Dividing fractions example 2

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.640
Divide and write the answer
as a mixed number.

00:00:04.640 --> 00:00:07.770
And we have 3/5 divided
by 1/2.

00:00:07.770 --> 00:00:10.220
Now, whenever you're dividing
any fractions, you just have

00:00:10.220 --> 00:00:14.200
to remember that dividing by a
fraction is the same thing as

00:00:14.200 --> 00:00:16.970
multiplying by its reciprocal.

00:00:16.970 --> 00:00:22.810
So this thing right here is the
same thing as 3/5 times--

00:00:22.810 --> 00:00:25.780
so this is our 3/5 right here,
and instead of a division

00:00:25.780 --> 00:00:28.120
sign, you want a multiplication
sign, and

00:00:28.120 --> 00:00:31.880
instead of a 1/2, you want to
take the reciprocal of 1/2,

00:00:31.880 --> 00:00:36.510
which would be 2/1--
so times 2/1.

00:00:36.510 --> 00:00:39.770
So dividing by 1/2 is the
exact same thing as

00:00:39.770 --> 00:00:42.360
multiplying by 2/1.

00:00:42.360 --> 00:00:45.000
And we just do this as a
straightforward multiplication

00:00:45.000 --> 00:00:46.080
problem now.

00:00:46.080 --> 00:00:49.840
3 times 2 is 6, so our
new numerator is 6.

00:00:49.840 --> 00:00:52.950
5 times 1 is 5.

00:00:52.950 --> 00:00:57.360
So 3/5 divided by 1/2 as an
improper fraction is 6/5.

00:00:57.360 --> 00:01:01.220
Now, they want us to write
it as at mixed number.

00:01:01.220 --> 00:01:04.450
So we divide the 5 into
the 6, figure out how

00:01:04.450 --> 00:01:05.300
many times it goes.

00:01:05.300 --> 00:01:07.530
That'll be the whole number
part of the mixed number.

00:01:07.530 --> 00:01:10.290
And then whatever's left over
will be the remaining

00:01:10.290 --> 00:01:13.620
numerator over 5.

00:01:13.620 --> 00:01:18.160
So what we'll do is
take 5 into 6.

00:01:18.160 --> 00:01:20.600
5 goes into 6 one time.

00:01:20.600 --> 00:01:22.700
1 times 5 is 5.

00:01:22.700 --> 00:01:23.360
Subtract.

00:01:23.360 --> 00:01:25.960
You have a remainder of 1.

00:01:25.960 --> 00:01:34.460
So 6/5 is equal to one whole,
or 5/5, and 1/5.

00:01:38.910 --> 00:01:42.610
This 1 comes from whatever
is left over.

00:01:42.610 --> 00:01:43.630
And now we're done!

00:01:43.630 --> 00:01:46.730
3/5 divided by 1/2
is 1 and 1/5.

00:01:46.730 --> 00:01:49.350
Now, the one thing that's not
obvious is why did this work?

00:01:49.350 --> 00:01:53.660
Why is dividing by 1/2 the
same thing as multiplying

00:01:53.660 --> 00:01:54.880
essentially by 2.

00:01:54.880 --> 00:01:57.000
2/1 is the same thing as 2.

00:01:57.000 --> 00:01:59.890
And to do that, I'll do a little
side-- fairly simple--

00:01:59.890 --> 00:02:03.870
example, but hopefully, it
gets the point across.

00:02:03.870 --> 00:02:05.910
Let me take four objects.

00:02:05.910 --> 00:02:08.850
So we have four objects:
one, two, three, four.

00:02:08.850 --> 00:02:13.790
So I have four objects, and if
I were to divide into groups

00:02:13.790 --> 00:02:17.300
of two, so I want to divide
it into groups of two.

00:02:17.300 --> 00:02:21.210
So that is one group of two and
then that is another group

00:02:21.210 --> 00:02:23.600
of two, how many groups
do I have?

00:02:23.600 --> 00:02:27.400
Well, 4 divided by 2, I have two
groups of two, so that is

00:02:27.400 --> 00:02:28.900
equal to 2.

00:02:28.900 --> 00:02:31.460
Now, what if I took those
same four objects:

00:02:31.460 --> 00:02:33.990
one, two, three, four.

00:02:33.990 --> 00:02:36.160
So I'm taking those
same four objects.

00:02:36.160 --> 00:02:38.890
Instead of dividing them into
groups of two, I want to

00:02:38.890 --> 00:02:44.610
divide them into groups of 1/2,
which means each group

00:02:44.610 --> 00:02:47.080
will have half of
an object in it.

00:02:47.080 --> 00:02:49.960
So let's say that would be
one group right there.

00:02:49.960 --> 00:02:51.900
That is a second group.

00:02:51.900 --> 00:02:53.140
That is a third group.

00:02:53.140 --> 00:02:56.690
I think you see each group has
half of a circle in it.

00:02:56.690 --> 00:02:58.280
That is the fourth.

00:02:58.280 --> 00:03:00.070
That's the fifth.

00:03:00.070 --> 00:03:01.390
That's the sixth.

00:03:01.390 --> 00:03:03.790
That's the seventh, and then
that's the eighth.

00:03:03.790 --> 00:03:08.660
You have eight groups of 1/2,
so this is equal to 8.

00:03:08.660 --> 00:03:12.920
And notice, now each of the
objects became two groups.

00:03:12.920 --> 00:03:14.660
So you could say how many
groups do you have?

00:03:14.660 --> 00:03:16.860
Well, you have four objects
and each of

00:03:16.860 --> 00:03:21.290
them became two groups.

00:03:21.290 --> 00:03:22.270
I'm looking for a
different color.

00:03:22.270 --> 00:03:24.520
Each of them became
two groups, and so

00:03:24.520 --> 00:03:26.840
you also have eight.

00:03:26.840 --> 00:03:30.710
So dividing by 1/2 is the same
thing as multiplying by 2.

00:03:30.710 --> 00:03:32.250
And you could think about it
with other numbers, but

00:03:32.250 --> 00:03:35.020
hopefully, that gives you a
little bit of an intuition.


Reciprocal of a mixed number

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.960
Find the reciprocal
of 2 and 4/5.

00:00:03.960 --> 00:00:06.840
So if we want to invert this
fraction, we have to write it

00:00:06.840 --> 00:00:09.780
as a pure fraction first, or
as an improper fraction.

00:00:09.780 --> 00:00:11.610
Right now, it's just
a mixed number.

00:00:11.610 --> 00:00:13.530
So let's just write it as
an improper fraction.

00:00:13.530 --> 00:00:20.710
So 2 and 4/5, if we write it as
an improper fraction, it's

00:00:20.710 --> 00:00:24.410
still going to be over 5, but
the numerator is now going to

00:00:24.410 --> 00:00:30.840
be 5 times 2 plus 4, and that's
because 2 is 10/5, or 2

00:00:30.840 --> 00:00:32.619
times 5 fifths.

00:00:32.619 --> 00:00:33.750
So let's write it like that.

00:00:33.750 --> 00:00:40.605
So it's going to be
2 times 5 plus 4.

00:00:43.610 --> 00:00:45.330
2 times 5 is 10.

00:00:45.330 --> 00:00:49.590
10/5 is the same thing as 2, and
then the 4/5 is this 4/5

00:00:49.590 --> 00:00:50.440
right there.

00:00:50.440 --> 00:00:53.790
So this is going to be equal
to 2 times 5 is 10

00:00:53.790 --> 00:00:58.670
plus 4 is 14: 14/5.

00:00:58.670 --> 00:01:00.930
So these are equivalent
fractions.

00:01:00.930 --> 00:01:02.480
This is written as
a mixed number.

00:01:02.480 --> 00:01:04.459
This is written as an
improper fraction.

00:01:04.459 --> 00:01:07.590
It's improper because the
numerator is bigger than the

00:01:07.590 --> 00:01:08.310
denominator.

00:01:08.310 --> 00:01:10.630
Now we want to find
the reciprocal.

00:01:10.630 --> 00:01:13.780
When you find the reciprocal,
you just switch the numerator

00:01:13.780 --> 00:01:14.280
and the denominator.

00:01:14.280 --> 00:01:16.270
You just flip the fraction.

00:01:16.270 --> 00:01:18.940
So if you want to take the
reciprocal-- let me do this in

00:01:18.940 --> 00:01:20.230
a new color.

00:01:20.230 --> 00:01:29.850
If we want to take the
reciprocal of this, we just

00:01:29.850 --> 00:01:32.160
swap the 5 and the 14.

00:01:32.160 --> 00:01:34.600
So this is going to be--
instead of 14/5,

00:01:34.600 --> 00:01:38.080
it's going to be 5/14.

00:01:38.080 --> 00:01:39.630
And we're done!


Dividing mixed numbers and fractions

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.570
Divide.

00:00:01.570 --> 00:00:07.440
Simplify the answer and write
as a mixed number.

00:00:07.440 --> 00:00:11.720
So we have 4 divided by 2/3, and
we've seen when you divide

00:00:11.720 --> 00:00:14.900
by the fraction, it's the same
thing as multiplying by its

00:00:14.900 --> 00:00:15.940
reciprocal.

00:00:15.940 --> 00:00:21.260
So this is going to be the
same thing as 4 times the

00:00:21.260 --> 00:00:25.840
reciprocal of 2/3, which is 3/2,
just swapping the bottom

00:00:25.840 --> 00:00:29.050
and the top number, the
denominator and the numerator.

00:00:29.050 --> 00:00:32.220
Now, 4, we've seen this before,
where we're saying,

00:00:32.220 --> 00:00:34.330
OK, well, we know how to
multiply fractions.

00:00:34.330 --> 00:00:36.660
You multiply the numerators
times each other and the

00:00:36.660 --> 00:00:38.170
denominators times
each other, but 4

00:00:38.170 --> 00:00:39.410
doesn't look like a fraction.

00:00:39.410 --> 00:00:41.880
Well, you always have to
remember: a whole number can

00:00:41.880 --> 00:00:44.650
be written as that same
whole number over one.

00:00:44.650 --> 00:00:47.610
4 is the same thing as 4/1.

00:00:47.610 --> 00:00:51.740
And now this becomes kind of
a plain vanilla multiplying

00:00:51.740 --> 00:00:53.140
fractions problem.

00:00:53.140 --> 00:00:54.710
And so we can just
multiply it out.

00:00:54.710 --> 00:00:57.900
We could just say this is 4
times 3, which would be 12

00:00:57.900 --> 00:01:01.040
over 1 times 2, which
would be 2.

00:01:01.040 --> 00:01:03.390
Or we could divide right now.

00:01:03.390 --> 00:01:05.190
Well, let's do it both
ways, actually.

00:01:05.190 --> 00:01:07.320
So we could say this is 4 times
3 in the numerator,

00:01:07.320 --> 00:01:11.800
which is 12, over 1 times 2 in
the denominator, which is 2.

00:01:11.800 --> 00:01:14.150
Now, what is 12 divided by 2?

00:01:14.150 --> 00:01:17.780
12/2 is the same thing
as 12 divided by 2.

00:01:17.780 --> 00:01:21.310
Well, this is equal to 6.

00:01:21.310 --> 00:01:24.740
Or another way we could think
about it is divide the

00:01:24.740 --> 00:01:26.230
numerator by 2.

00:01:26.230 --> 00:01:27.420
Well, that would give you 6.

00:01:27.420 --> 00:01:29.310
And divide the denominator
by 2.

00:01:29.310 --> 00:01:30.340
That'll give you 1.

00:01:30.340 --> 00:01:33.900
6/1, which is the
same thing as 6.

00:01:33.900 --> 00:01:35.720
Now, the other thing we could've
done at this stage

00:01:35.720 --> 00:01:37.870
right here is say, well, look,
we have things in the

00:01:37.870 --> 00:01:40.570
numerator divisible by 2, we
have things in the denominator

00:01:40.570 --> 00:01:43.480
divisible by 2, so let's
divide both by 2.

00:01:43.480 --> 00:01:45.960
So 4 divided by 2 is 2.

00:01:45.960 --> 00:01:47.770
2 divided by 2 is 1.

00:01:47.770 --> 00:01:53.930
So it becomes 2 times 3, which
is 6, over 1 times 1 over 1,

00:01:53.930 --> 00:01:56.220
or this is equal to 6.

00:01:56.220 --> 00:01:58.520
Now, this could be a fun one
to visualize so let's think

00:01:58.520 --> 00:01:59.490
about what this is.

00:01:59.490 --> 00:02:01.780
4 divided by 2/3.

00:02:01.780 --> 00:02:05.270
Let's draw four boxes here.

00:02:05.270 --> 00:02:05.890
Let me do this.

00:02:05.890 --> 00:02:10.979
So let's say I have one,
two-- let me separate

00:02:10.979 --> 00:02:11.460
them a little bit.

00:02:11.460 --> 00:02:17.090
One, two, three, and four.

00:02:17.090 --> 00:02:20.120
And you can view 4 divided by
2/3 as let's divide these four

00:02:20.120 --> 00:02:23.570
boxes into groups of 2/3 of a
box and figure out how many

00:02:23.570 --> 00:02:24.910
groups do we have.

00:02:24.910 --> 00:02:26.390
So where are our groups?

00:02:26.390 --> 00:02:29.800
So let's see the first group is
going to have 2/3 of a box

00:02:29.800 --> 00:02:32.320
in it, so the first group is
going to look like this.

00:02:32.320 --> 00:02:35.140
It's going to have
2/3 of a box.

00:02:35.140 --> 00:02:37.670
Now, the next group is going to
have 2/3 of a box, so it'll

00:02:37.670 --> 00:02:40.560
have 1/3 from here and
then it'll have 1/3

00:02:40.560 --> 00:02:42.090
from the next box.

00:02:42.090 --> 00:02:44.160
The next group is going to
have 2/3 of the box.

00:02:44.160 --> 00:02:46.770
There's 2/3 left right
over here.

00:02:46.770 --> 00:02:50.520
Then the group after that will
have another 2/3 of a box.

00:02:50.520 --> 00:02:52.870
That is that right over there.

00:02:52.870 --> 00:02:55.450
Then the next will have 1/3 from
here and 1/3 from here,

00:02:55.450 --> 00:02:58.510
so it'll have 1/3 from here
and 1/3 from here.

00:02:58.510 --> 00:02:59.630
That'll be 2/3.

00:02:59.630 --> 00:03:02.170
And then the last group will
have 2/3, the leftover on this

00:03:02.170 --> 00:03:03.500
box right here.

00:03:03.500 --> 00:03:06.150
Now, how many groups
of 2/3 do we have?

00:03:06.150 --> 00:03:13.190
We have one, two, three, four,
five, six, so we have six.

00:03:13.190 --> 00:03:16.190
Four can be divided into
six groups of 2/3.

00:03:16.190 --> 00:03:18.290
So it's just like regular
division, sometimes a little

00:03:18.290 --> 00:03:20.970
harder to visualize because
we're dealing with fractions.


Dividing mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:01.230
Divide.

00:00:01.230 --> 00:00:04.770
Simplify the answer and write
as a mixed number.

00:00:04.770 --> 00:00:08.230
And we have 2 and 1/4 divided
by 1 and 3/4.

00:00:08.230 --> 00:00:10.150
So the first thing we want to
do since both of these are

00:00:10.150 --> 00:00:12.260
mixed numbers is to
convert them both

00:00:12.260 --> 00:00:13.500
into improper fractions.

00:00:13.500 --> 00:00:15.740
So let's start with 2 and 1/4.

00:00:15.740 --> 00:00:18.850
So we're still going to have
4 in the denominator, but

00:00:18.850 --> 00:00:21.030
instead of 2 and 1/4,
remember, 2 is the

00:00:21.030 --> 00:00:22.680
same thing as 8/4.

00:00:22.680 --> 00:00:25.340
So we have 8/4, and then
we have another 1/4.

00:00:25.340 --> 00:00:27.230
That gives us 9/4.

00:00:27.230 --> 00:00:31.900
Or another way to come up with
this 9, you take 4 times 2,

00:00:31.900 --> 00:00:33.990
which is 8, plus 1.

00:00:33.990 --> 00:00:35.370
That gives you 9.

00:00:35.370 --> 00:00:38.720
And then the 1 and 3/4,
same process.

00:00:38.720 --> 00:00:41.880
You're going to have 4 in the
denominator, and then the

00:00:41.880 --> 00:00:46.780
numerator is going to be 4 times
1, which is 4, plus 3,

00:00:46.780 --> 00:00:48.450
which is 7.

00:00:48.450 --> 00:00:50.990
So this is the exact
same problem here.

00:00:50.990 --> 00:00:55.370
2 and 1/4 divided by 1 and 3/4
is the same thing as 9/4

00:00:55.370 --> 00:00:57.860
divided by 7/4.

00:00:57.860 --> 00:01:01.200
And we saw in several videos
already that dividing by a

00:01:01.200 --> 00:01:05.000
fraction is the same thing as
multiplying by its reciprocal.

00:01:05.000 --> 00:01:09.900
So this is equivalent to-- so
these are all equivalent.

00:01:09.900 --> 00:01:15.960
This is equivalent to 9/4 times
the reciprocal of this.

00:01:15.960 --> 00:01:18.833
We're changing the division
operation to a multiplication,

00:01:18.833 --> 00:01:21.390
and we're taking the reciprocal
of the 7/4.

00:01:21.390 --> 00:01:24.490
For the reciprocal of 7/4, you
swap the numerator and

00:01:24.490 --> 00:01:26.100
denominator, or the top
number and the bottom

00:01:26.100 --> 00:01:29.730
number, and you get 4/7.

00:01:29.730 --> 00:01:31.950
Now, we could just
multiply these.

00:01:31.950 --> 00:01:35.380
We could just say this is 9
times 4, which would be 36,

00:01:35.380 --> 00:01:38.590
over 4 times 7, which is 28,
and then try to put it in

00:01:38.590 --> 00:01:40.900
lowest terms, or we could do it
right now because it would

00:01:40.900 --> 00:01:41.290
be simpler.

00:01:41.290 --> 00:01:42.820
We have a 4 in the numerator.

00:01:42.820 --> 00:01:44.560
We have a 4 in the denominator,
that'll

00:01:44.560 --> 00:01:47.080
eventually be in the
denominator, so let's divide

00:01:47.080 --> 00:01:49.280
our eventual numerators
and our eventual

00:01:49.280 --> 00:01:50.960
denominators both by 4.

00:01:50.960 --> 00:01:54.040
So you divide this 4
by 4, you get 1.

00:01:54.040 --> 00:01:56.070
This 4 by 4, you get 1.

00:01:56.070 --> 00:01:59.990
So now when you multiply it, you
get 9 times 1, which is 9,

00:01:59.990 --> 00:02:03.660
over 1 times 7, which is 7.

00:02:03.660 --> 00:02:05.680
So we have our answer,
but right now,

00:02:05.680 --> 00:02:07.150
it's an improper fraction.

00:02:07.150 --> 00:02:10.160
They want us to write it
as a mixed number.

00:02:10.160 --> 00:02:12.560
And to figure it out as a mixed
number, we can do it in

00:02:12.560 --> 00:02:13.160
our heads now.

00:02:13.160 --> 00:02:15.160
I think we've seen this
enough times.

00:02:15.160 --> 00:02:20.190
We say how many times
does 7 go into 9?

00:02:20.190 --> 00:02:24.020
Well, it goes into it exactly
one time, but when you take 7

00:02:24.020 --> 00:02:26.720
into 9 one time, what do
you have left over?

00:02:26.720 --> 00:02:28.190
Well, you're going to have
2 left over, right?

00:02:28.190 --> 00:02:31.650
7 times 1 is 7, and you're going
to have 2 left over.

00:02:31.650 --> 00:02:33.015
You need 2 more to get to 9.

00:02:33.015 --> 00:02:37.920
So you're going to have 2 left
over, so this is 1 and 2/7.

00:02:37.920 --> 00:02:39.520
And we're done!


Dividing whole numbers and fractions: t-shirts

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:06.150
A baby's T-shirt requires 4/5
yards of fabric, or 4/5 of a

00:00:06.150 --> 00:00:07.470
yard of fabric.

00:00:07.470 --> 00:00:11.190
How many T-shirts can be
made from 48 yards?

00:00:11.190 --> 00:00:13.250
So what we want to do is we
essentially want to say how

00:00:13.250 --> 00:00:18.430
many groups of 4/5 of a yard
can we make with 48 yards?

00:00:18.430 --> 00:00:24.680
So you literally view this as
we want to take 48 yards and

00:00:24.680 --> 00:00:29.810
divide it into groups of 4/5
of a yard, and say how many

00:00:29.810 --> 00:00:30.610
groups are there?

00:00:30.610 --> 00:00:32.430
Because each of those
groups can make

00:00:32.430 --> 00:00:33.740
one to baby's T-shirt.

00:00:33.740 --> 00:00:37.110
If you give me 4/5 of a yard,
one baby's T-shirt, so the

00:00:37.110 --> 00:00:40.720
number of groups of 4/5 is the
number of babies' T-shirts.

00:00:40.720 --> 00:00:43.340
Now, when we divide by a
fraction, we just have to

00:00:43.340 --> 00:00:45.970
remember that that is the same
thing, that is completely

00:00:45.970 --> 00:00:51.030
equivalent to multiplying
times the

00:00:51.030 --> 00:00:52.365
reciprocal of the fraction.

00:00:52.365 --> 00:00:56.700
So if we have 4/5 here,
that'll be 5/4, the

00:00:56.700 --> 00:00:57.660
reciprocal.

00:00:57.660 --> 00:00:59.960
Now, you still might say, hey, I
have a whole number here and

00:00:59.960 --> 00:01:02.470
a fraction, and you just have
to remember any whole number

00:01:02.470 --> 00:01:04.099
can be written as a fraction.

00:01:04.099 --> 00:01:11.340
This is the same thing
as 48/1 times 5/4.

00:01:11.340 --> 00:01:14.040
Now, we could just multiply it
out at this point and figure

00:01:14.040 --> 00:01:19.380
out what 48 times 5 is and
that'll be over 4, but that'll

00:01:19.380 --> 00:01:21.650
get big numbers and it'll be
hard to kind of divide and all

00:01:21.650 --> 00:01:23.700
that, but we could divide at
this stage right here.

00:01:23.700 --> 00:01:26.230
We could divide the numerator
and the denominator by 4.

00:01:26.230 --> 00:01:30.240
Or we could say, look, this is
going to be equal to 48 times

00:01:30.240 --> 00:01:34.070
5, whatever that is, over 4.

00:01:34.070 --> 00:01:36.370
Now, let's divide the
numerator by 4.

00:01:36.370 --> 00:01:40.500
Well, we could divide 48 by 4,
and we will get to 12, and

00:01:40.500 --> 00:01:42.090
whatever we did to the
numerator, we have to do to

00:01:42.090 --> 00:01:45.750
the denominator, so if we
divide 4 by 4, we get 1.

00:01:45.750 --> 00:01:51.090
So then we're left with 12 times
5, which is equal to 60.

00:01:51.090 --> 00:01:56.820
12 times 5, which is equal to
60/1, which is the exact same

00:01:56.820 --> 00:01:58.540
thing as 60.

00:01:58.540 --> 00:02:02.960
So you can actually make 60
children's or babies' T-shirts

00:02:02.960 --> 00:02:07.300
from 48 yards if each of
them use 4/5 of a yard.


Multiplying two fractions: example

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:04.340
We're asked to multiply 5/6
times 2/3 and then simplify

00:00:04.340 --> 00:00:05.570
our answer.

00:00:05.570 --> 00:00:07.450
So let's just multiply
these two numbers.

00:00:07.450 --> 00:00:13.090
So we have 5/6 times 2/3.

00:00:13.090 --> 00:00:15.030
Now when you're multiplying
fractions, it's actually a

00:00:15.030 --> 00:00:17.470
pretty straightforward
process.

00:00:17.470 --> 00:00:20.190
The new numerator, or the
numerator of the product, is

00:00:20.190 --> 00:00:22.880
just the product of the two
numerators, or your new top

00:00:22.880 --> 00:00:25.340
number is a product of the
other two top numbers.

00:00:25.340 --> 00:00:29.240
So the numerator in our product
is just 5 times 2.

00:00:29.240 --> 00:00:37.250
So it's equal to 5 times 2 over
6 times 3, which is equal

00:00:37.250 --> 00:00:43.490
to-- 5 times 2 is 10 and
6 times 3 is 18, so

00:00:43.490 --> 00:00:44.710
it's equal to 10/18.

00:00:44.710 --> 00:00:50.820
And you could view this as
either 2/3 of 5/6 or 5/6 of

00:00:50.820 --> 00:00:53.640
2/3, depending on how you
want to think about it.

00:00:53.640 --> 00:00:54.750
And this is the right answer.

00:00:54.750 --> 00:00:57.220
It is 10/18, but when you look
at these two numbers, you

00:00:57.220 --> 00:00:59.460
immediately or you might
immediately see that they

00:00:59.460 --> 00:01:01.500
share some common factors.

00:01:01.500 --> 00:01:03.990
They're both divisible by 2,
so if we want it in lowest

00:01:03.990 --> 00:01:07.020
terms, we want to divide
them both by 2.

00:01:07.020 --> 00:01:12.800
So divide 10 by 2, divide 18 by
2, and you get 10 divided

00:01:12.800 --> 00:01:17.510
by 2 is 5, 18 divided
by 2 is 9.

00:01:17.510 --> 00:01:19.920
Now, you could have essentially
done this step

00:01:19.920 --> 00:01:20.630
earlier on.

00:01:20.630 --> 00:01:22.530
You could've done it actually
before we did the

00:01:22.530 --> 00:01:23.220
multiplication.

00:01:23.220 --> 00:01:24.450
You could've done
it over here.

00:01:24.450 --> 00:01:26.450
You could've said, well, I have
a 2 in the numerator and

00:01:26.450 --> 00:01:29.260
I have something divisible by 2
into the denominator, so let

00:01:29.260 --> 00:01:32.710
me divide the numerator by
2, and this becomes a 1.

00:01:32.710 --> 00:01:37.090
Let me divide the denominator
by 2, and this becomes a 3.

00:01:37.090 --> 00:01:42.070
And then you have 5 times 1
is 5, and 3 times 3 is 9.

00:01:42.070 --> 00:01:44.200
So it's really the same thing
we did right here.

00:01:44.200 --> 00:01:47.370
We just did it before we
actually took the product.

00:01:47.370 --> 00:01:49.220
You could actually
do it right here.

00:01:49.220 --> 00:01:53.890
So if you did it right over
here, you'd say, well, look, 6

00:01:53.890 --> 00:01:56.190
times 3 is eventually going
to be the denominator.

00:01:56.190 --> 00:02:00.030
5 times 2 is eventually going
to be the numerator.

00:02:00.030 --> 00:02:03.660
So let's divide the numerator by
2, so this will become a 1.

00:02:03.660 --> 00:02:05.180
Let's divide the denominator
by 2.

00:02:05.180 --> 00:02:07.550
This is divisible by 2,
so that'll become a 3.

00:02:07.550 --> 00:02:13.630
And it'll become 5 times 1
is 5 and 3 times 3 is 9.

00:02:13.630 --> 00:02:15.210
So either way you do
it, it'll work.

00:02:15.210 --> 00:02:18.450
If you do it this way, you get
to see the things factored out

00:02:18.450 --> 00:02:20.910
a little bit more, so it's
usually easier to recognize

00:02:20.910 --> 00:02:23.200
what's divisible by what, or you
could do it at the end and

00:02:23.200 --> 00:02:25.400
put things in lowest terms.


Multiplying fractions and whole numbers: mixed number answer

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.310
Multiply 6 times 1/4.

00:00:03.310 --> 00:00:07.160
Simplify your answer and write
it as a mixed number.

00:00:07.160 --> 00:00:09.170
So let's just do the
multiplication.

00:00:09.170 --> 00:00:12.470
So at first when you try to
multiply 6 times 1/4, you'll

00:00:12.470 --> 00:00:14.340
be like, well, gee, I know
how to multiply a

00:00:14.340 --> 00:00:15.720
fraction times a fraction.

00:00:15.720 --> 00:00:18.110
I know how to multiply a whole
number times a whole number,

00:00:18.110 --> 00:00:20.580
but what about a whole number
times a fraction?

00:00:20.580 --> 00:00:23.110
And kind of the key insight
you need here is that any

00:00:23.110 --> 00:00:25.430
whole number can be written
as a fraction.

00:00:25.430 --> 00:00:31.720
We can rewrite 6
as 6/1, right?

00:00:31.720 --> 00:00:34.020
6 divided by 1 is 6.

00:00:34.020 --> 00:00:36.220
6 ones is 6.

00:00:36.220 --> 00:00:38.360
Depending how you think about
it, this is exactly the same

00:00:38.360 --> 00:00:39.720
thing as 6.

00:00:39.720 --> 00:00:42.210
So we just rewrote our whole
number as a fraction.

00:00:42.210 --> 00:00:43.320
You can do it for any number.

00:00:43.320 --> 00:00:45.610
10 is the same thing as 10/1.

00:00:45.610 --> 00:00:54.930
So this become 6/1 times
1/4, and then we just

00:00:54.930 --> 00:00:56.160
multiply the fraction.

00:00:56.160 --> 00:01:00.740
We multiply the numerators, so
this is equal to 6 times 1 as

00:01:00.740 --> 00:01:01.590
our numerator.

00:01:01.590 --> 00:01:03.519
Let me do that in
another color.

00:01:03.519 --> 00:01:08.270
So this becomes 6 times 1 for
our numerator and 1 times 4

00:01:08.270 --> 00:01:11.190
for our denominator, for the
number on the bottom.

00:01:11.190 --> 00:01:14.260
And so this will become
6 over 4.

00:01:14.260 --> 00:01:17.000
And right now, it's just as an
improper fraction and it's

00:01:17.000 --> 00:01:19.900
also not in lowest terms. You
immediately see 6 and 4 are

00:01:19.900 --> 00:01:23.190
both divisible by 2, so let's
divide them both by 2.

00:01:23.190 --> 00:01:25.334
If you divide 6 by 2--
and I'll do it

00:01:25.334 --> 00:01:26.370
in a new color again.

00:01:26.370 --> 00:01:29.515
If you divide 6 by
2, you get 3.

00:01:29.515 --> 00:01:35.290
If you divide 4 by 2, you get
2, so this is equal to 3/2.

00:01:35.290 --> 00:01:37.300
So it's still written as
an improper fraction.

00:01:37.300 --> 00:01:39.570
We now have to write it
as a mixed a number.

00:01:39.570 --> 00:01:42.350
And the process for writing it
as a mixed number, you just

00:01:42.350 --> 00:01:45.820
divide the denominator into the
numerator, so this just

00:01:45.820 --> 00:01:50.920
becomes 2 into 3.

00:01:50.920 --> 00:01:53.820
Divide 2 into 3.

00:01:53.820 --> 00:01:55.560
2 goes into 3 one time.

00:01:55.560 --> 00:01:56.880
1 times 2 is 2.

00:01:56.880 --> 00:01:57.930
You subtract.

00:01:57.930 --> 00:02:00.680
You have a remainder of 1.

00:02:00.680 --> 00:02:10.474
So this will become one whole
and 1/2 left over.

00:02:14.760 --> 00:02:16.610
So this is 1 and 1/2.

00:02:16.610 --> 00:02:17.900
So that's our right answer.

00:02:17.900 --> 00:02:20.040
We've just simplified the answer
and wrote it is a mixed

00:02:20.040 --> 00:02:22.810
number or we could simplify
it at this stage.

00:02:22.810 --> 00:02:25.830
We could say right here, well,
look, we could divide what's

00:02:25.830 --> 00:02:29.310
eventually going to be in the
numerator by 2 and get a 3

00:02:29.310 --> 00:02:31.660
there, and divide what's
eventually in the denominator

00:02:31.660 --> 00:02:33.990
by 2 and get a 2 there.

00:02:33.990 --> 00:02:36.370
3 times 1 is 3.

00:02:36.370 --> 00:02:38.985
1 times 2 is 2, so it's 3/2.

00:02:38.985 --> 00:02:40.830
And you do this exact
same process.

00:02:40.830 --> 00:02:44.040
You say that 3/2 is the same
thing as 1 and 1/2.

00:02:44.040 --> 00:02:47.120
Either one of those will work.

00:02:47.120 --> 00:02:48.910
Now let's think about why
this makes sense.

00:02:48.910 --> 00:02:52.500
Let's think about what
6 times 1/4 is.

00:02:52.500 --> 00:02:54.410
Let me draw 1/4.

00:02:54.410 --> 00:02:57.130
Let's say that that is
1/4 right there, and

00:02:57.130 --> 00:02:58.120
let's do six of them.

00:02:58.120 --> 00:03:04.450
So that's 1/4, that's 2/4 that's
3/4, that's 4/4, which

00:03:04.450 --> 00:03:08.150
would be a whole, and then
you have 5/4, and

00:03:08.150 --> 00:03:09.320
then you have 6/4.

00:03:09.320 --> 00:03:10.710
So this is 6 times 1/4.

00:03:13.580 --> 00:03:15.590
This right here is 4/4.

00:03:15.590 --> 00:03:21.090
This right here is 4 over 4,
which is equal to a 1, so this

00:03:21.090 --> 00:03:22.600
is equal to 1.

00:03:22.600 --> 00:03:25.580
And then this right here
is two 1/4's, or this

00:03:25.580 --> 00:03:28.710
right here is 2/4.

00:03:28.710 --> 00:03:31.600
You can imagine this is two out
of a potential, if a whole

00:03:31.600 --> 00:03:34.300
has to have two more of them,
has to have four of them. so

00:03:34.300 --> 00:03:42.860
this is 1 and-- let me do it
in the same colors-- 1 and

00:03:42.860 --> 00:03:45.270
1/2, right?

00:03:45.270 --> 00:03:49.720
2 out of 4 is the same thing as
1/2, so this right here is

00:03:49.720 --> 00:03:55.930
one out of a possible
one, and then two.

00:03:55.930 --> 00:04:02.990
So this is 1 and 1/2, which is
exactly what we got before.


How to multiply mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.550
Multiply 1 and 3/4
times 7 and 1/5.

00:00:04.550 --> 00:00:08.760
Simplify your answer and write
it as a mixed fraction.

00:00:08.760 --> 00:00:12.080
So the first thing we want to
do is rewrite each of these

00:00:12.080 --> 00:00:14.990
mixed numbers as improper
fractions.

00:00:14.990 --> 00:00:17.790
It's very difficult, or at least
it's not easy for me, to

00:00:17.790 --> 00:00:19.680
directly multiply
mixed numbers.

00:00:19.680 --> 00:00:22.560
One can do it, but it's much
easier if you just make them

00:00:22.560 --> 00:00:24.180
improper fractions.

00:00:24.180 --> 00:00:25.620
So let's convert each of them.

00:00:25.620 --> 00:00:32.380
So 1 and 3/4 is equal to-- it's
still going to be over 4,

00:00:32.380 --> 00:00:34.810
so you're still going to have
the same denominator, but your

00:00:34.810 --> 00:00:37.710
numerator as an improper
fraction is going to be 4

00:00:37.710 --> 00:00:39.615
times 1 plus 3.

00:00:43.390 --> 00:00:48.750
And the reason why this makes
sense is 1 is 4/4, or 1 is 4

00:00:48.750 --> 00:00:50.790
times 1 fourths, right?

00:00:50.790 --> 00:00:53.480
1 is the same thing as 4/4, and
then you have three more

00:00:53.480 --> 00:00:56.670
fourths, so 4/4 plus 3/4
will give you 7/4.

00:00:59.760 --> 00:01:02.670
So that's the same thing
as 1 and 3/4.

00:01:02.670 --> 00:01:04.319
Now, let's do 7 and 1/5.

00:01:09.590 --> 00:01:12.580
Same exact process.

00:01:12.580 --> 00:01:14.860
We're going to still be talking
in terms of fifths.

00:01:14.860 --> 00:01:16.710
That's going to be
the denominator.

00:01:16.710 --> 00:01:19.600
You take 5 times 7, because
think about it.

00:01:19.600 --> 00:01:22.960
7 is the same thing as 35/5.

00:01:22.960 --> 00:01:29.150
So you take 5 times 7 plus this
numerator right here.

00:01:29.150 --> 00:01:32.670
So 7 is 35/5, then you have
one more fifth, so you're

00:01:32.670 --> 00:01:38.870
going to have 35 plus 1,
which is equal to 36/5.

00:01:38.870 --> 00:01:42.320
So this product is the exact
same thing as taking the

00:01:42.320 --> 00:01:51.350
product of 7/4 times 36/5.

00:01:51.350 --> 00:01:53.220
And we could multiply
it out right now.

00:01:53.220 --> 00:01:57.100
Take the 7 times 36 as our new
numerator, 4 times 5 as our

00:01:57.100 --> 00:01:59.780
new denominator, but that'll
give us large numbers.

00:01:59.780 --> 00:02:02.370
I can't multiply 7 and
36 in my head, or I

00:02:02.370 --> 00:02:03.890
can't do it too easily.

00:02:03.890 --> 00:02:06.590
So let's see if we can
simplify this first.

00:02:06.590 --> 00:02:09.320
Both our numerator and our
denominator have numbers that

00:02:09.320 --> 00:02:12.220
are divisible by 4, so let's
divide both the numerator and

00:02:12.220 --> 00:02:13.740
the denominator by 4.

00:02:13.740 --> 00:02:20.100
So in the numerator, we can
divide the 36 by 4 and get 9.

00:02:20.100 --> 00:02:22.360
If you divide something in the
numerator by 4, you need to

00:02:22.360 --> 00:02:25.020
divide something in the
denominator by 4, and the 4 is

00:02:25.020 --> 00:02:28.890
the obvious guy, so 4
divided by 4 is 1.

00:02:28.890 --> 00:02:33.700
So now this becomes 7 times 9,
and what's the 7 times 9?

00:02:33.700 --> 00:02:39.900
It's 63, over 1 times 5.

00:02:39.900 --> 00:02:42.920
So now we have our answer as an
improper fraction, but they

00:02:42.920 --> 00:02:46.020
want it as a mixed number
or as a mixed fraction.

00:02:46.020 --> 00:02:48.120
So what are 63/5?

00:02:48.120 --> 00:02:53.120
So to figure that out-- let me
pick a nice color here-- we

00:02:53.120 --> 00:02:57.260
take 5 into 63.

00:02:57.260 --> 00:02:59.420
5 goes into 6 one time.

00:02:59.420 --> 00:03:01.320
1 times 5 is 5.

00:03:01.320 --> 00:03:02.260
You subtract.

00:03:02.260 --> 00:03:06.260
6 minus 5 is 1.

00:03:06.260 --> 00:03:08.330
Bring down the 3.

00:03:08.330 --> 00:03:11.220
5 goes into 13 two times.

00:03:11.220 --> 00:03:14.160
And you could have immediately
said 5 goes into 63 twelve

00:03:14.160 --> 00:03:17.480
times, but this way, at
least to me, it's a

00:03:17.480 --> 00:03:18.960
little bit more obvious.

00:03:18.960 --> 00:03:27.305
And then 2 times 5 is 12,
and then we have sorry!

00:03:27.305 --> 00:03:32.390
2 times 5 is 10.

00:03:32.390 --> 00:03:33.660
That tells you not to
switch gears in the

00:03:33.660 --> 00:03:34.560
middle of a math problem.

00:03:34.560 --> 00:03:37.600
2 times 5 is 10, and then you
subtract, and you have a

00:03:37.600 --> 00:03:39.165
remainder of 3.

00:03:42.040 --> 00:03:51.070
So 63/5 is the same thing as 12
wholes and 3 left over, or

00:03:51.070 --> 00:03:55.800
3/5 left over.

00:03:55.800 --> 00:03:58.460
And if you wanted to go back
from this to that, just think:

00:03:58.460 --> 00:04:03.030
12 is the same thing as
60 fifths, or 60/5.

00:04:03.030 --> 00:04:07.950
60/5 plus 3/5 is 63/5,
so these two things

00:04:07.950 --> 00:04:09.720
are the same thing.

00:04:09.720 --> 00:04:10.810
These two things
are equivalent.

00:04:10.810 --> 00:04:12.980
This is as an improper
fraction.

00:04:12.980 --> 00:04:16.390
This is as a mixed number
or a mixed fraction.

00:04:16.390 --> 00:04:21.130
But this is our answer right
there: 12 and 3/5.


Multiplying fractions word problem: banana oat muffin recipe

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:05.830
A recipe for banana oat muffins
calls for 3/4 of a cup

00:00:05.830 --> 00:00:07.820
of old-fashioned oats.

00:00:07.820 --> 00:00:10.740
You are making 1/2
of the recipe.

00:00:10.740 --> 00:00:13.230
How much oats should you use?

00:00:13.230 --> 00:00:16.550
So if the whole recipe requires
3/4 of a cup and

00:00:16.550 --> 00:00:18.630
you're making half of
the recipe, you want

00:00:18.630 --> 00:00:20.520
half of 3/4, right?

00:00:20.520 --> 00:00:23.100
You want half of the number of
old-fashioned oats as the

00:00:23.100 --> 00:00:24.330
whole recipe.

00:00:24.330 --> 00:00:28.440
So you want 1/2 of 3/4.

00:00:31.010 --> 00:00:35.860
So you just multiply 1/2 times
3/4, and this is equal to--

00:00:35.860 --> 00:00:37.550
you multiply the numerators.

00:00:37.550 --> 00:00:40.590
1 times 3 is 3.

00:00:40.590 --> 00:00:42.960
2 times 4 is 8.

00:00:42.960 --> 00:00:43.820
And we're done!

00:00:43.820 --> 00:00:51.885
You need 3/8 of a cup of
old-fashioned oats.

00:00:51.885 --> 00:00:54.220
And let's visualize that a
little bit, just so it makes a

00:00:54.220 --> 00:00:55.040
little bit more sense.

00:00:55.040 --> 00:00:59.560
Let me draw what 3/4 looks like,
or essentially how much

00:00:59.560 --> 00:01:02.400
oats you would need in a normal
situation, or if you're

00:01:02.400 --> 00:01:04.060
doing the whole recipe.

00:01:04.060 --> 00:01:06.950
So let me draw.

00:01:06.950 --> 00:01:11.270
Let's say this represents a
whole cup, and if we put it

00:01:11.270 --> 00:01:15.040
into fourths-- let me divide
it a little bit better.

00:01:15.040 --> 00:01:23.690
So if we put it into fourths,
3/4 would represent three of

00:01:23.690 --> 00:01:28.290
these, so it would represent
one, two, three.

00:01:28.290 --> 00:01:30.740
It would represent
that many oats.

00:01:30.740 --> 00:01:33.240
Now, you want half
of this, right?

00:01:33.240 --> 00:01:35.110
Because you're going to make
half of the recipe.

00:01:35.110 --> 00:01:39.320
So we can just split
this in half.

00:01:39.320 --> 00:01:41.700
Let me do this with
a new color.

00:01:41.700 --> 00:01:44.600
So you would normally use this
orange amount of oats, but

00:01:44.600 --> 00:01:46.270
we're going to do half the
recipe, so you'd want

00:01:46.270 --> 00:01:47.645
half as many oats.

00:01:50.440 --> 00:01:54.930
So you would want
this many oats.

00:01:54.930 --> 00:01:56.790
Now, let's think about
what that is

00:01:56.790 --> 00:01:59.190
relative to a whole cup.

00:01:59.190 --> 00:02:02.520
Well, one way we can do it is
to turn each of these four

00:02:02.520 --> 00:02:05.360
buckets, or these four pieces,
or these four sections of a

00:02:05.360 --> 00:02:07.630
cup into eight sections
of a cup.

00:02:07.630 --> 00:02:09.000
Let's see what happens
when we do that.

00:02:09.000 --> 00:02:13.260
So we're essentially turning
each piece, each fourth, into

00:02:13.260 --> 00:02:14.530
two pieces.

00:02:14.530 --> 00:02:16.510
So let's divide each
of them into two.

00:02:16.510 --> 00:02:19.340
So this is the first piece.

00:02:19.340 --> 00:02:22.810
We're going to divide it into
two right there, so now it is

00:02:22.810 --> 00:02:24.250
two pieces.

00:02:24.250 --> 00:02:26.300
And then this is the second
piece right here.

00:02:26.300 --> 00:02:30.390
We divide it into one piece
and then two pieces.

00:02:30.390 --> 00:02:32.980
This is the third piece, so
we divide it into one, two

00:02:32.980 --> 00:02:35.805
pieces, and this is the fourth
piece, or the fourth section,

00:02:35.805 --> 00:02:37.950
and we divide it into
two sections.

00:02:37.950 --> 00:02:45.490
Now, what is this as a fraction
of the whole?

00:02:45.490 --> 00:02:47.840
Well, we have eight
pieces now, right?

00:02:47.840 --> 00:02:50.810
One, two, three, four, five,
six, seven, eight, because we

00:02:50.810 --> 00:02:53.380
turned each of the four, we
split them again into eight,

00:02:53.380 --> 00:02:56.650
so we have 8 as the denominator,
and we took half

00:02:56.650 --> 00:02:57.980
of the 3/4, right?

00:02:57.980 --> 00:03:00.430
Remember, 3/4 was in orange.

00:03:00.430 --> 00:03:02.170
Let me make this very
clear because this

00:03:02.170 --> 00:03:04.110
drawing can get confusing.

00:03:04.110 --> 00:03:06.280
This was 3/4 right there.

00:03:06.280 --> 00:03:09.090
So that is 3/4.

00:03:09.090 --> 00:03:19.620
This area in this purple color
is 1/2 of the 3/4.

00:03:19.620 --> 00:03:22.020
But let's think about it
in terms of the eights.

00:03:22.020 --> 00:03:24.180
How many of these sections
of eight is it?

00:03:24.180 --> 00:03:26.910
Well, you have one section of
eight here, two sections of

00:03:26.910 --> 00:03:30.820
eight there, three sections
of eight, so it is 3/8.

00:03:30.820 --> 00:03:33.460
So hopefully that makes some
sense or gives you a more

00:03:33.460 --> 00:03:35.260
tangible feel for what
it means when

00:03:35.260 --> 00:03:36.800
you take 1/2 of 3/4.


Recognizing divisibility

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.920
Determine whether 380 is
divisible by 2, 3,

00:00:03.920 --> 00:00:07.210
4, 5, 6, 9 or 10.

00:00:07.210 --> 00:00:09.010
They skipped 7 and 8
so we don't have

00:00:09.010 --> 00:00:10.460
to worry about those.

00:00:10.460 --> 00:00:13.000
So let's think about 2.

00:00:13.000 --> 00:00:16.524
So are we divisible by 2?

00:00:16.524 --> 00:00:20.220
Let me write the 2 here.

00:00:20.220 --> 00:00:22.880
Well, in order for something to
be divisible by 2, it has

00:00:22.880 --> 00:00:28.160
to be an even number, and to be
an even number, your ones

00:00:28.160 --> 00:00:31.560
digit-- so let me rewrite 380.

00:00:31.560 --> 00:00:35.450
To be even, your ones digit
has to be even, so

00:00:35.450 --> 00:00:38.780
this has to be even.

00:00:38.780 --> 00:00:43.270
And for this to be even, it has
to be 0, 2, 4, 6 or 8, and

00:00:43.270 --> 00:00:51.970
this is 0, so 380 is even, which
means it is divisible by

00:00:51.970 --> 00:00:54.360
2, so it works with 2.

00:00:54.360 --> 00:00:55.710
So 2 works out.

00:00:55.710 --> 00:00:58.530
Let's think about the
situation for 3.

00:00:58.530 --> 00:01:01.130
Now, a quick way to think about
3-- so let me write just

00:01:01.130 --> 00:01:06.010
3 question mark-- is to add
the digits of your number.

00:01:06.010 --> 00:01:09.890
And if the sum that you get is
divisible by 3, then you are

00:01:09.890 --> 00:01:10.980
divisible by 3.

00:01:10.980 --> 00:01:12.110
So let's try to do that.

00:01:12.110 --> 00:01:14.900
So 380, let's add the digits.

00:01:14.900 --> 00:01:22.290
3 plus 8 plus 0 is equal to--
3 plus 8 is 11 plus 0,

00:01:22.290 --> 00:01:23.770
so it's just 11.

00:01:23.770 --> 00:01:26.150
And if you have trouble figuring
out whether this is

00:01:26.150 --> 00:01:28.420
divisible by 3, you could then
just add these two numbers

00:01:28.420 --> 00:01:31.240
again, so you can actually add
the 1 plus 1 again, and you

00:01:31.240 --> 00:01:32.480
would get a 2.

00:01:32.480 --> 00:01:36.100
Regardless of whether you look
at the 1 or the 2, neither of

00:01:36.100 --> 00:01:37.730
these are divisible by 3.

00:01:37.730 --> 00:01:45.700
So not divisible by 3, and maybe
in a future video, I'll

00:01:45.700 --> 00:01:48.860
explain why this works, and
maybe you want to think about

00:01:48.860 --> 00:01:50.590
why this works.

00:01:50.590 --> 00:01:56.710
So these aren't divisible by
3, so 380 is not divisible.

00:01:56.710 --> 00:02:05.560
380, not divisible by 3,
so 3 does not work.

00:02:05.560 --> 00:02:07.100
We are not divisible by 3.

00:02:07.100 --> 00:02:12.880
Now, I'll think about the
situation for 4, so we're

00:02:12.880 --> 00:02:14.880
thinking about 4 divisibility.

00:02:14.880 --> 00:02:17.200
So let me write it in orange.

00:02:17.200 --> 00:02:20.430
So we are wondering about 4.

00:02:20.430 --> 00:02:23.930
Now, something you may or may
not already realize is that

00:02:23.930 --> 00:02:26.310
100 is divisible by 4.

00:02:26.310 --> 00:02:27.580
It goes evenly.

00:02:27.580 --> 00:02:30.130
So this is 380.

00:02:30.130 --> 00:02:33.940
So the 300 is divisible by 4, so
we just have to figure out

00:02:33.940 --> 00:02:37.030
whether the leftover, whether
the 80, is divisible by 4.

00:02:37.030 --> 00:02:46.050
Another way to think about it
is are the last two digits

00:02:46.050 --> 00:02:49.220
divisible by 4?

00:02:49.220 --> 00:02:52.410
And this comes from the fact
that 100 is divisible by 4, so

00:02:52.410 --> 00:02:54.870
everything, the hundreds place
or above, it's going to be

00:02:54.870 --> 00:02:55.450
divisible by 4.

00:02:55.450 --> 00:02:57.760
You just have to worry
about the last part.

00:02:57.760 --> 00:03:04.230
So in this situation, is
80 divisible by 4?

00:03:04.230 --> 00:03:05.530
Now, you could eyeball that.

00:03:05.530 --> 00:03:07.760
You could say, well, 8 is
definitely divisible by 4.

00:03:10.260 --> 00:03:11.700
8 divided by 4 is 2.

00:03:11.700 --> 00:03:15.340
80 divided by 4 is 20,
so this works.

00:03:15.340 --> 00:03:16.740
Yes!

00:03:16.740 --> 00:03:17.540
Yes!

00:03:17.540 --> 00:03:20.910
So since 80 is divisible
by 4, 380 is also

00:03:20.910 --> 00:03:24.290
divisible by 4, so 4 works.

00:03:24.290 --> 00:03:25.855
So let's do 5.

00:03:25.855 --> 00:03:27.910
I'll actually scroll
down a little bit.

00:03:27.910 --> 00:03:29.370
Let's try 5.

00:03:29.370 --> 00:03:32.280
So what's the pattern when
something is divisible by 5?

00:03:32.280 --> 00:03:33.580
Let's do the multiple of 5?

00:03:33.580 --> 00:03:38.930
5, 10, 15, 20, 25.

00:03:38.930 --> 00:03:45.270
So if something's divisible by
5-- I could keep going-- that

00:03:45.270 --> 00:03:48.690
means it ends with either
a 5 or a 0, right?

00:03:48.690 --> 00:03:52.215
Every multiple of 5 either has
a 5 or a 0 in the ones place.

00:03:59.230 --> 00:04:03.860
Now 380 has a 0 in the
ones place, so it is

00:04:03.860 --> 00:04:06.040
divisible by 5.

00:04:06.040 --> 00:04:08.800
Now, let's think about
the situation for 6.

00:04:08.800 --> 00:04:11.450
Let's think about what
happens with 6.

00:04:11.450 --> 00:04:13.780
So we want to know are
we divisible by 6?

00:04:13.780 --> 00:04:17.769
So to be divisible by 6, you
have to be divisible by the

00:04:17.769 --> 00:04:19.010
things that make up 6.

00:04:19.010 --> 00:04:23.960
Remember, 6 is equal
to 2 times 3.

00:04:23.960 --> 00:04:26.950
So if you're divisible by 6,
that means you are divisible

00:04:26.950 --> 00:04:30.120
by 2 and you are
divisible by 3.

00:04:30.120 --> 00:04:32.830
If you're divisible by both
2 and 3, you'll be

00:04:32.830 --> 00:04:34.500
divisible by 6.

00:04:34.500 --> 00:04:38.680
Now, 380 is divisible by 2, but
we've already established

00:04:38.680 --> 00:04:40.990
that it is not divisible by 3.

00:04:40.990 --> 00:04:45.780
If it's not divisible by 3, it
cannot be divisible by 6, so

00:04:45.780 --> 00:04:47.780
this gets knocked out.

00:04:47.780 --> 00:04:50.710
We are not divisible by 6.

00:04:50.710 --> 00:04:51.960
Now, let's go to 9.

00:04:54.600 --> 00:04:56.790
So divisibility by 9.

00:04:56.790 --> 00:05:00.100
So you can make a similar
argument here that if

00:05:00.100 --> 00:05:02.950
something is not divisible by
3, there's no way it's going

00:05:02.950 --> 00:05:08.270
to be divisible by 9 because
9 is equal to 3 times 3.

00:05:08.270 --> 00:05:11.940
So to be divisible by 9, you
have to be divisible by 3 at

00:05:11.940 --> 00:05:13.090
least twice.

00:05:13.090 --> 00:05:16.060
At least two 3's have to go
into your number, and this

00:05:16.060 --> 00:05:19.120
isn't the case, so you could
already knock 9 out.

00:05:19.120 --> 00:05:21.870
But if we didn't already know
that we're not divisible by 3,

00:05:21.870 --> 00:05:25.870
the other way to do it is a very
similar way to figure out

00:05:25.870 --> 00:05:28.050
divisibility by 3.

00:05:28.050 --> 00:05:29.670
We can add the digits.

00:05:29.670 --> 00:05:34.300
So you add 3 plus 8 plus
0, and you get 11.

00:05:34.300 --> 00:05:36.710
And you say is this
divisible by 9?

00:05:36.710 --> 00:05:43.170
And you say this is not
divisible by 9, so 380 must

00:05:43.170 --> 00:05:45.550
not be divisible by 9.

00:05:45.550 --> 00:05:47.720
And for 3, you do the same
thing, but you test whether

00:05:47.720 --> 00:05:49.540
the sum is divisible by 3.

00:05:49.540 --> 00:05:52.320
For 9, you test whether
it's divisible by 9.

00:05:52.320 --> 00:05:57.090
So lastly, we have
the number 10.

00:05:57.090 --> 00:05:59.230
We have the number 10, and
this is on some level the

00:05:59.230 --> 00:06:00.110
easiest one.

00:06:00.110 --> 00:06:02.050
What do all the multiples
of 10 look like?

00:06:02.050 --> 00:06:07.190
10, 20, 30, 40, we could just
keep going on and on.

00:06:07.190 --> 00:06:08.790
They all end with zero.

00:06:08.790 --> 00:06:12.030
Or if something ends with zero,
it is divisible by 10.

00:06:12.030 --> 00:06:17.470
380 does end with zero, or its
ones place does have a zero on

00:06:17.470 --> 00:06:20.080
it, so it is divisible by 10.

00:06:20.080 --> 00:06:22.210
So we're divisible by all
of these numbers except

00:06:22.210 --> 00:06:25.050
for 3, 6 and 9.


Numerator and denominator of a fraction

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.725
We're asked to identify the
numerator and denominator in

00:00:04.725 --> 00:00:08.400
the fraction 3 over 4, or 3/4.

00:00:08.400 --> 00:00:11.140
So let's rewrite this just
so it's nice and big.

00:00:11.140 --> 00:00:12.580
So let me just write
the fraction.

00:00:12.580 --> 00:00:20.470
So we have 3 over 4, 3/4.

00:00:20.470 --> 00:00:22.880
Now, they want us to identify
the numerator and the

00:00:22.880 --> 00:00:24.130
denominator.

00:00:25.950 --> 00:00:29.040
So the numerator is just the
number on top, so the

00:00:29.040 --> 00:00:31.600
numerator is the
3 right there.

00:00:31.600 --> 00:00:34.080
And then they want us to
find the denominator.

00:00:34.080 --> 00:00:37.160
The denominator is just the
number on the bottom.

00:00:37.160 --> 00:00:38.050
It's the 4.

00:00:38.050 --> 00:00:39.180
So if they say what's
the numerator?

00:00:39.180 --> 00:00:39.810
3.

00:00:39.810 --> 00:00:41.310
What's the denominator?

00:00:41.310 --> 00:00:43.220
It's 4, just the number
on the bottom.

00:00:43.220 --> 00:00:44.910
They could've just called this
the number on the bottom.

00:00:44.910 --> 00:00:47.340
They could've just called
this the number on top.

00:00:47.340 --> 00:00:49.680
Now to think about what this
represents, what this fraction

00:00:49.680 --> 00:00:52.600
represents, you can think
of it as three out of

00:00:52.600 --> 00:00:54.270
four pieces of a pie.

00:00:54.270 --> 00:00:55.380
That's how I think about it.

00:00:55.380 --> 00:00:58.330
So you can imagine, the
denominator tells us, what are

00:00:58.330 --> 00:01:01.680
we taking a fraction out of or
how many pieces are there?

00:01:01.680 --> 00:01:03.950
So let's imagine a
pie like this.

00:01:03.950 --> 00:01:07.240
So we could draw like
a square pie.

00:01:11.250 --> 00:01:13.230
So this is what the denominator
represents.

00:01:13.230 --> 00:01:17.100
This is what the number on
the bottom represents.

00:01:17.100 --> 00:01:20.510
And then 3 says, we are
representing three of those

00:01:20.510 --> 00:01:21.950
four pieces.

00:01:21.950 --> 00:01:25.390
So this 3 tells us that out of
4 possible ones, I guess you

00:01:25.390 --> 00:01:28.580
could think of it, we are using
three, or maybe we're

00:01:28.580 --> 00:01:30.360
eating three.

00:01:30.360 --> 00:01:33.205
So you can imagine if someone
says I ate three-fourths of a

00:01:33.205 --> 00:01:35.880
pie-- this would be read as
three-fourths-- they're eating

00:01:35.880 --> 00:01:38.740
the blue portion of the pie
if we cut it this way.

00:01:38.740 --> 00:01:42.380
If we imagine a round pie,
it would look like this.

00:01:42.380 --> 00:01:43.905
Let me draw a round pie.

00:01:43.905 --> 00:01:47.190
So that is my round pie.

00:01:47.190 --> 00:01:50.400
Let me cut it into four
equal pieces or

00:01:50.400 --> 00:01:52.320
roughly equal pieces.

00:01:52.320 --> 00:01:58.030
And if someone says I ate
three-fourths of this pie,

00:01:58.030 --> 00:02:01.070
where the 3 is the numerator,
and then the 4, and you'd read

00:02:01.070 --> 00:02:04.650
that as three-fourths, the 4 is
the denominator, they would

00:02:04.650 --> 00:02:06.190
eat this much of the pie.

00:02:06.190 --> 00:02:08.480
They would eat 3 of
the 4 pieces.

00:02:12.910 --> 00:02:19.520
So this is is one piece, this
is two pieces, and this is

00:02:19.520 --> 00:02:20.720
three pieces.

00:02:20.720 --> 00:02:23.200
So you could imagine the 4, the
denominator represents the

00:02:23.200 --> 00:02:27.730
total number of pieces in the
pie, and then the 3 represents

00:02:27.730 --> 00:02:30.200
how many of those we ate.


Identifying fraction parts

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:06.100
For an art project, a pentagon
made of construction paper is

00:00:06.100 --> 00:00:09.180
cut into five equal slices.

00:00:09.180 --> 00:00:11.210
Two of the slices are removed.

00:00:11.210 --> 00:00:14.540
Write the remaining portion of
the pentagon as a fraction.

00:00:14.540 --> 00:00:16.360
So let's draw ourselves
a pentagon.

00:00:16.360 --> 00:00:20.060
A pentagon is just a five-sided
figure, so

00:00:20.060 --> 00:00:22.010
it looks like this.

00:00:22.010 --> 00:00:24.770
It's also where the Department
of Defense is located, a

00:00:24.770 --> 00:00:26.450
building that's actually
in this shape.

00:00:26.450 --> 00:00:28.950
That's why they call
it the Pentagon.

00:00:28.950 --> 00:00:32.420
Let me draw it a little
nicer than that.

00:00:32.420 --> 00:00:36.050
It looks something like this.

00:00:36.050 --> 00:00:37.400
Eh!

00:00:37.400 --> 00:00:39.980
My pentagon drawing skills
need some work.

00:00:43.070 --> 00:00:43.840
There you go.

00:00:43.840 --> 00:00:45.990
That's a pretty decent
shot at a pentagon.

00:00:45.990 --> 00:00:48.480
So that's the pentagon made
out of construction paper.

00:00:48.480 --> 00:00:51.140
Notice it has one, two, three,
four, five sides.

00:00:51.140 --> 00:00:52.310
That's why it's called
a pentagon.

00:00:52.310 --> 00:00:57.840
And it's cut into five equals
slices, so we could do that.

00:00:57.840 --> 00:01:00.560
Maybe that's the center
of the pentagon.

00:01:00.560 --> 00:01:03.270
Here this is one slice
right there.

00:01:03.270 --> 00:01:08.690
That is two slices, three
slices, four slices, and then

00:01:08.690 --> 00:01:10.070
a fifth slice, so you
can imagine these

00:01:10.070 --> 00:01:11.970
are all equal slices.

00:01:11.970 --> 00:01:14.736
Now, they're saying two of
the slices are removed.

00:01:18.890 --> 00:01:21.310
So let's get rid of two
of these slices.

00:01:21.310 --> 00:01:25.510
Let's say we remove that
slice up there.

00:01:25.510 --> 00:01:27.750
Let's say we remove the
slice right next to

00:01:27.750 --> 00:01:29.990
it right over there.

00:01:29.990 --> 00:01:33.740
And then they want us to write
the remaining portion of the

00:01:33.740 --> 00:01:34.990
pentagon as a fraction.

00:01:39.120 --> 00:01:41.110
So what are the remaining
slices?

00:01:41.110 --> 00:01:45.340
Well, I have this slice right
there, that slice right over

00:01:45.340 --> 00:01:47.390
here and then this slice.

00:01:47.390 --> 00:01:54.960
So you have three slices
remaining out of a total

00:01:54.960 --> 00:01:56.610
possible of how many?

00:01:56.610 --> 00:02:00.190
How many slices are in
the entire pentagon?

00:02:00.190 --> 00:02:05.870
So if you look at the entire
pentagon, if you consider all

00:02:05.870 --> 00:02:08.630
of the slices, we have
five slices.

00:02:08.630 --> 00:02:12.120
So if you consider the entire
pentagon, it is made up of

00:02:12.120 --> 00:02:14.680
five total slices.

00:02:14.680 --> 00:02:21.670
So it's three remaining out of
five total slices, so you

00:02:21.670 --> 00:02:35.590
could say that 3/5 of
the pentagon remain.

00:02:35.590 --> 00:02:38.170
Or you could say 2/5
were removed.

00:02:38.170 --> 00:02:42.070
That's two of the five slices
were removed, and then three

00:02:42.070 --> 00:02:46.410
are remaining, or 3/5 of
the pentagon remain.


Proper and Improper Fractions

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:05.380
Identify 15 over 8,
or 15/8, as a

00:00:05.380 --> 00:00:08.260
proper or improper fraction.

00:00:08.260 --> 00:00:12.330
So proper fraction just means
that the numerator is equal or

00:00:12.330 --> 00:00:14.960
less than, or less than or
equal to the denominator.

00:00:14.960 --> 00:00:21.040
So proper fraction, numerator,
the number on top, is less

00:00:21.040 --> 00:00:23.710
than or equal to the
denominator.

00:00:23.710 --> 00:00:30.520
In an improper fraction, the
numerator is greater than the

00:00:30.520 --> 00:00:31.490
denominator.

00:00:31.490 --> 00:00:34.810
And we see in our example right
here our numerator is

00:00:34.810 --> 00:00:40.290
greater than our denominator,
so we are dealing with an

00:00:40.290 --> 00:00:42.200
improper fraction.

00:00:42.200 --> 00:00:46.160
Now, a more interesting question
is what does 15/8

00:00:46.160 --> 00:00:46.945
even represent?

00:00:46.945 --> 00:00:49.410
So let's think about
that a little bit.

00:00:49.410 --> 00:00:52.250
We saw in the last couple of
videos that the denominator

00:00:52.250 --> 00:00:55.520
tells you essentially how many
slices of pie there are, I

00:00:55.520 --> 00:00:56.780
guess is one way to
think about it.

00:00:56.780 --> 00:00:59.690
So let me draw something
with eight--

00:00:59.690 --> 00:01:00.810
let me divide something.

00:01:00.810 --> 00:01:03.780
Maybe we could view this as a
square pizza or a square pie

00:01:03.780 --> 00:01:06.030
of some kind.

00:01:06.030 --> 00:01:09.450
So let's say that's
the entire pie.

00:01:09.450 --> 00:01:12.750
I'm dividing it into two.

00:01:12.750 --> 00:01:15.800
That would divide it into four,
and then to divide it

00:01:15.800 --> 00:01:18.180
into eight, I'd be
just like that.

00:01:18.180 --> 00:01:22.080
So this would be
one whole pie.

00:01:22.080 --> 00:01:25.890
Now, what does one of these
boxes represent?

00:01:25.890 --> 00:01:28.890
What does that box right
there represent?

00:01:28.890 --> 00:01:31.550
Well, we've seen this
is one of the eight

00:01:31.550 --> 00:01:32.980
total pieces of pie.

00:01:32.980 --> 00:01:37.290
So this represents 1/8.

00:01:37.290 --> 00:01:47.060
Now, if I were to shade in blue
this box and that box

00:01:47.060 --> 00:01:51.120
right there, and I were to say
what fraction of my pie is

00:01:51.120 --> 00:01:52.630
shaded in blue?

00:01:52.630 --> 00:01:55.900
Well, you'd say I have two boxes
shaded in blue: one,

00:01:55.900 --> 00:02:01.860
two, out of a total
of eight boxes.

00:02:05.170 --> 00:02:12.640
This, these two right here,
are 2/8 of the entire pie.

00:02:12.640 --> 00:02:17.740
Or another way to think about
it, this is equal to two of

00:02:17.740 --> 00:02:19.200
these pieces, right?

00:02:19.200 --> 00:02:24.740
So you could say this is also
equal to 2 times 1/8, because

00:02:24.740 --> 00:02:29.970
each of these boxes is 1/8.

00:02:29.970 --> 00:02:30.960
We saw that right there.

00:02:30.960 --> 00:02:35.590
It is one of the total eight
pieces, so two of the eight,

00:02:35.590 --> 00:02:39.560
that's really just two of
these one-eighth boxes.

00:02:39.560 --> 00:02:42.400
Now, if everything I've just
said makes sense, let's think

00:02:42.400 --> 00:02:45.100
a about what 15/8 mean.

00:02:45.100 --> 00:02:47.090
So if you imagine these
boxes like that,

00:02:47.090 --> 00:02:49.020
let's try to do 15/8.

00:02:49.020 --> 00:02:51.970
So if you were to draw, that's
1/8 right there.

00:02:51.970 --> 00:02:53.610
I'm going to do it in the same
size as these boxes.

00:02:53.610 --> 00:02:56.090
Let me copy and paste this.

00:02:56.090 --> 00:02:59.570
So let me copy and then
let me paste.

00:02:59.570 --> 00:03:03.980
So then that would be 2/8.

00:03:03.980 --> 00:03:10.360
Let me do 3/8 and then 4.

00:03:10.360 --> 00:03:11.535
We're going to go all
the way up to 15.

00:03:11.535 --> 00:03:14.010
I hope it doesn't bore you.

00:03:14.010 --> 00:03:24.540
That is 5, 6, 7, 8, and now
we've made a whole pie, but

00:03:24.540 --> 00:03:25.220
we're only at 8.

00:03:25.220 --> 00:03:27.390
We have to get to 15.

00:03:27.390 --> 00:03:29.220
So let's keep going.

00:03:29.220 --> 00:03:30.590
So then we're going to
start another pie.

00:03:30.590 --> 00:03:36.030
So it's 1, 2, 3.

00:03:36.030 --> 00:03:36.910
So far, how many do we have?

00:03:36.910 --> 00:03:50.190
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14 and 15.

00:03:50.190 --> 00:03:55.520
So what have I just drawn here
with all these pink squares?

00:03:55.520 --> 00:03:56.990
Each of these is 1/8.

00:03:56.990 --> 00:04:00.510
Each of these boxes is 1/8 of
a pie, I guess is how you

00:04:00.510 --> 00:04:01.190
could think about it.

00:04:01.190 --> 00:04:03.270
This is 1/8.

00:04:03.270 --> 00:04:10.950
And I've just drawn 15 times
1/8, so all of these combined

00:04:10.950 --> 00:04:17.220
are 15 times 1/8, or we could
say that this is also equal

00:04:17.220 --> 00:04:21.440
to, if we just use the logic
we did before, 15/8.

00:04:21.440 --> 00:04:23.700
Now, how many pies
do I have here?

00:04:23.700 --> 00:04:31.130
Well, these first 8 of the 15,
this right here, that is 8 out

00:04:31.130 --> 00:04:36.070
of 8, or 8 one-eighths, and
that's equal to one whole pie.

00:04:36.070 --> 00:04:38.710
So this right here
is equal to one.

00:04:38.710 --> 00:04:41.160
That right there is
equal to one.

00:04:41.160 --> 00:04:42.640
And what do we have over here?

00:04:42.640 --> 00:04:46.440
We have one whole pie, 8 out
of 8 pieces of a potential

00:04:46.440 --> 00:04:50.480
pie, or 8 out of 8 one-eighths
of a pie, I guess you could

00:04:50.480 --> 00:04:54.030
call it, and then over here we
have 7/8 of a pie, right?

00:04:54.030 --> 00:04:58.000
You could imagine a whole pie
would involve another grid

00:04:58.000 --> 00:05:01.170
right there, would involve one
more over here, but this is 7

00:05:01.170 --> 00:05:02.260
out of the 8.

00:05:02.260 --> 00:05:05.580
So this right here-- let me do
this in another color-- this

00:05:05.580 --> 00:05:13.800
piece right here represents
7, 7 times 1/8, or 7/8.

00:05:13.800 --> 00:05:17.760
So if you were to say what 15/8
kind of represents, it

00:05:17.760 --> 00:05:22.090
represents one whole pie, and
then 7/8 of another pie.

00:05:22.090 --> 00:05:24.990
You can even write that down
as a mixed number.

00:05:24.990 --> 00:05:27.020
And it's a mixed number because
it has a whole number

00:05:27.020 --> 00:05:27.780
and a fraction.

00:05:27.780 --> 00:05:35.100
So this is equal to one whole
pie right there and 7/8 of

00:05:35.100 --> 00:05:36.350
another pie.

00:05:39.970 --> 00:05:41.660
And that's all it represents
when you think

00:05:41.660 --> 00:05:43.400
about what 15/8 means.

00:05:43.400 --> 00:05:47.170
Now, I had to draw it out like
this, but a quick way you can

00:05:47.170 --> 00:05:51.590
do to solve this is to
divide 8 into 15.

00:05:51.590 --> 00:05:53.730
So if you were to take--
let me do this

00:05:53.730 --> 00:05:54.690
with the new colors.

00:05:54.690 --> 00:05:56.780
So if you were to take that
8 right there, you

00:05:56.780 --> 00:06:04.330
divide 8 into 15.

00:06:04.330 --> 00:06:09.060
You see that it's equal to--
well, 8 goes into 15 one time,

00:06:09.060 --> 00:06:11.030
because 2 times 8 is 16,
so that's too big.

00:06:11.030 --> 00:06:13.290
So 1 times 8 is 8.

00:06:13.290 --> 00:06:18.650
You subtract 15 minus 8 is 7.

00:06:18.650 --> 00:06:23.240
So to get 1 and 7/8, this 1
right here-- so you get one

00:06:23.240 --> 00:06:24.320
whole 8 in there.

00:06:24.320 --> 00:06:26.710
If you have 15/8, you get a
whole 8 in there and then you

00:06:26.710 --> 00:06:30.650
have 7 left over, but these are
7/8, so that's where that

00:06:30.650 --> 00:06:31.800
7/8 come from.

00:06:31.800 --> 00:06:35.660
You have to write the
remainder over 8.

00:06:35.660 --> 00:06:37.680
Hopefully, you found
out helpful.


Mixed numbers: changing to improper fractions

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:05.190
Write 5 and 1/4 as an
improper fraction.

00:00:05.190 --> 00:00:08.770
An improper fraction is just
a pure fraction where the

00:00:08.770 --> 00:00:11.510
numerator is greater than
the denominator.

00:00:11.510 --> 00:00:13.270
This right here, it's
not a pure fraction.

00:00:13.270 --> 00:00:16.079
We have a whole number mixed
with a fraction, so we call

00:00:16.079 --> 00:00:17.720
this a mixed number.

00:00:17.720 --> 00:00:20.720
So let's think about what 5 and
1/4 represents, and let me

00:00:20.720 --> 00:00:21.400
rewrite it.

00:00:21.400 --> 00:00:27.780
So if we're talking about 5 and
1/4, and you can literally

00:00:27.780 --> 00:00:32.860
think of this as 5 and 1/4 or 5
plus 1/4, that's what 5 and

00:00:32.860 --> 00:00:34.760
1/4 represents.

00:00:34.760 --> 00:00:35.750
So let's think about 5.

00:00:35.750 --> 00:00:39.250
Five is 5 wholes, or if you're
thinking of pie, we could draw

00:00:39.250 --> 00:00:42.710
literally five pies.

00:00:42.710 --> 00:00:45.250
Let me just cut up the pies
from the get go into four

00:00:45.250 --> 00:00:48.330
pieces since we're dealing
with fourths.

00:00:48.330 --> 00:00:51.750
So let me just cut up the
pies right over here.

00:00:51.750 --> 00:00:54.050
So that's one pie right
over there.

00:00:54.050 --> 00:00:56.090
Let me copy and paste this.

00:00:56.090 --> 00:00:57.900
Copy and paste.

00:00:57.900 --> 00:01:04.040
So I have two pies, and then I
have three pies, and then I

00:01:04.040 --> 00:01:08.990
have four pies, and then
I have five pies.

00:01:08.990 --> 00:01:11.230
So this is what the
5 represents.

00:01:11.230 --> 00:01:13.570
5 literally represents--
so let me

00:01:13.570 --> 00:01:14.780
circle all of this together.

00:01:14.780 --> 00:01:17.110
That is the 5 part
right there.

00:01:17.110 --> 00:01:19.680
That is what 5 literally
represents.

00:01:19.680 --> 00:01:22.265
It represents five whole pies.

00:01:26.850 --> 00:01:32.170
Now, I have cut up the pies into
four pieces, so you can

00:01:32.170 --> 00:01:35.940
imagine each piece represents
a fourth.

00:01:35.940 --> 00:01:39.700
Now, how many pieces do I
have in these five pies?

00:01:39.700 --> 00:01:41.930
Well, I have four
pieces per pie.

00:01:50.970 --> 00:01:51.790
Let me just right it here.

00:01:51.790 --> 00:02:05.380
4 pieces per pie times 5 pies
is equal to 20 pieces.

00:02:05.380 --> 00:02:09.100
Or another way to think of it,
since each piece is a fourth,

00:02:09.100 --> 00:02:14.580
this is also equal to 20 times
1/4, or you could just write

00:02:14.580 --> 00:02:18.750
this as being equal to 20/4.

00:02:18.750 --> 00:02:25.640
So we have 5 whole pies is
equal to 20 fourths.

00:02:25.640 --> 00:02:26.830
Let me write it like that.

00:02:26.830 --> 00:02:29.460
20 fourths.

00:02:29.460 --> 00:02:35.230
Or we could write it as 20/4.

00:02:35.230 --> 00:02:36.700
I've kind of done the
same thing twice.

00:02:36.700 --> 00:02:39.020
So that's what the five
pies represent.

00:02:39.020 --> 00:02:43.160
20/4 or 20 pieces, where
each piece is 1/4.

00:02:43.160 --> 00:02:46.790
Now, the 1/4 right here
represents literally one more

00:02:46.790 --> 00:02:50.270
fourth of a pie or one more
piece of a pie, so let me draw

00:02:50.270 --> 00:02:51.520
another pie here.

00:02:54.410 --> 00:02:57.090
So that is another pie.

00:02:57.090 --> 00:02:59.670
Cut it into four pieces.

00:02:59.670 --> 00:03:04.080
But this 1/4 only represents
one of these pieces, right?

00:03:04.080 --> 00:03:06.750
This is one of the
four pieces.

00:03:06.750 --> 00:03:09.080
The denominator tells
us how many pieces.

00:03:09.080 --> 00:03:12.140
The 1 tells us how many of those
pieces we're dealing

00:03:12.140 --> 00:03:16.180
with, so it's just this
one piece over here.

00:03:16.180 --> 00:03:19.550
That right there is the 1/4.

00:03:19.550 --> 00:03:26.100
Now, if we write 5 and 1/4,
we just saw that the 5

00:03:26.100 --> 00:03:29.290
right here is 20/4.

00:03:29.290 --> 00:03:30.690
So we could rewrite this.

00:03:30.690 --> 00:03:31.440
Let me write it like this.

00:03:31.440 --> 00:03:44.560
5 and 1/4 can be rewritten as
the same thing as 5 plus 1/4,

00:03:44.560 --> 00:03:48.450
which is the same thing as--
we just saw that five whole

00:03:48.450 --> 00:03:53.960
pies is the same
thing as 20/4.

00:03:53.960 --> 00:03:55.650
And to see that these are the
same thing, you literally just

00:03:55.650 --> 00:03:57.120
divide 4 into 20.

00:03:57.120 --> 00:03:59.700
You get 5, and nothing
is left over.

00:03:59.700 --> 00:04:03.300
So 5 is the same thing as 20/4,
and then this plus 1/4

00:04:03.300 --> 00:04:05.050
is the same thing as plus 1/4.

00:04:05.050 --> 00:04:10.350
So if I have 20 fourths and I
add one more fourth to it, how

00:04:10.350 --> 00:04:12.380
many fourths do I have?

00:04:12.380 --> 00:04:14.640
Well, I have 21.

00:04:14.640 --> 00:04:17.120
I have 21 fourths.

00:04:17.120 --> 00:04:20.740
Or another way of thinking about
it, this 5 is-- so this

00:04:20.740 --> 00:04:27.990
right here is 20
pieces of pie.

00:04:27.990 --> 00:04:28.770
You can even count it.

00:04:28.770 --> 00:04:33.470
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17,

00:04:33.470 --> 00:04:34.640
18, 19, 20.

00:04:34.640 --> 00:04:36.570
But a quicker way is to say,
well, we have five pies.

00:04:36.570 --> 00:04:37.960
Each of them have four pieces.

00:04:37.960 --> 00:04:39.820
5 times 4 is 20.

00:04:39.820 --> 00:04:46.020
This 1/4 right here represents
one piece plus one piece, so

00:04:46.020 --> 00:04:51.060
total we're going to
have 21 pieces.

00:04:51.060 --> 00:04:56.150
So we have 21 pieces, where each
piece is 1/4, so we could

00:04:56.150 --> 00:05:02.030
say we have 21 times 1/4 or
21 fourths pieces of pie.

00:05:02.030 --> 00:05:03.270
However you want to
think of it, but

00:05:03.270 --> 00:05:05.490
we've solved the problem.

00:05:05.490 --> 00:05:07.330
We're at an improper fraction.

00:05:07.330 --> 00:05:09.850
We've written 5 and 1/4 as
an improper fraction.

00:05:09.850 --> 00:05:13.490
Now, I've gone through great
pains to give you the

00:05:13.490 --> 00:05:17.410
intuition of what 5 and 1/4
means, but there is a fairly

00:05:17.410 --> 00:05:20.710
straightforward process for
getting straight to the

00:05:20.710 --> 00:05:21.960
improper fraction.

00:05:24.510 --> 00:05:25.970
Let me color code it.

00:05:25.970 --> 00:05:35.150
So if you have 5 and 1 over 4,
to convert it into an improper

00:05:35.150 --> 00:05:39.020
fraction, you're going to keep
the same denominator, so

00:05:39.020 --> 00:05:41.680
you're going to have
the over 4 there.

00:05:41.680 --> 00:05:47.220
But your numerator is going to
be your numerator of the

00:05:47.220 --> 00:05:48.370
fraction part before.

00:05:48.370 --> 00:05:53.510
So it's going to be 1 plus your
whole number times your

00:05:53.510 --> 00:05:54.820
denominator.

00:05:54.820 --> 00:05:57.620
So 1 plus-- or actually, let
me do it the way I tend to

00:05:57.620 --> 00:05:58.290
think of it.

00:05:58.290 --> 00:06:00.240
What I do is I take 4 times 5.

00:06:00.240 --> 00:06:02.700
So let me write that down and
I want to color code it.

00:06:02.700 --> 00:06:12.830
4 times 5, and then to that,
I add this numerator.

00:06:12.830 --> 00:06:16.560
So I literally do 4 times 5 plus
1, which is-- so this is

00:06:16.560 --> 00:06:21.530
equal to 4 times 5 is 20, plus
1 is 21, and then that's over

00:06:21.530 --> 00:06:23.680
4, so it's 21/4.

00:06:23.680 --> 00:06:25.410
And all of this is kind of
a fast way to do it.

00:06:25.410 --> 00:06:27.910
We're literally doing the exact
same thing that we did

00:06:27.910 --> 00:06:29.420
here in kind of a slower way.

00:06:29.420 --> 00:06:33.550
We're saying, OK, 5 wholes is
the same thing as 20 fourths,

00:06:33.550 --> 00:06:37.220
so you take 5, and I figure that
out, 5 times 4, and then

00:06:37.220 --> 00:06:41.870
I have one more fourth there, so
4 times 5 plus 1 gives 21.


Finding factors of a number

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.180
Find all of the factors
of 120.

00:00:04.180 --> 00:00:06.160
Or another way to think about
it, find all of the whole

00:00:06.160 --> 00:00:09.650
numbers that 120 is
divisible by.

00:00:09.650 --> 00:00:12.040
So the first one, that's
maybe obvious.

00:00:12.040 --> 00:00:14.560
All whole numbers are
divisible by 1.

00:00:14.560 --> 00:00:21.090
So we could write 120 is equal
to is to 1 times 120.

00:00:21.090 --> 00:00:22.990
So let's write a factors
list over here.

00:00:26.530 --> 00:00:28.390
So this is going to be our
factors list over here.

00:00:28.390 --> 00:00:29.900
So we just found two factors.

00:00:29.900 --> 00:00:31.910
We said, well, is it
divisible by 1?

00:00:31.910 --> 00:00:33.940
Well, every whole number
is divisible by 1.

00:00:33.940 --> 00:00:37.630
This is a whole number, so 1
is a factor at the low end.

00:00:37.630 --> 00:00:38.490
1 is a factor.

00:00:38.490 --> 00:00:40.580
That's its actual smallest
factor, and its

00:00:40.580 --> 00:00:42.330
largest factor is 120.

00:00:42.330 --> 00:00:46.580
You can't have something larger
than 120 dividing

00:00:46.580 --> 00:00:49.500
evenly into 120.

00:00:49.500 --> 00:00:52.400
121 will not go into 120.

00:00:52.400 --> 00:00:54.710
So the largest factor
on our factors list

00:00:54.710 --> 00:00:57.080
is going to be 120.

00:00:57.080 --> 00:00:58.470
Now let's think about others.

00:00:58.470 --> 00:01:02.200
Let's think about whether
is 2 divisible into 120?

00:01:02.200 --> 00:01:06.910
So there's 120 equals
2 times something?

00:01:06.910 --> 00:01:09.680
Well, when you look here,
maybe you immediately

00:01:09.680 --> 00:01:12.760
recognize that 120 is
an even number.

00:01:12.760 --> 00:01:15.000
It's ones place is a 0.

00:01:15.000 --> 00:01:18.430
As as long as its ones place is
a 0, 2, 4, 6 or 8, as long

00:01:18.430 --> 00:01:21.110
as it's an even number, the
whole number is even and the

00:01:21.110 --> 00:01:23.540
whole number is divisible
by 2.

00:01:23.540 --> 00:01:26.440
And to figure out what you have
to multiply by 2 to get

00:01:26.440 --> 00:01:33.690
120, well, you can think of 120
as 12 times 10, or another

00:01:33.690 --> 00:01:36.480
way to think about it,
it's 2 times 6 times

00:01:36.480 --> 00:01:38.890
10, or 2 times 60.

00:01:38.890 --> 00:01:40.340
You could divide it
out if you want.

00:01:40.340 --> 00:01:43.690
You could say, OK,
2 goes into 120.

00:01:43.690 --> 00:01:45.420
2 goes into 1 no times.

00:01:45.420 --> 00:01:47.240
2 goes into 12 six times.

00:01:47.240 --> 00:01:49.270
6 times 2 is 12.

00:01:49.270 --> 00:01:50.410
Subtract.

00:01:50.410 --> 00:01:51.100
You get 0.

00:01:51.100 --> 00:01:52.090
Bring down the 0.

00:01:52.090 --> 00:01:53.910
2 goes into 0 zero times.

00:01:53.910 --> 00:01:58.010
0 times 2 is 0, and you get no
remainder there, so it goes

00:01:58.010 --> 00:01:59.430
sixty times.

00:01:59.430 --> 00:02:02.050
So we have two more factors
right here.

00:02:02.050 --> 00:02:04.420
So we have the factors.

00:02:04.420 --> 00:02:08.070
So we've established the next
lowest one is 2, and the next

00:02:08.070 --> 00:02:10.110
highest factor, if we're
starting from the large end,

00:02:10.110 --> 00:02:13.310
is going to be 60.

00:02:13.310 --> 00:02:14.880
Now let's think about three.

00:02:14.880 --> 00:02:19.780
Is 120 equal to 3
times something?

00:02:19.780 --> 00:02:22.060
Well, we could just try to test
and divide it from the

00:02:22.060 --> 00:02:24.410
get go, but hopefully,
you already know the

00:02:24.410 --> 00:02:25.630
divisibility rule.

00:02:25.630 --> 00:02:29.220
To figure out if something is
divisible by 3, you add up its

00:02:29.220 --> 00:02:30.910
digits, and if the
sum is divisible

00:02:30.910 --> 00:02:32.600
by 3, we're in business.

00:02:32.600 --> 00:02:38.540
So if you take 120-- let
me do it over here.

00:02:38.540 --> 00:02:44.180
1 plus 2 plus 0, well, that's
equal to 1 plus 2 is 3 plus 0

00:02:44.180 --> 00:02:48.700
is 3, and 3 is definitely
divisible by 3.

00:02:48.700 --> 00:02:52.610
So 120 is going to be
divisible by 3.

00:02:52.610 --> 00:02:56.050
To figure what that number that
you have to multiply by 3

00:02:56.050 --> 00:02:57.840
is, you could do it
in your head.

00:02:57.840 --> 00:03:01.140
You could say, well, 3 goes into
12 four times, and then

00:03:01.140 --> 00:03:04.440
you-- well, let me just do it
out, just in case, just for

00:03:04.440 --> 00:03:06.030
those of you who want to
see it worked out.

00:03:06.030 --> 00:03:08.090
3 goes into 12 four times.

00:03:08.090 --> 00:03:10.570
4 times 3 is 12.

00:03:10.570 --> 00:03:11.460
You subtract.

00:03:11.460 --> 00:03:12.690
You're left with nothing here.

00:03:12.690 --> 00:03:14.680
You bring down this 0.

00:03:14.680 --> 00:03:16.730
3 goes into 0 zero times.

00:03:16.730 --> 00:03:18.940
0 times 3 is 0.

00:03:18.940 --> 00:03:20.510
Nothing left over.

00:03:20.510 --> 00:03:22.077
So it goes into it
forty times.

00:03:24.690 --> 00:03:28.110
And the way to think of it in
your head is this is the same

00:03:28.110 --> 00:03:29.860
thing as 12 times 10.

00:03:29.860 --> 00:03:34.280
12 divided by 3 is 4, but this
is going to be 4 times 10,

00:03:34.280 --> 00:03:35.630
because you have that
10 left over.

00:03:35.630 --> 00:03:36.740
Whatever works for you.

00:03:36.740 --> 00:03:40.070
Or you can just ignore the 0,
divide by 3, you get a 4, and

00:03:40.070 --> 00:03:41.290
then put the 0 back there.

00:03:41.290 --> 00:03:42.370
Whatever works.

00:03:42.370 --> 00:03:43.650
So we have two more factors.

00:03:43.650 --> 00:03:50.780
At the low end, we have 3, and
at the high end, we have a 40.

00:03:50.780 --> 00:03:53.600
Now, let's see if 4 divisible
into 120.

00:03:53.600 --> 00:03:57.030
Now we saw the divisibility
rule for 4 is you ignore

00:03:57.030 --> 00:03:59.300
everything beyond the tens
places and you just look at

00:03:59.300 --> 00:04:01.040
the last two digits.

00:04:01.040 --> 00:04:05.700
So if we're going to to think
about whether 4 is divisible,

00:04:05.700 --> 00:04:07.130
you just look at the
last two digits.

00:04:07.130 --> 00:04:09.130
The last two digits are 20.

00:04:09.130 --> 00:04:13.430
20 is definitely divisible
by 4, so 120 will be

00:04:13.430 --> 00:04:14.220
divisible by 4.

00:04:14.220 --> 00:04:16.180
4 is going to be a factor.

00:04:16.180 --> 00:04:19.250
And to figure out what we have
to multiply 4 by to get 120,

00:04:19.250 --> 00:04:20.100
you could do it in your head.

00:04:20.100 --> 00:04:23.430
You could say 12 divided
by 4 is 3, so 120

00:04:23.430 --> 00:04:27.210
divided by 4 is 30.

00:04:27.210 --> 00:04:29.890
So we have two more
factors: 4 and 30.

00:04:29.890 --> 00:04:32.670
And you could work this out in
long division if you want to

00:04:32.670 --> 00:04:35.940
make sure that this works out,
so let's keep going.

00:04:35.940 --> 00:04:40.750
And then we have 120 is equal
to-- is 5 a factor?

00:04:40.750 --> 00:04:44.630
Is 5 times something
equal to 120?

00:04:44.630 --> 00:04:46.750
Well, you can't do that simple--
well, first of all,

00:04:46.750 --> 00:04:48.550
we could just test
is it divisible?

00:04:48.550 --> 00:04:50.650
And 120 ends with a 0.

00:04:50.650 --> 00:04:53.400
If you end with a 0 or a 5,
you are divisible by 5.

00:04:53.400 --> 00:04:55.340
So 5 definitely goes into it.

00:04:55.340 --> 00:04:56.690
Let's figure out
how many times.

00:04:56.690 --> 00:04:59.575
So 5 goes into 120.

00:04:59.575 --> 00:05:00.830
It doesn't go into 1.

00:05:00.830 --> 00:05:02.750
It goes into 12 two times.

00:05:02.750 --> 00:05:04.780
2 times 5 is 10.

00:05:04.780 --> 00:05:05.850
Subtract.

00:05:05.850 --> 00:05:07.130
You get 2.

00:05:07.130 --> 00:05:08.830
Bring down the 0.

00:05:08.830 --> 00:05:11.290
5 goes into 20 four times.

00:05:11.290 --> 00:05:18.620
4 times 5 is 20, and then you
subtract, and you have no left

00:05:18.620 --> 00:05:21.120
over, as we expect, because
it should go in evenly.

00:05:21.120 --> 00:05:24.760
This number ends with
a 0 or a 5.

00:05:24.760 --> 00:05:27.640
Let me delete all of this so we
can have our scratch space

00:05:27.640 --> 00:05:29.680
to work with later on.

00:05:29.680 --> 00:05:33.810
So 5 times 24 is also equal
to 120, we have two more

00:05:33.810 --> 00:05:37.950
factors: 5 and 24.

00:05:37.950 --> 00:05:40.400
Let me clear up some space here
because I think we're

00:05:40.400 --> 00:05:42.510
going to be dealing with
a lot of factors.

00:05:42.510 --> 00:05:45.230
So let me move this
right here.

00:05:45.230 --> 00:05:50.400
Let me cut it and then let me
paste it and move this over

00:05:50.400 --> 00:05:53.680
here so we have more space
for our factors.

00:05:53.680 --> 00:05:55.580
So we have 5 and 24.

00:05:55.580 --> 00:05:58.590
Let's move on to 6.

00:05:58.590 --> 00:06:02.470
So 120 is equal to
6 times what?

00:06:02.470 --> 00:06:05.050
Now, to be divisible by
6, you have to be

00:06:05.050 --> 00:06:07.370
divisible by 2 and 3.

00:06:07.370 --> 00:06:09.630
Now, we know that we're already
divisible by 2 and 3,

00:06:09.630 --> 00:06:12.530
so we're definitely going to
be divisible by 6, and you

00:06:12.530 --> 00:06:14.140
should hopefully be able to
do this one in your head.

00:06:14.140 --> 00:06:17.300
5 was a little bit harder to do
in your head. but 120, you

00:06:17.300 --> 00:06:21.930
could say, well, 12 divided by 6
is 2, and then you have that

00:06:21.930 --> 00:06:26.140
0 there, so 120 divided
by 6 would be 20.

00:06:26.140 --> 00:06:28.550
And you could work it out in
long division if you like.

00:06:28.550 --> 00:06:30.925
So 6 times 20 are two
more factors.

00:06:33.590 --> 00:06:35.850
Now let's think about 7.

00:06:35.850 --> 00:06:37.230
Let's think about 7 here.

00:06:37.230 --> 00:06:40.490
7 is a very bizarre number, and
just to test it, you could

00:06:40.490 --> 00:06:41.900
think of other ways to do it.

00:06:41.900 --> 00:06:45.130
Let's just try to divide
7 into 120.

00:06:45.130 --> 00:06:46.300
7 doesn't go into 1.

00:06:46.300 --> 00:06:48.070
It goes into 12 one time.

00:06:48.070 --> 00:06:50.130
1 times 7 is 7.

00:06:50.130 --> 00:06:51.020
You subtract.

00:06:51.020 --> 00:06:53.150
12 minus 7 is 5.

00:06:53.150 --> 00:06:56.000
Bring down the 0.

00:06:56.000 --> 00:06:59.610
7 times 7 is 49, so it goes
into it seven times.

00:06:59.610 --> 00:07:01.650
7 times 7 is 49.

00:07:01.650 --> 00:07:02.480
Subtract.

00:07:02.480 --> 00:07:05.820
You have a remainder, so it
does not divide evenly.

00:07:05.820 --> 00:07:07.515
So 7 does not work.

00:07:10.700 --> 00:07:12.770
Now let's think about 8.

00:07:12.770 --> 00:07:15.680
Let's think about
whether 8 works.

00:07:15.680 --> 00:07:17.360
Let's think about 8.

00:07:17.360 --> 00:07:18.850
I'll do the same process.

00:07:18.850 --> 00:07:26.540
Let's take 8 into 120.

00:07:26.540 --> 00:07:27.890
Let's just work it out.

00:07:27.890 --> 00:07:29.640
And just as a little bit
of a hint-- well, I'll

00:07:29.640 --> 00:07:30.270
just work it out.

00:07:30.270 --> 00:07:33.390
8 goes into 12-- it doesn't
go into 1, so it

00:07:33.390 --> 00:07:35.500
goes into 12 one time.

00:07:35.500 --> 00:07:38.250
1 times 8 is 8.

00:07:38.250 --> 00:07:39.240
Subtract there.

00:07:39.240 --> 00:07:41.160
12 minus 8 is 4.

00:07:41.160 --> 00:07:43.150
Bring down the 0.

00:07:43.150 --> 00:07:45.280
8 goes into 40 five times.

00:07:45.280 --> 00:07:49.190
5 times 8 is 40, and you're left
with no remainder, so it

00:07:49.190 --> 00:07:49.940
goes evenly.

00:07:49.940 --> 00:07:53.250
So 120-- let me get
rid of that.

00:07:53.250 --> 00:08:02.630
120 is equal to 8 times 15, so
let's add that to our factor

00:08:02.630 --> 00:08:09.440
list. We now have an 8
and now we have a 15.

00:08:09.440 --> 00:08:11.920
Now, is it divisible by 9?

00:08:11.920 --> 00:08:13.870
Is 120 divisible by 9?

00:08:13.870 --> 00:08:16.300
To test that out, you just
add up the digits.

00:08:16.300 --> 00:08:20.430
1 plus 2 plus 0 is equal to 3.

00:08:20.430 --> 00:08:24.350
Well, that'll satisfy our 3
divisibility rule, but 3 is

00:08:24.350 --> 00:08:27.320
not divisible by 9, so our
number will not be

00:08:27.320 --> 00:08:28.700
divisible by 9.

00:08:28.700 --> 00:08:31.380
So 9 will not work out.

00:08:31.380 --> 00:08:32.950
9 does not work out.

00:08:32.950 --> 00:08:34.730
So let's move on to 10.

00:08:34.730 --> 00:08:36.450
Well, this is pretty
straightforward.

00:08:36.450 --> 00:08:39.679
It ends in 0, so we will
be divisible by 10.

00:08:39.679 --> 00:08:41.559
So let me write that down.

00:08:41.559 --> 00:08:46.610
120 is equal to 10 times--
and this is pretty

00:08:46.610 --> 00:08:49.780
straightforward-- 10 times 12.

00:08:49.780 --> 00:08:51.560
This is exactly what 120 is.

00:08:51.560 --> 00:08:53.800
It's 10 times 12, so let's
write those factors down.

00:08:53.800 --> 00:08:56.500
10 and 12.

00:08:56.500 --> 00:08:58.220
And then we have one
number left.

00:08:58.220 --> 00:08:58.740
We have 11.

00:08:58.740 --> 00:09:00.440
We don't have to go above 11,
because we already went

00:09:00.440 --> 00:09:02.670
through 12, and we know that
there aren't any factors above

00:09:02.670 --> 00:09:07.270
that, because we were going in
descending order, so we've

00:09:07.270 --> 00:09:08.610
really filled in all the gaps.

00:09:08.610 --> 00:09:09.830
You could try 11.

00:09:09.830 --> 00:09:12.040
We could try it by hand,
if you like.

00:09:12.040 --> 00:09:15.370
11 goes into 120-- now you know,
if with you know your

00:09:15.370 --> 00:09:17.880
multiplication tables through
11, that this won't work, but

00:09:17.880 --> 00:09:18.900
I'll just show you.

00:09:18.900 --> 00:09:21.250
11 goes into 12 one time.

00:09:21.250 --> 00:09:23.250
1 times 11 is 11.

00:09:23.250 --> 00:09:24.630
Subtract.

00:09:24.630 --> 00:09:26.460
1, bring down the 0.

00:09:26.460 --> 00:09:29.010
11 goes into 10 zero times.

00:09:29.010 --> 00:09:30.960
0 times 11 is 0.

00:09:30.960 --> 00:09:33.500
you're left with a
remainder of 10.

00:09:33.500 --> 00:09:36.150
So 11 goes into 20 ten times
with a remainder of 10.

00:09:36.150 --> 00:09:37.900
It definitely does
not go in evenly.

00:09:37.900 --> 00:09:45.220
So we have all of our factors
here: 1, 2, 3, 4, 5, 6, 8, 10,

00:09:45.220 --> 00:09:51.260
12, 15, 20, 24, 30,
40, 60 and 120.

00:09:51.260 --> 00:09:52.750
And we're done!


Mixed numbers: changing from an improper fraction

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.290
Write 7/4 as a mixed number.

00:00:04.290 --> 00:00:06.210
So right now it's an
improper fraction.

00:00:06.210 --> 00:00:07.580
7 is larger than 4.

00:00:07.580 --> 00:00:09.160
Let's write it is
a mixed number.

00:00:09.160 --> 00:00:11.110
So first I'm just going
to show you a fairly

00:00:11.110 --> 00:00:12.510
straightforward way of doing
it and then we're going to

00:00:12.510 --> 00:00:15.830
think a little bit about
what it actually means.

00:00:15.830 --> 00:00:20.210
So to figure out what 7/4
represents as a mixed number,

00:00:20.210 --> 00:00:22.560
let me write it in
different colors.

00:00:22.560 --> 00:00:26.045
So this is going to be equal
to-- the easiest way I do it

00:00:26.045 --> 00:00:29.540
is you say, well, you
divide 4 it 7.

00:00:36.280 --> 00:00:38.910
If we're dealing with fourths,
4 goes into 7 a

00:00:38.910 --> 00:00:40.240
total of one time.

00:00:43.370 --> 00:00:44.620
Let me do this in
another color.

00:00:47.100 --> 00:00:50.220
1 times 4 is 4.

00:00:50.220 --> 00:00:52.600
And then what is
our remainder?

00:00:52.600 --> 00:00:55.160
7 minus 4 is 3.

00:00:55.160 --> 00:00:57.680
So if we wanted to write this
in plain-- well, let me just

00:00:57.680 --> 00:00:59.150
do the problem, and then
we'll think about what

00:00:59.150 --> 00:01:00.460
it means in a second.

00:01:00.460 --> 00:01:05.019
So you see that 4 goes into 7
one time, so you have one

00:01:05.019 --> 00:01:10.200
whole here, and then how much
do you have left over?

00:01:10.200 --> 00:01:15.810
Well, you have 3 left
over, and that comes

00:01:15.810 --> 00:01:16.700
from right over there.

00:01:16.700 --> 00:01:19.630
That is the remainder when
you divide 4 into 7.

00:01:19.630 --> 00:01:26.000
3 left over, but it's 3 of
your 4, or 3/4 left over.

00:01:26.000 --> 00:01:28.880
So that's the way we just
converted it from an improper

00:01:28.880 --> 00:01:30.200
fraction to a mixed number.

00:01:30.200 --> 00:01:31.650
Now, it might seem
a little bit like

00:01:31.650 --> 00:01:33.190
voodoo what I just did.

00:01:33.190 --> 00:01:36.780
I divided 4 into 7, it goes
one time, and then the

00:01:36.780 --> 00:01:40.530
remainder is 3, so
I got 1 and 3/4.

00:01:40.530 --> 00:01:42.480
But why does that make sense?

00:01:42.480 --> 00:01:44.480
Why does that actually
makes sense?

00:01:44.480 --> 00:01:46.230
So let's draw fourths.

00:01:46.230 --> 00:01:53.540
Let's draw literally 7 fourths
and maybe it'll become clear.

00:01:53.540 --> 00:01:56.460
So let's do a little
square as a fourth.

00:01:56.460 --> 00:02:03.750
So let's say I have a square
like that, and that is 1/4.

00:02:03.750 --> 00:02:07.170
Now, let's think about what
seven of those mean, so let me

00:02:07.170 --> 00:02:08.410
copy and paste that.

00:02:08.410 --> 00:02:11.500
Copy and then paste it.

00:02:11.500 --> 00:02:17.510
So here I have 2 one-fourths,
or you could see I have 2/4.

00:02:17.510 --> 00:02:20.740
Now I have 3 one-fourths.

00:02:20.740 --> 00:02:22.950
Now, I have 4 one-fourths.

00:02:22.950 --> 00:02:24.430
Now this is a whole, right?

00:02:24.430 --> 00:02:25.800
I have 4 one-fourths.

00:02:25.800 --> 00:02:27.310
This is a whole.

00:02:27.310 --> 00:02:28.910
So let me start on
another whole.

00:02:28.910 --> 00:02:31.500
So now I have 5.

00:02:31.500 --> 00:02:37.890
Now I have 6 one-fourths, and
now I have 7 one-fourths.

00:02:37.890 --> 00:02:39.410
Now, what does this look like?

00:02:39.410 --> 00:02:43.490
So all I did is I rewrote
7/4, or 7 one-fourths.

00:02:43.490 --> 00:02:45.630
I just kind of drew
it for you.

00:02:45.630 --> 00:02:47.350
Now, what does this represent?

00:02:47.350 --> 00:02:52.110
Well, I have 4 fourths
here, so this is 4/4.

00:02:52.110 --> 00:02:55.750
This right here is 3/4.

00:02:55.750 --> 00:03:01.980
Notice, 7/4 is 4/4 with
3/4 left over.

00:03:01.980 --> 00:03:03.110
So let me write it this way.

00:03:03.110 --> 00:03:09.330
7/4 is 4/4 with 3/4 left over.

00:03:09.330 --> 00:03:10.580
Now what is 4/4?

00:03:13.350 --> 00:03:16.320
4/4 is one whole.

00:03:16.320 --> 00:03:20.970
So you have one whole with
3/4 left over, so you end

00:03:20.970 --> 00:03:23.400
up with 1 and 3/4.

00:03:23.400 --> 00:03:27.470
So that is the 3/4 part and
that is your one whole.

00:03:27.470 --> 00:03:29.590
Hopefully that makes sense and
hopefully you understand why

00:03:29.590 --> 00:03:30.440
it connects.

00:03:30.440 --> 00:03:32.760
Because you say, well, how
many wholes do you have?

00:03:32.760 --> 00:03:36.100
When you're dividing the 4 into
the 7 and getting the

00:03:36.100 --> 00:03:38.550
one, you're essentially saying
how many wholes?

00:03:42.250 --> 00:03:47.070
So the number of wholes, or you
can imagine, the number of

00:03:47.070 --> 00:03:48.190
whole pies.

00:03:48.190 --> 00:03:50.680
And then how many pieces
do we have left over?

00:03:50.680 --> 00:03:53.770
Well, we have 3 pieces and each
piece is 1/4, so we have

00:03:53.770 --> 00:03:55.430
3/4 left over.

00:03:55.430 --> 00:03:59.080
So we have one whole pie and
three pieces, which are each a

00:03:59.080 --> 00:04:01.200
fourth left over.


Prime factorization

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.160
Write the prime factorization
of 75.

00:00:04.160 --> 00:00:07.390
Write your answer using
exponential notation.

00:00:07.390 --> 00:00:08.970
So we have a couple of
interesting things here.

00:00:08.970 --> 00:00:12.410
Prime factorization, and they
say exponential notation.

00:00:12.410 --> 00:00:15.460
We'll worry about the
exponential notation later.

00:00:15.460 --> 00:00:18.560
So the first thing we have to
worry about is what is even a

00:00:18.560 --> 00:00:19.380
prime number?

00:00:19.380 --> 00:00:22.240
And just as a refresher, a
prime number is a number

00:00:22.240 --> 00:00:26.130
that's only divisible by itself
and one, so examples of

00:00:26.130 --> 00:00:28.880
prime numbers-- let me write
some numbers down.

00:00:28.880 --> 00:00:31.030
Prime, not prime.

00:00:34.760 --> 00:00:36.840
So 2 is a prime number.

00:00:36.840 --> 00:00:39.850
It's only divisible
by 1 and 2.

00:00:39.850 --> 00:00:42.490
3 is another prime number.

00:00:42.490 --> 00:00:46.790
Now, 4 is not prime,
because this is

00:00:46.790 --> 00:00:49.790
divisible by 1, 2 and 4.

00:00:49.790 --> 00:00:50.580
We could keep going.

00:00:50.580 --> 00:00:56.220
5, well, 5 is only divisible
by 1 and 5, so 5 is prime.

00:00:56.220 --> 00:00:59.920
6 is not prime, because it's
divisible by 2 and 3.

00:00:59.920 --> 00:01:01.590
I think you get the
general idea.

00:01:01.590 --> 00:01:04.160
You move to 7, 7 is prime.

00:01:04.160 --> 00:01:06.470
It's only divisible
by 1 and 7.

00:01:06.470 --> 00:01:08.220
8 is not prime.

00:01:08.220 --> 00:01:11.440
9 you might be tempted to say
is prime, but remember, it's

00:01:11.440 --> 00:01:15.420
also divisible by 3,
so 9 is not prime.

00:01:15.420 --> 00:01:18.970
Prime is not the same thing
as odd numbers.

00:01:18.970 --> 00:01:21.400
Then if you move to 10,
10 is also not prime,

00:01:21.400 --> 00:01:23.560
divisible by 2 and 5.

00:01:23.560 --> 00:01:27.220
11, it's only divisible
by 1 and 11, so 11 is

00:01:27.220 --> 00:01:28.240
then a prime number.

00:01:28.240 --> 00:01:29.780
And we could keep going
on like this.

00:01:29.780 --> 00:01:31.570
People have written computer
programs looking for the

00:01:31.570 --> 00:01:33.260
highest prime and all of that.

00:01:33.260 --> 00:01:35.220
So now that we know what
a prime is, a prime

00:01:35.220 --> 00:01:39.240
factorization is breaking up
a number, like 75, into a

00:01:39.240 --> 00:01:41.620
product of prime numbers.

00:01:41.620 --> 00:01:43.180
So let's try to do that.

00:01:43.180 --> 00:01:45.530
So we're going to start with
75, and I'm going to do it

00:01:45.530 --> 00:01:49.080
using what we call a
factorization tree.

00:01:49.080 --> 00:01:51.750
So we first try to find just the
smallest prime number that

00:01:51.750 --> 00:01:53.890
will go into 75.

00:01:53.890 --> 00:01:55.430
Now, the smallest prime
number is 2.

00:01:55.430 --> 00:01:57.390
Does 2 go into 75?

00:01:57.390 --> 00:02:00.705
Well, 75 is an odd number, or
the number in the ones place,

00:02:00.705 --> 00:02:02.280
this 5, is an odd number.

00:02:02.280 --> 00:02:06.580
5 is not divisible by 2, so
2 will not go into 75.

00:02:06.580 --> 00:02:08.090
So then we could try 3.

00:02:08.090 --> 00:02:09.639
Does 3 go into 75?

00:02:09.639 --> 00:02:12.440
Well, 7 plus 5 is 12.

00:02:12.440 --> 00:02:15.480
12 is divisible by 3, so
3 will go into it.

00:02:15.480 --> 00:02:20.440
So 75 is 3 times
something else.

00:02:20.440 --> 00:02:22.990
And if you've ever dealt with
change, you know that if you

00:02:22.990 --> 00:02:25.890
have three quarters, you have
75 cents, or if you have 3

00:02:25.890 --> 00:02:28.930
times 25, you have 75.

00:02:28.930 --> 00:02:31.560
So this is 3 times 25.

00:02:31.560 --> 00:02:33.720
And you can multiply this out
if you don't believe me.

00:02:33.720 --> 00:02:35.960
Multiple out 3 times 25.

00:02:35.960 --> 00:02:40.470
Now, is 25 divisible by--
you can give up on 2.

00:02:40.470 --> 00:02:44.910
If 75 wasn't divisible by 2,
25's not going to be divisible

00:02:44.910 --> 00:02:46.000
by 2 either.

00:02:46.000 --> 00:02:48.730
But maybe 25 is divisible
by 3 again.

00:02:48.730 --> 00:02:52.290
So if you take the digits
2 plus 5, you get 7.

00:02:52.290 --> 00:02:57.700
7 is not divisible by 3, so 25
will not be divisible by 3.

00:02:57.700 --> 00:02:59.480
So we keep moving up: 5.

00:02:59.480 --> 00:03:01.430
Is 25 divisible by 5?

00:03:01.430 --> 00:03:01.980
Well, sure.

00:03:01.980 --> 00:03:03.590
It's 5 times 5.

00:03:03.590 --> 00:03:08.330
So 25 is 5 times 5.

00:03:08.330 --> 00:03:11.730
And we're done with our prime
factorization because now we

00:03:11.730 --> 00:03:13.390
have all prime numbers here.

00:03:13.390 --> 00:03:18.270
So we can write that 75
is 3 times 5 times 5.

00:03:18.270 --> 00:03:25.640
So 75 is equal to 3
times 5 times 5.

00:03:25.640 --> 00:03:27.350
We can say it's 3 times 25.

00:03:27.350 --> 00:03:29.400
25 is 5 times 5.

00:03:29.400 --> 00:03:33.370
3 times 25, 25 is 5 times 5.

00:03:33.370 --> 00:03:36.460
So this is a prime
factorization, but they want

00:03:36.460 --> 00:03:41.690
us to write our answer using
exponential notation.

00:03:41.690 --> 00:03:44.560
So that just means, if we have
repeated primes, we can write

00:03:44.560 --> 00:03:45.920
those as an exponent.

00:03:45.920 --> 00:03:48.480
So what is 5 times 5?

00:03:48.480 --> 00:03:52.380
5 times 5 is 5 multiplied
by itself two times.

00:03:52.380 --> 00:03:56.310
This is the same thing as
5 to the second power.

00:03:56.310 --> 00:03:58.380
So if we want to write our
answer using exponential

00:03:58.380 --> 00:04:03.420
notation, we could say this is
equal to 3 times 5 to the

00:04:03.420 --> 00:04:08.110
second power, which is the
same thing as 5 times 5.