13.370
48, and you could rewrite
48 as a fraction 48/1.

00:04:13.370 --> 00:04:16.510
It literally represents
48 wholes.

00:04:16.510 --> 00:04:18.380
And when you multiply fractions,
you can just

00:04:18.380 --> 00:04:20.470
multiply the numerators.

00:04:20.470 --> 00:04:25.370
So this is equal to 48 over--
and then you just multiply the

00:04:25.370 --> 00:04:26.300
denominators.

00:04:26.300 --> 00:04:29.970
48/3, 1 times 48 is 48.

00:04:29.970 --> 00:04:31.560
We'll see this in more
detail in the future.

00:04:31.560 --> 00:04:33.480
Don't worry about it
if it confuses you.

00:04:33.480 --> 00:04:38.350
In the denominator, 3 times 1 is
3, and 48 divided by 3, or

00:04:38.350 --> 00:04:41.320
48/3, is equal to 16.

00:04:41.320 --> 00:04:47.960
So 1/3 of 48 is 16,
or 16/48 is 1/3.

00:04:47.960 --> 00:04:50.310
Hopefully, that make
sense to you.


Fractions in lowest terms

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Kind: captions
Language: en

00:00:00.610 --> 00:00:04.480
Simplify 48/64 to
lowest terms.

00:00:04.480 --> 00:00:06.440
Let's see if we can
visualize this.

00:00:06.440 --> 00:00:08.180
So we have 64.

00:00:08.180 --> 00:00:12.200
I guess 64 would be a whole,
so let's draw a whole here.

00:00:12.200 --> 00:00:13.400
So let's say that's a whole.

00:00:13.400 --> 00:00:16.020
Maybe we're talking
about a candy bar.

00:00:16.020 --> 00:00:17.910
Let me draw the whole.

00:00:17.910 --> 00:00:20.230
We're talking about
a whole candy bar.

00:00:20.230 --> 00:00:21.790
That would be 64 fourths.

00:00:21.790 --> 00:00:24.230
It would be the whole candy
bar right there.

00:00:24.230 --> 00:00:28.360
And 48 of the 64, you could
imagine splitting this up into

00:00:28.360 --> 00:00:29.930
64 super-small pieces.

00:00:29.930 --> 00:00:33.180
You wouldn't be able to see what
I drew here, but if we

00:00:33.180 --> 00:00:41.090
had 48 of them, it would get us
about that much of them, so

00:00:41.090 --> 00:00:47.960
that would be the 48
out of the 64.

00:00:47.960 --> 00:00:50.370
So this whole blue area is 64.

00:00:50.370 --> 00:00:53.550
The 48 is this purple area
right over here.

00:00:53.550 --> 00:00:54.360
So let me write it over here.

00:00:54.360 --> 00:01:00.700
48/64, and we want to write it
in lowest terms, and we'll

00:01:00.700 --> 00:01:04.110
talk more about what lowest
terms even means.

00:01:04.110 --> 00:01:08.820
Now, is there a way to group
these 48 or these 64 into

00:01:08.820 --> 00:01:10.890
groups of numbers that
will maybe simplify

00:01:10.890 --> 00:01:11.890
them a little bit?

00:01:11.890 --> 00:01:14.740
And to think about that, you'd
have to think about what is

00:01:14.740 --> 00:01:20.100
the largest factor that is
common to both 48 and 64?

00:01:20.100 --> 00:01:21.550
Or you can think of it
as what is their

00:01:21.550 --> 00:01:24.230
greatest common divisor?

00:01:24.230 --> 00:01:26.640
Well, the largest number that I
can think of that goes into

00:01:26.640 --> 00:01:30.650
48-- you could do it either by
just thinking about it or you

00:01:30.650 --> 00:01:32.200
could actually write out
all of its factors.

00:01:32.200 --> 00:01:34.310
But if you were to write all the
factors for 48 and all the

00:01:34.310 --> 00:01:37.630
factors for 64, the one that
pops out at me as the largest

00:01:37.630 --> 00:01:39.930
that goes into both is 16.

00:01:39.930 --> 00:01:43.860
So you could say that 48 is
equal to-- well, what is it?

00:01:43.860 --> 00:01:53.770
It's 3 times 16, and
64 is 4 times 16.

00:01:53.770 --> 00:01:55.080
Now this is interesting.

00:01:55.080 --> 00:01:59.135
So this 48 that we drew in
magenta right here, we could

00:01:59.135 --> 00:02:01.500
view this as three
groups of 16.

00:02:01.500 --> 00:02:05.190
So that's one, two--
let me make them a

00:02:05.190 --> 00:02:06.210
little bit more even.

00:02:06.210 --> 00:02:12.730
So one, two, three groups of 16,
so that's 16, that's 16,

00:02:12.730 --> 00:02:14.140
and that is 16.

00:02:14.140 --> 00:02:15.170
That would be 48.

00:02:15.170 --> 00:02:17.830
I could draw 16 bars here so
that we have 16 pieces, but

00:02:17.830 --> 00:02:21.040
that's a group of 16, a group
of 16 and a group of 16.

00:02:21.040 --> 00:02:22.790
That's what 48 is.

00:02:22.790 --> 00:02:25.960
Now, 64 is four groups of 16.

00:02:25.960 --> 00:02:30.050
So we could make, if you look at
the 64, that is a 16, that

00:02:30.050 --> 00:02:34.750
is a 16, that is a 16, and
then that is another 16.

00:02:34.750 --> 00:02:36.585
These should all be
the same length.

00:02:36.585 --> 00:02:38.300
I drew it a little bit off.

00:02:38.300 --> 00:02:42.750
So what is 48/64 in
lowest terms?

00:02:42.750 --> 00:02:45.590
We want to write this in as
simple as possible fraction.

00:02:45.590 --> 00:02:49.390
Well, if we make each of the
pieces equal to 16 of our old

00:02:49.390 --> 00:02:55.450
pieces, if we make this into one
piece, if we turn 16 into

00:02:55.450 --> 00:02:59.790
one, then we are talking about
instead of 48/64, we're

00:02:59.790 --> 00:03:03.310
talking about three.

00:03:03.310 --> 00:03:07.920
So this is one piece, two
pieces, three pieces of a

00:03:07.920 --> 00:03:09.320
total of four.

00:03:09.320 --> 00:03:13.840
So this is going to
be equal to 3/4.

00:03:13.840 --> 00:03:16.820
And hopefully, you see kind
of a mathematical way of

00:03:16.820 --> 00:03:17.890
immediately thinking about it.

00:03:17.890 --> 00:03:20.710
If you can factor this out and
you can actually factor out

00:03:20.710 --> 00:03:26.510
its greatest common factor, so
48 is 3 times 16, 64 is 4

00:03:26.510 --> 00:03:29.930
times 16, and then these
cancel each other out.

00:03:29.930 --> 00:03:42.430
view This is equivalent
to 3/4 times 16/16.

00:03:42.430 --> 00:03:43.810
This is the same
thing as that.

00:03:43.810 --> 00:03:49.290
And 16/16 is 1, and you're
just left with 3/4.

00:03:49.290 --> 00:03:52.520
Now, if you didn't immediately
recognize that 16 goes into

00:03:52.520 --> 00:03:57.300
both 48 and 64, you could
do it step by step.

00:03:57.300 --> 00:04:02.440
So let's say we started
off with 48/64.

00:04:02.440 --> 00:04:04.680
Now, the key thing to remember
with any fraction, whatever

00:04:04.680 --> 00:04:07.780
you do to the numerator, you
have to do to the denominator.

00:04:07.780 --> 00:04:17.589
So let's say we divide the
numerator by 2, we also have

00:04:17.589 --> 00:04:21.470
to divide the denominator
by 2, so we could get 2.

00:04:21.470 --> 00:04:23.010
We know that these are
both divisible by 2.

00:04:23.010 --> 00:04:23.800
They're both even.

00:04:23.800 --> 00:04:28.160
So that would get us to 24/32.

00:04:28.160 --> 00:04:30.780
And we'll say, well, look, these
two numbers, those are

00:04:30.780 --> 00:04:32.020
both divisible by 2.

00:04:32.020 --> 00:04:33.240
Well, see if we can think
of a larger number.

00:04:33.240 --> 00:04:35.830
Well, actually, they're both
divisible by 4, so maybe you

00:04:35.830 --> 00:04:38.180
don't realize that they're
also both divisible by 8.

00:04:38.180 --> 00:04:39.770
So let's say you
did it with 4.

00:04:39.770 --> 00:04:42.800
So now we divide the top by 4.

00:04:42.800 --> 00:04:44.580
So we're going to divide by 4.

00:04:44.580 --> 00:04:45.680
We get 6.

00:04:45.680 --> 00:04:47.880
You have to do the same to the
bottom, to the denominator.

00:04:47.880 --> 00:04:50.520
Divide by 4, you get 8.

00:04:50.520 --> 00:04:53.880
So 48/64 is the same thing
as 24/32, which is the

00:04:53.880 --> 00:04:55.560
same thing as 6/8.

00:04:55.560 --> 00:04:59.140
And these are both divisible
by 2, so if you divide the

00:04:59.140 --> 00:05:01.170
numerator by 2, you get 3.

00:05:01.170 --> 00:05:04.860
You divide that the denominator
by 2, you get 4.

00:05:04.860 --> 00:05:08.030
And so this is the simplest
possible terms, because 3 and

00:05:08.030 --> 00:05:13.400
4 share no common factors
greater than 1, so we're in

00:05:13.400 --> 00:05:14.920
lowest possible terms.

00:05:14.920 --> 00:05:15.800
So however you want to do it.

00:05:15.800 --> 00:05:18.520
The easiest way or the fastest
way is to say, hey, 16 is the

00:05:18.520 --> 00:05:20.290
biggest number that goes
into both of these.

00:05:20.290 --> 00:05:22.120
Divide both by 16.

00:05:22.120 --> 00:05:23.400
You get 3/4.

00:05:23.400 --> 00:05:25.660
And really when you're dividing
the numerator and

00:05:25.660 --> 00:05:29.770
denominator by 16, you're
turning groups of 16 into one

00:05:29.770 --> 00:05:33.460
piece or 16 super-small pieces
of the pie into one bigger

00:05:33.460 --> 00:05:34.480
piece of the pie.

00:05:34.480 --> 00:05:37.310
So this goes from 64
pieces to 4 pieces.

00:05:37.310 --> 00:05:40.630
This goes from 48 pieces
to 3 pieces.


Comparing Fractions 2

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.410
Use less than, greater than or
equal to compare the two

00:00:04.410 --> 00:00:08.560
fractions 21/28 and 6/9.

00:00:08.560 --> 00:00:10.920
Now, your impulse might just
be to look at these numbers

00:00:10.920 --> 00:00:13.110
and say, oh, look, these are
larger numbers than those.

00:00:13.110 --> 00:00:14.120
This must be bigger.

00:00:14.120 --> 00:00:15.440
You cannot do that.

00:00:15.440 --> 00:00:18.300
And we'll talk about in future
videos on why that is, or I

00:00:18.300 --> 00:00:21.780
encourage you to think about why
you cannot do that, but to

00:00:21.780 --> 00:00:24.700
do it properly, at least in my
mind, the best way is to get

00:00:24.700 --> 00:00:27.740
these both in simplified forms
and to just think about what

00:00:27.740 --> 00:00:28.630
they represent.

00:00:28.630 --> 00:00:30.850
And in the future when we
learn to convert them to

00:00:30.850 --> 00:00:33.680
decimals, that's actually the
best way to do it, or actually

00:00:33.680 --> 00:00:34.810
for me, the easiest way.

00:00:34.810 --> 00:00:36.980
But let's try to do what
we can so far.

00:00:36.980 --> 00:00:44.310
So 21/28, we want to put this
in simplified form.

00:00:44.310 --> 00:00:47.500
21 is the same thing
as 3 times 7.

00:00:47.500 --> 00:00:49.710
28 is the same thing
as 4 times 7.

00:00:49.710 --> 00:00:51.640
They're both divisible by 7.

00:00:51.640 --> 00:00:54.460
Now, if you divide the numerator
and the denominator

00:00:54.460 --> 00:01:00.270
by 7, 3 times 7 divided by
7 is just going to be 3.

00:01:00.270 --> 00:01:03.980
4 times 7 divided by 7 is
just going to be 4.

00:01:03.980 --> 00:01:10.140
So this guy right here is 3/4,
is equivalent to 3/4.

00:01:10.140 --> 00:01:12.490
Now, let's think about
what 6/9 is.

00:01:17.800 --> 00:01:19.840
Since these numbers are smaller,
this is actually a

00:01:19.840 --> 00:01:21.710
good chance to visualize it.

00:01:21.710 --> 00:01:27.010
So let's visualize if we have
ninths, let me draw the 9

00:01:27.010 --> 00:01:34.120
ninths first. So if I were to
split it into thirds, I just

00:01:34.120 --> 00:01:36.060
want to make it so
it's reasonably--

00:01:36.060 --> 00:01:38.170
oh, that's not right.

00:01:38.170 --> 00:01:42.020
Maybe there and there, and then
divide each of these into

00:01:42.020 --> 00:01:47.260
thirds, so just like that, a
little bit more space to that

00:01:47.260 --> 00:01:50.630
one, and then we have it
like that, and then

00:01:50.630 --> 00:01:52.990
this one into thirds.

00:01:52.990 --> 00:01:55.750
Let me draw it like that.

00:01:55.750 --> 00:01:58.900
So here we have 9 pieces of a
candy bar, whatever you want

00:01:58.900 --> 00:02:00.740
to think about it, or 9/9.

00:02:00.740 --> 00:02:12.010
6/9 literally means one, two,
three, four, five six.

00:02:12.010 --> 00:02:17.160
6/9 references 6 of the 9
potential pieces, 6 of these

00:02:17.160 --> 00:02:19.940
one-ninth of a candy bar piece,
however you want to

00:02:19.940 --> 00:02:20.830
think about it.

00:02:20.830 --> 00:02:28.920
Now, 6 is the same thing as 2
times 3, so you can view it as

00:02:28.920 --> 00:02:32.290
two groups of 3.

00:02:32.290 --> 00:02:33.440
Let me do it this way.

00:02:33.440 --> 00:02:36.440
You can view it as
two groups of 3.

00:02:36.440 --> 00:02:38.970
So this is one group of 3.

00:02:38.970 --> 00:02:43.410
This is one group of 3 right
there, and that is another

00:02:43.410 --> 00:02:44.170
group of 3.

00:02:44.170 --> 00:02:47.640
These are two groups of 3.

00:02:47.640 --> 00:02:50.370
Now, what is 9 equivalent to?

00:02:50.370 --> 00:02:58.250
9 is equivalent to 3 times
3, or three groups of 3.

00:02:58.250 --> 00:03:02.870
So 9 is equivalent to that's
one group of 3, that is two

00:03:02.870 --> 00:03:08.760
groups of 3, and now this
is three groups of 3.

00:03:08.760 --> 00:03:13.010
So we could redraw this as--
so if we just draw just the

00:03:13.010 --> 00:03:15.155
groups of 3, it would
look like this.

00:03:22.190 --> 00:03:27.410
And these two groups of 3 that
the 6 represents, if we just

00:03:27.410 --> 00:03:33.110
draw the groups of three, it
would be that guy and that the

00:03:33.110 --> 00:03:35.960
guy right over there.

00:03:35.960 --> 00:03:41.500
So how many thirds do we have,
if you consider each of those

00:03:41.500 --> 00:03:42.760
groups of three as a third?

00:03:42.760 --> 00:03:43.890
Well, we have 2/3.

00:03:43.890 --> 00:03:45.570
This is equivalent to 2/3.

00:03:45.570 --> 00:03:49.880
But doing the process that we're
used to doing, we could

00:03:49.880 --> 00:03:52.790
imagine 6 is 2 times 3.

00:03:52.790 --> 00:03:54.600
9 is 3 times 3.

00:03:54.600 --> 00:03:58.230
If you divide 2 times
3 by 3, you get 2.

00:03:58.230 --> 00:04:01.660
If you divide 3 times 3
by 3, you get the 3.

00:04:01.660 --> 00:04:12.870
So 6/9 is the same thing as 2/3,
two of the thirds, of the

00:04:12.870 --> 00:04:13.870
one-thirds.

00:04:13.870 --> 00:04:15.940
That's what 6/9 is
equivalent to.

00:04:15.940 --> 00:04:18.760
Now, let's think about
what's larger.

00:04:18.760 --> 00:04:22.640
2/3, which is the same
thing as 6/9, or 3/4?

00:04:22.640 --> 00:04:24.350
I didn't do 21/28 because
it would've

00:04:24.350 --> 00:04:25.730
taken forever to draw.

00:04:25.730 --> 00:04:27.990
So what does 3/4 look like?

00:04:27.990 --> 00:04:29.110
What is 3/4?

00:04:29.110 --> 00:04:31.610
So if we were to take this
exact same square-- so

00:04:31.610 --> 00:04:33.040
let me redraw it.

00:04:33.040 --> 00:04:37.550
So if we were to take that same
exact candy bar, so these

00:04:37.550 --> 00:04:41.070
should be the same size, instead
of cutting it up into

00:04:41.070 --> 00:04:45.370
three pieces, if we were to cut
it up into four pieces,

00:04:45.370 --> 00:04:46.030
what do we get?

00:04:46.030 --> 00:04:49.920
So let's see, that's cutting
up into two pieces, and now

00:04:49.920 --> 00:04:53.300
let me cut it up into
four pieces.

00:04:53.300 --> 00:04:55.900
So now this one has fourths,
this one has thirds.

00:04:55.900 --> 00:04:58.940
3 in the denominator here-- so
let me make it very clear.

00:04:58.940 --> 00:05:02.540
This represents 2/3.

00:05:02.540 --> 00:05:05.500
Now, this is going to represent,
we have four

00:05:05.500 --> 00:05:10.300
pieces, and if we're coloring in
three of them, so if we're

00:05:10.300 --> 00:05:13.950
talking about 3/4, that's one--
let me draw it like

00:05:13.950 --> 00:05:15.100
this, actually.

00:05:15.100 --> 00:05:20.610
That is one, two-- let
me scroll up a

00:05:20.610 --> 00:05:22.490
little bit-- three.

00:05:22.490 --> 00:05:24.990
This is three of the fourths.

00:05:24.990 --> 00:05:29.270
And the way I drew it, it is not
completely obvious to you.

00:05:29.270 --> 00:05:31.110
Actually, the way I drew it
because-- let me draw it a

00:05:31.110 --> 00:05:33.360
little bit neater than this.

00:05:33.360 --> 00:05:34.870
Let me draw it a little bit
neater and it'll become a

00:05:34.870 --> 00:05:35.830
little bit more obvious.

00:05:35.830 --> 00:05:37.570
I didn't draw the fourths.

00:05:37.570 --> 00:05:41.050
Here, drawing neatly
actually matters.

00:05:41.050 --> 00:05:44.070
So let me draw it like
that and then split

00:05:44.070 --> 00:05:45.755
this group-- nope.

00:05:45.755 --> 00:05:50.870
I want to make it even, so I
want to make it like this.

00:05:50.870 --> 00:05:53.280
OK, so if those are the fourths,
if we're talking

00:05:53.280 --> 00:06:01.490
about three of the fourths,
we're talking about one, two,

00:06:01.490 --> 00:06:07.060
three fourths, and you can see
that 3/4 is larger than 2/3.

00:06:07.060 --> 00:06:10.750
2/3 gets us to this far of the
candy bar, while 3/4 is a

00:06:10.750 --> 00:06:12.070
little bit more of
the candy bar.

00:06:12.070 --> 00:06:15.910
2/3 is about that far, 3/4
is even more, so 3/4

00:06:15.910 --> 00:06:18.190
is larger than 2/3.

00:06:18.190 --> 00:06:19.570
And we'll show you in the
future how to do it

00:06:19.570 --> 00:06:20.010
numerically.

00:06:20.010 --> 00:06:22.890
You can use a calculator, if you
like, or you could do it

00:06:22.890 --> 00:06:30.030
by hand, but you can see from
this visually that 3/4 is

00:06:30.030 --> 00:06:31.620
greater than 2/3.

00:06:34.420 --> 00:06:36.810
Now, they want to compare
these two fractions.

00:06:36.810 --> 00:06:39.080
So 21/28 is the same
thing as 3/4.

00:06:39.080 --> 00:06:44.200
So 21/28 is going to be greater
than 2/3, which is the

00:06:44.200 --> 00:06:47.690
same thing as 6/9.

00:06:47.690 --> 00:06:50.110
And so we are done.


Comparing fractions

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.820
Determine whether
30/45 and 54/81

00:00:04.820 --> 00:00:06.090
are equivalent fractions.

00:00:06.090 --> 00:00:08.039
Well, the easiest way I can
think of doing this is to put

00:00:08.039 --> 00:00:11.040
both of these fractions into
lowest possible terms, and

00:00:11.040 --> 00:00:12.460
then if they're the same
fraction, then they're

00:00:12.460 --> 00:00:13.460
equivalent.

00:00:13.460 --> 00:00:21.510
So 30/45, what's the largest
factor of both 30 and 45?

00:00:21.510 --> 00:00:22.850
15 will go into 30.

00:00:22.850 --> 00:00:24.070
It'll also go into 45.

00:00:24.070 --> 00:00:25.030
So this is the same thing.

00:00:25.030 --> 00:00:32.310
30 is 2 times 15 and
45 is 3 times 15.

00:00:32.310 --> 00:00:36.170
So we can divide both the
numerator and the

00:00:36.170 --> 00:00:37.810
denominator by 15.

00:00:37.810 --> 00:00:40.640
So if we divide both the
numerator and the denominator

00:00:40.640 --> 00:00:43.690
by 15, what happens?

00:00:43.690 --> 00:00:47.740
Well, this 15 divided by 15,
they cancel out, this 15

00:00:47.740 --> 00:00:52.520
divided by 15 cancel out, and
we'll just be left with 2/3.

00:00:52.520 --> 00:00:56.490
So 30/45 is the same
thing as 2/3.

00:00:56.490 --> 00:00:57.880
It's equivalent to 2/3.

00:00:57.880 --> 00:01:02.550
2/3 is in lowest possible terms,
or simplified form,

00:01:02.550 --> 00:01:03.770
however you want to
think about it.

00:01:03.770 --> 00:01:12.210
Now, let's try to do 54/81.

00:01:12.210 --> 00:01:13.320
Now, let's see.

00:01:13.320 --> 00:01:15.020
Nothing really jumps
out at me.

00:01:15.020 --> 00:01:17.580
Let's see, 9 is divisible
into both of these.

00:01:17.580 --> 00:01:22.840
We could write 54 as being 6
times 9, and 81 is the same

00:01:22.840 --> 00:01:24.870
thing as 9 times 9.

00:01:24.870 --> 00:01:28.470
You can divide the numerator
and the denominator by 9.

00:01:28.470 --> 00:01:30.970
So we could divide both
of them by 9.

00:01:30.970 --> 00:01:34.530
9 divided by 9 is 1, 9 divided
by 9 is 1, so we get this as

00:01:34.530 --> 00:01:36.900
being equal to 6/9.

00:01:36.900 --> 00:01:37.170
Now, let's see.

00:01:37.170 --> 00:01:41.590
6 is the same thing
as 2 times 3.

00:01:41.590 --> 00:01:44.610
9 is the same thing
as 3 times 3.

00:01:44.610 --> 00:01:47.280
We could just cancel these 3's
out, or you could imagine this

00:01:47.280 --> 00:01:50.530
is the same thing as dividing
both the numerator and the

00:01:50.530 --> 00:01:53.880
denominator by 3, or multiplying
both the numerator

00:01:53.880 --> 00:01:56.380
and the denominator by 1/3.

00:01:56.380 --> 00:01:57.930
These are all equivalent.

00:01:57.930 --> 00:02:01.400
I could write divide by
3 or multiply by 1/3.

00:02:01.400 --> 00:02:03.832
Actually, let me write
divide by 3.

00:02:03.832 --> 00:02:06.860
Let me write divide
by 3 for now.

00:02:06.860 --> 00:02:08.440
I don't want to assume you
know how to multiply

00:02:08.440 --> 00:02:11.070
fractions, because we're going
to learn that in the future.

00:02:11.070 --> 00:02:12.900
So we're going to divide by 3.

00:02:12.900 --> 00:02:14.830
3 divided by 3 is just 1.

00:02:14.830 --> 00:02:18.530
3 divided by 3 is 1, and
you're left with 2/3.

00:02:18.530 --> 00:02:21.305
So both of these fractions, when
you simplify them, when

00:02:21.305 --> 00:02:25.470
you put them in simplified form,
both end up being 2/3,

00:02:25.470 --> 00:02:29.350
so they are equivalent
fractions.


Exponent example 2

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.930
Write 6 times 6 times 6 times
6 times 6 times 6 times 6

00:00:05.930 --> 00:00:11.640
times 6 in exponential
notation.

00:00:11.640 --> 00:00:13.260
So what's going on over here?

00:00:13.260 --> 00:00:16.590
Well, we have 6 multiplied
by itself how many times?

00:00:16.590 --> 00:00:20.030
Let's see, that's 6 times 1,
then 6 times 2 is that.

00:00:20.030 --> 00:00:22.330
That's 6 times-- well,
it's not 6 times 2.

00:00:22.330 --> 00:00:24.490
It's 6 times itself two times.

00:00:24.490 --> 00:00:28.150
Remember, 6 times itself
two times would be 36.

00:00:28.150 --> 00:00:30.360
6 times 2 would only be 12.

00:00:30.360 --> 00:00:36.720
So we have one 6, two 6, three
6, four 6, five 6's, six 6's,

00:00:36.720 --> 00:00:39.240
seven 6's, eight 6's.

00:00:39.240 --> 00:00:43.470
So we're multiplying 6 by
itself eight times.

00:00:43.470 --> 00:00:46.380
To write this in exponential
notation, we would say that

00:00:46.380 --> 00:00:55.180
this is equal to 6 to the
eighth power, which is

00:00:55.180 --> 00:00:58.550
literally equal to 6 times 6
times 6 times 6 times 6 times

00:00:58.550 --> 00:01:00.220
6 times 6 times 6.

00:01:00.220 --> 00:01:04.670
Now, I want to make it very
clear this is NOT, this is

00:01:04.670 --> 00:01:10.500
NOT, this is NOT equal
to 6 times 8.

00:01:13.110 --> 00:01:15.930
6 times 8 will only be 48.

00:01:15.930 --> 00:01:19.680
6 to the eighth power is
a super huge number.

00:01:19.680 --> 00:01:23.580
6 times 6 is 36, and you're
going to multiply that times

00:01:23.580 --> 00:01:24.880
6, which is what?

00:01:24.880 --> 00:01:27.360
That's like 36 times--
that's what?

00:01:27.360 --> 00:01:31.370
216, and you keep multiplying
it by 6, you get some huge

00:01:31.370 --> 00:01:31.990
number here.

00:01:31.990 --> 00:01:33.690
This number right here.

00:01:33.690 --> 00:01:37.120
It's worthwhile to point out
this number is huge.

00:01:40.320 --> 00:01:43.870
This number right here,
not so huge.

00:01:43.870 --> 00:01:45.250
It is not huge.

00:01:45.250 --> 00:01:46.320
So don't get confused.

00:01:46.320 --> 00:01:50.310
If you see 6 to the eighth
power, it's 6 times itself

00:01:50.310 --> 00:01:54.110
eight times, not 6 times 8.


Understanding square roots

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.730
We're asked to find the
square root of 100.

00:00:03.730 --> 00:00:05.130
Let me write this down bigger.

00:00:05.130 --> 00:00:09.060
So the square root is this
big check-looking thing.

00:00:09.060 --> 00:00:11.880
The square root of 100.

00:00:11.880 --> 00:00:13.650
When you see it like this,
this means the

00:00:13.650 --> 00:00:15.120
positive square root.

00:00:15.120 --> 00:00:16.850
If you're familiar with negative
numbers, you know

00:00:16.850 --> 00:00:19.490
that there's also a negative
square root, but when you just

00:00:19.490 --> 00:00:22.700
see this symbol, that means
the positive square root.

00:00:22.700 --> 00:00:24.570
So let's think about what
this is saying.

00:00:24.570 --> 00:00:28.090
This is asking us find the
number, the positive number,

00:00:28.090 --> 00:00:34.350
that when I multiply that number
by itself, I get 100.

00:00:34.350 --> 00:00:37.700
So what number when I multiply
it by itself do I get 100?

00:00:37.700 --> 00:00:40.220
Well, let's see, if I multiply
9 by itself, that's

00:00:40.220 --> 00:00:41.580
only going to be 81.

00:00:41.580 --> 00:00:44.280
If I multiply 10 by itself,
that is 100.

00:00:44.280 --> 00:00:46.800
So this is equal to-- and let
me write it this way.

00:00:46.800 --> 00:00:48.740
Normally, you could
skip this step.

00:00:48.740 --> 00:00:51.470
But you could write this as
the square root of-- and

00:00:51.470 --> 00:00:56.610
instead of 100, 100 is the same
thing as 10 times 10.

00:00:56.610 --> 00:00:58.960
And then you know, the square
root of something times

00:00:58.960 --> 00:01:00.890
itself, that's just going
to be that something.

00:01:00.890 --> 00:01:03.240
This is just equal to 10.

00:01:03.240 --> 00:01:06.720
So the square root
of 100 is 10.

00:01:06.720 --> 00:01:09.330
Or another way you could write,
I guess, this same

00:01:09.330 --> 00:01:16.520
truth is that 10 squared, which
is equal to 10 times 10,

00:01:16.520 --> 00:01:20.000
is equal to 100.


The distributive law of multiplication over addition

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.960
Rewrite the expression 4 times,
and then in parentheses

00:00:03.960 --> 00:00:07.810
we have 8 plus 3, using the
distributive law of

00:00:07.810 --> 00:00:09.960
multiplication over addition.

00:00:09.960 --> 00:00:11.880
Then simplify the expression.

00:00:11.880 --> 00:00:16.050
So let's just try to solve
this or evaluate this

00:00:16.050 --> 00:00:18.080
expression, then we'll talk
a little bit about the

00:00:18.080 --> 00:00:20.720
distributive law of
multiplication over addition,

00:00:20.720 --> 00:00:23.100
usually just called the
distributive law.

00:00:23.100 --> 00:00:34.250
So we have 4 times
8 plus 8 plus 3.

00:00:34.250 --> 00:00:36.070
Now there's two ways to do it.

00:00:36.070 --> 00:00:38.630
Normally, when you have
parentheses, your inclination

00:00:38.630 --> 00:00:41.020
is, well, let me just evaluate
what's in the parentheses

00:00:41.020 --> 00:00:43.430
first and then worry about
what's outside of the

00:00:43.430 --> 00:00:46.110
parentheses, and we can do
that fairly easily here.

00:00:46.110 --> 00:00:48.520
We can evaluate what
8 plus 3 is.

00:00:48.520 --> 00:00:49.940
8 plus 3 is 11.

00:00:49.940 --> 00:00:52.030
So if we do that-- let me do
that in this direction.

00:00:52.030 --> 00:00:58.820
So if we do that, we get 4
times, and in parentheses we

00:00:58.820 --> 00:01:00.120
have an 11.

00:01:00.120 --> 00:01:04.090
8 plus 3 is 11, and then this
is going to be equal to--

00:01:04.090 --> 00:01:08.820
well, 4 times 11 is just
44, so you can

00:01:08.820 --> 00:01:09.880
evaluate it that way.

00:01:09.880 --> 00:01:11.790
But they want us to use the
distributive law of

00:01:11.790 --> 00:01:12.510
multiplication.

00:01:12.510 --> 00:01:14.960
We did not use the distributive
law just now.

00:01:14.960 --> 00:01:17.270
We just evaluated
the expression.

00:01:17.270 --> 00:01:20.470
We used the parentheses first,
then multiplied by 4.

00:01:20.470 --> 00:01:24.330
In the distributive law, we
multiply by 4 first. And it's

00:01:24.330 --> 00:01:27.080
called the distributive law
because you distribute the 4,

00:01:27.080 --> 00:01:29.040
and we're going to think
about what that means.

00:01:29.040 --> 00:01:33.200
So in the distributive law, what
this will become, it'll

00:01:33.200 --> 00:01:37.120
become 4 times 8 plus 4 times
3, and we're going to think

00:01:37.120 --> 00:01:38.490
about why that is in a second.

00:01:38.490 --> 00:01:52.620
So this is going to be equal to
4 times 8 plus 4 times 3.

00:01:52.620 --> 00:01:54.920
A lot of people's first instinct
is just to multiply

00:01:54.920 --> 00:01:56.680
the 4 times the 8, but no!

00:01:56.680 --> 00:01:58.110
You have to distribute the 4.

00:01:58.110 --> 00:02:03.250
You have to multiply it times
the 8 and times the 3.

00:02:03.250 --> 00:02:04.040
This is right here.

00:02:04.040 --> 00:02:07.200
This is the distributive
property in action right here.

00:02:07.200 --> 00:02:10.139
Distributive property
in action.

00:02:10.139 --> 00:02:11.770
And then when you evaluate it--
and I'm going to show you

00:02:11.770 --> 00:02:14.190
in kind of a visual way
why this works.

00:02:14.190 --> 00:02:17.680
But then when you evaluate it,
4 times 8-- I'll do this in a

00:02:17.680 --> 00:02:23.010
different color-- 4 times 8 is
32, and then so we have 32

00:02:23.010 --> 00:02:25.780
plus 4 times 3.

00:02:25.780 --> 00:02:32.840
4 times 3 is 12 and 32 plus
12 is equal to 44.

00:02:32.840 --> 00:02:36.040
That is also equal to 44, so
you can get it either way.

00:02:36.040 --> 00:02:38.860
But when they want us to use
the distributive law, you'd

00:02:38.860 --> 00:02:41.130
distribute the 4 first.
Now let's think

00:02:41.130 --> 00:02:42.290
about why that happens.

00:02:42.290 --> 00:02:45.670
Let's visualize just
what 8 plus 3 is.

00:02:45.670 --> 00:02:47.400
Let me draw eight
of something.

00:02:47.400 --> 00:02:53.930
So one, two, three,
four, five, six,

00:02:53.930 --> 00:02:55.380
seven, eight, right?

00:02:58.790 --> 00:03:04.380
And then we're going to add to
that three of something, of

00:03:04.380 --> 00:03:05.710
maybe the same thing.

00:03:05.710 --> 00:03:07.880
One, two, three.

00:03:07.880 --> 00:03:10.760
So you can imagine this is what
we have inside of the

00:03:10.760 --> 00:03:11.350
parentheses.

00:03:11.350 --> 00:03:14.240
We have 8 circles
plus 3 circles.

00:03:14.240 --> 00:03:16.900
Now, when we're multiplying this
whole thing, this whole

00:03:16.900 --> 00:03:20.120
thing times 4, what
does that mean?

00:03:20.120 --> 00:03:22.330
Well, that means we're just
going to add this to itself

00:03:22.330 --> 00:03:23.690
four times.

00:03:23.690 --> 00:03:26.750
Let me do that with
a copy and paste.

00:03:26.750 --> 00:03:28.470
Copy and paste.

00:03:28.470 --> 00:03:31.860
Let me copy and then
let me paste.

00:03:31.860 --> 00:03:32.410
There you go.

00:03:32.410 --> 00:03:33.240
That's two.

00:03:33.240 --> 00:03:39.780
That's one, two, three, and then
we have four, and we're

00:03:39.780 --> 00:03:41.460
going to add them
all together.

00:03:41.460 --> 00:03:42.690
So this is literally what?

00:03:42.690 --> 00:03:45.020
Four times, right?

00:03:45.020 --> 00:03:47.290
Let me go back to the
drawing tool.

00:03:47.290 --> 00:03:50.630
We have it one, two, three, four
times this expression,

00:03:50.630 --> 00:03:52.750
which is 8 plus 3.

00:03:52.750 --> 00:03:54.750
Now, what is this
thing over here?

00:04:00.270 --> 00:04:03.120
If you were to count all of this
stuff, you would get 44.

00:04:03.120 --> 00:04:05.710
But what is this thing
over here?

00:04:05.710 --> 00:04:09.140
Well, that's 8 added to
itself four times.

00:04:09.140 --> 00:04:11.800
You could imagine you're
adding all of these.

00:04:11.800 --> 00:04:14.120
So what's 8 added to
itself four times?

00:04:14.120 --> 00:04:16.279
That is 4 times 8.

00:04:16.279 --> 00:04:22.780
So this is 4 times 8,
and what is this

00:04:22.780 --> 00:04:24.710
over here in the orange?

00:04:24.710 --> 00:04:27.030
We have one, two, three,
four times.

00:04:27.030 --> 00:04:28.330
Well, each time we have three.

00:04:28.330 --> 00:04:32.620
So it's 4 times this
right here.

00:04:32.620 --> 00:04:37.580
This right here is 4 times 3.

00:04:37.580 --> 00:04:39.620
So you see why the distributive
property works.

00:04:39.620 --> 00:04:44.170
If you do 4 times 8 plus 3, you
have to multiply-- when

00:04:44.170 --> 00:04:46.570
you, I guess you could imagine,
duplicate the thing

00:04:46.570 --> 00:04:50.450
four times, both the 8 and the
3 is getting duplicated four

00:04:50.450 --> 00:04:53.000
times or it's being added to
itself four times, and that's

00:04:53.000 --> 00:04:55.290
why we distribute the 4.


Order of operations example

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.750
We're asked to simplify 8 plus
5 times 4 minus, and then in

00:00:04.750 --> 00:00:09.010
parentheses, 6 plus 10
divided by 2 plus 44.

00:00:09.010 --> 00:00:12.120
Whenever you see some type of
crazy expression like this

00:00:12.120 --> 00:00:15.360
where you have parentheses and
addition and subtraction and

00:00:15.360 --> 00:00:17.760
division, you always want
to keep the order of

00:00:17.760 --> 00:00:19.000
operations in mind.

00:00:19.000 --> 00:00:20.970
Let me write them
down over here.

00:00:20.970 --> 00:00:23.420
So when you're doing order of
operations, or really when

00:00:23.420 --> 00:00:24.820
you're evaluating any
expression, you should have

00:00:24.820 --> 00:00:29.520
this in the front of your brain
that the top priority

00:00:29.520 --> 00:00:30.770
goes to parentheses.

00:00:37.850 --> 00:00:40.060
And those are these little
brackets over here, or however

00:00:40.060 --> 00:00:40.670
you want to call them.

00:00:40.670 --> 00:00:42.320
Those are the parentheses
right there.

00:00:42.320 --> 00:00:44.030
That gets top priority.

00:00:44.030 --> 00:00:46.120
Then after that, you want to
worry about exponents.

00:00:46.120 --> 00:00:48.240
There are no exponents in this
expression, but I'll just

00:00:48.240 --> 00:00:52.150
write it down just for future
reference: exponents.

00:00:52.150 --> 00:00:55.000
One way I like to think about it
is parentheses always takes

00:00:55.000 --> 00:00:58.730
top priority, but then after
that, we go in descending

00:00:58.730 --> 00:01:01.610
order, or I guess we should
say in-- well, yeah, in

00:01:01.610 --> 00:01:05.400
descending order of how fast
that computation is.

00:01:05.400 --> 00:01:07.520
When I say fast, how
fast it grows.

00:01:07.520 --> 00:01:09.800
When I take something to an
exponent, when I'm taking

00:01:09.800 --> 00:01:13.270
something to a power, it grows
really fast. Then it grows a

00:01:13.270 --> 00:01:15.730
little bit slower or shrinks
a little bit slower if I

00:01:15.730 --> 00:01:17.760
multiply or divide,
so that comes

00:01:17.760 --> 00:01:21.800
next: multiply or divide.

00:01:21.800 --> 00:01:24.910
Multiplication and division
comes next, and then last of

00:01:24.910 --> 00:01:27.550
all comes addition
and subtraction.

00:01:27.550 --> 00:01:29.950
So these are kind of the
slowest operations.

00:01:29.950 --> 00:01:31.450
This is a little bit faster.

00:01:31.450 --> 00:01:33.610
This is the fastest operation.

00:01:33.610 --> 00:01:35.290
And then the parentheses,
just no matter

00:01:35.290 --> 00:01:36.980
what, always take priority.

00:01:36.980 --> 00:01:38.180
So let's apply it over here.

00:01:38.180 --> 00:01:40.290
Let me rewrite this
whole expression.

00:01:40.290 --> 00:01:47.960
So it's 8 plus 5 times 4 minus,
in parentheses, 6 plus

00:01:47.960 --> 00:01:53.150
10 divided by 2 plus 44.

00:01:53.150 --> 00:01:55.730
So we're going to want to do the
parentheses first. We have

00:01:55.730 --> 00:02:00.050
parentheses there and there.

00:02:00.050 --> 00:02:02.170
Now this parentheses is pretty
straightforward.

00:02:02.170 --> 00:02:05.120
Well, inside the parentheses
is already evaluated, so we

00:02:05.120 --> 00:02:08.009
could really just view
this as 5 times 4.

00:02:08.009 --> 00:02:11.000
So let's just evaluate that
right from the get go.

00:02:11.000 --> 00:02:14.935
So this is going to result in
8 plus-- and really, when

00:02:14.935 --> 00:02:16.650
you're evaluating the
parentheses, if your evaluate

00:02:16.650 --> 00:02:18.980
this parentheses, you literally
just get 5, and you

00:02:18.980 --> 00:02:21.070
evaluate that parentheses, you
literally just get 4, and then

00:02:21.070 --> 00:02:22.930
they're next to each other,
so you multiply them.

00:02:22.930 --> 00:02:28.170
So 5 times 4 is 20 minus--
let me stay

00:02:28.170 --> 00:02:29.960
consistent with the colors.

00:02:29.960 --> 00:02:34.290
Now let me write the next
parenthesis right there, and

00:02:34.290 --> 00:02:38.050
then inside of it, we'd evaluate
this first. Let me

00:02:38.050 --> 00:02:40.080
close the parenthesis
right there.

00:02:40.080 --> 00:02:43.280
And then we have plus 44.

00:02:43.280 --> 00:02:46.250
So what is this thing right here
evaluate to, this thing

00:02:46.250 --> 00:02:47.800
inside the parentheses?

00:02:47.800 --> 00:02:49.170
Well, you might be tempted
to say, well, let me

00:02:49.170 --> 00:02:50.160
just go left to right.

00:02:50.160 --> 00:02:53.770
6 plus 10 is 16 and then
divide by 2 and

00:02:53.770 --> 00:02:54.780
you would get 8.

00:02:54.780 --> 00:02:57.080
But remember: order
of operations.

00:02:57.080 --> 00:03:01.760
Division takes priority over
addition, so you actually want

00:03:01.760 --> 00:03:04.850
to do the division first, and
we could actually write it

00:03:04.850 --> 00:03:05.790
here like this.

00:03:05.790 --> 00:03:07.960
You could imagine putting
some more parentheses.

00:03:07.960 --> 00:03:09.790
Let me do it in that
same purple.

00:03:09.790 --> 00:03:13.200
You could imagine putting some
more parentheses right here to

00:03:13.200 --> 00:03:16.110
really emphasize the fact that
you're going to do the

00:03:16.110 --> 00:03:17.760
division first.

00:03:17.760 --> 00:03:24.740
So 10 divided by 2 is 5, so this
will result in 6, plus 10

00:03:24.740 --> 00:03:27.560
divided by 2, is 5.

00:03:27.560 --> 00:03:28.730
6 plus 5.

00:03:28.730 --> 00:03:32.230
Well, we still have to evaluate
this parentheses, so

00:03:32.230 --> 00:03:34.600
this results-- what's
6 plus 5?

00:03:34.600 --> 00:03:36.050
Well, that's 11.

00:03:36.050 --> 00:03:38.630
So we're left with
the 20-- let me

00:03:38.630 --> 00:03:39.630
write it all down again.

00:03:39.630 --> 00:03:44.050
We're left with 8 plus
20 minus 6 plus 5,

00:03:44.050 --> 00:03:47.000
which is 11, plus 44.

00:03:47.000 --> 00:03:50.820
And now that we have everything
at this level of

00:03:50.820 --> 00:03:53.580
operations, we can just
go left to right.

00:03:53.580 --> 00:03:58.770
So 8 plus 20 is 28, so you
can view this as 28

00:03:58.770 --> 00:04:01.890
minus 11 plus 44.

00:04:01.890 --> 00:04:06.030
28 minus 11-- 28 minus 10
would be 18, so this

00:04:06.030 --> 00:04:07.386
is going to be 17.

00:04:07.386 --> 00:04:10.630
It's going to be 17 plus 44.

00:04:10.630 --> 00:04:16.209
And then 17 plus 44-- I'll
scroll down a little bit.

00:04:16.209 --> 00:04:21.160
7 plus 44 would be 51, so
this is going to be 61.

00:04:21.160 --> 00:04:24.940
So this is going to
be equal to 61.

00:04:24.940 --> 00:04:26.460
And we're done!


Exponent example 1

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.960
Find the value of 5 to
the third power.

00:00:03.960 --> 00:00:05.020
Let me rewrite that.

00:00:05.020 --> 00:00:07.910
We have 5 to the third power.

00:00:07.910 --> 00:00:10.340
Now, it's important to remember,
this does not

00:00:10.340 --> 00:00:12.630
mean 5 times 3.

00:00:12.630 --> 00:00:17.770
This means 5 times itself three
times, so this is equal

00:00:17.770 --> 00:00:23.690
to 5 times 5 times 5.

00:00:23.690 --> 00:00:26.930
5 times 3, just as a bit of a
refresher so you realize the

00:00:26.930 --> 00:00:31.680
difference, 5 times 3-- let
me write it over here.

00:00:31.680 --> 00:00:41.370
5 times 3 is equal to
5 plus 5 plus 5.

00:00:41.370 --> 00:00:44.820
So when you multiply by 3,
you're adding the number to

00:00:44.820 --> 00:00:46.690
itself three times.

00:00:46.690 --> 00:00:49.520
When you take it to the third
power, you're multiplying the

00:00:49.520 --> 00:00:51.980
number by itself three times.

00:00:51.980 --> 00:00:55.720
So 5 times 3, you've seen
that before, that's 15.

00:00:55.720 --> 00:01:00.650
But 5 to the third power, 5
times itself three times, is

00:01:00.650 --> 00:01:07.400
equal to-- well, 5 times 5 is
25, and then 25 times 5 is

00:01:07.400 --> 00:01:12.760
125, so this is equal to 125.

00:01:12.760 --> 00:01:14.270
And we're done!


Dividing Whole Numbers and Applications 2

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.000
Four people earn $180
for working at an

00:00:04.000 --> 00:00:05.560
after-school program.

00:00:05.560 --> 00:00:09.580
If they divide this money up
equally, how much will each

00:00:09.580 --> 00:00:10.520
person receive?

00:00:10.520 --> 00:00:15.970
So we want to divide $180 into
four equal groups so that each

00:00:15.970 --> 00:00:18.190
of the four people can
have the same amount.

00:00:18.190 --> 00:00:26.310
So we want to divide 180 by 4.

00:00:26.310 --> 00:00:30.300
We want to take 180
divided by 4.

00:00:30.300 --> 00:00:33.044
Now, most of us don't know this
off the top of our heads,

00:00:33.044 --> 00:00:34.493
so we need to solve for it.

00:00:34.493 --> 00:00:36.290
We need to do a little
bit of division.

00:00:36.290 --> 00:00:39.100
So another way of writing this
exact same expression is to

00:00:39.100 --> 00:00:44.430
write 4 divided into 180.

00:00:44.430 --> 00:00:47.200
And when you write it like this,
it's easier to do it if

00:00:47.200 --> 00:00:49.350
either of them actually has
more than one digit,

00:00:49.350 --> 00:00:52.170
especially if the number we're
dividing into has more than

00:00:52.170 --> 00:00:54.990
one or two or three
or four digits.

00:00:54.990 --> 00:00:56.460
So let's do it.

00:00:56.460 --> 00:00:59.320
So the first thing you say is,
well, does 4 go into 1?

00:00:59.320 --> 00:01:00.670
Well, no, 4 does
not go into 1.

00:01:00.670 --> 00:01:01.940
It's larger than 1.

00:01:01.940 --> 00:01:05.300
So you could put a 0 up here,
but instead of doing that,

00:01:05.300 --> 00:01:07.150
let's just move on to the right
and say, well, OK, it

00:01:07.150 --> 00:01:08.190
doesn't go into 1.

00:01:08.190 --> 00:01:10.540
Does 4 go into 18?

00:01:10.540 --> 00:01:14.380
Well, 4 times 4 is 16,
so that goes into 18

00:01:14.380 --> 00:01:15.350
at least four times.

00:01:15.350 --> 00:01:18.110
4 times 5-- let me
write this down.

00:01:18.110 --> 00:01:22.920
4 times 4 is equal to 16,
so that goes into 18.

00:01:22.920 --> 00:01:28.200
4 times 5 is equal to 20, so
that does not go into 18, so

00:01:28.200 --> 00:01:31.050
this is going to go into
18 four times.

00:01:31.050 --> 00:01:34.900
4 times 4 is 16, and
now we subtract.

00:01:34.900 --> 00:01:36.850
We essentially find the
remainder here.

00:01:36.850 --> 00:01:40.050
18 minus 6 is 2.

00:01:40.050 --> 00:01:44.290
Now we want to bring down this
0, because we really said 4

00:01:44.290 --> 00:01:46.440
goes into 180 40 times, because

00:01:46.440 --> 00:01:47.820
it's in the tens place.

00:01:47.820 --> 00:01:50.030
40 times 4 is 160.

00:01:50.030 --> 00:01:53.040
180 minus 160 is 20.

00:01:53.040 --> 00:01:54.310
That's why we bring
down the 0.

00:01:54.310 --> 00:01:57.430
But if you just think about
the process, you say, OK,

00:01:57.430 --> 00:02:00.440
what's the smallest-- if I just
go with the 1, 4 doesn't

00:02:00.440 --> 00:02:01.700
go into it, so let me move on.

00:02:01.700 --> 00:02:03.850
Let me look at the
1 and the 8, 18.

00:02:03.850 --> 00:02:06.240
4 goes into that four times.

00:02:06.240 --> 00:02:07.800
4 times 4 is 16.

00:02:07.800 --> 00:02:11.620
Subtract from 18, you get
a 2, bring down the 0.

00:02:11.620 --> 00:02:13.980
Now 4 goes into 20
how many times?

00:02:13.980 --> 00:02:15.120
Well, we just wrote
it over here.

00:02:15.120 --> 00:02:19.110
4 times 5 is equal to 20.

00:02:19.110 --> 00:02:21.800
So 4 goes into 20 five times.

00:02:21.800 --> 00:02:24.130
5 times 4 is 20.

00:02:24.130 --> 00:02:27.450
You subtract, and you have
no remainder left over.

00:02:27.450 --> 00:02:32.360
So 180 divided by 4
is equal to 45.

00:02:32.360 --> 00:02:35.790
So if these four people were
to split the money equally,

00:02:35.790 --> 00:02:38.210
they'll each get $45.


Dividing Whole Numbers and Applications 3

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:05.510
We're asked to compute
347 divided by 5.

00:00:05.510 --> 00:00:06.980
So let me just rewrite
it like this.

00:00:06.980 --> 00:00:13.490
We have 347-- let me
write it bigger.

00:00:13.490 --> 00:00:21.530
We have 347, and we want to
divide it by-- so I'll put the

00:00:21.530 --> 00:00:24.150
division sign here, maybe the
long division sign, depending

00:00:24.150 --> 00:00:29.160
on how you want to view it,
and then I'll put the 5 in

00:00:29.160 --> 00:00:32.299
blue-- divided by 5.

00:00:32.299 --> 00:00:35.040
So the first thing we do is
say does 5 go into 3?

00:00:35.040 --> 00:00:37.440
No, it doesn't, so let's
add another digit here.

00:00:37.440 --> 00:00:39.590
Does 5 go into 34?

00:00:39.590 --> 00:00:44.590
Well, 5 times 6 is equal to 30,
so that goes into it, and

00:00:44.590 --> 00:00:49.290
if we go above that, 5 times 7
is equal to 35, so that's too

00:00:49.290 --> 00:00:53.180
big, so it goes into
34 six times.

00:00:53.180 --> 00:00:55.260
Because if we went with 7,
we would've gotten 35.

00:00:55.260 --> 00:00:56.350
That's too big.

00:00:56.350 --> 00:00:59.340
So 5 goes into 34 six times.

00:00:59.340 --> 00:01:03.560
6 times 5 is 30-- we
saw that right over

00:01:03.560 --> 00:01:05.960
there-- and we subtract.

00:01:05.960 --> 00:01:07.850
34 minus 30.

00:01:07.850 --> 00:01:09.670
Let me do this in a new color.

00:01:09.670 --> 00:01:11.090
34 minus 30.

00:01:11.090 --> 00:01:13.460
Well, 4 minus 0 is 4.

00:01:13.460 --> 00:01:19.040
3 minus 3 is just 0, and then we
can bring down this 7 here.

00:01:19.040 --> 00:01:22.310
5 goes into 47 how many times?

00:01:22.310 --> 00:01:27.890
Let's see, 5 times 8 is equal to
40, so that would work, but

00:01:27.890 --> 00:01:30.240
let's see if we can go
in even more times.

00:01:30.240 --> 00:01:33.200
5 times 9 is equal to 45.

00:01:33.200 --> 00:01:33.780
That'll work.

00:01:33.780 --> 00:01:36.310
That's still less than 47.

00:01:36.310 --> 00:01:43.350
Then we have 5 times 10 is
50, so this is too big.

00:01:43.350 --> 00:01:47.410
So we want to go with 5 times
9 is equal to 45, or 5 goes

00:01:47.410 --> 00:01:50.290
into 47 nine times.

00:01:50.290 --> 00:01:55.310
9 times 5 is 45, and then
you subtract again.

00:01:55.310 --> 00:01:59.410
47 minus 45 is 2.

00:01:59.410 --> 00:02:00.040
And then we're done.

00:02:00.040 --> 00:02:02.400
There's nothing else
to bring down here.

00:02:02.400 --> 00:02:05.480
One day, we'll learn about
dividing it into decimals and

00:02:05.480 --> 00:02:07.720
getting some kind of a decimal
representation, but we're not

00:02:07.720 --> 00:02:09.280
going to worry about
that right now.

00:02:09.280 --> 00:02:11.670
We have nothing left
to bring down.

00:02:11.670 --> 00:02:17.025
So 347 divided by 5 is 69.

00:02:17.025 --> 00:02:20.560
It is 69.

00:02:20.560 --> 00:02:25.930
It is equal to 69, but it
doesn't go evenly into 347.

00:02:25.930 --> 00:02:27.470
There's something left over.

00:02:27.470 --> 00:02:29.050
There's a remainder.

00:02:29.050 --> 00:02:30.960
There is a remainder
right here.

00:02:30.960 --> 00:02:33.710
So it equals 69, and sometimes
you'll see it written as

00:02:33.710 --> 00:02:37.100
remainder 2, or sometimes
it's a capital R.

00:02:37.100 --> 00:02:40.640
Sometimes people will write out
the word "remainder." So 5

00:02:40.640 --> 00:02:44.430
goes into 347 sixty-nine times,
but you're going to

00:02:44.430 --> 00:02:46.180
have 2 left over.

00:02:46.180 --> 00:02:51.580
Or another way to think about
it is 5 times 69 is 345, so

00:02:51.580 --> 00:02:53.820
you're always going to have
2 left over in this

00:02:53.820 --> 00:02:54.890
circumstance.

00:02:54.890 --> 00:02:57.700
So that is our answer.


Dividing Whole Numbers and Applications 4

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:05.490
We're asked to compute
3,060 divided by 36.

00:00:05.490 --> 00:00:09.860
We want to figure out how many
times does 36-- I don't need

00:00:09.860 --> 00:00:11.045
to write it that big.

00:00:11.045 --> 00:00:21.400
We need to figure out how many
times does 36 go into 3,060.

00:00:21.400 --> 00:00:23.910
Now, this is interesting because
we're dividing by a

00:00:23.910 --> 00:00:26.080
two-digit number, by 36.

00:00:26.080 --> 00:00:28.450
We're going to see in this video
that the process is the

00:00:28.450 --> 00:00:29.670
exact same way.

00:00:29.670 --> 00:00:31.870
There's just going to be a
little bit more mental

00:00:31.870 --> 00:00:34.900
estimation going on, but we'll
do it explicitly here, so

00:00:34.900 --> 00:00:37.230
hopefully, it won't
be too bad.

00:00:37.230 --> 00:00:40.490
So we first look at 36, and we
say, well, does 36 go into 3?

00:00:40.490 --> 00:00:41.010
Well, no.

00:00:41.010 --> 00:00:43.550
3 is smaller than 36, so
it won't go into 3.

00:00:43.550 --> 00:00:45.750
Does 36 go into 30?

00:00:45.750 --> 00:00:49.350
No, 30 is still smaller than 36,
so it won't go into that.

00:00:49.350 --> 00:00:53.350
Does 36 go into 306?

00:00:53.350 --> 00:00:54.030
Well, sure.

00:00:54.030 --> 00:01:00.130
306 is larger than 36, and if
we were to estimate it, 30

00:01:00.130 --> 00:01:05.750
would go into 300 ten times, but
this is larger than 30, so

00:01:05.750 --> 00:01:08.100
it's going to go fewer times.

00:01:08.100 --> 00:01:09.150
Maybe it's 9.

00:01:09.150 --> 00:01:09.740
I'm not sure.

00:01:09.740 --> 00:01:10.410
Let's try it out.

00:01:10.410 --> 00:01:11.570
Let me try it out over
on the side.

00:01:11.570 --> 00:01:16.390
What is 36 times 9?

00:01:16.390 --> 00:01:18.610
And this is kind of the art of
doing these problems when

00:01:18.610 --> 00:01:21.360
you're dividing a two-digit
number into something.

00:01:21.360 --> 00:01:27.280
So 6 times 9 is 54.

00:01:27.280 --> 00:01:29.320
Regroup, or carry the 5.

00:01:29.320 --> 00:01:33.540
3 times 9 is 27, plus 5.

00:01:33.540 --> 00:01:36.880
27 plus 5 is 32.

00:01:36.880 --> 00:01:39.710
So 36 times 9 is 324.

00:01:39.710 --> 00:01:43.860
That's still larger than 306,
so we're going to have to go

00:01:43.860 --> 00:01:45.310
36 less than that,
so we're going to

00:01:45.310 --> 00:01:47.075
have to do eight times.

00:01:47.075 --> 00:01:50.180
So we're going to have to
go into it eight times.

00:01:50.180 --> 00:01:53.320
And remember, the way I thought
about that is I said

00:01:53.320 --> 00:01:55.630
my first guess was maybe it goes
in nine times, but when I

00:01:55.630 --> 00:01:59.640
tried out 36 times 9, that was
still larger than 306.

00:01:59.640 --> 00:02:01.380
That got me 324.

00:02:01.380 --> 00:02:04.860
Eight times should work, because
if you take 36 away,

00:02:04.860 --> 00:02:07.040
if you only do 8 times
36, that's going to

00:02:07.040 --> 00:02:09.110
take us below 300.

00:02:09.110 --> 00:02:15.080
342, 136 away from that, or
another way, 324 minus 36 is

00:02:15.080 --> 00:02:17.080
going to get us below 300.

00:02:17.080 --> 00:02:19.200
So let's try this out
and just make sure.

00:02:19.200 --> 00:02:24.310
8 times 6 is 48.

00:02:24.310 --> 00:02:27.170
Put the 8 there; carry the 4.

00:02:27.170 --> 00:02:32.600
8 times 3 is 24 plus 4 is 28.

00:02:32.600 --> 00:02:35.650
So 8 times 36 is 288
and it fits.

00:02:35.650 --> 00:02:37.610
It's less than 306.

00:02:37.610 --> 00:02:38.960
9 would've been too much.

00:02:38.960 --> 00:02:41.060
It would've been greater
than 306, so

00:02:41.060 --> 00:02:42.310
now we can just subtract.

00:02:45.570 --> 00:02:47.520
So we have 6 minus 8.

00:02:47.520 --> 00:02:49.220
We can't just subtract
8 from 6.

00:02:49.220 --> 00:02:50.890
We have to do a little
bit of regrouping.

00:02:50.890 --> 00:02:53.760
Let's get a 10 from the ones
place right here, but we can't

00:02:53.760 --> 00:02:56.205
borrow anything from here or
regroup anything from there,

00:02:56.205 --> 00:02:59.670
so let's go to the
hundreds place.

00:02:59.670 --> 00:03:03.000
Let's take 1 from the hundreds
place, and so this will become

00:03:03.000 --> 00:03:08.170
a 2, and then this 0 here
will become a 10.

00:03:08.170 --> 00:03:14.370
Another way to think about it
is this is now 10 tens.

00:03:14.370 --> 00:03:16.200
We took 100 from there,
and we're now

00:03:16.200 --> 00:03:18.910
writing 100 as 10 tens.

00:03:18.910 --> 00:03:21.940
Or another way to think about it
is you borrowed a 1, or you

00:03:21.940 --> 00:03:23.430
took a 1, really.

00:03:23.430 --> 00:03:25.040
You took a 1 from the
hundreds place.

00:03:25.040 --> 00:03:27.290
That made it into a 2, and
then you put the 1 out in

00:03:27.290 --> 00:03:28.510
front of the 0.

00:03:28.510 --> 00:03:31.450
That's kind of the process
way of thinking about it.

00:03:31.450 --> 00:03:34.380
Now, we have this 10 here,
and now we can

00:03:34.380 --> 00:03:36.140
borrow from this 10.

00:03:36.140 --> 00:03:37.780
So let's borrow from this 10.

00:03:37.780 --> 00:03:40.290
So this 10 can become a 9.

00:03:40.290 --> 00:03:45.310
1 is really taken from it, and
then this 6 becomes a 16.

00:03:45.310 --> 00:03:49.600
Because if you think about it,
this 10 was really 10 tens.

00:03:49.600 --> 00:03:54.310
It's in the tens place, so when
you put a 10 there, it

00:03:54.310 --> 00:03:55.990
really means 10 tens.

00:03:55.990 --> 00:03:59.730
So if you take 1 away from it,
this is actually now 90, and

00:03:59.730 --> 00:04:03.600
we have 10 to give to this 6,
and 10 plus 6 is 16, so that's

00:04:03.600 --> 00:04:05.090
what we're really doing.

00:04:05.090 --> 00:04:08.340
But now we're ready to subtract,
so we have 16 minus

00:04:08.340 --> 00:04:15.060
8 is 8, 9 minus 8 is 1,
and 2 minus 2 is 0.

00:04:15.060 --> 00:04:19.149
So we have 18 left over, and now
we can bring down this 0

00:04:19.149 --> 00:04:20.339
right here.

00:04:20.339 --> 00:04:21.440
Bring down that 0.

00:04:21.440 --> 00:04:23.880
So how many times does
36 go into 180?

00:04:23.880 --> 00:04:26.020
And once again, this is going
to be estimation.

00:04:26.020 --> 00:04:27.790
It's not going to
be six times.

00:04:27.790 --> 00:04:31.240
6 times 30 would be 180.

00:04:31.240 --> 00:04:34.800
6 times 36 would be too big,
because 36 is bigger than 30,

00:04:34.800 --> 00:04:36.020
so let's try 5.

00:04:36.020 --> 00:04:37.790
Let's see if 5 works.

00:04:37.790 --> 00:04:38.770
So let's see.

00:04:38.770 --> 00:04:40.750
5 times 6-- and there's
a little trial

00:04:40.750 --> 00:04:41.760
and error here sometimes.

00:04:41.760 --> 00:04:44.860
5 times 6 is 30.

00:04:44.860 --> 00:04:47.310
Put the 0; carry the 3.

00:04:47.310 --> 00:04:47.990
We don't need that.

00:04:47.990 --> 00:04:49.330
That's from last time.

00:04:49.330 --> 00:04:53.410
5 times 3 is 15 plus 3 is 18.

00:04:53.410 --> 00:04:54.840
It looks like it works out.

00:04:54.840 --> 00:04:58.020
You subtract 180 from
180, your get zero,

00:04:58.020 --> 00:05:00.410
so we have no remainder.

00:05:03.470 --> 00:05:09.000
So we're left with 36 goes in
3,060 eighty-five times.

00:05:09.000 --> 00:05:14.000
So this is equal to 85.


Dividing Whole Numbers and Applications 5

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.150
We're asked to compute 4,800,
or four thousand eight

00:00:04.150 --> 00:00:07.720
hundred, divided by 80, and then
they want us to check our

00:00:07.720 --> 00:00:09.780
answer, or they say
check your answer.

00:00:09.780 --> 00:00:12.480
So I'll first do it just the
traditional long division way,

00:00:12.480 --> 00:00:14.500
and then we'll think about if
there's maybe a faster way to

00:00:14.500 --> 00:00:16.920
do this or if we can even
do it in our heads.

00:00:16.920 --> 00:00:19.380
So let's just do it the
traditional way first. So we

00:00:19.380 --> 00:00:28.460
have 80 and we want to
divide it into 4,800.

00:00:28.460 --> 00:00:31.980
Four thousand eight hundred.

00:00:31.980 --> 00:00:35.100
Or if we want to know the
terminology, 80 is the

00:00:35.100 --> 00:00:40.480
divisor, 4,800 is the dividend,
and in the number

00:00:40.480 --> 00:00:45.230
that we get, the number of times
80 goes into 4,800, we

00:00:45.230 --> 00:00:46.990
could call that the quotient.

00:00:46.990 --> 00:00:50.230
I always get confused when I
start using words like that,

00:00:50.230 --> 00:00:54.430
so we just think of it as 80 is
being divided into 4,800.

00:00:54.430 --> 00:00:58.840
Now, the first thing we say:
Well, does 80 go into 4?

00:00:58.840 --> 00:00:59.500
No.

00:00:59.500 --> 00:01:01.320
Does 80 go into 48?

00:01:01.320 --> 00:01:02.710
No, it's bigger than 48.

00:01:02.710 --> 00:01:06.040
Does 80 going into 480?

00:01:06.040 --> 00:01:07.260
Well, that looks interesting.

00:01:07.260 --> 00:01:12.310
Let's see, 6 times 8 is 48,
so 6 times 80 is 480.

00:01:12.310 --> 00:01:14.315
We could do that right here.

00:01:14.315 --> 00:01:16.180
80 times 6.

00:01:16.180 --> 00:01:18.000
6 times 0 is 0.

00:01:18.000 --> 00:01:20.150
6 times 8 is 48.

00:01:20.150 --> 00:01:22.800
Or another way you could have
thought about it is 6 times 8

00:01:22.800 --> 00:01:25.470
is 48, and we have this
extra 0 here, so you

00:01:25.470 --> 00:01:26.910
add the 0 right there.

00:01:26.910 --> 00:01:33.020
So 80 goes into 480 six times,
because we're only

00:01:33.020 --> 00:01:34.820
considering the 480.

00:01:34.820 --> 00:01:38.980
We put the 6 right in 480's one
spot, which is really the

00:01:38.980 --> 00:01:41.750
tens place, but if you just
think about the 480, it's

00:01:41.750 --> 00:01:43.360
right above the zero.

00:01:43.360 --> 00:01:50.520
Now 6 times 80 is-- we already
saw-- 480, and then we can

00:01:50.520 --> 00:01:53.830
subtract, and we get the
zero, and we bring

00:01:53.830 --> 00:01:56.090
down another zero.

00:01:56.090 --> 00:01:58.710
80 goes into 0 how many times?

00:01:58.710 --> 00:02:00.530
Well, it goes into
it zero times.

00:02:00.530 --> 00:02:03.140
0 times 80 is 0.

00:02:03.140 --> 00:02:03.960
You subtract.

00:02:03.960 --> 00:02:06.710
You get zeroes, and there's
nothing left to bring down, so

00:02:06.710 --> 00:02:07.630
we're done.

00:02:07.630 --> 00:02:11.100
4,800 divided by 80 is 60.

00:02:11.100 --> 00:02:15.930
When the dividend is 4,800,
the divisor-- let me write

00:02:15.930 --> 00:02:16.660
these words down.

00:02:16.660 --> 00:02:18.090
This is divisor.

00:02:21.020 --> 00:02:22.270
This is dividend.

00:02:25.320 --> 00:02:30.200
Then our quotient is 60.

00:02:30.200 --> 00:02:32.690
Now they want us to check our
answer, so we just have to

00:02:32.690 --> 00:02:37.520
confirm that 60 times 80 is
4,800 so let's do that.

00:02:37.520 --> 00:02:41.960
60 times 80-- now there's
two ways you can do it.

00:02:41.960 --> 00:02:45.190
You could just literally say 6
times 8 is 48, and then it'll

00:02:45.190 --> 00:02:47.320
have two zeroes here,
one, two.

00:02:47.320 --> 00:02:48.900
That's 4,800.

00:02:48.900 --> 00:02:50.120
Or we could do it by hand.

00:02:50.120 --> 00:02:52.090
0 times 0 is 0.

00:02:52.090 --> 00:02:54.320
0 times 6 is 0.

00:02:54.320 --> 00:02:56.220
Then we'll put a zero
here because we're

00:02:56.220 --> 00:02:58.340
now in the tens spot.

00:02:58.340 --> 00:02:59.320
Put the zero here.

00:02:59.320 --> 00:03:00.820
8 times 0 is 0.

00:03:00.820 --> 00:03:04.130
8 times 6 is 48.

00:03:04.130 --> 00:03:05.200
Add all of this up.

00:03:05.200 --> 00:03:08.570
You get 4,800, so it works.

00:03:08.570 --> 00:03:13.170
A faster way would just have
been to say 8 times 6 is 48,

00:03:13.170 --> 00:03:16.640
and you have one, two zeroes
here, so one, two

00:03:16.640 --> 00:03:18.430
zeroes right there.

00:03:18.430 --> 00:03:23.720
Now, a quick way to think about
4,800 divided by 80 is

00:03:23.720 --> 00:03:28.050
to divide both of these numbers
by 10 first. And just

00:03:28.050 --> 00:03:31.090
to think about it this way, what
we're doing, this could

00:03:31.090 --> 00:03:36.230
be written as 4,800/80.

00:03:36.230 --> 00:03:40.220
Or this could be written as--
let me write it down here.

00:03:40.220 --> 00:03:52.130
4,800/80 could be written as 480
times 10 over 8 times 10.

00:03:52.130 --> 00:04:00.500
4,800 is just 480 times 10,
and 80 is just 8 times 10.

00:04:00.500 --> 00:04:01.890
Now what happens here?

00:04:01.890 --> 00:04:13.940
We could view this as being
equal to 480/8 times this 10

00:04:13.940 --> 00:04:16.769
up here divided by this 10.

00:04:16.769 --> 00:04:18.200
Now, what's 10 divided by 10?

00:04:18.200 --> 00:04:21.829
What's any non-zero number
divided by itself?

00:04:21.829 --> 00:04:23.420
Well, this is just
going to be 1.

00:04:23.420 --> 00:04:26.900
10 goes into 10 one time,
so we're just left

00:04:26.900 --> 00:04:28.670
with 480 over 8.

00:04:28.670 --> 00:04:30.940
So essentially, you can
just knock off.

00:04:30.940 --> 00:04:33.570
You can knock off the same
number of zeroes off of the

00:04:33.570 --> 00:04:35.620
divisor as the dividend.

00:04:35.620 --> 00:04:39.310
So we knock off one zero here,
one zero there, it becomes 480

00:04:39.310 --> 00:04:41.960
divided by 8, and that's
an easier problem.

00:04:41.960 --> 00:04:47.000
8 goes into 480, and you say 8
goes into-- it doesn't go into

00:04:47.000 --> 00:04:49.440
4, so it goes into
48 six times.

00:04:49.440 --> 00:04:51.590
6 times 8 is 48.

00:04:51.590 --> 00:04:52.920
You have no remainder.

00:04:52.920 --> 00:04:54.260
Bring down this zero.

00:04:54.260 --> 00:04:56.770
8 goes into 0 zero times.

00:04:56.770 --> 00:04:58.190
0 times 8 is 0.

00:04:58.190 --> 00:04:59.710
No remainder.

00:04:59.710 --> 00:05:02.230
And we get 60 again, so either
way would have worked.

00:05:02.230 --> 00:05:03.950
This way is a little
bit faster.

00:05:03.950 --> 00:05:05.960
You might be able to do it in
your head, but no matter which

00:05:05.960 --> 00:05:08.690
way you do it, you get 60 as our
answer, and we confirmed

00:05:08.690 --> 00:05:10.390
that it really works.


Dividing Whole Numbers and Applications 6

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.510
Paulo wants to earn $1500
by mowing lawns.

00:00:04.510 --> 00:00:08.980
He will charge $35 for
each lawn he mows.

00:00:08.980 --> 00:00:13.860
What is the minimum number of
lawns the Paulo needs to mow

00:00:13.860 --> 00:00:17.790
to reach his goal of $1500?

00:00:17.790 --> 00:00:21.770
So we need to figure out the
minimum number of lawns that

00:00:21.770 --> 00:00:23.700
Paulo needs to mow.

00:00:23.700 --> 00:00:28.470
Now, whatever that number is,
that number times $35, because

00:00:28.470 --> 00:00:30.580
that's how much he's going
to get per lawn.

00:00:30.580 --> 00:00:35.470
So the number that he has to
mow times $35 needs to be

00:00:35.470 --> 00:00:37.450
greater than 1,500.

00:00:37.450 --> 00:00:40.640
So to figure out that number,
let's just divide 1,500

00:00:40.640 --> 00:00:41.820
divided by 35.

00:00:41.820 --> 00:00:44.460
Maybe it goes evenly, and
that'll be the number of lawn

00:00:44.460 --> 00:00:45.740
he needs to mow.

00:00:45.740 --> 00:00:57.210
So we're going to take 1,500
and divide it by 35 because

00:00:57.210 --> 00:00:59.070
that's how much he gets
paid per lawn.

00:01:02.530 --> 00:01:06.060
Now 35 does not go into 1, so we
have to add another digit.

00:01:06.060 --> 00:01:08.570
35 does not go into 15.

00:01:08.570 --> 00:01:11.570
35 goes into 150.

00:01:11.570 --> 00:01:14.170
Well, 30 goes into
150 five times.

00:01:14.170 --> 00:01:16.700
35 is bigger so it won't
go into it five times.

00:01:16.700 --> 00:01:17.620
Let's try out 4.

00:01:17.620 --> 00:01:20.620
Let's see if it goes
into it four times.

00:01:20.620 --> 00:01:23.630
4 times 5 is 20.

00:01:23.630 --> 00:01:25.880
Let's regroup the 2,
or carry the 2.

00:01:25.880 --> 00:01:29.900
Four times 3 is 12.

00:01:29.900 --> 00:01:35.580
12 plus 2 is 14.

00:01:35.580 --> 00:01:36.480
Now we subtract.

00:01:36.480 --> 00:01:37.930
0 minus 0 is 0.

00:01:37.930 --> 00:01:40.160
5 minus 4 is 1.

00:01:40.160 --> 00:01:43.500
The ones cancel out, and
then we bring down

00:01:43.500 --> 00:01:45.940
this zero right here.

00:01:45.940 --> 00:01:47.440
35 goes into 100.

00:01:47.440 --> 00:01:51.140
Well, 35 times 2 is 70, and if
you add another-- 35 times 3

00:01:51.140 --> 00:01:55.440
is 105, so that's too big, so
it goes into it two times.

00:01:55.440 --> 00:01:57.850
2 times 5 is 10.

00:01:57.850 --> 00:01:59.250
Let's carry the 1.

00:01:59.250 --> 00:02:04.680
2 times 3 is 6 plus 1 is
7, or 35 times 2 is 70.

00:02:04.680 --> 00:02:06.190
You subtract.

00:02:06.190 --> 00:02:08.370
0 minus 0 is 0.

00:02:08.370 --> 00:02:09.960
0 minus 7, you can't do it.

00:02:09.960 --> 00:02:12.360
You could already view it as 10
minus 7, or if you want to

00:02:12.360 --> 00:02:15.580
go in the traditional way, this
0 can take or borrow 1

00:02:15.580 --> 00:02:17.810
from this 1, so that
becomes a 0.

00:02:17.810 --> 00:02:19.330
This becomes a 10.

00:02:19.330 --> 00:02:22.060
10 minus 7 is 30.

00:02:22.060 --> 00:02:24.810
And so there's nothing
left to bring down.

00:02:24.810 --> 00:02:29.510
so we now know that 35 goes into
1,500 forty-two times,

00:02:29.510 --> 00:02:33.950
but there's a remainder of 30.

00:02:33.950 --> 00:02:39.550
If he mowed 42 lawns, he'd
be 30 short of 1,500.

00:02:39.550 --> 00:02:44.320
35 times 42 is 30
less than 1,500.

00:02:44.320 --> 00:02:45.090
What would it be?

00:02:45.090 --> 00:02:47.240
1,470.

00:02:47.240 --> 00:02:50.300
So in order to make 1,500, he's
going to have to go over

00:02:50.300 --> 00:02:52.350
it, so he's going to have to
do one more lawn than this.

00:02:52.350 --> 00:02:55.750
So just to remind ourselves,
if we just divide-- let me

00:02:55.750 --> 00:02:56.260
write it here.

00:02:56.260 --> 00:03:06.160
1,500 divided by 35 is equal
to 42, remainder 30, so 42

00:03:06.160 --> 00:03:07.430
won't be enough lawns.

00:03:07.430 --> 00:03:10.630
We're going to have more to-- we
have 30 more bucks to earn,

00:03:10.630 --> 00:03:13.560
so he's going to have to mow one
more lawn to get another

00:03:13.560 --> 00:03:16.110
$35, so he's going to have
$5 left over after he

00:03:16.110 --> 00:03:17.470
makes up this 30.

00:03:17.470 --> 00:03:23.780
So he's going to have to mow 43
lawns if he really wants to

00:03:23.780 --> 00:03:26.640
cross this threshold of $1500.

00:03:26.640 --> 00:03:30.480
If he does mow the 43 lawns,
he'll have more than $1500.

00:03:30.480 --> 00:03:33.690
He'll actually have $5
more than $1500.


Identity property of 1

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:05.550
We're asked to multiply
65 times 1.

00:00:05.550 --> 00:00:08.760
So literally, we just need to
multiply 65-- we could write

00:00:08.760 --> 00:00:13.150
it is a times sign like that or
we could write it as a dot

00:00:13.150 --> 00:00:16.860
like that-- but this
means 65 times 1.

00:00:16.860 --> 00:00:19.080
And there's two ways
to interpret this.

00:00:19.080 --> 00:00:23.420
You could view this as the
number 65 one time or you

00:00:23.420 --> 00:00:25.980
could view this as the
number 1 sixty-five

00:00:25.980 --> 00:00:27.500
times, all added up.

00:00:27.500 --> 00:00:32.189
But either way, if you have one
65, this is literally just

00:00:32.189 --> 00:00:34.130
going to be 65.

00:00:34.130 --> 00:00:37.810
Anything times 1 is going
to be that anything,

00:00:37.810 --> 00:00:39.290
whatever this is.

00:00:39.290 --> 00:00:42.580
Whatever this is times
1 is going to be

00:00:42.580 --> 00:00:44.080
that same thing again.

00:00:44.080 --> 00:00:53.240
If I have just some kind of
placeholder here times 1, and

00:00:53.240 --> 00:00:57.190
I could even write it as the
times symbol times 1, that's

00:00:57.190 --> 00:00:58.600
going to be that same
placeholder.

00:01:01.420 --> 00:01:04.810
So if I have 3 times 1,
I'm going to get 3.

00:01:04.810 --> 00:01:09.630
If I have 5 times 1, I'm going
to get 5, because literally,

00:01:09.630 --> 00:01:13.940
all this is saying
is 5 one time.

00:01:13.940 --> 00:01:21.160
If I put-- I don't know-- 157
times 1, that'll be 157.

00:01:21.160 --> 00:01:23.160
I think you get the idea.


Properties of Whole Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.610
We're asked to add 50 plus 0.

00:00:05.610 --> 00:00:07.240
Well, zero is nothing.

00:00:07.240 --> 00:00:09.660
So you can imagine that we're
starting off with 50 of

00:00:09.660 --> 00:00:12.890
something, and if we add nothing
to that 50, we're

00:00:12.890 --> 00:00:16.320
still going to have 50
of that something.

00:00:16.320 --> 00:00:19.280
If I have three of something and
I added nothing, I'm still

00:00:19.280 --> 00:00:22.680
going to have three, so anything
plus zero is going to

00:00:22.680 --> 00:00:24.140
be whatever you started
off with.

00:00:24.140 --> 00:00:30.420
So 50 plus 0 is going
to still be 50.


Commutative law of addition

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.590
Use the commutative law of
addition-- let me underline

00:00:04.590 --> 00:00:07.970
that-- the commutative law
of addition to write the

00:00:07.970 --> 00:00:12.510
expression 5 plus 8 plus 5
in a different way and

00:00:12.510 --> 00:00:13.470
then find the sum.

00:00:13.470 --> 00:00:16.550
Now, this commutative law of
addition sounds like a very

00:00:16.550 --> 00:00:19.790
fancy thing, but all it means
is if you're just adding a

00:00:19.790 --> 00:00:23.680
bunch of numbers, it doesn't
matter what order you add the

00:00:23.680 --> 00:00:24.500
numbers in.

00:00:24.500 --> 00:00:29.090
So we could add it as
5 plus 8 plus 5.

00:00:29.090 --> 00:00:33.990
We could order it as
5 plus 5 plus 8.

00:00:33.990 --> 00:00:38.140
We could order it
8 plus 5 plus 5.

00:00:38.140 --> 00:00:40.750
These are all going to add up
to the same things, and it

00:00:40.750 --> 00:00:41.235
makes sense.

00:00:41.235 --> 00:00:43.750
If I have 5 of something and
then I add 8 more and then I

00:00:43.750 --> 00:00:46.250
add 5 more, I'm going to get
the same thing as if I had

00:00:46.250 --> 00:00:49.280
took 5 of something, then added
the 5, then added the 8.

00:00:49.280 --> 00:00:50.270
You could try all
of these out.

00:00:50.270 --> 00:00:51.770
You'll get the same thing.

00:00:51.770 --> 00:00:54.420
Now, they say in a different
way, and then find the sum.

00:00:54.420 --> 00:00:56.480
The easiest one to find the sum
of-- actually, let's do

00:00:56.480 --> 00:00:56.950
all of them.

00:00:56.950 --> 00:00:59.110
But the easiest one, just
because a lot of people

00:00:59.110 --> 00:01:02.660
immediately know that 5 plus 5
is 10, is to maybe start with

00:01:02.660 --> 00:01:04.150
the 5 plus 5.

00:01:04.150 --> 00:01:06.980
So if you have 5 plus
5, that's 10, plus

00:01:06.980 --> 00:01:09.690
8 is equal to 18.

00:01:09.690 --> 00:01:13.370
Now, let's verify that these two
are the same exact thing.

00:01:13.370 --> 00:01:17.040
Up here, 5 plus 8 is 13.

00:01:17.040 --> 00:01:20.840
13 plus 5 is also 18.

00:01:20.840 --> 00:01:22.520
That is also 18.

00:01:22.520 --> 00:01:25.550
If we go down here,
8 plus 5 is 13.

00:01:25.550 --> 00:01:28.720
13 plus 5 is also equal to 18.

00:01:28.720 --> 00:01:30.900
So no matter how you do it and
no matter what order you do it

00:01:30.900 --> 00:01:33.420
in-- and that's the commutative
law of addition.

00:01:33.420 --> 00:01:35.790
It sounds very fancy, but it
just means that order doesn't

00:01:35.790 --> 00:01:39.370
matter if you're adding
a bunch of things.


Commutative law of multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:08.119
Use the commutative law of
multiplication to write 2

00:00:08.119 --> 00:00:11.130
times 34 in a different way.

00:00:11.130 --> 00:00:13.550
Simplify both expressions
to show that they

00:00:13.550 --> 00:00:16.420
have identical results.

00:00:16.420 --> 00:00:19.250
So once again, this commutative
law just means

00:00:19.250 --> 00:00:20.540
that order doesn't matter.

00:00:20.540 --> 00:00:21.760
It sounds very fancy.

00:00:21.760 --> 00:00:23.910
Commutative law of
multiplication.

00:00:23.910 --> 00:00:27.000
But all that says is that it
doesn't matter whether we do 2

00:00:27.000 --> 00:00:32.439
times 34 or whether
we do 34 times 2.

00:00:32.439 --> 00:00:33.990
The order does not matter.

00:00:33.990 --> 00:00:38.590
We can commute the two terms.
Both of these are going to get

00:00:38.590 --> 00:00:40.510
you the same exact answer.

00:00:40.510 --> 00:00:41.340
So let's try it out.

00:00:41.340 --> 00:00:44.060
What is 2 times 34?

00:00:44.060 --> 00:00:45.540
And we could write it like
this, literally.

00:00:45.540 --> 00:00:48.230
You'll almost never see it
written like this, but it is

00:00:48.230 --> 00:00:51.020
literally 2 times 34.

00:00:51.020 --> 00:00:54.340
Almost always people write the
larger digit on top, or the

00:00:54.340 --> 00:00:56.690
digit with more digits,
or the number with

00:00:56.690 --> 00:00:58.415
more digits on top.

00:00:58.415 --> 00:00:59.970
But let's do it this way.

00:00:59.970 --> 00:01:04.300
4 times 2 is 8, and then
we'll put a 0.

00:01:04.300 --> 00:01:10.510
3 times 2 is 6, or you can view
it as 30 times 2 is 60.

00:01:10.510 --> 00:01:11.640
Add them together.

00:01:11.640 --> 00:01:13.710
8 plus 0 is 8.

00:01:13.710 --> 00:01:15.450
6, bring it down.

00:01:15.450 --> 00:01:16.560
It's not being added
to anything.

00:01:16.560 --> 00:01:17.810
You get 68.

00:01:17.810 --> 00:01:20.930
So 2 times 34 is 68.

00:01:20.930 --> 00:01:31.080
Now, if you do 34 times 2, 2
times 4 is 8, 2 times 3 is 6.

00:01:31.080 --> 00:01:34.100
That's why it's always nicer to
write the number with more

00:01:34.100 --> 00:01:35.280
digits on top.

00:01:35.280 --> 00:01:38.150
It also is equal to 68.

00:01:38.150 --> 00:01:42.150
So it doesn't matter whether you
have two groups of 34 or

00:01:42.150 --> 00:01:44.870
thirty-four groups of 2,
in either case, you're

00:01:44.870 --> 00:01:47.000
going to have 68.


Associative law of addition

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:03.750
Use the associative law of
addition to write the

00:00:03.750 --> 00:00:04.800
expression.

00:00:04.800 --> 00:00:08.700
We have a 77 plus 2 in
parentheses, plus 3, in a

00:00:08.700 --> 00:00:09.680
different way.

00:00:09.680 --> 00:00:11.860
Simplify both expressions
to show they

00:00:11.860 --> 00:00:13.710
have identical results.

00:00:13.710 --> 00:00:17.230
So this associative law of
addition, which sounds very

00:00:17.230 --> 00:00:20.460
fancy and complicated, literally
means that you can

00:00:20.460 --> 00:00:23.670
associate these three numbers
in different ways or you can

00:00:23.670 --> 00:00:25.500
add them in different orders.

00:00:25.500 --> 00:00:26.760
Now let me just make
that clear.

00:00:26.760 --> 00:00:31.920
So the way they wrote it right
here, they wrote it 77 plus 2

00:00:31.920 --> 00:00:35.090
in parentheses, and then
they wrote plus 3.

00:00:35.090 --> 00:00:38.530
These parentheses mean
do the 77 plus 2

00:00:38.530 --> 00:00:40.980
before you add the 3.

00:00:40.980 --> 00:00:43.280
So if you were to evaluate
this, you would evaluate

00:00:43.280 --> 00:00:45.930
what's in the parentheses first.
So you would say, well,

00:00:45.930 --> 00:00:50.550
77 plus 2, that's 79, so
everything in the parentheses

00:00:50.550 --> 00:00:52.730
just evaluates to 79.

00:00:52.730 --> 00:00:55.050
And then you still
have that plus 3.

00:00:55.050 --> 00:01:00.430
And 79 plus 3 is 82, so
this is equal to 82.

00:01:00.430 --> 00:01:02.690
That's if you just evaluate
it the way that

00:01:02.690 --> 00:01:03.570
they gave it to us.

00:01:03.570 --> 00:01:06.190
Now, the associative law of
addition tells us it doesn't

00:01:06.190 --> 00:01:10.080
matter whether we add 77 and 2
first or whether we add 2 and

00:01:10.080 --> 00:01:12.860
3 first. We can associate
them differently.

00:01:12.860 --> 00:01:15.590
So this is going to be
the exact same thing.

00:01:15.590 --> 00:01:18.250
This is the exact same
thing as-- we could

00:01:18.250 --> 00:01:19.610
write it this way.

00:01:19.610 --> 00:01:20.520
Let me write them all.

00:01:20.520 --> 00:01:23.765
77 plus 2 plus 3.

00:01:23.765 --> 00:01:26.090
If we have no parentheses here,
this is actually the

00:01:26.090 --> 00:01:28.750
same thing as this over here,
because we'd go 77 plus 2 is

00:01:28.750 --> 00:01:31.430
79 plus 3 is 82.

00:01:31.430 --> 00:01:33.510
But the associative law tells
as, well, you know what?

00:01:33.510 --> 00:01:36.380
I could do 77 plus 2 plus 3.

00:01:36.380 --> 00:01:39.430
I could add this first and then
add it to 77, and it's

00:01:39.430 --> 00:01:41.970
going to be the exact same thing
as if I added these two

00:01:41.970 --> 00:01:43.790
guys first and then add the 3.

00:01:43.790 --> 00:01:46.710
Let's verify that
for ourselves.

00:01:46.710 --> 00:01:56.240
So 2 plus 3 is 5, so this
evaluates to 77 plus 5.

00:01:56.240 --> 00:02:02.130
And 77 plus 5, once
again, is 82.

00:02:02.130 --> 00:02:04.840
So it doesn't matter how you
associate the numbers.

00:02:04.840 --> 00:02:06.570
Either way, you get 82.

00:02:06.570 --> 00:02:09.729
And that's the associative
law of addition.


Associative Law of Multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.950
Use the associative law of
multiplication to write-- and

00:00:04.950 --> 00:00:07.910
here they have 12 times 3 in
parentheses, and then they

00:00:07.910 --> 00:00:11.360
want us to multiply that times
10-- in a different way.

00:00:11.360 --> 00:00:13.730
Simplify both expressions
to show they

00:00:13.730 --> 00:00:16.090
have identical results.

00:00:16.090 --> 00:00:19.160
So the way that they wrote it
is-- let me just rewrite it.

00:00:19.160 --> 00:00:24.160
So they have 12 times 3 in
parentheses, and then they

00:00:24.160 --> 00:00:26.410
multiply that times 10.

00:00:26.410 --> 00:00:28.820
Now whenever something is in
parentheses, that means do

00:00:28.820 --> 00:00:33.390
that first. So this literally
says let's do the 12 times 3

00:00:33.390 --> 00:00:35.920
first. Now, what
is 12 times 3?

00:00:35.920 --> 00:00:37.670
It's 36.

00:00:37.670 --> 00:00:40.950
So this evaluates to 36, and
then we still have that times

00:00:40.950 --> 00:00:44.040
10 over there.

00:00:44.040 --> 00:00:44.870
And we know the trick.

00:00:44.870 --> 00:00:48.960
Whenever we multiply something
times a power of ten, we just

00:00:48.960 --> 00:00:51.930
add the number of zeroes that we
have at the back of it, so

00:00:51.930 --> 00:00:54.440
this is going to be 360.

00:00:54.440 --> 00:00:59.390
This is going to be
equal to 360.

00:00:59.390 --> 00:01:03.160
Now, the associative law of
multiplication, once again, it

00:01:03.160 --> 00:01:04.950
sounds like a very
fancy thing.

00:01:04.950 --> 00:01:08.630
All that means is it doesn't
matter how we associate the

00:01:08.630 --> 00:01:11.370
multiplication or it doesn't
matter how we put the

00:01:11.370 --> 00:01:14.310
parentheses, we're going to get
the same answer, so let me

00:01:14.310 --> 00:01:15.840
write it down again.

00:01:15.840 --> 00:01:22.020
If we were to do 12 times 3
times 10, if we just wrote it

00:01:22.020 --> 00:01:24.180
like this without parentheses,
if we just went left to right,

00:01:24.180 --> 00:01:27.000
that would essentially be
exactly what we just did here

00:01:27.000 --> 00:01:27.860
on the left.

00:01:27.860 --> 00:01:29.890
But the associative law
of multiplication

00:01:29.890 --> 00:01:30.910
says, you know what?

00:01:30.910 --> 00:01:34.560
We can multiply the 3 times 10
first and then multiply the

00:01:34.560 --> 00:01:38.260
12, and we're going to get the
exact same answer as if we

00:01:38.260 --> 00:01:40.710
multiplied the 12 times
the 3 and then the 10.

00:01:40.710 --> 00:01:43.620
So let's just verify
it for ourselves.

00:01:43.620 --> 00:01:50.580
So 3 times 10 is 30, and we
still want to multiply the 12

00:01:50.580 --> 00:01:51.490
times that.

00:01:51.490 --> 00:01:53.150
Now, what's 12 times 30?

00:01:53.150 --> 00:01:55.230
And we've seen this several
times before.

00:01:55.230 --> 00:01:59.390
You can view it as a 12 times
3, which is 36, but we still

00:01:59.390 --> 00:02:00.970
have this 0 here.

00:02:00.970 --> 00:02:03.340
So that is also equal to 360.

00:02:03.340 --> 00:02:06.720
So it didn't matter how we
associated the multiplication.

00:02:06.720 --> 00:02:09.570
You can do the 12 times 3 first
or you can do the 3

00:02:09.570 --> 00:02:14.340
times 10 first. Either way, they
both evaluated to 360.


Comparing Whole Numbers 2

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.420
Complete this statement using
less than or greater than--

00:00:04.420 --> 00:00:06.850
that's what those symbols
are-- for this kind of

00:00:06.850 --> 00:00:08.029
brackets there.

00:00:08.029 --> 00:00:10.780
So all they're asking is they
have a 5 and they have a 2,

00:00:10.780 --> 00:00:13.740
and they want us to put either
a less than sign for these

00:00:13.740 --> 00:00:16.500
brackets or a greater
than sign.

00:00:16.500 --> 00:00:19.400
So we just have a 5 and
a 2, and we're just

00:00:19.400 --> 00:00:20.660
comparing the two.

00:00:20.660 --> 00:00:22.250
Which number is greater?

00:00:22.250 --> 00:00:23.540
Well, 5 is greater.

00:00:23.540 --> 00:00:25.630
If you count to 5, you're
going to pass up 2.

00:00:25.630 --> 00:00:29.690
If you look at a number line
here, if you started at 0, 1,

00:00:29.690 --> 00:00:35.460
2, 3, 4, 5, you see that
2 comes before 5.

00:00:35.460 --> 00:00:37.120
2 is less than 5.

00:00:37.120 --> 00:00:43.310
If I have $2, that's less than
$5, or $5 is greater than $2.

00:00:43.310 --> 00:00:46.450
And so the symbol for greater
than, the way you remember

00:00:46.450 --> 00:00:47.910
it-- this is probably the
hardest part of how do you

00:00:47.910 --> 00:00:51.050
remember it-- is you want the
larger side of the symbol

00:00:51.050 --> 00:00:52.360
facing the larger number.

00:00:52.360 --> 00:00:54.830
So 5 is the larger number,
so we want the

00:00:54.830 --> 00:00:56.460
bigger side of the symbol.

00:00:56.460 --> 00:00:57.860
So we want it like this.

00:00:57.860 --> 00:01:00.040
The way I think about is that
this right here is the big

00:01:00.040 --> 00:01:02.980
side of the symbol, and this
right here, this point, is the

00:01:02.980 --> 00:01:07.110
small side of the symbol,
so 5 is greater than 2.

00:01:07.110 --> 00:01:11.800
If it was the other way, we
would write 2 is less than 5.

00:01:11.800 --> 00:01:14.860
Once again, though, we always
have the point pointing to the

00:01:14.860 --> 00:01:19.090
smaller number, and we have the
wide-open part, the big

00:01:19.090 --> 00:01:22.650
part of the symbol, pointing
to the larger number.


Adding Whole Numbers and Applications 2

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.410
Let's add 249 to 383.

00:00:04.410 --> 00:00:06.530
Now, the first thing you want
to do with any type of

00:00:06.530 --> 00:00:08.860
addition problem, especially
when you have multiple digits,

00:00:08.860 --> 00:00:11.490
is to write the two numbers
above and below each other.

00:00:11.490 --> 00:00:14.290
So we have 200-- let me write
a little bigger than that--

00:00:14.290 --> 00:00:20.050
249 plus 383.

00:00:20.050 --> 00:00:22.170
And you want to line them up
so that we have the same

00:00:22.170 --> 00:00:23.980
places above and below
each other.

00:00:23.980 --> 00:00:28.550
So the 9 and the 3 are both in
the ones place here, so we've

00:00:28.550 --> 00:00:29.470
lined them up.

00:00:29.470 --> 00:00:32.600
The 4 and the 8 are both in the
tens place, and the 2 and

00:00:32.600 --> 00:00:34.890
the 3 are both in the
hundreds place.

00:00:34.890 --> 00:00:36.770
If you had a thousands place in
one of them and not in the

00:00:36.770 --> 00:00:38.840
other, it would just kind of
be sitting out here by

00:00:38.840 --> 00:00:39.640
themselves.

00:00:39.640 --> 00:00:42.590
The easiest thing is just to
kind of both make them line up

00:00:42.590 --> 00:00:44.160
to the right, or the
ones places.

00:00:44.160 --> 00:00:46.870
Now I'll show you the process,
but while we do the process,

00:00:46.870 --> 00:00:49.520
we'll think a little bit about
why it makes sense.

00:00:49.520 --> 00:00:53.320
So we start at the ones place,
and we say what is 9 plus 3?

00:00:53.320 --> 00:00:55.000
So we say 9 plus 3.

00:00:55.000 --> 00:00:59.990
Well, that's equal to 12.

00:00:59.990 --> 00:01:03.740
I'm writing the 1 in 12 in
orange on purpose, because

00:01:03.740 --> 00:01:06.770
when you want to write the sum
down here, you can't write a

00:01:06.770 --> 00:01:07.660
two-digit number.

00:01:07.660 --> 00:01:10.560
You can only write a digit
between 0 and 9.

00:01:10.560 --> 00:01:13.625
So what we do is 9 plus
3 is 12, that's the

00:01:13.625 --> 00:01:15.270
equivalent of 10 plus 2.

00:01:15.270 --> 00:01:18.720
So we write the 2 in the ones
place, and the 1, which is

00:01:18.720 --> 00:01:21.740
really representing 10, we put
in the tens place, which

00:01:21.740 --> 00:01:22.710
hopefully makes sense.

00:01:22.710 --> 00:01:23.960
This now represents 10.

00:01:23.960 --> 00:01:27.960
And now we want to add this
10 plus 40 plus 80.

00:01:27.960 --> 00:01:30.710
It looks like we're adding a 1
to a 4 to an 8, but these are

00:01:30.710 --> 00:01:32.430
really in the tens place,
so this is really a

00:01:32.430 --> 00:01:33.920
10, a 40 and an 80.

00:01:33.920 --> 00:01:35.810
But if you just think about
the process, you just take

00:01:35.810 --> 00:01:42.050
your 1, you add it to your 4 and
your 8-- plus 4 plus 8--

00:01:42.050 --> 00:01:43.340
and what do you get?

00:01:43.340 --> 00:01:45.040
1 plus 4 plus 8 is what?

00:01:45.040 --> 00:01:49.370
5 plus 8, that is 13.

00:01:49.370 --> 00:01:53.200
I'll write it like this.

00:01:53.200 --> 00:01:54.860
Actually, this is in the tens
place, so it actually

00:01:54.860 --> 00:01:57.540
represents 130, but you don't
have to think about that.

00:01:57.540 --> 00:01:59.830
The process is fairly
straightforward.

00:01:59.830 --> 00:02:00.740
You have 13.

00:02:00.740 --> 00:02:04.700
You write the 3 down here, and
we regroup the 1 or, you might

00:02:04.700 --> 00:02:07.130
have learned, you carry
the 1, so then you

00:02:07.130 --> 00:02:08.650
carry the 1 up here.

00:02:08.650 --> 00:02:10.590
Same process now in the
hundreds place.

00:02:10.590 --> 00:02:12.730
This 1 actually represent a
hundred, because this was

00:02:12.730 --> 00:02:15.190
actually 130 because it
was in the tens place.

00:02:15.190 --> 00:02:22.900
So we have a 1 plus 2 plus
3, what is that?

00:02:22.900 --> 00:02:24.040
Well, that's just 6, right?

00:02:24.040 --> 00:02:27.060
1 plus 2 is 3 plus 3 is 6.

00:02:27.060 --> 00:02:28.715
So that is equal to 6.

00:02:28.715 --> 00:02:30.600
So we write it right
over here.

00:02:30.600 --> 00:02:35.680
So 249 plus 383 is 632.


Example: Adding two digit numbers (no carrying)

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:04.900
So we need to figure out
what 46 plus 43 is.

00:00:04.900 --> 00:00:06.800
Let me write it down again
and I'll write it like

00:00:06.800 --> 00:00:11.220
this: 46 plus 43.

00:00:11.220 --> 00:00:14.770
What we do here is we just first
look at the ones place.

00:00:14.770 --> 00:00:18.600
We literally have 6 ones plus 3
ones, or you could say 6 and

00:00:18.600 --> 00:00:21.400
3, and 6 plus 3 is just 9.

00:00:21.400 --> 00:00:24.660
6 plus 3 is 9, so
we have 9 ones.

00:00:24.660 --> 00:00:26.990
And then you go to the tens
place, and there's two ways to

00:00:26.990 --> 00:00:27.570
think about it.

00:00:27.570 --> 00:00:34.260
You could view this as 4 plus
4 is equal to 8, but really,

00:00:34.260 --> 00:00:36.655
the reality of what we're doing
since we're operating in

00:00:36.655 --> 00:00:41.370
the tens place, this is really
40 plus 40 is equal to 80.

00:00:41.370 --> 00:00:43.650
And we could do the same
problem expanded out.

00:00:43.650 --> 00:00:47.560
This is the same thing
as 40 plus 6.

00:00:47.560 --> 00:00:49.180
We've seen this before, right?

00:00:49.180 --> 00:00:51.420
That's what 46 is.

00:00:51.420 --> 00:00:57.650
And then 43 is the same
thing as 40 plus 3.

00:00:57.650 --> 00:00:59.690
We've expanded these
out before.

00:00:59.690 --> 00:01:02.290
And so when you add them,
when you add a 6 plus a

00:01:02.290 --> 00:01:05.556
3, you get a 9.

00:01:05.556 --> 00:01:11.040
When you add a 40 plus
a 40, you get an 80.

00:01:11.040 --> 00:01:13.740
So you get 80 plus
9, which is 89.

00:01:13.740 --> 00:01:16.200
And the whole reason why I did
this is I wanted to show you

00:01:16.200 --> 00:01:19.790
that when you're adding 4's in
the tens place, you're really

00:01:19.790 --> 00:01:20.990
adding 40's.

00:01:20.990 --> 00:01:23.800
The fact that it's in the tens
place represents-- or the fact

00:01:23.800 --> 00:01:25.890
that the 4 is in the tens
place shows that

00:01:25.890 --> 00:01:26.840
it represents 40.

00:01:26.840 --> 00:01:30.890
The fact that the 8's there
shows that it represents 80.