Collateralized debt obligation overview

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.350
We've seen that an
investment bank can

00:00:02.350 --> 00:00:04.860
buy a bunch of mortgages,
which essentially makes them

00:00:04.860 --> 00:00:06.840
the lender to the
homeowners, and then it

00:00:06.840 --> 00:00:09.690
could stick those mortgages
inside of a special purpose

00:00:09.690 --> 00:00:10.365
entity.

00:00:10.365 --> 00:00:12.740
And then it could sell the
shares in that special purpose

00:00:12.740 --> 00:00:14.198
entity, and that
these shares would

00:00:14.198 --> 00:00:16.170
be called mortgage-backed
securities.

00:00:16.170 --> 00:00:18.253
And let's just say, just
for the sake of argument,

00:00:18.253 --> 00:00:23.360
when it sells these shares
it sells them at $10 a share,

00:00:23.360 --> 00:00:26.580
and it promises
dividends at $10 a share,

00:00:26.580 --> 00:00:32.080
the equivalent of an 8% yield.

00:00:32.080 --> 00:00:34.240
So maybe the homeowners
here are paying

00:00:34.240 --> 00:00:36.060
a higher than 8%
interest, some of them

00:00:36.060 --> 00:00:37.872
default, once you
average everything out,

00:00:37.872 --> 00:00:40.080
and the investment bank
keeps a little bit for itself

00:00:40.080 --> 00:00:43.100
and to do all the operations
and all the overhead,

00:00:43.100 --> 00:00:46.450
and so it can actually give
the investors an 8% yield.

00:00:46.450 --> 00:00:48.710
This might be good for a
whole class of investors.

00:00:48.710 --> 00:00:51.795
They might like the safety
profile, the risk profile

00:00:51.795 --> 00:00:54.420
of the special purpose entity of
this mortgage-backed security,

00:00:54.420 --> 00:00:57.420
and they might like the return,
and they might go for it.

00:00:57.420 --> 00:00:59.660
But there might be a
class of investors,

00:00:59.660 --> 00:01:03.560
may be very risk-averse
investors like pensions,

00:01:03.560 --> 00:01:07.820
that says that this
mortgage-backed security is

00:01:07.820 --> 00:01:08.619
too risky.

00:01:08.619 --> 00:01:10.160
They've looked at
what we're holding,

00:01:10.160 --> 00:01:12.743
and they are like, hey, some of
these are sub-prime mortgages,

00:01:12.743 --> 00:01:15.330
some of these are shady, some
of these are to risky borrowers.

00:01:15.330 --> 00:01:16.780
I don't like where
this is going.

00:01:16.780 --> 00:01:18.530
And even if they can't
look under the hood

00:01:18.530 --> 00:01:20.610
to see what this is,
the investment bank

00:01:20.610 --> 00:01:22.700
might have hired
a ratings agency.

00:01:22.700 --> 00:01:29.630
So maybe a ratings agency to
essentially look under the hood

00:01:29.630 --> 00:01:32.340
and tell investors what's there.

00:01:32.340 --> 00:01:34.980
So the ratings agency might look
at this special purpose entity

00:01:34.980 --> 00:01:36.521
and look at these
securities and say,

00:01:36.521 --> 00:01:42.140
look, I would say that these
securities should be rated BB.

00:01:42.140 --> 00:01:45.640
So not super safe, but
not super risky either,

00:01:45.640 --> 00:01:48.670
but for pensions that
is not safe enough.

00:01:48.670 --> 00:01:51.760
Now on the other hand, you might
have people who want more risk.

00:01:51.760 --> 00:01:54.300
So you might have, maybe
there's some risky hedge funds,

00:01:54.300 --> 00:01:55.940
and not all hedge
funds are risky,

00:01:55.940 --> 00:01:58.106
but let's say that there
are some risky hedge funds,

00:01:58.106 --> 00:02:00.000
and then they say that
this yield is too low.

00:02:03.690 --> 00:02:05.220
And the investment
bankers, they're

00:02:05.220 --> 00:02:07.840
very creative people,
they say, well, look,

00:02:07.840 --> 00:02:10.100
here's some people who
want to buy securities,

00:02:10.100 --> 00:02:12.515
but these securities
are too risky for them.

00:02:12.515 --> 00:02:13.890
And there's other
people who want

00:02:13.890 --> 00:02:16.330
to buy securities who are
able to take on more risk,

00:02:16.330 --> 00:02:18.730
but they say the
yield is too low.

00:02:18.730 --> 00:02:20.830
So instead of losing
out on these investors,

00:02:20.830 --> 00:02:23.150
why don't I just split
up this special purpose

00:02:23.150 --> 00:02:24.750
entity in a different way?

00:02:24.750 --> 00:02:27.780
Why don't I split
it up into tranches?

00:02:27.780 --> 00:02:30.670
So instead of all of the
securities being the same,

00:02:30.670 --> 00:02:32.160
why don't I put
them into classes?

00:02:36.030 --> 00:02:40.900
And they're often called
the senior tranche,

00:02:40.900 --> 00:02:43.400
the mezzanine tranche,
I'll just write "Mez"

00:02:43.400 --> 00:02:47.186
for short or the middle tranche,
and then the equity tranche.

00:02:47.186 --> 00:02:49.310
And the way it works, in
a mortgage-backed security

00:02:49.310 --> 00:02:51.470
everyone gets paid
the same amount.

00:02:51.470 --> 00:02:54.890
In this situation, when
you split it this way,

00:02:54.890 --> 00:02:57.110
the holders of the
senior tranche securities

00:02:57.110 --> 00:02:58.910
are going to get paid first.

00:02:58.910 --> 00:03:00.650
Only when they
are made whole are

00:03:00.650 --> 00:03:02.620
the owners of the
mezzanine tranche security

00:03:02.620 --> 00:03:06.200
is going to get paid, and only
when they are made whole will

00:03:06.200 --> 00:03:10.770
the owners of the equity
tranche security will get paid.

00:03:10.770 --> 00:03:12.460
And this scenario
right over here

00:03:12.460 --> 00:03:17.300
is called a collateralized
debt obligation, CDO.

00:03:17.300 --> 00:03:20.800
And it's really a derivative
security from the mortgages.

00:03:20.800 --> 00:03:23.855
We've sliced it and diced it
in a slightly different way.

00:03:23.855 --> 00:03:26.230
Now you might be saying, how
does this solve the problem?

00:03:26.230 --> 00:03:28.160
Well, now the ratings
agency will say,

00:03:28.160 --> 00:03:29.618
well, look if the
senior people are

00:03:29.618 --> 00:03:31.300
going to get paid
before everyone else,

00:03:31.300 --> 00:03:33.133
then I'm going to give
them a higher rating.

00:03:33.133 --> 00:03:35.960
And they can even get insurance
on this and get a credit

00:03:35.960 --> 00:03:39.190
default swap, and then maybe
they'll give it a AAA rating,

00:03:39.190 --> 00:03:42.390
and which means that the
pensions can now buy the senior

00:03:42.390 --> 00:03:44.932
rated CDO's.

00:03:44.932 --> 00:03:46.390
But they'll pay
them less interest.

00:03:46.390 --> 00:03:48.110
Maybe they'll pay them 5%.

00:03:48.110 --> 00:03:50.240
Maybe the mezzanine,
they get paid next,

00:03:50.240 --> 00:03:53.920
they'll get maybe still a BB
rating, and they'll get the 8%.

00:03:53.920 --> 00:03:57.350
And then the equity tranche,
they'll get a higher interest.

00:03:57.350 --> 00:04:01.890
So they'll get say a
15% interest in exchange

00:04:01.890 --> 00:04:04.690
for being the last
person to get paid.

00:04:04.690 --> 00:04:06.600
And maybe they don't
get any ratings at all.

00:04:06.600 --> 00:04:08.360
So you could almost view
this as a junk rating

00:04:08.360 --> 00:04:09.693
if you want to view it that way.

00:04:09.693 --> 00:04:11.230
But that makes
both people happy.

00:04:11.230 --> 00:04:14.050
Pensions get something
safe, lower yield.

00:04:14.050 --> 00:04:18.720
Hedge funds get something risky,
but it has a higher yield.


Mortgage-backed security overview

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.110
Let's say I'm some type of the Investment Bank.

00:00:03.110 --> 00:00:06.580
Can I go out there and I buy a whole bunch of mortgages.

00:00:06.950 --> 00:00:10.470
So right here are a whole bunch of people's mortgages.

00:00:10.780 --> 00:00:12.510
And when I say buy a mortgage

00:00:12.510 --> 00:00:15.450
it means the home-owners who borrow the money

00:00:15.710 --> 00:00:19.350
instead of the money going to the mortgage broker

00:00:19.350 --> 00:00:22.560
whom the home-owner think they got the loan from the servicer

00:00:22.560 --> 00:00:24.120
who they think they are paying the check to.

00:00:24.470 --> 00:00:27.280
Since I now have bought these mortgages,

00:00:27.280 --> 00:00:30.010
I've essentially become the lender to the home-owner,

00:00:30.010 --> 00:00:31.970
and so the home-owner's morgage payment

00:00:31.970 --> 00:00:38.970
will now come to me the Investment Bank .

00:00:38.970 --> 00:00:41.640
So I do this for a bunch of home-owners right over there.

00:00:41.640 --> 00:00:45.550
But I don't want to be the end holder of the paper.

00:00:45.550 --> 00:00:47.630
I really want to be the intermediary.

00:00:47.630 --> 00:00:53.630
So what I do is I set up a Special Purpose Entity,

00:00:53.630 --> 00:00:56.100
which is really just a corporation that I set up.

00:00:56.100 --> 00:00:58.520
And I stick all of the mortgages inside

00:00:58.520 --> 00:01:00.300
that Special Purpose Entity.

00:01:00.300 --> 00:01:02.900
So I stick them all inside of that.

00:01:02.900 --> 00:01:06.770
So now the Special Purpose Entity is the owner of the mortgages .

00:01:06.770 --> 00:01:08.340
The investment bank at least starting off

00:01:08.340 --> 00:01:11.810
with all of the shares of the Special Purpose Entity.

00:01:11.810 --> 00:01:15.130
And so the Investment Bank could then sell shares.

00:01:15.130 --> 00:01:16.710
It could split the Special Purpose Entity

00:01:16.710 --> 00:01:19.400
into a million or ten million shares

00:01:19.400 --> 00:01:25.180
and then sell those shares to investors.

00:01:25.180 --> 00:01:28.860
And these shares would be called Mortgage-Backed Securities.

00:01:28.860 --> 00:01:31.000
or sometimes MBSes.

00:01:31.000 --> 00:01:34.060
There are a part of a general class called Asset-Backed Securities

00:01:34.060 --> 00:01:36.840
because what is going to happen now is that all of this money

00:01:37.310 --> 00:01:42.410
flowing from home-owners will go the mortgages

00:01:42.410 --> 00:01:44.160
which is now on the Specail Purpose Entity.

00:01:44.160 --> 00:01:46.850
So all ended up inside the Special Purpose Entities .

00:01:46.850 --> 00:01:49.440
And some of them will default,some of them won't.

00:01:49.440 --> 00:01:51.800
But on average this Special Purpose Entity

00:01:51.800 --> 00:01:53.910
will then be able to pay a dividend.

00:01:53.910 --> 00:01:56.080
Essentially you can almost fueled on interest

00:01:56.480 --> 00:02:01.280
to the owner of the Mortgage-Backed Security.


Credit default swaps (CDS) intro

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:04.070
Let's say that I run some
type of a pension fund,

00:00:04.070 --> 00:00:08.340
and I have $1 billion that
I need to invest someplace.

00:00:08.340 --> 00:00:10.530
And I dig around a
little bit, and there's

00:00:10.530 --> 00:00:13.280
this Company A over
here, and it needs

00:00:13.280 --> 00:00:15.640
to borrow a billion
dollars, and it's

00:00:15.640 --> 00:00:19.280
willing to give 10%
interest in exchange

00:00:19.280 --> 00:00:21.240
for borrowing that
billion dollars.

00:00:21.240 --> 00:00:23.770
So if I give these
characters over here

00:00:23.770 --> 00:00:27.330
my billion dollars, if I lend
it to them, if I lend them

00:00:27.330 --> 00:00:29.680
the billion dollars,
on an annual basis,

00:00:29.680 --> 00:00:32.770
they're going to
give me 10% interest.

00:00:32.770 --> 00:00:34.510
There's a problem here though.

00:00:34.510 --> 00:00:39.190
The rating agencies, the
ones that in theory should

00:00:39.190 --> 00:00:44.666
be independent, they've only
given Company A a BB rating.

00:00:44.666 --> 00:00:45.790
And this is a pension fund.

00:00:45.790 --> 00:00:48.870
I have to only invest in the
safest of safest of securities.

00:00:48.870 --> 00:00:51.780
I have to invest only
in things that are AA.

00:00:51.780 --> 00:00:55.930
And so here might enter
a character like AIG,

00:00:55.930 --> 00:00:59.450
and obviously things have
changed since their heyday.

00:00:59.450 --> 00:01:03.470
But AIG, based on at
least Moody's perspective,

00:01:03.470 --> 00:01:05.440
has a AA rating.

00:01:05.440 --> 00:01:08.810
So AIG could say, hey, look
pension fund, why don't you

00:01:08.810 --> 00:01:10.430
lend them the money,
and what we're

00:01:10.430 --> 00:01:14.120
going to do is enter into
a credit default swap.

00:01:14.120 --> 00:01:17.030
And what that essentially
is, is a form of insurance.

00:01:17.030 --> 00:01:21.810
Of that 10% every year, why
don't you give us 1% of that?

00:01:21.810 --> 00:01:23.760
And in financial
lingo, that's sometimes

00:01:23.760 --> 00:01:26.370
referred to as 100 basis points.

00:01:26.370 --> 00:01:29.740
And in exchange for that, in
exchange for that 100 basis

00:01:29.740 --> 00:01:32.590
points a year, you could view
this as the insurance premium,

00:01:32.590 --> 00:01:41.110
we are going to ensure
that if for whatever reason

00:01:41.110 --> 00:01:46.480
Company A defaults on that
debt, you can give us the debt,

00:01:46.480 --> 00:01:50.180
you can give us
that debt security,

00:01:50.180 --> 00:01:53.060
and we will just give you
back your billion dollars.

00:01:53.060 --> 00:01:55.680
Now from the pension's point of
view, this sounds pretty good.

00:01:55.680 --> 00:02:01.200
They're now getting a net 9%
of interest, 10% minus the 1%.

00:02:01.200 --> 00:02:05.230
And they're essentially getting
to lend to a BB company,

00:02:05.230 --> 00:02:08.229
but it's in effect
like lending to a AA,

00:02:08.229 --> 00:02:11.950
because as long as AIG
is good for the money,

00:02:11.950 --> 00:02:16.380
then the pension fund is
going to get their money back

00:02:16.380 --> 00:02:18.400
one way or the other.

00:02:18.400 --> 00:02:21.130
Now where this gets
a little bit shady

00:02:21.130 --> 00:02:24.460
is AIG right here didn't
have to do anything.

00:02:24.460 --> 00:02:27.830
What's interesting about
credit default swaps--

00:02:27.830 --> 00:02:29.580
credit default swaps
sometimes referred to

00:02:29.580 --> 00:02:33.740
as CDSs-- is that even
though they are insurance,

00:02:33.740 --> 00:02:36.800
for all purposes
they are insurance,

00:02:36.800 --> 00:02:39.170
they are not regulated
like insurance.

00:02:39.170 --> 00:02:40.559
So typically an
insurance company

00:02:40.559 --> 00:02:42.100
when they insure
something, they have

00:02:42.100 --> 00:02:45.820
to set aside some money, in case
that thing actually happens.

00:02:45.820 --> 00:02:48.450
And they have to work out the
probabilities and all of that.

00:02:48.450 --> 00:02:51.190
Credit default swaps were
not regulated in that way.

00:02:51.190 --> 00:02:53.310
So AIG could do
this without having

00:02:53.310 --> 00:02:55.290
to set aside any type of money.

00:02:55.290 --> 00:02:56.970
And they could do
this over and over

00:02:56.970 --> 00:02:59.700
and over and over again,
kind of snowballing

00:02:59.700 --> 00:03:01.640
all of their
potential liabilities.

00:03:01.640 --> 00:03:04.070
And so you could imagine,
it's really good money

00:03:04.070 --> 00:03:06.210
while no one is
defaulting, but all

00:03:06.210 --> 00:03:09.020
of a sudden when people
start defaulting, then

00:03:09.020 --> 00:03:11.530
all of a sudden AIG is
going to be in trouble.

00:03:11.530 --> 00:03:13.670
And all the people who
thought they had, in effect,

00:03:13.670 --> 00:03:18.656
AA debt because AIG was
insuring it, might not.


Futures fair value in the pre-market

WEBVTT
Kind: captions
Language: en

00:00:00.965 --> 00:00:05.770
Voiceover: The fair value of a futures
contract is the price of the contract

00:00:05.770 --> 00:00:09.461
at which a buyer of the
stock would be neutral

00:00:09.461 --> 00:00:12.665
between buying it on in
an actual stock exchange

00:00:12.665 --> 00:00:16.497
and actually buying the stock and
agreeing to buy the futures contract.

00:00:16.497 --> 00:00:20.114
Or a seller of the stock would be
neutral between selling it right now

00:00:20.114 --> 00:00:24.257
in the actual exchange
versus agreeing to go short

00:00:24.257 --> 00:00:27.284
or agreeing to be on the selling
side for a futures contract.

00:00:27.284 --> 00:00:30.821
The fair value attends to be
quoted for the front month

00:00:30.821 --> 00:00:33.463
or the next expiring futures contract.

00:00:33.463 --> 00:00:37.045
Next expiring futures contract.

00:00:37.045 --> 00:00:40.645
To see how this works, why
someone would be neutral

00:00:40.645 --> 00:00:46.577
between say, buying something now
for a $100 and agreeing to buy it

00:00:46.593 --> 00:00:50.031
maybe next month at a
$102, think of it this way.

00:00:50.031 --> 00:00:54.006
If they want to just hold that stock
a month, or two, or three from now,

00:00:54.006 --> 00:00:58.964
they could pay a $100 right now
or they could take their $100

00:00:58.964 --> 00:01:01.411
so they could either just buy it right now

00:01:01.411 --> 00:01:04.466
or they could stick it maybe
in the money market account.

00:01:04.466 --> 00:01:07.643
They could get some interest
for the next few weeks

00:01:07.643 --> 00:01:10.327
and then they could buy
the futures contract.

00:01:10.327 --> 00:01:16.654
If they got interest of $2, if they kept
that $100 say in a money market account

00:01:16.654 --> 00:01:21.437
for the next, until the actual
contract date for that futures contract

00:01:21.437 --> 00:01:22.486
they'd be neutral.

00:01:22.486 --> 00:01:26.491
I could spend a $100 today or I could
put a $100 in a money market account,

00:01:26.491 --> 00:01:30.653
get $2 in interest and I'm not going
to worry about the dividends right now.

00:01:30.653 --> 00:01:34.088
If a dividend happens then, obviously
you would want to pay less for this thing

00:01:34.088 --> 00:01:35.735
where you didn't get the dividend.

00:01:35.735 --> 00:01:39.324
Assuming no dividends, you say well or
I'd be willing to pay a $102 for that.

00:01:39.324 --> 00:01:44.854
If this was only a $101, if the price
of the futures contract was a $101,

00:01:44.854 --> 00:01:48.233
then you say, "Wait, if the price
of the futures contract is a $101,"

00:01:48.233 --> 00:01:53.182
"I would rather put my money in a money
market account, get $2 of interest."

00:01:53.182 --> 00:01:55.824
"And buy it in a month for a $101."

00:01:55.824 --> 00:01:58.730
I'll essentially get an
extra dollar if I do that.

00:01:58.730 --> 00:02:07.400
If the price is at a $103, you say,
"Well gee, if I want to hold that stock"

00:02:07.400 --> 00:02:10.671
"a month or two from now. I'm
way better buying it right now."

00:02:10.671 --> 00:02:14.412
Or another way to think about
it if the price is at a $103

00:02:14.412 --> 00:02:15.913
and I'm a seller of the stock.

00:02:15.913 --> 00:02:17.963
Instead of selling it today for a $100,

00:02:17.963 --> 00:02:20.858
let's say I really need a $100 right now.

00:02:20.858 --> 00:02:23.565
I'm better off borrowing a $100 right now.

00:02:23.565 --> 00:02:29.405
Paying maybe $2 in interest and then
selling it a month later for a $103.

00:02:29.405 --> 00:02:33.641
The fair value is the price which
a buyer or seller is neutral

00:02:33.641 --> 00:02:37.603
between buying and selling the stock
or entering into a futures contract.

00:02:37.603 --> 00:02:40.403
I'm going to go a little bit
more detail in the next video

00:02:40.403 --> 00:02:44.921
but it's main use is an
indicator of what's likely
to happen once a market opens

00:02:44.921 --> 00:02:50.845
and that's because futures markets
trade have much longer trading hours,

00:02:50.845 --> 00:02:53.699
sometimes 24 hours than
traditional stock markets.

00:02:53.699 --> 00:02:57.895
If you could imagine, if the
price of a futures contract

00:02:57.895 --> 00:03:00.004
is trading below its fair value,

00:03:00.004 --> 00:03:01.631
the only reason why that would happen

00:03:01.631 --> 00:03:05.467
is if a lot of people are actually
expecting, once the market opens

00:03:05.467 --> 00:03:07.739
that the stock price is going to go down.


Fraction to decimal

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.670
Write 7/8 as a decimal.

00:00:03.670 --> 00:00:05.670
And so the main
realization here is

00:00:05.670 --> 00:00:13.190
that 7/8 is the same
thing as 7 divided by 8,

00:00:13.190 --> 00:00:16.890
which is the same thing
as 7 divided by 8.

00:00:16.890 --> 00:00:21.180
These are all different ways
of writing the same thing.

00:00:21.180 --> 00:00:23.119
So let's actually
divide 8 into 7.

00:00:23.119 --> 00:00:25.660
And I'll do it down here just
so I have some more real estate

00:00:25.660 --> 00:00:26.720
to work with.

00:00:26.720 --> 00:00:29.500
I'm going to divide 8 into 7.

00:00:29.500 --> 00:00:31.250
And I'm going to add
a decimal point here,

00:00:31.250 --> 00:00:32.875
just because we know
that this value is

00:00:32.875 --> 00:00:34.010
going to be less than 1.

00:00:34.010 --> 00:00:35.560
7/8 is less than 1.

00:00:35.560 --> 00:00:39.990
We're going to have
some digits to the right

00:00:39.990 --> 00:00:41.260
of the decimal point.

00:00:41.260 --> 00:00:43.350
And let me put the decimal
point right up here,

00:00:43.350 --> 00:00:45.470
right above the
decimal point in 7.

00:00:45.470 --> 00:00:46.770
And then we start dividing.

00:00:46.770 --> 00:00:49.210
And now this really turns
into a long division problem.

00:00:49.210 --> 00:00:52.260
And we just have to make sure we
keep track of the decimal sign.

00:00:52.260 --> 00:00:56.090
So 8 goes into-- it
doesn't go into 7 at all,

00:00:56.090 --> 00:00:58.590
but it does go into 70.

00:00:58.590 --> 00:01:03.220
So 8 goes into 70 eight times.

00:01:03.220 --> 00:01:07.040
So it goes into 70 eight times.

00:01:07.040 --> 00:01:09.830
8 times 8 is 64.

00:01:09.830 --> 00:01:11.290
And then you subtract.

00:01:11.290 --> 00:01:14.540
70 minus 64 is 6.

00:01:14.540 --> 00:01:17.042
And then bring down
another 0 because we still

00:01:17.042 --> 00:01:17.750
have a remainder.

00:01:17.750 --> 00:01:19.840
We want to get to the point
that we have no remainders.

00:01:19.840 --> 00:01:21.839
Assuming that this thing
doesn't repeat forever.

00:01:21.839 --> 00:01:24.160
And there's other ways
we can deal with that.

00:01:24.160 --> 00:01:26.134
8 goes into 60?

00:01:26.134 --> 00:01:26.800
Well, let's see.

00:01:26.800 --> 00:01:30.120
It doesn't go into it eight
times because that's 64.

00:01:30.120 --> 00:01:33.310
8 goes into 60 seven times.

00:01:33.310 --> 00:01:35.210
7 times 8 is 56.

00:01:38.050 --> 00:01:39.300
And then we subtract again.

00:01:39.300 --> 00:01:42.740
60 minus 56 is 4.

00:01:42.740 --> 00:01:46.170
And now, we can bring down
another 0 right over here.

00:01:46.170 --> 00:01:47.370
And 8 goes into 40?

00:01:47.370 --> 00:01:50.340
Well, it goes into 40
exactly five times.

00:01:50.340 --> 00:01:52.480
5 times 8 is 40.

00:01:52.480 --> 00:01:54.090
And we have nothing.

00:01:54.090 --> 00:01:56.230
We have nothing left over.

00:01:56.230 --> 00:01:57.320
And so we're done.

00:01:57.320 --> 00:02:01.190
7 divided by 8 or 7/8 is
equal to 7 divided by 8,

00:02:01.190 --> 00:02:04.060
which is equal to 0.875.

00:02:04.060 --> 00:02:07.000
But I'll put a leading 0 here
just so it makes it clear

00:02:07.000 --> 00:02:09.509
that this is where
the decimal is.

00:02:09.509 --> 00:02:13.950
0.875.

00:02:13.950 --> 00:02:16.830
And we are done.


Comparing Rational Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.813 --> 00:00:03.460
- We're told to look at
the rational numbers below,

00:00:03.460 --> 00:00:05.596
order them from least to greatest.

00:00:05.596 --> 00:00:07.094
They really didn't have to
tell us this first sentence,

00:00:07.094 --> 00:00:09.311
I would have known to look
at the rational numbers

00:00:09.311 --> 00:00:11.110
to order them from least to greatest.

00:00:11.110 --> 00:00:13.722
Well anyway, they tell
us 1/2, negative five,

00:00:13.722 --> 00:00:17.460
three point three, zero, 21 over 12,

00:00:17.460 --> 00:00:20.247
negative five point
five, and two and 1/8ths.

00:00:20.247 --> 00:00:22.708
So the easiest way to
visualize this might just be to

00:00:22.708 --> 00:00:25.865
make a number line that's long enough that

00:00:25.865 --> 00:00:28.384
it actually can contain
all of these numbers,

00:00:28.384 --> 00:00:30.625
and then we can think about
how we can compare them.

00:00:30.625 --> 00:00:34.224
So let me just draw a huge
number line over here.

00:00:34.224 --> 00:00:38.043
So, take up almost all the entire screen.

00:00:38.043 --> 00:00:40.446
I'll stick with, we have
some negative numbers here,

00:00:40.446 --> 00:00:42.524
we go as low as negative five point five,

00:00:42.524 --> 00:00:44.498
and we have some positive numbers here,

00:00:44.498 --> 00:00:47.087
looks like we go as high
as three point three.

00:00:47.087 --> 00:00:48.979
This thing is still a
little less than two,

00:00:48.979 --> 00:00:50.685
so we go about as high
as three point three,

00:00:50.685 --> 00:00:53.506
so I can put, I can safely I think put

00:00:53.506 --> 00:00:55.561
zero right here in the middle,

00:00:55.561 --> 00:00:57.024
I can go a little bit to
the right since we have

00:00:57.024 --> 00:00:59.183
our negative numbers go more negative.

00:00:59.183 --> 00:01:01.702
So zero, and let's make this

00:01:01.702 --> 00:01:06.601
negative one, negative two,
negative three, negative four,

00:01:06.601 --> 00:01:10.386
negative five, and well
that should be enough.

00:01:10.386 --> 00:01:12.882
Negative five, and then in the
positive direction we have,

00:01:12.882 --> 00:01:17.455
one, two, three, in
the positive direction.

00:01:17.455 --> 00:01:19.418
And let's see if we can plot these.

00:01:19.418 --> 00:01:24.015
So, to start off, to start
off let's look at 1/2.

00:01:24.015 --> 00:01:26.360
Where does 1/2 sit, so it sits,

00:01:26.360 --> 00:01:30.562
let me actually make the
scale a little bit better.

00:01:30.562 --> 00:01:35.490
So this is one, two and three, and four.

00:01:35.490 --> 00:01:38.294
Alright, so let's start with 1/2.

00:01:38.294 --> 00:01:41.997
1/2 is directly in-between zero and one,

00:01:41.997 --> 00:01:43.402
it is half of a whole.

00:01:43.402 --> 00:01:46.049
This right here would be one whole.

00:01:46.049 --> 00:01:48.103
This would be one whole,
let me label that.

00:01:48.103 --> 00:01:50.123
This over here is one.

00:01:50.123 --> 00:01:52.944
So 1/2 is directly between zero and one.

00:01:52.944 --> 00:01:56.021
So 1/2 is gonna sit right over here.

00:01:56.021 --> 00:01:59.562
So that is, let me write
that a little bit bigger,

00:01:59.562 --> 00:02:01.047
you probably have trouble reading that,

00:02:01.047 --> 00:02:03.079
Alright, one over two,

00:02:03.079 --> 00:02:05.366
which is also zero point five.

00:02:05.366 --> 00:02:07.398
So this is also zero point five,

00:02:07.398 --> 00:02:09.185
anyway that's where it sits.

00:02:09.185 --> 00:02:11.893
Then we have negative five.

00:02:11.893 --> 00:02:14.839
Negative five, well this is
negative one, negative two,

00:02:14.839 --> 00:02:17.625
negative three, negative
four, negative five.

00:02:17.625 --> 00:02:21.096
Negative five sits right over there.

00:02:21.096 --> 00:02:24.440
And then we have three point three.

00:02:24.440 --> 00:02:27.597
Positive three point three,
I'll do that in blue.

00:02:27.597 --> 00:02:29.106
Positive three point three.

00:02:29.106 --> 00:02:31.579
So this is one, two, three,

00:02:31.579 --> 00:02:33.738
and then we want to do
another point three.

00:02:33.738 --> 00:02:36.385
So point three is about
a third of the way,

00:02:36.385 --> 00:02:37.836
a little less than a third of the way,

00:02:37.836 --> 00:02:40.181
it would be three point
three three three forever,

00:02:40.181 --> 00:02:41.563
if it was a third of the way.

00:02:41.563 --> 00:02:43.084
So a third of the way,

00:02:43.084 --> 00:02:45.185
that looks like about right over here.

00:02:45.185 --> 00:02:48.261
This is three, this
right over here would be

00:02:48.261 --> 00:02:51.721
three point three, let me label.

00:02:51.721 --> 00:02:53.717
What I'm gonna do is I'm
gonna label the numbers

00:02:53.717 --> 00:02:57.734
on the number line up here
so it's one, two, three,

00:02:57.734 --> 00:03:01.855
four, this is zero,
negative one, negative two,

00:03:01.855 --> 00:03:05.721
negative three, negative
four, negative five,

00:03:05.721 --> 00:03:07.393
and so on and so forth.

00:03:07.393 --> 00:03:09.633
And then we get to zero,
which is one of the numbers

00:03:09.633 --> 00:03:11.177
that we've already written down.

00:03:11.177 --> 00:03:14.382
Zero is obviously right over
there on the number line,

00:03:14.382 --> 00:03:16.436
so I'll just write this zero
in orange to make it clear,

00:03:16.436 --> 00:03:17.737
it's this zero.

00:03:17.737 --> 00:03:20.997
Then we have 21 over 12,
which is an improper fraction,

00:03:20.997 --> 00:03:23.019
and to think about where
we should place that

00:03:23.019 --> 00:03:24.818
on the number line,

00:03:24.818 --> 00:03:27.256
to think about where to
place it on the number line,

00:03:27.256 --> 00:03:29.137
let me do this in this blue color.

00:03:29.137 --> 00:03:30.692
To think about where to
place this on the number line

00:03:30.692 --> 00:03:32.759
let's change it into a mixed
number, makes it a little bit

00:03:32.759 --> 00:03:34.895
easier to visualize,
at least for my brain.

00:03:34.895 --> 00:03:39.399
So 12 goes into 21, well
it goes into it one time.

00:03:39.399 --> 00:03:41.512
One times 12 is 12.

00:03:41.512 --> 00:03:43.857
If you subtract you get a remainder of,

00:03:43.857 --> 00:03:46.794
well we could actually
regroup here, or borrow,

00:03:46.794 --> 00:03:48.454
if you don't want to do this in your head,

00:03:48.454 --> 00:03:50.835
you would get nine, but let's do this.

00:03:50.835 --> 00:03:54.409
So if we borrow one from the
two, the two becomes a one,

00:03:54.409 --> 00:03:57.114
this becomes 11, or we're
really regrouping a 10.

00:03:57.114 --> 00:04:00.713
Anyway, 11 minus two is
nine, one minus one is zero.

00:04:00.713 --> 00:04:02.234
So we have a remainder of nine.

00:04:02.234 --> 00:04:05.438
So this thing, written as
a mixed number, 21 over 12

00:04:05.438 --> 00:04:09.536
written as a mixed number
is one and 9/12ths.

00:04:09.536 --> 00:04:11.115
You get one 12/12ths in there

00:04:11.115 --> 00:04:13.193
and then you get 9/12ths left over.

00:04:13.193 --> 00:04:15.712
So one and 9/12ths we can also write that,

00:04:15.712 --> 00:04:18.289
actually we could've simplified
this right from the get go,

00:04:18.289 --> 00:04:22.631
cause both 21 and 12
are divisible by three,

00:04:22.631 --> 00:04:26.372
but now we can just divide
nine, we can simplify

00:04:26.372 --> 00:04:29.040
9/12ths, divide both the numerator

00:04:29.040 --> 00:04:31.693
and the denominator by three,

00:04:31.693 --> 00:04:36.016
we then get one and three
over four, one and 3/4ths.

00:04:36.016 --> 00:04:37.514
And just to make it clear,
I could have simplified

00:04:37.514 --> 00:04:40.734
this right from the get
go, 21 divided by three,

00:04:40.734 --> 00:04:44.328
is equal to seven, and
12 divided by three,

00:04:44.328 --> 00:04:46.093
is equal to four.

00:04:46.093 --> 00:04:48.345
So this is the same thing as 7/4ths,

00:04:48.345 --> 00:04:51.491
and if you were to divide four into seven,

00:04:51.491 --> 00:04:53.429
four goes into seven one time,

00:04:53.429 --> 00:04:55.566
subtract, one times four is four,

00:04:55.566 --> 00:04:59.025
subtract to get a remainder
of three, one and 3/4ths.

00:04:59.025 --> 00:05:01.533
So going back to where do we plot this?

00:05:01.533 --> 00:05:05.073
Well it is, it's one,
and then we have 3/4ths,

00:05:05.073 --> 00:05:06.443
we're going to go three
fourths of the way.

00:05:06.443 --> 00:05:09.891
This is half way, this is
one fourths, two fourths,

00:05:09.891 --> 00:05:12.526
three fourths, would be right over there.

00:05:12.526 --> 00:05:17.526
So this is our 21 over 12,
which is the same thing as

00:05:18.246 --> 00:05:21.465
7/4ths, which is the same
thing as one and 3/4ths.

00:05:21.465 --> 00:05:24.228
And then we have negative five point five.

00:05:24.228 --> 00:05:27.833
Negative five point five,
I'll do that in magenta again,

00:05:27.833 --> 00:05:29.081
running out of colors.

00:05:29.081 --> 00:05:31.089
Negative five point five,
well this is negative five,

00:05:31.089 --> 00:05:32.784
so negative five point
five is going to be between

00:05:32.784 --> 00:05:35.106
negative five and negative six.

00:05:35.106 --> 00:05:37.381
So let me add negative
six to our number line,

00:05:37.381 --> 00:05:39.448
right here just to make it clear.

00:05:39.448 --> 00:05:41.781
So let me go a little bit further,

00:05:41.781 --> 00:05:44.080
let's say that this is negative six.

00:05:44.080 --> 00:05:46.401
Negative six, and our
number line will keep going

00:05:46.401 --> 00:05:48.201
to smaller values.

00:05:48.201 --> 00:05:50.279
Let me scroll to the left a little bit.

00:05:50.279 --> 00:05:52.682
Negative six, so if we go
to negative five point five,

00:05:52.682 --> 00:05:56.722
it's smack dab in-between
negative five and negative six.

00:05:56.722 --> 00:06:00.123
So this is negative five point five,

00:06:00.123 --> 00:06:02.317
right over there.

00:06:02.317 --> 00:06:05.390
And then finally we have two and 1/8ths.

00:06:05.390 --> 00:06:08.505
I'll do that in orange again,

00:06:08.505 --> 00:06:09.782
or I'll do it in blue.

00:06:09.782 --> 00:06:13.984
Two and 1/8ths, so it's
two and then 1/8th.

00:06:13.984 --> 00:06:15.958
And so if we want to find the exact place

00:06:15.958 --> 00:06:17.664
we could divide this into eighths,

00:06:17.664 --> 00:06:21.019
this would be 4/8ths,
this would be 2/8ths,

00:06:21.019 --> 00:06:23.260
and that would be 6/8ths,

00:06:23.260 --> 00:06:25.744
and then 1/8th would sit right over here.

00:06:25.744 --> 00:06:29.239
So that right over there is two and 1/8th.

00:06:29.239 --> 00:06:31.502
So we've actually plotted,
as best as we could,

00:06:31.502 --> 00:06:32.559
the exact locations.

00:06:32.559 --> 00:06:33.940
You didn't have to plot
the exact locations

00:06:33.940 --> 00:06:35.415
if you were just trying to order them,

00:06:35.415 --> 00:06:39.002
but it doesn't hurt to
see exactly where they sit

00:06:39.002 --> 00:06:40.464
when we order them.

00:06:40.464 --> 00:06:41.939
So now we've essentially
ordered them cause we

00:06:41.939 --> 00:06:43.831
stuck them all on this number line.

00:06:43.831 --> 00:06:46.466
The order is negative five
point five is the smallest,

00:06:46.466 --> 00:06:49.345
then negative five, then a zero,

00:06:49.345 --> 00:06:54.345
and then positive 1/2, then 21
over 12, then two and 1/8th,

00:06:54.813 --> 00:06:57.228
and then three point three.

00:06:57.228 --> 00:06:58.679
And we're done.


Comparing decimals: place value difference

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:08.060
We're asked to compare
156.378 to 156.348.

00:00:08.060 --> 00:00:12.550
So I assume that they're asking
us which of these is greater.

00:00:12.550 --> 00:00:14.430
And so let's rewrite
these numbers.

00:00:14.430 --> 00:00:23.960
156.378, and we want to
compare that to 156.348.

00:00:23.960 --> 00:00:25.650
And it might already
be obvious to you

00:00:25.650 --> 00:00:26.733
which of these is greater.

00:00:26.733 --> 00:00:28.710
But I'll work through
it step-by-step, just

00:00:28.710 --> 00:00:30.330
in case it's not.

00:00:30.330 --> 00:00:32.629
So let's first go to
the hundreds place.

00:00:32.629 --> 00:00:34.420
And you could view this
as, to some degree,

00:00:34.420 --> 00:00:36.760
the most significant
place, because this

00:00:36.760 --> 00:00:39.690
is where they have the largest--
the 1 in the hundreds place

00:00:39.690 --> 00:00:43.019
represents a lot more than
a 1 in the tens place or a 1

00:00:43.019 --> 00:00:43.810
in the six's place.

00:00:43.810 --> 00:00:46.010
A 1 in the hundreds
place represents 100.

00:00:46.010 --> 00:00:47.730
It's this many hundreds.

00:00:47.730 --> 00:00:50.570
But we have the same number of
hundreds in both situations.

00:00:50.570 --> 00:00:53.140
We have 1 hundred and
we have 1 hundred.

00:00:53.140 --> 00:00:54.740
Let's go to the tens place.

00:00:54.740 --> 00:00:57.040
We have 5 tens
here, which is 50.

00:00:57.040 --> 00:00:58.610
And we have 50 right over here.

00:00:58.610 --> 00:01:02.380
So, so far they're
equal, so far,

00:01:02.380 --> 00:01:04.670
if you just look at the
hundreds and the tens place.

00:01:04.670 --> 00:01:05.880
Let's go the ones place.

00:01:05.880 --> 00:01:08.240
We have a 6 here in
the ones place, 6 ones,

00:01:08.240 --> 00:01:09.720
which is really just 6.

00:01:09.720 --> 00:01:11.140
And you also have 6 ones.

00:01:11.140 --> 00:01:13.460
So, so far, they're equal again.

00:01:13.460 --> 00:01:17.850
Let's go to the tenths
place-- 3 tenths, 3 tenths.

00:01:17.850 --> 00:01:19.280
Let's go to the
hundredths place.

00:01:19.280 --> 00:01:22.090
So we're going into
smaller and smaller places.

00:01:22.090 --> 00:01:26.400
So then you have
the 7 hundredths.

00:01:26.400 --> 00:01:28.100
And now we have 4
hundredths here.

00:01:28.100 --> 00:01:31.399
So this has less hundredths
than this does over here.

00:01:31.399 --> 00:01:33.690
And if we go to the thousandths
place, these are equal.

00:01:33.690 --> 00:01:35.773
But it doesn't matter,
even if these were unequal.

00:01:35.773 --> 00:01:39.110
It doesn't matter, even if this
one was larger than this one,

00:01:39.110 --> 00:01:41.150
because in the more
significant place,

00:01:41.150 --> 00:01:43.770
this number was larger
than this number.

00:01:43.770 --> 00:01:45.120
So 7 is greater than 4.

00:01:45.120 --> 00:01:47.550
We have 7 hundredths here.

00:01:47.550 --> 00:01:49.080
We have 4 hundredths here.

00:01:49.080 --> 00:01:53.860
So we get this number is greater
than this number down here.

00:01:53.860 --> 00:02:02.950
Or 156.378 is
greater than 156.348.

00:02:02.950 --> 00:02:07.167
And in particular, it's because
of the hundredths place.


Comparing two decimal numbers using a number line (example)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.790
Use a number line to
compare 11.5 and 11.7.

00:00:04.790 --> 00:00:06.630
So let's draw a
number line here.

00:00:06.630 --> 00:00:09.310
And I'm going to focus
between 11 and 12,

00:00:09.310 --> 00:00:11.430
because that's where our
two numbers are sitting.

00:00:11.430 --> 00:00:14.550
They're 11, and then something
else, some number of 10ths.

00:00:14.550 --> 00:00:16.390
So this right here is 11.

00:00:16.390 --> 00:00:19.700
And this right here would be 12.

00:00:19.700 --> 00:00:21.710
And then let me draw the 10ths.

00:00:21.710 --> 00:00:23.970
So this would be
smack dab in between.

00:00:23.970 --> 00:00:28.190
So that would be 11 and
5/10, or that would be 11.5.

00:00:28.190 --> 00:00:30.150
Well, I've already
done the first part.

00:00:30.150 --> 00:00:33.090
I've figured out where 11.5 is.

00:00:33.090 --> 00:00:35.720
It's smack dab in
between 11 and 12.

00:00:35.720 --> 00:00:36.907
It's 11 and 5/10.

00:00:36.907 --> 00:00:38.240
But let me find everything else.

00:00:38.240 --> 00:00:40.240
Let me mark everything
else on this number line.

00:00:40.240 --> 00:00:47.200
So that's 1/10, 2/10, 3/10,
4/10, 5/10, 6/10, 7/10, 8/10,

00:00:47.200 --> 00:00:50.622
9/10, and then 10/10,
right on the 12.

00:00:50.622 --> 00:00:52.080
It's not completely
drawn to scale.

00:00:52.080 --> 00:00:53.621
I'm hand-drawing it
as good as I can.

00:00:53.621 --> 00:00:56.510
So where is 11.7 going to be?

00:00:56.510 --> 00:01:00.676
Well, this is 11.5, this
is 11.6, this is 11.7.

00:01:00.676 --> 00:01:01.970
11 and 7/10.

00:01:01.970 --> 00:01:06.450
1/10, 2/10, 3/10,
4/10, 5/10, 6/10, 7/10.

00:01:06.450 --> 00:01:09.110
This is 11.7.

00:01:09.110 --> 00:01:10.900
And the way we've
drawn our number line,

00:01:10.900 --> 00:01:13.370
we are increasing as
we go to the right.

00:01:13.370 --> 00:01:15.980
11.7 is to the right of 11.5.

00:01:15.980 --> 00:01:19.510
It's clearly greater than 11.5.

00:01:19.510 --> 00:01:21.920
11.7 is greater than 11.5.

00:01:21.920 --> 00:01:23.870
And really,
seriously, you didn't

00:01:23.870 --> 00:01:26.200
have to draw a number
line to figure that out.

00:01:26.200 --> 00:01:28.260
They're both 11
and something else.

00:01:28.260 --> 00:01:29.650
This is 11 and 5/10.

00:01:29.650 --> 00:01:31.780
This is 11 and 7/10.

00:01:31.780 --> 00:01:34.330
So clearly, this one
is going to be greater.

00:01:34.330 --> 00:01:38.462
You both have 11, but this
has 7/10, as opposed to 5/10.


Example: Evaluating expressions with 2 variables

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.880
We're asked to evaluate
the expression a squared

00:00:03.880 --> 00:00:11.860
plus 10b minus 8 when a is
equal to 7 and b is equal to 4.

00:00:11.860 --> 00:00:13.800
So to evaluate the
expression, we really just

00:00:13.800 --> 00:00:18.620
have to substitute a with 7 and
substitute b with negative 4

00:00:18.620 --> 00:00:21.090
because they're saying evaluate
it when a is equal to 7

00:00:21.090 --> 00:00:22.610
and b is equal to negative 4.

00:00:22.610 --> 00:00:23.940
So let's do that.

00:00:23.940 --> 00:00:26.860
So a everywhere we see
an a in the expression,

00:00:26.860 --> 00:00:27.930
we should put a 7 there.

00:00:27.930 --> 00:00:34.920
So instead of a squared, we
should write 7 squared plus--

00:00:34.920 --> 00:00:39.170
I'll do it in that same
color-- plus 10 times b.

00:00:39.170 --> 00:00:42.030
But instead of a b
there, we are now

00:00:42.030 --> 00:00:45.110
going to substitute it with
b is equal to negative 4.

00:00:45.110 --> 00:00:52.610
So 10 times negative 4 instead
of the b right over there.

00:00:52.610 --> 00:00:55.120
And then we have the minus 8.

00:00:55.120 --> 00:00:57.770
And now we just have
to evaluate this thing.

00:00:57.770 --> 00:01:01.780
7 squared is 49.

00:01:01.780 --> 00:01:05.110
And then 10 times negative 4.

00:01:05.110 --> 00:01:07.180
Remember, order of
operations, multiplication

00:01:07.180 --> 00:01:08.480
comes before addition.

00:01:08.480 --> 00:01:09.960
So we have to multiply this.

00:01:09.960 --> 00:01:12.990
10 times negative
4 is negative 40.

00:01:12.990 --> 00:01:16.670
So it's negative 40.

00:01:16.670 --> 00:01:21.410
And then we have minus
8 back over here.

00:01:21.410 --> 00:01:24.030
And so we get 49
plus negative 40,

00:01:24.030 --> 00:01:27.560
which is really the same
thing as 49 minus 40

00:01:27.560 --> 00:01:30.030
is going to be 9.

00:01:30.030 --> 00:01:32.220
And then we're going to
subtract 8 from that.

00:01:32.220 --> 00:01:34.340
And so we get 1.

00:01:34.340 --> 00:01:38.450
49 minus 40 is 9 minus 8 is 1.

00:01:38.450 --> 00:01:39.710
And we are done.

00:01:39.710 --> 00:01:42.630
We've evaluated the expression
when a is equal to 7

00:01:42.630 --> 00:01:45.520
and b is equal to negative 4.


Finding common denominators

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.680
We're asked to rewrite
the following two

00:00:02.680 --> 00:00:06.720
fractions as fractions with
a least common denominator.

00:00:10.960 --> 00:00:13.260
So a least common
denominator for two fractions

00:00:13.260 --> 00:00:17.250
is really just going to be the
least common multiple of both

00:00:17.250 --> 00:00:19.630
of these denominators over here.

00:00:19.630 --> 00:00:21.480
And the value of
doing that is then

00:00:21.480 --> 00:00:24.530
if you can make these
a common denominator,

00:00:24.530 --> 00:00:26.400
then you can add
the two fractions.

00:00:26.400 --> 00:00:28.046
And we'll see that
in other videos.

00:00:28.046 --> 00:00:30.545
But first of all, let's just
find the least common multiple.

00:00:33.330 --> 00:00:35.920
Let me write it out
because sometimes LCD

00:00:35.920 --> 00:00:37.400
could meet other things.

00:00:37.400 --> 00:00:48.160
So least common denominator
of these two things

00:00:48.160 --> 00:00:51.360
is going to be the same thing
as the least common multiple

00:00:51.360 --> 00:00:53.580
of the two
denominators over here.

00:00:53.580 --> 00:00:57.342
The least common
multiple of 8 and 6.

00:00:57.342 --> 00:00:59.800
And a couple of ways to think
about least common multiple--

00:00:59.800 --> 00:01:02.380
you literally could just
take the multiples of 8 and 6

00:01:02.380 --> 00:01:05.370
and see what they're
smallest common multiple is.

00:01:05.370 --> 00:01:07.180
So let's do it that way first.

00:01:07.180 --> 00:01:13.760
So multiples of six
are 6, 12, 18, 24 30.

00:01:13.760 --> 00:01:17.050
And I could keep going if we
don't find any common multiples

00:01:17.050 --> 00:01:20.360
out of this group here with
any of the multiples in eight.

00:01:20.360 --> 00:01:25.687
And the multiples of
eight are 8, 16, 24,

00:01:25.687 --> 00:01:26.895
and it looks like we're done.

00:01:26.895 --> 00:01:29.150
And we could keep
going obviously-- 32,

00:01:29.150 --> 00:01:30.490
so on and so forth.

00:01:30.490 --> 00:01:32.290
But I found a common
multiple and this

00:01:32.290 --> 00:01:33.740
is their smallest
common multiple.

00:01:33.740 --> 00:01:38.050
They have other common
multiples-- 48 and 72,

00:01:38.050 --> 00:01:40.050
and we could keep adding
more and more multiple.

00:01:40.050 --> 00:01:41.841
But this is their
smallest common multiple,

00:01:41.841 --> 00:01:44.400
their least common multiple.

00:01:44.400 --> 00:01:47.550
So it is 24.

00:01:47.550 --> 00:01:50.420
Another way that you could have
found at least common multiple

00:01:50.420 --> 00:01:52.910
is you could have taken the
prime factorization of six

00:01:52.910 --> 00:01:55.330
and you say, hey,
that's 2, and 3.

00:01:55.330 --> 00:02:00.810
So the least common multiple has
to have at least 1, 2, and 1, 3

00:02:00.810 --> 00:02:02.700
in its prime factorization
in order for it

00:02:02.700 --> 00:02:04.440
to be divisible by 6.

00:02:04.440 --> 00:02:07.610
And you could have said, what's
the prime factorization of 8?

00:02:07.610 --> 00:02:11.190
It is 2 times 4
and 4 is 2 times 2.

00:02:11.190 --> 00:02:12.820
So in order to be
divisible by 8,

00:02:12.820 --> 00:02:16.760
you have to have at least three
2's in the prime factorization.

00:02:16.760 --> 00:02:21.607
So to be divisible by 6, you
have to have a 2 times a 3.

00:02:21.607 --> 00:02:24.190
And then to be divisible by 8,
you have to have at least three

00:02:24.190 --> 00:02:25.900
2's.

00:02:25.900 --> 00:02:27.810
You have to have two
times itself three times

00:02:27.810 --> 00:02:28.690
I should say.

00:02:28.690 --> 00:02:32.070
Well, we have one 2 and
let's throw in a couple more.

00:02:32.070 --> 00:02:34.830
So then you have another
2 and then another 2.

00:02:34.830 --> 00:02:38.190
So this part right over here
makes it divisible by 8.

00:02:38.190 --> 00:02:41.260
And this part right over
here makes it divisible by 6.

00:02:41.260 --> 00:02:48.206
If I take 2 times 2 times 2
times 3, that does give me 24.

00:02:48.206 --> 00:02:49.872
So our least common
multiple of 8 and 6,

00:02:49.872 --> 00:02:52.590
which is also the least common
denominator of these two

00:02:52.590 --> 00:02:54.790
fractions is going to be 24.

00:02:54.790 --> 00:02:57.200
So what we want to do is
rewrite each of these fractions

00:02:57.200 --> 00:02:59.570
with 24 as the denominator.

00:02:59.570 --> 00:03:01.790
So I'll start with 2 over 8.

00:03:01.790 --> 00:03:04.790
And I want to write that
as something over 24.

00:03:08.790 --> 00:03:11.180
Well, to get the
denominator be 24,

00:03:11.180 --> 00:03:13.350
we have to multiply it by 3.

00:03:13.350 --> 00:03:15.126
8 times 3 is 24.

00:03:15.126 --> 00:03:16.500
And so if we don't
want to change

00:03:16.500 --> 00:03:17.920
the value of the
fraction, we have

00:03:17.920 --> 00:03:21.560
to multiply the numerator and
denominator by the same thing.

00:03:21.560 --> 00:03:24.740
So let's multiply the
numerator by 3 as well.

00:03:24.740 --> 00:03:26.870
2 times 3 is 6.

00:03:26.870 --> 00:03:29.936
So 2/8 is the exact
same thing as 6/24.

00:03:29.936 --> 00:03:31.310
To see that a
little bit clearer,

00:03:31.310 --> 00:03:37.040
you say, look, if I have 2/8,
and if I multiply this times 3

00:03:37.040 --> 00:03:39.635
over 3, that gives me 6/24.

00:03:42.370 --> 00:03:45.970
And this are the same
fraction because 3 over 3

00:03:45.970 --> 00:03:47.970
is really just 1.

00:03:47.970 --> 00:03:49.540
It's one whole.

00:03:49.540 --> 00:03:53.600
So 2/8 is 6/24 let's do
the same thing with 5/6.

00:03:56.590 --> 00:04:03.150
So 5 over 6 is equal
to something over 24.

00:04:03.150 --> 00:04:05.740
Let me do that in
a different color.

00:04:05.740 --> 00:04:07.430
I'll do it in blue.

00:04:07.430 --> 00:04:09.590
Something over 24.

00:04:09.590 --> 00:04:11.910
To get the denominator
from 6 to 24,

00:04:11.910 --> 00:04:14.230
we have to multiply it by 4.

00:04:14.230 --> 00:04:16.240
So if we don't want to
change the value of 5/6,

00:04:16.240 --> 00:04:18.281
we have to multiply the
numerator and denominator

00:04:18.281 --> 00:04:19.190
by the same thing.

00:04:19.190 --> 00:04:22.190
So let's multiply the
numerator times 4.

00:04:22.190 --> 00:04:24.610
5 times 4 is 20.

00:04:24.610 --> 00:04:26.820
5/6 is the same thing as 20/24.

00:04:26.820 --> 00:04:27.700
So we're done.

00:04:27.700 --> 00:04:31.902
We've written 2/8 as 6/24 and
we've written 5/6 as 20/24.

00:04:31.902 --> 00:04:34.110
If we wanted to add them
now, we could literally just

00:04:34.110 --> 00:04:36.849
add 6/24 to 20/24.

00:04:36.849 --> 00:04:38.390
And I'll leave you
there because they

00:04:38.390 --> 00:04:41.140
didn't ask us to
actually do that.


Subtracting Whole Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.280 --> 00:00:01.475
We're asked to subtract.

00:00:01.475 --> 00:00:04.900
And we have the
problem 68 minus 42.

00:00:04.900 --> 00:00:06.790
And what I want
to do here is one,

00:00:06.790 --> 00:00:09.050
just show you how I
would do this problem,

00:00:09.050 --> 00:00:12.520
and then talk a little bit
about why it actually works.

00:00:12.520 --> 00:00:15.800
So if I have 68 minus
42, one way to do it--

00:00:15.800 --> 00:00:17.670
and this is probably
the most typical way--

00:00:17.670 --> 00:00:20.690
is to write the number you're
subtracting from on top,

00:00:20.690 --> 00:00:23.650
and write the number that you're
subtracting from that number

00:00:23.650 --> 00:00:24.410
below that.

00:00:24.410 --> 00:00:27.200
So we're going to
have 68 minus 42.

00:00:27.200 --> 00:00:30.050
And the thing you really
want to make sure you do

00:00:30.050 --> 00:00:32.340
is that you want to align
it according to place.

00:00:32.340 --> 00:00:33.930
So the 2 is below the 8.

00:00:33.930 --> 00:00:35.610
They're both in the ones place.

00:00:35.610 --> 00:00:37.085
And the 4 is below the 6.

00:00:37.085 --> 00:00:38.526
They're both in the tens place.

00:00:38.526 --> 00:00:40.400
And later in this video,
it should be clear--

00:00:40.400 --> 00:00:42.100
or I'll hopefully
make it clear--

00:00:42.100 --> 00:00:44.700
why that's a good thing to do.

00:00:44.700 --> 00:00:47.460
So then we go to the ones
place, and we see an 8.

00:00:47.460 --> 00:00:49.630
And from that, we're
going to subtract a 2.

00:00:49.630 --> 00:00:52.217
So 8 minus 2 is 6.

00:00:52.217 --> 00:00:53.300
I'll write that over here.

00:00:53.300 --> 00:00:54.880
Let me write that over here.

00:00:54.880 --> 00:00:55.920
8 minus 2 is 6.

00:00:55.920 --> 00:00:59.180
8 minus 2 is equal to 6.

00:01:02.120 --> 00:01:03.505
And then we go to
the tens place.

00:01:03.505 --> 00:01:05.177
We have 6 minus 4.

00:01:05.177 --> 00:01:06.760
And since they're
[? in ?] tens place,

00:01:06.760 --> 00:01:08.580
this is really 60 minus 40.

00:01:08.580 --> 00:01:11.520
But 6 minus 4 is 2.

00:01:11.520 --> 00:01:15.437
6 minus 4 is equal to 2.

00:01:15.437 --> 00:01:17.020
And since these are
in the tens place,

00:01:17.020 --> 00:01:19.424
this is really saying 60
minus 40 is equal to 20.

00:01:19.424 --> 00:01:21.590
And I'll make that a little
bit clearer in a second.

00:01:21.590 --> 00:01:22.810
So we're actually done here.

00:01:22.810 --> 00:01:26.411
68 minus 42 is equal to 26.

00:01:26.411 --> 00:01:27.910
And you can check
that for yourself.

00:01:27.910 --> 00:01:31.510
If you add 26 plus
42, you should get 68.

00:01:31.510 --> 00:01:35.960
Or if you subtract 26 from
68, you should get 42.

00:01:35.960 --> 00:01:39.270
So I encourage you do that in
your own time after this video.

00:01:39.270 --> 00:01:46.272
Verify that 42 plus
26 is equal to 68.

00:01:46.272 --> 00:01:50.900
And also verify what 68
minus 26 is equal to.

00:01:50.900 --> 00:01:52.880
And you should see
that it is equal to 42.

00:01:52.880 --> 00:01:56.134
So these are two things for
you to check on your own.

00:01:56.134 --> 00:01:58.050
Now, the last thing I
want to do in this video

00:01:58.050 --> 00:02:00.190
is just to explain in
a little bit more depth

00:02:00.190 --> 00:02:01.800
why this actually works.

00:02:01.800 --> 00:02:05.272
And at least in my mind,
I like to visualize 68.

00:02:05.272 --> 00:02:06.980
You won't have to
write it out like this,

00:02:06.980 --> 00:02:08.240
but it's one way to
make sure you really

00:02:08.240 --> 00:02:09.573
understand what's going on here.

00:02:09.573 --> 00:02:15.440
68 is the same thing as
60 plus 8, or 60 and 8.

00:02:15.440 --> 00:02:17.600
And from that, we're
subtracting 42.

00:02:17.600 --> 00:02:21.070
But 42 is the same
thing as 40 plus 2.

00:02:21.070 --> 00:02:25.760
So we're subtracting 40,
and we're subtracting 2.

00:02:25.760 --> 00:02:26.980
We're subtracting 42.

00:02:26.980 --> 00:02:29.370
We're subtracting 40 and 2.

00:02:29.370 --> 00:02:32.780
And so you could view this as
really two separate problems.

00:02:32.780 --> 00:02:35.072
In the ones place,
you have an 8 minus 2.

00:02:35.072 --> 00:02:36.905
And that's is exactly
what we did over here.

00:02:36.905 --> 00:02:38.240
You have a 6.

00:02:38.240 --> 00:02:41.460
And then in the tens place,
you have a 60 minus 40,

00:02:41.460 --> 00:02:44.270
which is 20.

00:02:44.270 --> 00:02:46.810
So you end up with 20 plus 6.

00:02:46.810 --> 00:02:52.840
And you know that that's
the same exact thing as 26.


Subtraction Word Problem

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:08.761
A farmer grows 531 tomatoes
and is able to sell 176 of them

00:00:08.761 --> 00:00:15.460
in three days, so is able to
sell 176 of them in three days.

00:00:15.460 --> 00:00:20.670
Given that his supply of
tomatoes decreases by 176,

00:00:20.670 --> 00:00:23.520
how many tomatoes
does he have remaining

00:00:23.520 --> 00:00:25.260
at the end of three days?

00:00:25.260 --> 00:00:29.570
So he starts with 531 tomatoes.

00:00:29.570 --> 00:00:33.230
Let me give myself a little
bit more space to work with.

00:00:33.230 --> 00:00:39.280
He starts with 531 tomatoes,
and he's able to sell 176.

00:00:39.280 --> 00:00:42.910
So he's essentially going to
subtract the 176 that he's

00:00:42.910 --> 00:00:46.470
selling if we want to figure
out how many he's left with.

00:00:46.470 --> 00:00:48.996
So we're going to subtract 176.

00:00:48.996 --> 00:00:50.620
That's how many he
sells in three days.

00:00:50.620 --> 00:00:52.120
And they're asking
us, how much does

00:00:52.120 --> 00:00:53.810
he have left at the
end of three days?

00:00:53.810 --> 00:00:56.860
We just have to subtract
those 176 from the amount

00:00:56.860 --> 00:00:57.730
that he grew.

00:00:57.730 --> 00:01:00.310
So it turns into this
straight-up subtraction

00:01:00.310 --> 00:01:03.030
problem, so let's
see if we can do it.

00:01:03.030 --> 00:01:05.390
So if we go straight
to the ones place

00:01:05.390 --> 00:01:06.989
right over here--
and actually, let

00:01:06.989 --> 00:01:08.530
me do it in parallel,
because I think

00:01:08.530 --> 00:01:10.140
that might be
interesting over here.

00:01:10.140 --> 00:01:11.920
I'm going to do it the way
you traditionally do it

00:01:11.920 --> 00:01:13.470
here on the left,
and then I'm going

00:01:13.470 --> 00:01:16.040
to show you what's happening
over here on the right.

00:01:16.040 --> 00:01:21.640
So 531 is the same thing
as 500 plus 30 plus 1.

00:01:21.640 --> 00:01:24.550
And if you subtract 176,
that's the same thing

00:01:24.550 --> 00:01:27.840
as subtracting 100,
subtracting another 70,

00:01:27.840 --> 00:01:29.350
and subtracting another 6.

00:01:29.350 --> 00:01:33.250
And I wrote it this way,
because the 5 in 531

00:01:33.250 --> 00:01:34.880
is the same thing as 500.

00:01:34.880 --> 00:01:37.660
The 3 in 531 is
in the tens place,

00:01:37.660 --> 00:01:39.530
so it's really representing 30.

00:01:39.530 --> 00:01:43.890
The 1 in 531 is in the ones
place, so it represents 1.

00:01:43.890 --> 00:01:45.770
And now it'll be a
little bit clearer

00:01:45.770 --> 00:01:48.370
what we're doing when we're
borrowing or regrouping

00:01:48.370 --> 00:01:50.490
on this problem right over here.

00:01:50.490 --> 00:01:53.453
And so let's start off
with the ones place.

00:01:53.453 --> 00:01:56.500
1 is less than 6.

00:01:56.500 --> 00:01:59.430
It'd be great if we can
regroup some of the value

00:01:59.430 --> 00:02:02.120
from the rest of the places
so we could go straight

00:02:02.120 --> 00:02:03.160
to the tens place.

00:02:03.160 --> 00:02:06.220
The tens place, we can
borrow or regroup 10 from it.

00:02:06.220 --> 00:02:09.280
So if we take 10 from
here, this becomes 20.

00:02:09.280 --> 00:02:12.300
We're going to take that
10 and add it to the 1,

00:02:12.300 --> 00:02:14.720
so this will become 11.

00:02:14.720 --> 00:02:16.870
We just added 10.

00:02:16.870 --> 00:02:20.660
We moved over 10 from the
tens place to the ones place.

00:02:20.660 --> 00:02:22.820
If you look at it over
here, you could say, look,

00:02:22.820 --> 00:02:25.240
we're taking 10 from the 30.

00:02:25.240 --> 00:02:29.357
That becomes a 20, and
then the 1 becomes an 11.

00:02:29.357 --> 00:02:31.690
The way when I was first in
school, people would say you

00:02:31.690 --> 00:02:33.760
would borrow 1 from the 3.

00:02:33.760 --> 00:02:36.040
And you'd kind of just stick
that 1 right over here.

00:02:36.040 --> 00:02:38.970
But what you're really doing
is you're taking a 10 from a 30

00:02:38.970 --> 00:02:42.725
and making it into a 20, and
you're adding that 10 to the 1

00:02:42.725 --> 00:02:43.880
and getting 11.

00:02:43.880 --> 00:02:47.120
But either way, you'll end
up with 11 in the ones place.

00:02:47.120 --> 00:02:48.390
And now you can subtract.

00:02:48.390 --> 00:02:50.556
11 minus 6 is 5.

00:02:53.100 --> 00:02:55.340
Now we go over to
the tens place.

00:02:55.340 --> 00:02:59.250
In the tens place we
now have 2 minus 7,

00:02:59.250 --> 00:03:03.670
which is really
representing 20 minus 7.

00:03:03.670 --> 00:03:06.980
Well, that 70 is
bigger than 20, so we

00:03:06.980 --> 00:03:08.826
want to add some more
to the tens place.

00:03:08.826 --> 00:03:10.450
Well, we could go to
the hundreds place

00:03:10.450 --> 00:03:13.380
to find some more
value to regroup.

00:03:13.380 --> 00:03:14.920
So let's see if we can do that.

00:03:14.920 --> 00:03:15.900
We have 500 here.

00:03:15.900 --> 00:03:18.250
What happens if we
take 100 from that?

00:03:18.250 --> 00:03:20.560
So we're just left
with 400 here.

00:03:20.560 --> 00:03:24.370
And we take that 100, and
we put it in the tens place.

00:03:24.370 --> 00:03:29.250
So instead of 20,
we now have 120.

00:03:29.250 --> 00:03:31.260
If you look at it
on this problem,

00:03:31.260 --> 00:03:33.690
since we're using
the places here,

00:03:33.690 --> 00:03:38.160
we're going to take 100
from the 500 and have 400.

00:03:38.160 --> 00:03:40.340
And then we're going
to take that 100

00:03:40.340 --> 00:03:42.090
that we took and put
it to the tens place.

00:03:42.090 --> 00:03:43.540
Well, 100 is 10 tens.

00:03:43.540 --> 00:03:45.530
So we're going to
add 10 to this,

00:03:45.530 --> 00:03:47.450
so this is going to become a 12.

00:03:47.450 --> 00:03:50.670
Once again, the more
mechanical or rote way

00:03:50.670 --> 00:03:54.100
of thinking about it is that,
oh, you took 1 away from the 4.

00:03:54.100 --> 00:03:57.090
And you stick that 1
onto the front of the 2.

00:03:57.090 --> 00:04:02.080
But you're really taking 100
from the 500 to make it 400

00:04:02.080 --> 00:04:05.890
and then adding that 100 to
the 20 here and making it 120.

00:04:05.890 --> 00:04:09.070
But you write it here as
a 12 because it's 12 tens.

00:04:09.070 --> 00:04:10.175
You're in the tens place.

00:04:10.175 --> 00:04:11.050
Let me write it down.

00:04:11.050 --> 00:04:12.670
This is the ones place.

00:04:12.670 --> 00:04:16.954
This is the tens place, and
this is the hundreds place.

00:04:19.630 --> 00:04:24.245
So now that our number
on top in the tens place

00:04:24.245 --> 00:04:26.620
is bigger than our number on
the bottom, we can subtract.

00:04:26.620 --> 00:04:29.640
So we get 120 minus 70.

00:04:29.640 --> 00:04:33.320
That is 50, or 12 minus 7 is 5.

00:04:33.320 --> 00:04:37.130
But that 5 is in the tens place,
so it's really representing 50.

00:04:37.130 --> 00:04:38.860
Let me circle it
with the same color

00:04:38.860 --> 00:04:42.860
so you recognize that
this 5 is representing 50.

00:04:42.860 --> 00:04:47.350
Then finally, we're
in the hundreds place.

00:04:47.350 --> 00:04:50.654
400 minus 100 is 300.

00:04:50.654 --> 00:04:53.860
4 minus 1 is 3, but
this 3 represents 300.

00:04:53.860 --> 00:04:55.420
This 5 represents 50.

00:04:55.420 --> 00:04:58.250
This 5 represents 5,
and so we're done.

00:04:58.250 --> 00:04:59.810
We get 355.

00:04:59.810 --> 00:05:03.200
The farmer is left
with 355 tomatoes

00:05:03.200 --> 00:05:08.070
at the end of three days, or
300 plus 50 plus 5 tomatoes.


Comparing with multiplication: age

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:05.500
We're told that Jared is twice
the age of his brother, Peter.

00:00:05.500 --> 00:00:09.470
So Jared is twice the age
of his brother, Peter.

00:00:09.470 --> 00:00:11.370
Peter is 4 years old.

00:00:14.210 --> 00:00:17.760
And Jared is twice the
age of his brother, Peter.

00:00:17.760 --> 00:00:21.260
So Jared is twice
Peter's age of 4.

00:00:21.260 --> 00:00:26.550
And then they tell us, Talia's
age is 3 times Jared's age.

00:00:26.550 --> 00:00:28.432
How old are Talia and Jared?

00:00:28.432 --> 00:00:34.390
So Talia's age is 3
times Jared's age.

00:00:34.390 --> 00:00:35.560
So let's write this down.

00:00:35.560 --> 00:00:37.430
We have Peter is 4 years old.

00:00:37.430 --> 00:00:45.050
So Peter's age,
he's 4 years old.

00:00:45.050 --> 00:00:50.090
Jared is twice the age
of his brother, Peter.

00:00:50.090 --> 00:00:53.475
Let me do that in yellow.

00:00:53.475 --> 00:00:54.225
That's not yellow.

00:00:58.390 --> 00:01:02.454
Jared is twice the
age of his brother.

00:01:02.454 --> 00:01:09.495
Twice is the same thing as 2
times the age of his brother.

00:01:09.495 --> 00:01:11.620
You could say Jared is
twice the age of his brother

00:01:11.620 --> 00:01:14.320
or Jared is 2 times
the age of his brother.

00:01:14.320 --> 00:01:15.670
Well, his brother is 4.

00:01:15.670 --> 00:01:17.610
They told us that
right over there.

00:01:17.610 --> 00:01:21.485
So Jared is going to
be 2 times 4 years old.

00:01:24.440 --> 00:01:26.280
And if you know your
multiplication tables,

00:01:26.280 --> 00:01:28.770
you know that's the
same thing as 8.

00:01:28.770 --> 00:01:29.466
2 times 4 is 8.

00:01:29.466 --> 00:01:30.910
4 times 2 is 8.

00:01:30.910 --> 00:01:33.710
Or you know this is the
same thing as Peter's age

00:01:33.710 --> 00:01:37.380
twice, which is the
same thing as 4 plus 4.

00:01:37.380 --> 00:01:41.060
That would also get you 8.

00:01:41.060 --> 00:01:44.370
So Jared's age is equal to 8.

00:01:44.370 --> 00:01:47.030
And then, finally, they
say Talia's age-- so

00:01:47.030 --> 00:01:51.120
let me write Talia
right over here.

00:01:51.120 --> 00:01:54.060
Talia's age and they tell
it to us right over here.

00:01:54.060 --> 00:01:56.500
It's 3 times Jared's age.

00:01:56.500 --> 00:02:00.105
So Talia's age is 3
times Jared's age.

00:02:00.105 --> 00:02:01.980
Well, we were able to
figure out Jared's age.

00:02:01.980 --> 00:02:03.800
It is 8.

00:02:03.800 --> 00:02:06.465
It is 3 times Jared's age.

00:02:06.465 --> 00:02:08.090
If you know your
multiplication tables,

00:02:08.090 --> 00:02:09.810
this should maybe
jump out at you.

00:02:09.810 --> 00:02:12.700
And your multiplication tables
really are one of those things

00:02:12.700 --> 00:02:16.192
in mathematics that you should
really just know back and forth

00:02:16.192 --> 00:02:18.150
just because it will make
the rest of your life

00:02:18.150 --> 00:02:20.090
very, very, very simple.

00:02:20.090 --> 00:02:23.620
Or you could view 3 times 8
as literally 8 three times.

00:02:23.620 --> 00:02:26.810
So it's 8 plus 8 plus 8.

00:02:26.810 --> 00:02:30.250
8 plus 8 is 16, 16 plus 8 is 24.

00:02:30.250 --> 00:02:31.890
So 3 times 8 is 24.

00:02:31.890 --> 00:02:37.450
So Peter is 4, Jared
is 8, and Talia is 24.


Least common multiple exercise: 3 numbers

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.020
What is the least common
multiple, abbreviated as LCM,

00:00:05.020 --> 00:00:07.850
of 15, 6, and 10?

00:00:07.850 --> 00:00:09.949
So the least common
multiple is exactly what

00:00:09.949 --> 00:00:10.740
the word is saying.

00:00:10.740 --> 00:00:14.054
It's the least common
multiple of these numbers.

00:00:14.054 --> 00:00:15.970
And I know that probably
didn't help you much.

00:00:15.970 --> 00:00:17.620
But let's actually work
through this problem.

00:00:17.620 --> 00:00:20.161
So to do that, let's just think
about the different multiples

00:00:20.161 --> 00:00:23.880
of the 15, 6, and 10 and
then find at the smallest

00:00:23.880 --> 00:00:27.160
multiple, the least multiple,
they have in common.

00:00:27.160 --> 00:00:30.170
So let's find the
multiples of 15.

00:00:30.170 --> 00:00:32.580
So you have 1 times 15 is 15.

00:00:32.580 --> 00:00:35.280
2 times 15 is 30.

00:00:35.280 --> 00:00:37.740
Then if you add 15
again, you get 45.

00:00:37.740 --> 00:00:39.880
You add 15 again, you get 60.

00:00:39.880 --> 00:00:44.100
You add 15 again, you get 75.

00:00:44.100 --> 00:00:46.370
You add 15 again, you get 90.

00:00:46.370 --> 00:00:49.560
You add 15 again, you get 105.

00:00:49.560 --> 00:00:52.745
And if still none of
these are common multiples

00:00:52.745 --> 00:00:55.245
with these guys over here, then
we might have to go further.

00:00:55.245 --> 00:00:56.640
But I'll stop there for now.

00:00:56.640 --> 00:01:00.310
So that's the multiples
of 15 up through 105.

00:01:00.310 --> 00:01:05.060
Obviously, we can
keep going from there.

00:01:05.060 --> 00:01:07.335
Now, let's do the
multiples of 6.

00:01:10.450 --> 00:01:13.160
1 times 6 is 6.

00:01:13.160 --> 00:01:14.700
2 times 6 is 12.

00:01:14.700 --> 00:01:16.200
3 times 6 is 18.

00:01:16.200 --> 00:01:18.250
4 times 6 is 24.

00:01:18.250 --> 00:01:20.486
5 times 6 is 30.

00:01:20.486 --> 00:01:23.640
6 times 6 is 36.

00:01:23.640 --> 00:01:25.680
7 times 6 is 42.

00:01:25.680 --> 00:01:28.120
8 times 6 is 48.

00:01:28.120 --> 00:01:30.030
9 times 6 is 54.

00:01:30.030 --> 00:01:32.490
10 times 6 is 60.

00:01:32.490 --> 00:01:34.800
60 already looks
interesting, because it

00:01:34.800 --> 00:01:37.985
is a common multiple
of both 15 and 60,

00:01:37.985 --> 00:01:39.610
although we have two
of them over here.

00:01:39.610 --> 00:01:41.150
We have a 30, and we have a 30.

00:01:41.150 --> 00:01:42.170
We have a 60 and a 60.

00:01:42.170 --> 00:01:44.760
So the smallest common
multiple, so if we only

00:01:44.760 --> 00:01:47.710
cared about the least
common multiple of 15 and 6,

00:01:47.710 --> 00:01:49.320
we would say it's 30.

00:01:49.320 --> 00:01:51.440
So let me write this
down as an intermediate.

00:01:51.440 --> 00:01:57.700
The LCM of 15 and 6, so
the least common multiple,

00:01:57.700 --> 00:01:59.870
the smallest multiple
that they have in common,

00:01:59.870 --> 00:02:00.710
we see over here.

00:02:00.710 --> 00:02:03.090
15 times 2 is 30.

00:02:03.090 --> 00:02:06.140
And 6 times 5 is 30.

00:02:06.140 --> 00:02:07.960
So this is definitely
a common multiple.

00:02:07.960 --> 00:02:10.251
And it's the smallest of all
of their common multiples.

00:02:10.251 --> 00:02:11.624
60 is also a common multiple.

00:02:11.624 --> 00:02:12.540
But it's a bigger one.

00:02:12.540 --> 00:02:15.160
This is the least
common multiple.

00:02:15.160 --> 00:02:16.660
So this is 30.

00:02:16.660 --> 00:02:18.600
Well, we haven't thought
about the 10 yet.

00:02:18.600 --> 00:02:20.010
So let's bring the 10 in there.

00:02:20.010 --> 00:02:22.660
And I think you already
see where this is going.

00:02:22.660 --> 00:02:24.470
Let's do the multiples of 10.

00:02:24.470 --> 00:02:27.730
They are 10, 20, 30, 40.

00:02:27.730 --> 00:02:29.350
Well, we already
went far enough,

00:02:29.350 --> 00:02:31.130
because we already got to 30.

00:02:31.130 --> 00:02:37.040
And 30 is a common
multiple of 15 and 6.

00:02:37.040 --> 00:02:39.250
And it's the smallest common
multiple of all of them.

00:02:39.250 --> 00:02:45.270
So it's actually the fact
that the LCM of 15, 6, and 10

00:02:45.270 --> 00:02:47.789
is equal to 30.

00:02:47.789 --> 00:02:50.080
Now, this is one way to find
the least common multiple.

00:02:50.080 --> 00:02:52.920
Literally just look at the
multiples of each of the number

00:02:52.920 --> 00:02:55.510
and then see what the
smallest multiple they have

00:02:55.510 --> 00:02:57.350
is in common.

00:02:57.350 --> 00:02:58.820
Another way to do
that is to look

00:02:58.820 --> 00:03:01.670
at the prime factorization
of each of these numbers.

00:03:01.670 --> 00:03:04.450
And the least common
multiple is the number

00:03:04.450 --> 00:03:07.550
that has all of the elements
of the prime factorizations

00:03:07.550 --> 00:03:09.140
of these and nothing else.

00:03:09.140 --> 00:03:11.450
So let me show you
what I mean by that.

00:03:11.450 --> 00:03:12.960
So you could do it this way.

00:03:12.960 --> 00:03:17.440
Or you could say 15 is the
same thing as 3 times 5.

00:03:17.440 --> 00:03:18.149
And that's it.

00:03:18.149 --> 00:03:19.440
That's its prime factorization.

00:03:19.440 --> 00:03:20.790
15 is 3 times 5.

00:03:20.790 --> 00:03:22.800
Both 3 and 5 are prime.

00:03:22.800 --> 00:03:27.346
We can say that 6 is the
same thing as 2 times 3.

00:03:27.346 --> 00:03:27.846
That's it.

00:03:27.846 --> 00:03:29.440
That's its prime factorization.

00:03:29.440 --> 00:03:31.830
Both 2 and 3 are prime.

00:03:31.830 --> 00:03:37.610
And then we can say that 10 is
the same thing as 2 times 5.

00:03:37.610 --> 00:03:39.220
Both 2 and 5 are prime.

00:03:39.220 --> 00:03:41.210
So we're done factoring it.

00:03:41.210 --> 00:03:49.510
And so the least common
multiple of 15, 6, and 10 just

00:03:49.510 --> 00:03:52.410
needs to have all of
these prime factors.

00:03:52.410 --> 00:03:55.880
And what I mean, to be clear, is
in order to be divisible by 15,

00:03:55.880 --> 00:03:59.180
it has to have at
least one 3 and one 5

00:03:59.180 --> 00:04:00.760
in its prime factorization.

00:04:00.760 --> 00:04:03.970
So it has to have at least
one 3 and at least one 5.

00:04:03.970 --> 00:04:06.520
By having a 3 times 5 in
its prime factorization,

00:04:06.520 --> 00:04:09.780
that ensures that this
number is divisible by 15.

00:04:09.780 --> 00:04:13.890
To be divisible by 6, it has to
have at least one 2 and one 3.

00:04:13.890 --> 00:04:15.870
So it has to have
at least one 2.

00:04:15.870 --> 00:04:17.500
And we already
have a 3 over here.

00:04:17.500 --> 00:04:18.529
So that's all we want.

00:04:18.529 --> 00:04:19.709
We just need one 3.

00:04:19.709 --> 00:04:22.310
So one 2 and one
3, this 2 times 3,

00:04:22.310 --> 00:04:25.319
ensures that we
are divisible by 6.

00:04:25.319 --> 00:04:26.360
And let me make it clear.

00:04:26.360 --> 00:04:29.760
This right here is the 15.

00:04:29.760 --> 00:04:32.090
And then to make sure that
we're divisible by 10,

00:04:32.090 --> 00:04:34.550
we have to have at
least one 2 and one 5.

00:04:37.820 --> 00:04:42.600
These two over here make sure
that we are divisible by 10.

00:04:42.600 --> 00:04:44.190
And so we have all of them.

00:04:44.190 --> 00:04:47.990
This 2 times 3 times 5 has
all of the prime factors

00:04:47.990 --> 00:04:50.990
of either 10, 6, or 15.

00:04:50.990 --> 00:04:53.060
So it is the least
common multiple.

00:04:53.060 --> 00:04:57.430
And so if you multiply this out,
you will get 2 times 3 is 6.

00:04:57.430 --> 00:05:00.370
6 times 5 is 30.

00:05:00.370 --> 00:05:03.740
So either way, hopefully, both
of these resonate with you.

00:05:03.740 --> 00:05:05.680
And you see why they make sense.

00:05:05.680 --> 00:05:09.960
This second way is
a little bit better

00:05:09.960 --> 00:05:13.920
if you're trying to do it for
really complex numbers, numbers

00:05:13.920 --> 00:05:16.940
where you might have to be
multiplying it for a long time.

00:05:16.940 --> 00:05:20.140
Either way, both of
these are valid ways

00:05:20.140 --> 00:05:22.604
of finding the least
common multiple.


Recognizing prime and composite numbers

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.950
Determine whether the following
numbers are prime, composite,

00:00:04.950 --> 00:00:06.510
or neither.

00:00:06.510 --> 00:00:09.200
So just as a bit of
review, a prime number

00:00:09.200 --> 00:00:12.560
is a natural number-- so one of
the counting numbers, 1, 2, 3,

00:00:12.560 --> 00:00:17.810
4, 5, 6, so on and so forth--
that has exactly two factors.

00:00:17.810 --> 00:00:25.580
So its factors are 1 and itself.

00:00:25.580 --> 00:00:28.080
So an example of a
prime factor is 3.

00:00:28.080 --> 00:00:30.090
There's only two
natural numbers that

00:00:30.090 --> 00:00:34.330
are divisible into 3-- 1 and 3.

00:00:34.330 --> 00:00:37.080
Or another way to think about
it is, the only way to get 3

00:00:37.080 --> 00:00:41.220
as a product of other
natural numbers is 1 times 3.

00:00:41.220 --> 00:00:45.280
So it only has 1 and itself.

00:00:45.280 --> 00:00:47.890
A composite number
is a natural number

00:00:47.890 --> 00:00:51.510
that has more than just
1 and itself as factors.

00:00:51.510 --> 00:00:54.150
And we'll see examples of
that and neither-- we'll

00:00:54.150 --> 00:00:57.050
see an interesting case
of that in this problem.

00:00:57.050 --> 00:00:59.082
So first let's think about 24.

00:00:59.082 --> 00:01:00.540
So let's think
about all of the-- I

00:01:00.540 --> 00:01:02.760
guess you could think of
it as the natural numbers

00:01:02.760 --> 00:01:04.739
or the whole numbers,
although 0 is also

00:01:04.739 --> 00:01:05.870
included in whole numbers.

00:01:05.870 --> 00:01:08.370
Let's think of all of the
natural counting numbers

00:01:08.370 --> 00:01:11.060
that we can actually
divide into 24

00:01:11.060 --> 00:01:12.550
without having any remainder.

00:01:12.550 --> 00:01:14.610
We'd consider those the factors.

00:01:14.610 --> 00:01:17.660
Well, clearly it is
divisible by 1 and 24.

00:01:17.660 --> 00:01:20.580
In fact, 1 times
24 is equal to 24.

00:01:20.580 --> 00:01:24.520
But it's also divisible by 2.

00:01:24.520 --> 00:01:28.540
2 times 12 is 24.

00:01:28.540 --> 00:01:30.890
So it's also divisible by 12.

00:01:30.890 --> 00:01:33.510
And it is also divisible by 3.

00:01:33.510 --> 00:01:38.110
3 times 8 is also equal to 24.

00:01:38.110 --> 00:01:39.910
And even at this point,
we don't actually

00:01:39.910 --> 00:01:43.390
have to find all of the factors
to realize that it's not prime.

00:01:43.390 --> 00:01:46.760
It clearly has more factors
than just 1 and itself.

00:01:46.760 --> 00:01:49.065
So then it is clearly
going to be composite.

00:01:52.015 --> 00:01:54.460
This is going to be composite.

00:01:54.460 --> 00:01:57.470
Now, let's just finish factoring
it just since we started it.

00:01:57.470 --> 00:01:59.290
It's also divisible by 4.

00:01:59.290 --> 00:02:02.560
And 4 times 6-- had just
enough space to do that.

00:02:02.560 --> 00:02:04.390
4 times 6 is also 24.

00:02:04.390 --> 00:02:07.660
So these are all of the
factors of 24, clearly

00:02:07.660 --> 00:02:10.669
more than just one and 24.

00:02:10.669 --> 00:02:12.660
Now let's think about 2.

00:02:12.660 --> 00:02:18.010
Well, the non-zero whole numbers
that are divisible into 2,

00:02:18.010 --> 00:02:21.430
well, 1 times 2
definitely works, 1 and 2.

00:02:21.430 --> 00:02:22.850
But there really
aren't any others

00:02:22.850 --> 00:02:25.370
that are divisible into 2.

00:02:25.370 --> 00:02:28.990
And so it only has two
factors, 1 and itself,

00:02:28.990 --> 00:02:31.650
and that's the definition
of a prime number.

00:02:31.650 --> 00:02:33.350
So 2 is prime.

00:02:37.210 --> 00:02:41.575
And 2 is interesting because it
is the only even prime number.

00:02:48.680 --> 00:02:50.760
And that might be
common sense you.

00:02:50.760 --> 00:02:56.930
Because by definition, an
even number is divisible by 2.

00:02:56.930 --> 00:02:58.720
So 2 is clearly divisible by 2.

00:02:58.720 --> 00:02:59.930
That's what makes it even.

00:02:59.930 --> 00:03:02.461
But it's only
divisible by 2 and 1.

00:03:02.461 --> 00:03:03.710
So that's what makes it prime.

00:03:03.710 --> 00:03:06.490
But anything else
that's even is going

00:03:06.490 --> 00:03:13.550
to be divisible by
1, itself, and 2.

00:03:13.550 --> 00:03:15.420
Any other number
that is even is going

00:03:15.420 --> 00:03:18.277
to be divisible by
1, itself, and 2.

00:03:18.277 --> 00:03:19.860
So by definition,
it's going to have 1

00:03:19.860 --> 00:03:21.480
and itself and something else.

00:03:21.480 --> 00:03:23.350
So it's going to be composite.

00:03:23.350 --> 00:03:24.440
So 2 is prime.

00:03:24.440 --> 00:03:27.880
Every other even number
other than 2 is composite.

00:03:27.880 --> 00:03:30.480
Now, here is an
interesting case.

00:03:30.480 --> 00:03:35.820
1-- 1 is only divisible by 1.

00:03:35.820 --> 00:03:41.570
So it is not prime,
technically, because it only

00:03:41.570 --> 00:03:46.070
has 1 as a factor.

00:03:46.070 --> 00:03:47.840
It does not have two factors.

00:03:47.840 --> 00:03:48.814
1 is itself.

00:03:48.814 --> 00:03:50.230
But in order to
be prime, you have

00:03:50.230 --> 00:03:52.850
to have exactly two factors.

00:03:52.850 --> 00:03:55.400
1 has only one factor.

00:03:55.400 --> 00:03:56.960
In order to be
composite, you have

00:03:56.960 --> 00:03:58.630
to have more than two factors.

00:03:58.630 --> 00:04:02.660
You have to have 1, yourself,
and some other things.

00:04:02.660 --> 00:04:03.860
So it's not composite.

00:04:03.860 --> 00:04:07.090
So 1 is neither
prime nor composite.

00:04:10.690 --> 00:04:13.438
And then finally we get to 17.

00:04:13.438 --> 00:04:20.250
17 Is divisible by 1 and 17.

00:04:20.250 --> 00:04:23.990
It's not divisible by 2,
not divisible by 3, 4, 5, 6.

00:04:23.990 --> 00:04:27.590
7, 8, 9 10, 11, 12,
13, 14, 15, or 16.

00:04:27.590 --> 00:04:33.000
So it has exactly two
factors-- 1 and itself.

00:04:33.000 --> 00:04:38.928
So 17 is once
again-- 17 is prime.


Comparing fractions with different denominators

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.420
Use less than,
greater than, or equal

00:00:03.420 --> 00:00:09.170
to compare the two fractions
21/28, or 21 over 28, and 6/9,

00:00:09.170 --> 00:00:11.060
or 6 over 9.

00:00:11.060 --> 00:00:13.110
So there's a bunch
of ways to do this.

00:00:13.110 --> 00:00:16.690
The easiest way is if they
had the same denominator,

00:00:16.690 --> 00:00:18.450
you could just compare
the numerators.

00:00:18.450 --> 00:00:22.040
Unlucky for us, we do not
have the same denominator.

00:00:22.040 --> 00:00:24.600
So what we could do is we
can find a common denominator

00:00:24.600 --> 00:00:26.840
for both of them and convert
both of these fractions

00:00:26.840 --> 00:00:30.250
to have the same denominator
and then compare the numerators.

00:00:30.250 --> 00:00:34.340
Or even more simply, we could
simplify them first and then

00:00:34.340 --> 00:00:34.970
try to do it.

00:00:34.970 --> 00:00:36.511
So let me do that
last one, because I

00:00:36.511 --> 00:00:39.190
have a feeling that'll be
the fastest way to do it.

00:00:39.190 --> 00:00:45.050
So 21/28-- you can see that
they are both divisible by 7.

00:00:45.050 --> 00:00:46.650
So let's divide
both the numerator

00:00:46.650 --> 00:00:48.570
and the denominator by 7.

00:00:48.570 --> 00:00:51.190
So we could divide 21 by 7.

00:00:51.190 --> 00:00:53.600
And we can divide-- so let
me make the numerator--

00:00:53.600 --> 00:00:55.489
and we can divide
the denominator by 7.

00:00:55.489 --> 00:00:57.280
We're doing the same
thing to the numerator

00:00:57.280 --> 00:00:58.655
and the denominator,
so we're not

00:00:58.655 --> 00:01:00.950
going to change the
value of the fraction.

00:01:00.950 --> 00:01:05.650
So 21 divided by 7 is 3,
and 28 divided by 7 is 4.

00:01:05.650 --> 00:01:09.510
So 21/28 is the exact
same fraction as 3/4.

00:01:09.510 --> 00:01:11.770
3/4 is the simplified
version of it.

00:01:11.770 --> 00:01:14.680
Let's do the same thing for 6/9.

00:01:14.680 --> 00:01:17.520
6 and 9 are both divisible by 3.

00:01:17.520 --> 00:01:19.410
So let's divide
them both by 3 so we

00:01:19.410 --> 00:01:21.430
can simplify this fraction.

00:01:21.430 --> 00:01:24.080
So let's divide
both of them by 3.

00:01:24.080 --> 00:01:30.380
6 divided by 3 is 2,
and 9 divided by 3 is 3.

00:01:30.380 --> 00:01:33.060
So 21/28 is 3/4.

00:01:33.060 --> 00:01:35.760
They're the exact same fraction,
just written a different way.

00:01:35.760 --> 00:01:37.700
This is the more
simplified version.

00:01:37.700 --> 00:01:41.580
And 6/9 is the exact
same fraction as 2/3.

00:01:41.580 --> 00:01:44.590
So we really can
compare 3/4 and 2/3.

00:01:44.590 --> 00:01:49.230
So this is really
comparing 3/4 and 2/3.

00:01:49.230 --> 00:01:51.990
And the real benefit of
doing this is now this

00:01:51.990 --> 00:01:55.870
is much easier to find a common
denominator for than 28 and 9.

00:01:55.870 --> 00:01:58.680
Then we would have to
multiply big numbers.

00:01:58.680 --> 00:02:00.850
Here we could do
fairly small numbers.

00:02:00.850 --> 00:02:03.780
The common denominator
of 3/4 and 2/3

00:02:03.780 --> 00:02:08.810
is going to be the least
common multiple of 4 and 3.

00:02:08.810 --> 00:02:13.570
And 4 and 3 don't share any
prime factors with each other.

00:02:13.570 --> 00:02:15.520
So their least common
multiple is really

00:02:15.520 --> 00:02:17.430
just going to be the
product of the two.

00:02:17.430 --> 00:02:20.890
So we can write 3/4
as something over 12.

00:02:20.890 --> 00:02:24.800
And we can write 2/3
as something over 12.

00:02:24.800 --> 00:02:27.570
And I got the 12 by
multiplying 3 times 4.

00:02:27.570 --> 00:02:29.150
They have no common factors.

00:02:29.150 --> 00:02:31.500
Another way you could
think about it is 4,

00:02:31.500 --> 00:02:34.470
if you do a prime
factorization, is 2 times 2.

00:02:34.470 --> 00:02:36.870
And 3-- it's already
a prime number,

00:02:36.870 --> 00:02:39.239
so you can't prime
factorize it any more.

00:02:39.239 --> 00:02:41.030
So what you want to do
is think of a number

00:02:41.030 --> 00:02:44.120
that has all of the
prime factors of 4 and 3.

00:02:44.120 --> 00:02:46.970
So it needs one 2,
another 2, and a 3.

00:02:46.970 --> 00:02:49.744
Well, 2 times 2 times 3 is 12.

00:02:49.744 --> 00:02:51.160
And either way you
think about it,

00:02:51.160 --> 00:02:53.243
that's how you would get
the least common multiple

00:02:53.243 --> 00:02:57.450
or the common
denominator for 4 and 3.

00:02:57.450 --> 00:03:04.360
Well, to get from 4 to 12,
you've got to multiply by 3.

00:03:04.360 --> 00:03:07.460
So we're multiplying the
denominator by 3 to get to 12.

00:03:07.460 --> 00:03:10.710
So we also have to multiply
the numerator by 3.

00:03:10.710 --> 00:03:13.740
So 3 times 3 is 9.

00:03:13.740 --> 00:03:15.640
Over here, to get
from 3 to 12, we

00:03:15.640 --> 00:03:18.320
have to multiply the
denominator by 4.

00:03:18.320 --> 00:03:21.080
So we also have to multiply
the numerator by 4.

00:03:21.080 --> 00:03:24.540
So we get 8.

00:03:24.540 --> 00:03:26.250
And so now when we
compare the fractions,

00:03:26.250 --> 00:03:27.770
it's pretty straightforward.

00:03:27.770 --> 00:03:33.270
21/28 is the exact
same thing as 9/12,

00:03:33.270 --> 00:03:40.140
and 6/9 is the exact
same thing as 8/12.

00:03:40.140 --> 00:03:43.960
So which of these is
a greater quantity?

00:03:43.960 --> 00:03:46.210
Well, clearly, we have the
same denominator right now.

00:03:46.210 --> 00:03:50.120
We have 9/12 is clearly
greater than 8/12.

00:03:50.120 --> 00:03:54.180
So 9/12 is clearly
greater than 8/12.

00:03:54.180 --> 00:03:56.450
Or if you go back
and you realize

00:03:56.450 --> 00:03:59.650
that 9/12 is the exact
same thing as 21/28,

00:03:59.650 --> 00:04:04.750
we could say 21/28 is definitely
greater than-- and 8/12

00:04:04.750 --> 00:04:10.130
is the same thing as 6/9-- is
definitely greater than 6/9.

00:04:10.130 --> 00:04:11.867
And we are done.

00:04:11.867 --> 00:04:13.450
Another way we could
have done it-- we

00:04:13.450 --> 00:04:15.370
didn't necessarily
have to simplify that.

00:04:15.370 --> 00:04:17.310
And let me show you
that just for fun.

00:04:17.310 --> 00:04:21.450
So if we were doing it
with-- if we didn't think

00:04:21.450 --> 00:04:24.920
to simplify our
two numbers first.

00:04:24.920 --> 00:04:27.270
I'm trying to find a
color I haven't used yet.

00:04:27.270 --> 00:04:32.270
So 21/28 and 6/9.

00:04:32.270 --> 00:04:35.460
So we could just find
a least common multiple

00:04:35.460 --> 00:04:37.980
in the traditional way
without simplifying first.

00:04:37.980 --> 00:04:40.340
So what's the prime
factorization of 28?

00:04:40.340 --> 00:04:42.720
It's 2 times 14.

00:04:42.720 --> 00:04:45.710
And 14 is 2 times 7.

00:04:45.710 --> 00:04:47.230
That's its prime factorization.

00:04:47.230 --> 00:04:50.490
Prime factorization
of 9 is 3 times 3.

00:04:50.490 --> 00:04:53.070
So the least common
multiple of 28 and 9

00:04:53.070 --> 00:04:56.570
have to contain a 2,
a 2, a 7, a 3 and a 3.

00:04:56.570 --> 00:04:59.830
Or essentially, it's
going to be 28 times 9.

00:04:59.830 --> 00:05:04.232
So let's over here
multiply 28 times 9.

00:05:04.232 --> 00:05:05.940
There's a couple of
ways you could do it.

00:05:05.940 --> 00:05:07.870
You could multiply
in your head 28 times

00:05:07.870 --> 00:05:11.079
10, which would be 280,
and then subtract 28

00:05:11.079 --> 00:05:12.370
from that, which would be what?

00:05:12.370 --> 00:05:14.140
252.

00:05:14.140 --> 00:05:16.850
Or we could just multiply
it out if that confuses you.

00:05:16.850 --> 00:05:18.580
So let's just do the second way.

00:05:18.580 --> 00:05:22.110
9 times 8 is 72.

00:05:22.110 --> 00:05:24.380
9 times 2 is 18.

00:05:24.380 --> 00:05:26.370
18 plus 7 is 25.

00:05:26.370 --> 00:05:28.880
So we get 252.

00:05:28.880 --> 00:05:31.660
So I'm saying the
common denominator here

00:05:31.660 --> 00:05:37.650
is going to be 252.

00:05:37.650 --> 00:05:40.270
Least common
multiple of 28 and 9.

00:05:40.270 --> 00:05:46.370
Well, to go from 28 to 252,
we had to multiply it by 9.

00:05:46.370 --> 00:05:48.590
We had to multiply 28 times 9.

00:05:48.590 --> 00:05:50.570
So we're multiplying 28 times 9.

00:05:50.570 --> 00:05:53.240
So we also have to multiply
the numerator times 9.

00:05:53.240 --> 00:05:55.336
So what is 21 times 9?

00:05:55.336 --> 00:05:56.710
That's easier to
do in your head.

00:05:56.710 --> 00:05:59.080
20 times 9 is 180.

00:05:59.080 --> 00:06:00.710
And then 1 times 9 is 9.

00:06:00.710 --> 00:06:04.080
So this is going to be 189.

00:06:04.080 --> 00:06:09.100
To go from 9 to 252, we
had to multiply by 28.

00:06:09.100 --> 00:06:11.630
So we also have to multiply
the numerator by 28

00:06:11.630 --> 00:06:14.290
if we don't want to change
the value of the fraction.

00:06:14.290 --> 00:06:18.512
So 6 times 28-- 6
times 20 is 120.

00:06:18.512 --> 00:06:22.420
6 times 8 is 48.

00:06:22.420 --> 00:06:24.414
So we get 168.

00:06:24.414 --> 00:06:25.830
Let me write that
out just to make

00:06:25.830 --> 00:06:27.290
sure I didn't make a mistake.

00:06:27.290 --> 00:06:36.180
So 28 times 6-- 8 times 6 is 48.

00:06:36.180 --> 00:06:40.040
2 times 6 is 12, plus 4 is 16.

00:06:40.040 --> 00:06:41.770
So right, 168.

00:06:41.770 --> 00:06:44.819
So now we have a common
denominator here.

00:06:44.819 --> 00:06:46.860
And so we can really just
compare the numerators.

00:06:46.860 --> 00:06:50.380
And 189 is clearly
greater than 168.

00:06:50.380 --> 00:06:56.110
So 189/252 is clearly
greater than 168/252.

00:06:56.110 --> 00:06:58.710
Or that's the same
thing as saying 21/28,

00:06:58.710 --> 00:07:01.580
because that's what
this is over here.

00:07:01.580 --> 00:07:04.400
The left-hand side is
21/28, is clearly greater

00:07:04.400 --> 00:07:07.710
than the right-hand side,
which is really 6/9.


Converting mixed numbers to improper fractions

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.510
Write 5 and 1/4 as
an improper fraction.

00:00:04.510 --> 00:00:07.710
So just as a reminder,
an improper fraction

00:00:07.710 --> 00:00:10.550
is one where the
numerator is greater

00:00:10.550 --> 00:00:12.550
than or equal to
the denominator.

00:00:12.550 --> 00:00:15.160
So improper fraction means that
the numerator-- and actually,

00:00:15.160 --> 00:00:17.900
I should say the absolute
value of the numerator--

00:00:17.900 --> 00:00:21.360
is greater than or equal
to the absolute value

00:00:21.360 --> 00:00:24.690
of the denominator.

00:00:24.690 --> 00:00:26.960
I could have just shortened
it, but you get the idea.

00:00:26.960 --> 00:00:29.077
The numerator is greater
than or equal to--

00:00:29.077 --> 00:00:30.660
or the absolute value
of the numerator

00:00:30.660 --> 00:00:32.600
is greater than or equal
to the absolute value

00:00:32.600 --> 00:00:33.820
of the denominator.

00:00:33.820 --> 00:00:36.570
In this situation right
here, it's a mixed number.

00:00:36.570 --> 00:00:38.750
We have it as a combination
of 5, which is just

00:00:38.750 --> 00:00:42.180
a whole number, and 1/4,
which is a proper fraction,

00:00:42.180 --> 00:00:44.550
because the numerator is
less than the denominator.

00:00:44.550 --> 00:00:46.030
The absolute value
of the numerator

00:00:46.030 --> 00:00:49.430
is less than the absolute
value of the denominator.

00:00:49.430 --> 00:00:51.507
So to write it as an
improper fraction--

00:00:51.507 --> 00:00:53.840
and I'll show you kind of the
methodology for doing it--

00:00:53.840 --> 00:00:56.190
and then we'll talk a
little bit about why

00:00:56.190 --> 00:00:57.830
that methodology works.

00:00:57.830 --> 00:01:01.590
So 5 and 1/4, the methodology
is pretty straightforward.

00:01:01.590 --> 00:01:05.050
You say, OK, 5,
the whole number 5,

00:01:05.050 --> 00:01:08.570
that is the same
thing as 20 over 4.

00:01:08.570 --> 00:01:12.250
And so 20 over 4 plus
1 over 4 is 21 over 4.

00:01:12.250 --> 00:01:16.900
Or another way to think about
it is, 5 times 4 is 20, plus 1

00:01:16.900 --> 00:01:17.400
is 21.

00:01:17.400 --> 00:01:20.857
21 over 4.

00:01:20.857 --> 00:01:21.940
So that's the methodology.

00:01:21.940 --> 00:01:22.970
Pretty straightforward.

00:01:22.970 --> 00:01:24.400
And I'm going to
draw it out so it

00:01:24.400 --> 00:01:25.660
makes a little bit more sense.

00:01:25.660 --> 00:01:27.930
So once again, you
take the mixed number,

00:01:27.930 --> 00:01:30.310
or you take the whole
part of the mixed number,

00:01:30.310 --> 00:01:33.030
you multiply it times the
denominator, you get 20.

00:01:33.030 --> 00:01:36.304
And then you add that 20
to 1, you get 21 over 4.

00:01:36.304 --> 00:01:37.720
And now it's an
improper fraction.

00:01:37.720 --> 00:01:39.178
The absolute value
of our numerator

00:01:39.178 --> 00:01:41.650
is greater than the absolute
value of our denominator.

00:01:41.650 --> 00:01:44.530
Now I do want to show
you why this works.

00:01:44.530 --> 00:01:47.240
So to see why this
works, let's think

00:01:47.240 --> 00:01:49.290
about what 5 and 1/4 means.

00:01:49.290 --> 00:01:51.320
It means we have 5 wholes.

00:01:51.320 --> 00:01:54.730
So let's say that
this is a whole.

00:01:54.730 --> 00:01:56.100
So that's 1 whole.

00:01:56.100 --> 00:01:57.955
And let me copy and
paste it five times.

00:02:05.600 --> 00:02:14.050
So that's 2, that's 3,
that's 4, and then that is 5.

00:02:14.050 --> 00:02:15.950
So we have five wholes.

00:02:15.950 --> 00:02:20.290
So that's this stuff
that I've drawn in green.

00:02:20.290 --> 00:02:22.540
That's 5 holes right there.

00:02:22.540 --> 00:02:23.610
And then we have 1/4.

00:02:27.450 --> 00:02:29.320
So I'll do 1/4 of a whole.

00:02:33.192 --> 00:02:35.150
And just to be clear that
it's part of a whole,

00:02:35.150 --> 00:02:37.890
I could have drawn the
whole in a dotted line,

00:02:37.890 --> 00:02:40.730
to say that we don't have that
whole, we have only 1/4 of it.

00:02:40.730 --> 00:02:41.570
And that's that 1/4.

00:02:41.570 --> 00:02:43.570
So this is 5 and 1/4.

00:02:43.570 --> 00:02:45.370
So to write it as an
improper fraction,

00:02:45.370 --> 00:02:48.360
you can really view this
5 as some fraction over 4.

00:02:48.360 --> 00:02:50.990
And to think of it that
way, divide each of these

00:02:50.990 --> 00:02:51.910
into fourths.

00:02:51.910 --> 00:02:53.680
Or this is one way
to think about it.

00:02:53.680 --> 00:03:00.040
So this right over here is
4/4, this is another 4/4,

00:03:00.040 --> 00:03:03.710
this is another 4/4-- I should
have copied and pasted this--

00:03:03.710 --> 00:03:09.490
this is another 4/4,
and this is another 4/4.

00:03:09.490 --> 00:03:12.452
So now how many
fourths do we have?

00:03:12.452 --> 00:03:13.160
How many fourths?

00:03:13.160 --> 00:03:18.530
We have 4/4 here, 4/4 here, 4/4
here, 4/4 here, and 4/4 here.

00:03:18.530 --> 00:03:19.910
So just what we have in green.

00:03:22.760 --> 00:03:24.125
We have 20 fourths.

00:03:27.294 --> 00:03:29.210
That's what we have in
green, right over here.

00:03:29.210 --> 00:03:31.870
That is the same thing as 5.

00:03:31.870 --> 00:03:38.540
Each of these are 4/4, so you
could view this as 5 times

00:03:38.540 --> 00:03:39.950
4/4, right?

00:03:39.950 --> 00:03:40.800
4/4 is 1.

00:03:40.800 --> 00:03:43.720
5 times 4/4 is 20 over 4.

00:03:43.720 --> 00:03:45.840
That's what we have
right over here.

00:03:45.840 --> 00:03:47.550
And now we can add
it to that 1/4.

00:03:51.010 --> 00:03:53.200
And you will get 21.

00:03:53.200 --> 00:03:54.460
We have the same denominator.

00:03:54.460 --> 00:03:56.030
So we can just add
the numerators.

00:03:56.030 --> 00:03:58.319
21 over 4.

00:03:58.319 --> 00:04:00.610
So that's the conceptual
understanding of why it works.

00:04:00.610 --> 00:04:02.026
But when you see
any mixed number,

00:04:02.026 --> 00:04:03.650
it's a pretty
straightforward process.

00:04:03.650 --> 00:04:06.646
Multiply the 5
times 4, you get 20.

00:04:06.646 --> 00:04:09.892
20 plus 1 is equal to 21 over 4.

00:04:09.892 --> 00:04:11.600
And actually, let me
just write that out.

00:04:11.600 --> 00:04:14.410
Just so you get clear
on what I'm doing.

00:04:14.410 --> 00:04:16.980
So 5 and 1/4, that
is the same thing--

00:04:16.980 --> 00:04:24.220
that's equal to 5
times 4 plus 1, over 4.

00:04:24.220 --> 00:04:26.320
And this is why.


Proper and improper fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.210
Identify 15/8, or 15
over 8, as a proper

00:00:05.210 --> 00:00:09.237
or an improper fraction.

00:00:09.237 --> 00:00:10.820
So this is actually
just a good review

00:00:10.820 --> 00:00:12.560
of proper and
improper fractions.

00:00:12.560 --> 00:00:14.970
And there's a pretty easy
way to identify them.

00:00:14.970 --> 00:00:18.720
You have a proper fraction if
your numerator-- so this right

00:00:18.720 --> 00:00:23.210
here is your numerator,
the number on top.

00:00:23.210 --> 00:00:26.465
Let me just label them,
numerator and denominator.

00:00:31.200 --> 00:00:36.620
You have a proper
fraction if your numerator

00:00:36.620 --> 00:00:38.276
is less than your denominator.

00:00:41.554 --> 00:00:43.220
And if you don't have
a proper fraction,

00:00:43.220 --> 00:00:44.470
you have an improper fraction.

00:00:44.470 --> 00:00:49.200
So your improper fraction
is if your numerator

00:00:49.200 --> 00:00:52.951
is greater than or equal
to your denominator.

00:00:52.951 --> 00:00:55.200
And over here, actually, all
the numbers are positive.

00:00:55.200 --> 00:00:56.950
But if you had some
negative numbers here,

00:00:56.950 --> 00:00:59.830
then you would actually say the
absolute value of the numerator

00:00:59.830 --> 00:01:02.180
is less than the absolute
value of the denominator,

00:01:02.180 --> 00:01:03.480
for a proper fraction.

00:01:03.480 --> 00:01:05.280
And the absolute
value of the numerator

00:01:05.280 --> 00:01:07.560
is greater than or equal
to the absolute value

00:01:07.560 --> 00:01:09.810
of the denominator for
an improper fraction.

00:01:09.810 --> 00:01:11.850
So let me just write
that down, just

00:01:11.850 --> 00:01:14.520
to be a little bit
more particular here,

00:01:14.520 --> 00:01:17.950
because we could have negative
numbers in either the numerator

00:01:17.950 --> 00:01:18.750
or the denominator.

00:01:18.750 --> 00:01:20.770
Anyway, let's go
back to this problem.

00:01:20.770 --> 00:01:25.960
15 is clearly greater
than or equal to 8.

00:01:25.960 --> 00:01:32.230
So 15 is greater
than or equal to 8,

00:01:32.230 --> 00:01:39.809
so we are dealing with
an improper fraction.