GMAT: Math 46

WEBVTT
Kind: captions
Language: en

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All right.

00:00:00.920 --> 00:00:02.960
I re-read the question,
and now I realize

00:00:02.960 --> 00:00:04.540
the error in my logic.

00:00:04.540 --> 00:00:06.350
I misread the question.

00:00:06.350 --> 00:00:09.440
I had misread it and I said,
oh, there are 60 siblings.

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No, there are 60
sibling pairs.

00:00:16.540 --> 00:00:19.853
So that means in each class
there-- so 60 siblings pairs,

00:00:19.853 --> 00:00:21.480
that means there are
120 siblings.

00:00:21.480 --> 00:00:23.760
But that tells us that in each
class there aren't 30, there

00:00:23.760 --> 00:00:27.310
are 60 members of a sibling
group, right?

00:00:27.310 --> 00:00:30.700
So let's go back through
our logic again.

00:00:30.700 --> 00:00:33.350
Let's say I pick someone from
the junior class first. What

00:00:33.350 --> 00:00:34.920
is the probability that
I even pick someone

00:00:34.920 --> 00:00:35.965
from a sibling pair?

00:00:35.965 --> 00:00:41.810
That's going to be 60/1,000,
which is the same thing as

00:00:41.810 --> 00:00:46.910
6/100, which is the same
thing as 3/50, right?

00:00:46.910 --> 00:00:47.360
Fair enough.

00:00:47.360 --> 00:00:49.890
So there's a 3 in 50 chance that
I pick someone who's a

00:00:49.890 --> 00:00:51.650
member of a sibling pair.

00:00:51.650 --> 00:00:54.340
Now, in order for me to have a
complete sibling pair when I

00:00:54.340 --> 00:00:57.900
go to the senior class and pick
someone, I have to pick

00:00:57.900 --> 00:01:01.760
this junior sibling, and that
junior only has 1 sibling in

00:01:01.760 --> 00:01:02.980
the senior class.

00:01:02.980 --> 00:01:04.670
So there's only a 1
in 800 chance that

00:01:04.670 --> 00:01:05.560
I pick their sibling.

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It doesn't matter that-- the
other 59 siblings in that are

00:01:09.060 --> 00:01:11.380
useless, because they're
not this guy's sibling.

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So I have a 1 in 800 chance
of doing that.

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You multiply them,
you get 3/40,000.

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And that's choice A.

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If you don't see your choice,
make sure to read

00:01:22.380 --> 00:01:24.042
the question again.

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Almost out of space so let
me clear all of this.

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Let me clear all of that.

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And we're on problem 218.

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Which of the following cannot
be the median of the three

00:01:41.350 --> 00:01:44.940
positive integers x, y, and z?

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So the median, the
middle number.

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They say A is x.

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Well, no, that's not true.

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What if this was the order?
x y, z, then x

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would be the median.

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So it's not A.

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B, z .

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No reason why z can't
be the median.

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The order can be y, z, x.

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It's not B.

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C, x plus z.

00:02:09.150 --> 00:02:15.456
Well, let's see, could x
plus z be the median?

00:02:15.456 --> 00:02:19.570
x plus z, could that
be the median?

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Let me think about it.

00:02:22.130 --> 00:02:31.380
If we have x, z, and y, I mean,
by definition one of

00:02:31.380 --> 00:02:33.180
these numbers has to
be the median.

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We have three numbers,
one of them has to

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be the middle number.

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Under what circumstances--
because x plus-- I mean, if x

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plus z is the median, that
implies that neither x nor z

00:02:45.440 --> 00:02:47.390
is the median, right?

00:02:47.390 --> 00:02:48.720
Do they tell us that
those are-- right,

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they're positive integers.

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None of these are zero.

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So in order for x plus z to be
the median, that tells us that

00:02:55.470 --> 00:02:58.400
neither x nor z is the
median, right?

00:02:58.400 --> 00:03:01.390
And if neither x nor z
is the median, then y

00:03:01.390 --> 00:03:02.750
has to be the median.

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And then that tells us that x
plus z has to be equal to y.

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But if y is the median, the
order is this: x, y, z.

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And there's no way that I can
add x and z and get a smaller

00:03:15.530 --> 00:03:17.430
number than z, because
all of these numbers

00:03:17.430 --> 00:03:19.310
are positive, right?

00:03:19.310 --> 00:03:21.110
Yeah, they said three
positive integers.

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So I go with choice C.

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That cannot happen.

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The median cannot be x plus z.

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Problem 219.

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What is the 25th digit to the
right of the decimal point in

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the decimal form of 6/11?

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And whenever they ask you for
something ridiculous, there

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must be some type of pattern we
can exploit to figure out

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what the 25th digit is.

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Let's just start dividing.

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11 goes into 6, so we're
going to have to have

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some decimals there.

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11 goes into 60 five times.

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5 times 11 is 55,
and remainder 5.

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And you're going to
50, you get a 4.

00:04:05.740 --> 00:04:08.800
4 times-- I already see
the pattern-- 44.

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Oh no, never mind.

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Wait, let me make sure
that's right.

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55, you get 50 remainder, 11
goes into 50 four times.

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You get 40.

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So then you get 60.

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11 goes into 60 five times,
and you repeat

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so on and so forth.

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I just want to make sure.

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Let's see, the first digit
to the right is 25.

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The second digit to
the right is 4.

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The third digit to
the right is 5.

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And so, you know, we
could keep going.

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The fourth digit to
the right is 4.

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So every odd digit to the
right of this is what?

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It's 5.

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So they want to know the 25th
digit to the right, and that's

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an odd number, 25.

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So it has to be 5 as well.

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So that's choice C.

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Next question.

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Oh, I thought I had
to erase that.

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OK, question 220.

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John and Mary were each paid x
dollars in advance to do a

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certain job together.

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Fair enough.

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John worked on the job for 10
hours, so John worked 10

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hours, and Mary worked 2
hours less than John.

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So Mary worked 8 hours, right?

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John worked on the job for 10
hours and Mary worked for 2

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hours less than John, so
she worked 8 hours.

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If Mary gave y dollars of her
payment so that they would

00:05:27.470 --> 00:05:31.860
have received the same hourly
wage, what was the dollar

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amount in terms of y that John
was paid in advance?

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OK, John and Mary were each paid
x dollars in advance to

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do a certain job together.

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OK, so how much money did they--
so John got x dollars

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and Mary got x dollars.

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They both got x dollars,
right?

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So if you think about
it, let's ignore

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who did what, right?

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We had how many total
hours of labor?

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We had 18 hours of labor,
and they were

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paid how many dollars?

00:06:06.300 --> 00:06:18.040
They were paid 2x dollars
for 18 hours of labor.

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So the average wage should be--
you could kind of do it

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the fair-- you know, per hourly
wage should be-- let's

00:06:27.375 --> 00:06:31.330
see, we could divide by 9, you
get a 1 and you get a 9.

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You get x/9 dollars per hour.

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That's the fair way.

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So let me re-read
the question.

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If Mary gave John y dollars of
her payment so that they would

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have had the same hourly wage,
what was the dollar amount in

00:06:55.180 --> 00:06:59.480
terms of y that John was
paid in advance?

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Actually, let's think
of it this way.

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So Mary pays John-- let
me ignore this.

00:07:04.280 --> 00:07:06.190
I just thought of a better
way of doing it.

00:07:06.190 --> 00:07:09.380
So they were each paid
x hours in advance to

00:07:09.380 --> 00:07:09.930
do a certain job.

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John worked on the job for 10
hours and Mary worked 2 hours

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less than John.

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If Mary paid John y dollars of
her payment so that they would

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have received the same hourly
wage-- so after Mary pays John

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y, what-- so that means John got
a total of y dollars and

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Mary has paid y dollars.

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So now John has x plus y, and
Mary has x minus y dollars.

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And so, if Mary paid of her
payment so that they would

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receive the same hourly wage,
what was the dollar amount in

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terms of y that John was
paid in advance?

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So let me think about
this a little bit.

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So they got the same-- they
received the same hourly wage.

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So Mary should have 8/10 the
money of John, right?

00:08:03.930 --> 00:08:13.581
So x minus y should be equal
to 0.8 times x plus y.

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Am I thinking about that right
or is there an easier way to

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think about it?

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Well, let's think about
it this way.

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John worked 2 hours more
than Mary, right?

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So Mary, essentially, if she
pays him for 1 hour-- no,

00:08:31.360 --> 00:08:34.260
yeah, this is the right thing.
x minus y is equal to 0.8

00:08:34.260 --> 00:08:36.880
times x plus y, but I'm
going to still end

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up with an x here.

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So you get x minus y is equal
to 0.8x plus 0.8y, right?

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And all the logic here I just
said is-- well, Mary's total

00:08:51.900 --> 00:08:55.395
money, if we're going to be
fair, should be 0.8 of John's

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total money, because she worked
0.8 as much hours.

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And so, let's see.

00:09:00.370 --> 00:09:02.575
Oh, actually, they want to know
what John was paid in

00:09:02.575 --> 00:09:03.680
advance, so that's x.

00:09:03.680 --> 00:09:04.860
So actually, OK, we're fine.

00:09:04.860 --> 00:09:07.000
So they just want to know
what x is in terms of y.

00:09:07.000 --> 00:09:07.710
So we're cool.

00:09:07.710 --> 00:09:10.320
So let's subtract 0.8x
from both sides.

00:09:10.320 --> 00:09:17.120
So you get 0.2x minus
y is equal to 0.8y.

00:09:17.120 --> 00:09:19.180
And then add y to both sides.

00:09:19.180 --> 00:09:22.990
You get 0.2x is equal to 1.8y.

00:09:25.680 --> 00:09:30.410
And then multiply both sides by
5, you get x is equal to--

00:09:30.410 --> 00:09:31.250
what is that?

00:09:31.250 --> 00:09:34.220
5 times 18 is 50 plus 40.

00:09:34.220 --> 00:09:37.450
Add a decimal point is
90 is equal to 9y.

00:09:37.450 --> 00:09:39.310
And that is choice E

00:09:39.310 --> 00:09:41.120
I got a little confused with
the wording of it when they

00:09:41.120 --> 00:09:43.530
say what was the dollar amount
in terms of y that John was

00:09:43.530 --> 00:09:44.820
paid in advance?

00:09:44.820 --> 00:09:46.920
They should've said
what was x?

00:09:46.920 --> 00:09:48.160
And that would've been
a little bit easier.

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Anyway, I'm all out of time.

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Sorry for bumbling through
that problem.

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See you in the next video.


GMAT: Math 44

WEBVTT
Kind: captions
Language: en

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We're on problem 210.

00:00:03.235 --> 00:00:07.290
A television manufacturer
produces 600 units of a

00:00:07.290 --> 00:00:15.140
certain model each month at a
cost to the manufacturer of

00:00:15.140 --> 00:00:20.730
$90 per unit.

00:00:20.730 --> 00:00:23.960
And all of the produced units
are sold each month.

00:00:23.960 --> 00:00:27.490
What is the minimum selling
price per unit that will

00:00:27.490 --> 00:00:31.010
ensure that the monthly profit--
revenue from sales

00:00:31.010 --> 00:00:33.750
minus manufacturer's cost to
produce-- on the sales of

00:00:33.750 --> 00:00:39.460
these units will be
at least $42,000.

00:00:39.460 --> 00:00:40.310
OK.

00:00:40.310 --> 00:00:43.260
So, they want to know the
minimum selling price.

00:00:43.260 --> 00:00:46.200
So, let's call that p.

00:00:46.200 --> 00:00:48.190
So how much are they going
to make per unit?

00:00:48.190 --> 00:00:50.900
p minus whatever
their cost was.

00:00:50.900 --> 00:00:53.280
p minus $90.

00:00:53.280 --> 00:00:55.600
That's how much they
make per unit.

00:00:55.600 --> 00:00:59.490
And they do that
for 600 units.

00:00:59.490 --> 00:01:03.460
And they say that this has
to be at least $42,000.

00:01:03.460 --> 00:01:08.820
So, this has to be greater
than or equal to 42,000.

00:01:08.820 --> 00:01:09.170
OK.

00:01:09.170 --> 00:01:15.050
So we get 600p minus--
what's 9 times 6?

00:01:15.050 --> 00:01:18.580
It's 54.

00:01:18.580 --> 00:01:20.400
And then we have one,
two, three 0's.

00:01:20.400 --> 00:01:22.760
54,000-- Right?

00:01:22.760 --> 00:01:23.930
9 times 6.

00:01:23.930 --> 00:01:24.940
Yep.

00:01:24.940 --> 00:01:28.540
--is greater than or
equal to 42,000.

00:01:28.540 --> 00:01:31.285
So then you get 600p.

00:01:31.285 --> 00:01:33.760
Let's add 54,000
to both sides.

00:01:33.760 --> 00:01:42.390
Has to be greater than or equal
to 54 plus 42 is 96,000.

00:01:42.390 --> 00:01:44.520
And just to simplify things,
let's divide

00:01:44.520 --> 00:01:45.890
both sides by 100.

00:01:45.890 --> 00:01:48.670
So, you get 6p.

00:01:48.670 --> 00:01:51.310
If you get two 0's here,
two 0's here.

00:01:51.310 --> 00:01:56.190
6p has to be greater than 960.

00:01:56.190 --> 00:01:58.630
And then p has to be greater
than or equal to-- we don't

00:01:58.630 --> 00:01:59.560
have to worry about
this because

00:01:59.560 --> 00:02:01.510
it's a positive number.

00:02:01.510 --> 00:02:08.180
6 goes into 96, 16 times, so it
goes into 960, 160 times.

00:02:08.180 --> 00:02:15.460
So the minimum selling price
is $160 which is choice d.

00:02:15.460 --> 00:02:19.430
Problem 211.

00:02:19.430 --> 00:02:31.980
If 4x plus 3y is equal to minus
2, and 3x plus 6 is

00:02:31.980 --> 00:02:34.850
equal to 0, what is
the value of y?

00:02:34.850 --> 00:02:35.690
Well this is a quick one.

00:02:35.690 --> 00:02:39.190
3x would be equal to minus 6.

00:02:39.190 --> 00:02:41.450
So x is equal to minus 2.

00:02:41.450 --> 00:02:42.550
Substitute back here.

00:02:42.550 --> 00:02:48.350
4 times minus 2, minus 8 plus
3y is equal to minus 2.

00:02:48.350 --> 00:02:50.080
Add 8 to both sides.

00:02:50.080 --> 00:02:54.170
3y is equal to minus 2 plus 8.

00:02:54.170 --> 00:02:58.250
Or 3y is equal to 6.

00:02:58.250 --> 00:03:01.100
y is equal to 2.

00:03:01.100 --> 00:03:03.025
And so that is choice e.

00:03:05.720 --> 00:03:07.090
Problem 212.

00:03:07.090 --> 00:03:09.560
I'll do it in a different
color.

00:03:09.560 --> 00:03:12.060
212.

00:03:12.060 --> 00:03:14.560
The data sets one, two, and
three above are ordered from

00:03:14.560 --> 00:03:17.690
greatest standard deviation to
least standard deviation in

00:03:17.690 --> 00:03:19.840
which of the following?

00:03:19.840 --> 00:03:20.280
OK.

00:03:20.280 --> 00:03:21.740
So they want us to order
them essentially.

00:03:21.740 --> 00:03:34.540
Greatest-- So, set one was
72, 73, 74, 75, 76.

00:03:34.540 --> 00:03:44.660
Set two is 74-- it says 74 a
bunch of times-- 74, 74.

00:03:44.660 --> 00:03:55.580
And then set three has 62, and
then 74, 74, 74, and then 89.

00:03:55.580 --> 00:03:58.400
So, standard deviation is how
much, on average, do the

00:03:58.400 --> 00:04:02.040
numbers vary from the mean.

00:04:02.040 --> 00:04:04.210
And all of these cases,
the mean is 74.

00:04:04.210 --> 00:04:07.290
I'm just eyeballing it.

00:04:07.290 --> 00:04:10.610
Let's see 89 is 15 more.

00:04:10.610 --> 00:04:13.120
Actually here the
mean isn't 74.

00:04:13.120 --> 00:04:16.040
So, first of all, every number
here is the mean.

00:04:16.040 --> 00:04:17.870
So, the standard deviation
is actually 0.

00:04:17.870 --> 00:04:20.839
So this is going to be-- they
want to know from greatest

00:04:20.839 --> 00:04:23.660
standard deviation to least
standard deviation.

00:04:23.660 --> 00:04:26.980
So, 2 is definitely the least
standard deviation.

00:04:26.980 --> 00:04:30.390
So that leaves us with choices
either b or d.

00:04:33.820 --> 00:04:35.860
And now have to figure out which
of these have a lower

00:04:35.860 --> 00:04:37.750
standard deviation.

00:04:37.750 --> 00:04:41.080
So here we can just eyeball
and say the average is 74.

00:04:41.080 --> 00:04:42.710
If you average this and this
you get 74, that and

00:04:42.710 --> 00:04:44.110
that you get 74.

00:04:44.110 --> 00:04:46.500
So, what's the difference
from 74?

00:04:46.500 --> 00:04:47.730
Here it's 2.

00:04:47.730 --> 00:04:48.760
Here it's 1.

00:04:48.760 --> 00:04:50.170
Here it's 0.

00:04:50.170 --> 00:04:52.450
Here it's 1.

00:04:52.450 --> 00:04:53.710
Here it's 2.

00:04:53.710 --> 00:04:55.570
You just want the magnitude,
how far away you are

00:04:55.570 --> 00:04:57.410
from 74 each time.

00:04:57.410 --> 00:04:59.740
And if you were to average these
numbers, you get 2 plus

00:04:59.740 --> 00:05:01.230
1, plus 1 plus 2.

00:05:01.230 --> 00:05:03.970
That's 6 divided by 5.

00:05:03.970 --> 00:05:06.160
So, you get 6 over 5.

00:05:06.160 --> 00:05:08.260
You know they square-- you could
even do the variance if

00:05:08.260 --> 00:05:10.000
you really cared about it, you
could square these numbers.

00:05:10.000 --> 00:05:12.270
But you just want to get some
measure of, on average, how

00:05:12.270 --> 00:05:14.000
far is each of the numbers
from the mean.

00:05:14.000 --> 00:05:15.970
So here's 6 over 5.

00:05:15.970 --> 00:05:17.700
Let's see 2 plus 1, 3.

00:05:17.700 --> 00:05:17.950
Yep.

00:05:17.950 --> 00:05:19.240
6-- and there's five numbers.

00:05:19.240 --> 00:05:22.450
And 6 over 5 is the standard
deviation there.

00:05:22.450 --> 00:05:23.410
What's the mean here?

00:05:23.410 --> 00:05:25.390
This is interesting.

00:05:25.390 --> 00:05:27.930
So, 74 to 89.

00:05:27.930 --> 00:05:28.880
Do I have those numbers right?

00:05:28.880 --> 00:05:29.070
Yeah.

00:05:29.070 --> 00:05:32.280
to go from 74 to 89 you
have to go plus-- let

00:05:32.280 --> 00:05:33.530
me add these up.

00:05:35.960 --> 00:05:44.100
You get 62-- actually what's
the average of 62 and 89?

00:05:44.100 --> 00:05:47.836
No, it's not 74.

00:05:47.836 --> 00:05:56.820
Let's see 2 plus 11 is-- well
I could just take the

00:05:56.820 --> 00:05:58.170
average-- I just want
to figure out the

00:05:58.170 --> 00:05:59.470
mean as fast as possible.

00:05:59.470 --> 00:06:02.630
So 89 and 62.

00:06:02.630 --> 00:06:05.620
9 plus 2 is 11.

00:06:05.620 --> 00:06:07.890
9 plus 6-- 151.

00:06:07.890 --> 00:06:10.400
You divide that by 2.

00:06:10.400 --> 00:06:16.110
The mean of these two
numbers are 75.5.

00:06:16.110 --> 00:06:19.090
So, at least if you're trying to
figure out the mean, that's

00:06:19.090 --> 00:06:23.330
going to be the same thing is
the mean of-- so actually

00:06:23.330 --> 00:06:24.110
let's just do this.

00:06:24.110 --> 00:06:25.530
It's 151.

00:06:25.530 --> 00:06:27.110
Let's just take the average
of everything.

00:06:27.110 --> 00:06:30.990
I tried to be cute, but it's
better to just be brute force.

00:06:30.990 --> 00:06:38.560
So, 62 plus 89 is 151, plus
74 times 3, divided by 5.

00:06:38.560 --> 00:06:38.840
Let's see.

00:06:38.840 --> 00:06:44.690
74 times 3.

00:06:44.690 --> 00:06:46.050
4 times 3 is 12.

00:06:46.050 --> 00:06:47.430
7 times 3 is 21.

00:06:47.430 --> 00:06:48.700
22.

00:06:48.700 --> 00:06:54.470
And then 222 plus 151 is 373.

00:06:54.470 --> 00:06:56.880
And then we're going to
divide that by 5.

00:06:56.880 --> 00:07:07.250
So, 5 goes into 373-- 7 times 5
is 35, 23, 5 goes into 23 4

00:07:07.250 --> 00:07:09.870
times, 4 times 5 is 20.

00:07:09.870 --> 00:07:11.100
3.

00:07:11.100 --> 00:07:13.790
74.6.

00:07:13.790 --> 00:07:16.340
So the mean here is 74.6.

00:07:16.340 --> 00:07:18.040
This is going to have the
largest standard deviation

00:07:18.040 --> 00:07:21.570
because this is 0.6 from the
means, this is 0.6 from the

00:07:21.570 --> 00:07:23.890
mean, this is 0.6
from the mean.

00:07:23.890 --> 00:07:28.920
This number is 12.6
from the mean.

00:07:28.920 --> 00:07:33.590
And then this number is
14.4 from the mean.

00:07:33.590 --> 00:07:37.590
So, if you're to average these
numbers, you're going to get a

00:07:37.590 --> 00:07:41.030
bigger number than 6/5, but
let's just confirm it.

00:07:41.030 --> 00:07:46.090
So, 12.6 plus 14.4-- oh yeah,
you're definitely going to get

00:07:46.090 --> 00:07:48.550
a much bigger number because
when you summed all these up,

00:07:48.550 --> 00:07:49.350
you got a 6.

00:07:49.350 --> 00:07:55.060
When you just sum up the 12 and
the 14.4 you get-- 15--

00:07:55.060 --> 00:07:56.070
you get 27.

00:07:56.070 --> 00:07:57.740
And then it's going
to get to 29.

00:07:57.740 --> 00:07:59.510
You divide that by 5.

00:07:59.510 --> 00:08:00.970
The standard deviation here
is going to be 4.

00:08:00.970 --> 00:08:03.970
So, 3 has the largest
standard deviation.

00:08:03.970 --> 00:08:08.220
2 has the least. So, the
correct answer is d.

00:08:08.220 --> 00:08:09.510
I should've been able
to eyeball that.

00:08:09.510 --> 00:08:11.660
I shouldn't have had to go
through all that computation.

00:08:11.660 --> 00:08:12.650
I took too much time.

00:08:12.650 --> 00:08:14.120
That wouldn't have been
acceptable on the GMAT.

00:08:16.910 --> 00:08:21.150
Problem 213.

00:08:21.150 --> 00:08:24.890
Which of the following is the
lowest positive integer that

00:08:24.890 --> 00:08:27.890
is the divisible by all
of those numbers?

00:08:27.890 --> 00:08:33.190
So, it's 2, 3, 4,
5, 6, 7, 8, 8.

00:08:33.190 --> 00:08:34.980
So the lowest positive
integer that is

00:08:34.980 --> 00:08:37.299
divisible by all of those.

00:08:37.299 --> 00:08:41.120
So, one way to think about it is
let's just try to minimize

00:08:41.120 --> 00:08:43.020
our multiplication.

00:08:43.020 --> 00:08:48.140
If we're divisible bye 8,
then we don't have to

00:08:48.140 --> 00:08:51.160
worry about 2 and 4.

00:08:51.160 --> 00:08:54.730
Being divisible by
8 is enough.

00:08:54.730 --> 00:08:57.670
Likewise if we're divisible by
9, we don't have to worry

00:08:57.670 --> 00:08:59.726
about being divisible by 3
because we're definitely going

00:08:59.726 --> 00:09:01.810
to be divisible by 3.

00:09:01.810 --> 00:09:04.340
We need to be divisible by 7.

00:09:04.340 --> 00:09:08.700
If we're divisible by 6-- well,
if we're divisible by 8

00:09:08.700 --> 00:09:12.880
and 9 then that means we're
already divisible by 2 and 3.

00:09:12.880 --> 00:09:16.400
And if we're divisible by 2 and
3 then we don't have to

00:09:16.400 --> 00:09:18.760
worry about 6.

00:09:18.760 --> 00:09:20.930
So, we only have to worry
about 5, 7, 8, and 9.

00:09:20.930 --> 00:09:23.350
So what are those numbers when
you multiply by each other?

00:09:23.350 --> 00:09:27.900
So if we do 8 times 9 is 72.

00:09:27.900 --> 00:09:35.240
8 times 9 is equal to 72, we got
rid of those, times 7 is

00:09:35.240 --> 00:09:38.270
equal to 2 times 7 is 14.

00:09:38.270 --> 00:09:41.320
7 times 7 is 49, plus 1 is 50.

00:09:41.320 --> 00:09:42.570
And that times 5.

00:09:45.280 --> 00:09:47.040
4 times 5 is 20.

00:09:47.040 --> 00:09:48.450
5 times 0 is 0.

00:09:48.450 --> 00:09:49.730
Bring down the 2.

00:09:49.730 --> 00:09:51.030
5 times 5 is 25.

00:09:51.030 --> 00:09:52.860
So, it's 2,520.

00:09:52.860 --> 00:09:55.480
And that is choice c.

00:09:55.480 --> 00:09:56.370
And I'm out of time.

00:09:56.370 --> 00:09:58.170
See you in the next video.


GMAT: Math 43

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.280
We're on problem 207.

00:00:04.280 --> 00:00:12.630
If n is equal to 4p, where p is
a prime number greater than

00:00:12.630 --> 00:00:18.350
2, how many different positive,
even divisors does n

00:00:18.350 --> 00:00:20.590
have including n?

00:00:20.590 --> 00:00:24.960
So how many integers go
into n, or go into 4p?

00:00:24.960 --> 00:00:29.220
Well, definitely 4 goes into it,
because this is going to

00:00:29.220 --> 00:00:30.690
be a multiple of 4.

00:00:30.690 --> 00:00:32.750
Then anything that goes into
4 will also go into it.

00:00:32.750 --> 00:00:35.070
So 2 goes into it.

00:00:35.070 --> 00:00:37.430
2 is also a divisor.

00:00:37.430 --> 00:00:40.800
p would go into it.

00:00:40.800 --> 00:00:42.520
Are there any other numbers
that go into p?

00:00:42.520 --> 00:00:43.410
No, p is prime.

00:00:43.410 --> 00:00:45.260
Well, you know 1 goes into p.

00:00:45.260 --> 00:00:47.950
But they wanted to know
the different

00:00:47.950 --> 00:00:49.860
positive, even divisors.

00:00:49.860 --> 00:00:51.740
So, the positive-- we're just
dealing in the positive world.

00:00:51.740 --> 00:00:53.570
That just lets us know this
isn't some trick question

00:00:53.570 --> 00:00:55.270
where we have to count
negative numbers.

00:00:55.270 --> 00:00:57.260
But they want even divisors.

00:00:57.260 --> 00:00:58.870
So, first of all,
1 is not even.

00:00:58.870 --> 00:01:00.185
So that can't be it.

00:01:00.185 --> 00:01:02.830
The second question,
is p even?

00:01:02.830 --> 00:01:06.300
Well, by definition, if p was
even, and if it's greater than

00:01:06.300 --> 00:01:08.070
2, it would be divisible by 2.

00:01:08.070 --> 00:01:09.880
So it wouldn't be able
to be prime.

00:01:09.880 --> 00:01:11.890
But they tell us that
p is prime.

00:01:11.890 --> 00:01:13.670
P is prime and it's
greater than 2.

00:01:13.670 --> 00:01:15.560
So it can't be divisible by 2.

00:01:15.560 --> 00:01:16.960
So it has to be odd.

00:01:16.960 --> 00:01:19.160
So this is also not
an even divisor.

00:01:19.160 --> 00:01:21.700
So there's only two even
divisors, 4 and 2.

00:01:21.700 --> 00:01:23.500
So the answer is 2.

00:01:23.500 --> 00:01:26.050
A.

00:01:26.050 --> 00:01:30.750
Problem 208.

00:01:30.750 --> 00:01:33.145
S is a set containing
9 different numbers.

00:01:35.700 --> 00:01:41.060
T is a set containing 8 numbers,
all of which are

00:01:41.060 --> 00:01:44.430
members of S.

00:01:44.430 --> 00:01:47.810
Which of the following
statements cannot be true?

00:01:47.810 --> 00:01:53.220
The mean of S is equal
to the mean of T.

00:01:53.220 --> 00:01:55.900
So, let's think about
this a little bit.

00:01:55.900 --> 00:01:57.830
So, choice A, that their
averages are

00:01:57.830 --> 00:01:58.530
equal to each other.

00:01:58.530 --> 00:02:07.330
Let's just say that set S is
equal to 1 through 9, 1, 2, 3,

00:02:07.330 --> 00:02:12.570
4, 5, 6, 7, 8, 9.

00:02:12.570 --> 00:02:14.850
So what would its mean be?

00:02:14.850 --> 00:02:16.460
Well its mean, if you average
all these numbers, if you

00:02:16.460 --> 00:02:19.100
average 1 and 9, you get 5.

00:02:19.100 --> 00:02:20.410
If you average 2 and
8, you get 5.

00:02:20.410 --> 00:02:21.880
If you average 3 and
7, you get 5.

00:02:21.880 --> 00:02:23.080
You average 4 and
6, you get 5.

00:02:23.080 --> 00:02:24.490
If you average 5, you get 5.

00:02:24.490 --> 00:02:25.845
So the average is 5.

00:02:29.180 --> 00:02:32.210
So if this is S, what
if T were this?

00:02:32.210 --> 00:02:36.400
What if T were just
this right here?

00:02:36.400 --> 00:02:37.170
Let me label that.

00:02:37.170 --> 00:02:38.420
That's T.

00:02:40.540 --> 00:02:41.610
That's not 7.

00:02:41.610 --> 00:02:43.360
That's T.

00:02:43.360 --> 00:02:45.150
Then S is everything.

00:02:45.150 --> 00:02:47.920
So, it's T plus the 5.

00:02:47.920 --> 00:02:50.380
So the average of all the
numbers in T, if you average

00:02:50.380 --> 00:02:56.140
1, 2, 3, 4, 6, 7, 8, 9, your
average is going to be 5.

00:02:56.140 --> 00:02:57.595
If you think about it, the
average of 4 and 6 is 5.

00:02:57.595 --> 00:02:58.740
The average of 3 and 7 is 5.

00:02:58.740 --> 00:02:59.450
I think you get the idea.

00:02:59.450 --> 00:03:01.830
You could work it
out if you like.

00:03:01.830 --> 00:03:05.490
If you were to take set S, which
is all of these numbers

00:03:05.490 --> 00:03:09.450
plus the number 5, if you add
5 to a set of numbers whose

00:03:09.450 --> 00:03:11.610
average is 5, the average
is still going to be 5.

00:03:11.610 --> 00:03:14.100
So A definitely works.

00:03:14.100 --> 00:03:19.870
B, The median of S is equal
to the median of T.

00:03:19.870 --> 00:03:23.450
Well, once again, that's
true with this set.

00:03:23.450 --> 00:03:25.040
This is a good set
to work with.

00:03:25.040 --> 00:03:27.050
It proves everything
we need to prove.

00:03:27.050 --> 00:03:28.280
What's the median of T.

00:03:28.280 --> 00:03:29.790
It's the middle number in T.

00:03:29.790 --> 00:03:32.000
There's 8 numbers, so
you have to average

00:03:32.000 --> 00:03:33.700
the middle two numbers.

00:03:33.700 --> 00:03:37.180
So, if you average the middle
two numbers in T, 4

00:03:37.180 --> 00:03:39.050
and 6, you get 5.

00:03:39.050 --> 00:03:42.100
So 5 is median of T.

00:03:42.100 --> 00:03:43.180
Then what's the median of S?

00:03:43.180 --> 00:03:44.390
Well, now S includes the 5.

00:03:44.390 --> 00:03:46.000
So you can put all these
numbers in order.

00:03:46.000 --> 00:03:48.210
So 5 is the middle number.

00:03:48.210 --> 00:03:51.170
So 5 is also the median of S.

00:03:54.140 --> 00:03:56.310
So, both of these can
happen, A or B.

00:03:56.310 --> 00:04:00.160
Choice C, the range of S is
equal to the range of T.

00:04:00.160 --> 00:04:03.000
The range of a set is just the
difference between the highest

00:04:03.000 --> 00:04:04.540
and lowest number.

00:04:04.540 --> 00:04:07.840
Then the range of S
is equal to what?

00:04:07.840 --> 00:04:09.250
The highest number in S is 9.

00:04:09.250 --> 00:04:11.510
The lowest number in S is 1.

00:04:11.510 --> 00:04:15.270
So that equals 9 minus 1,
which is equal to 8.

00:04:15.270 --> 00:04:17.640
What's the range of T?

00:04:17.640 --> 00:04:19.370
Well the highest number
in T is 9.

00:04:19.370 --> 00:04:20.769
The lowest number is 1.

00:04:20.769 --> 00:04:22.150
9 minus 1 is equal to 8.

00:04:22.150 --> 00:04:25.680
So this one can definitely
be true.

00:04:25.680 --> 00:04:34.390
Statement D, the mean of S is
greater than the mean of T.

00:04:34.390 --> 00:04:35.770
Well, sure.

00:04:35.770 --> 00:04:40.620
Instead of this being a 5,
what if this was a 50?

00:04:40.620 --> 00:04:43.660
Now all of a sudden the average
of T would still be 5.

00:04:43.660 --> 00:04:46.975
But if you threw this 50 in
there for S, your average is

00:04:46.975 --> 00:04:50.030
going to go a good
bit above 5.

00:04:50.030 --> 00:04:52.570
So statement D could definitely
work if you just

00:04:52.570 --> 00:04:56.550
make that extra number that's
in S a lot larger number.

00:04:56.550 --> 00:04:58.980
Then statement E, which I'm
guessing is the choice it

00:04:58.980 --> 00:05:01.180
cannot be true because we could
figure out a case for

00:05:01.180 --> 00:05:03.030
all of A through D.

00:05:03.030 --> 00:05:06.460
The range of S is less
than the range of T.

00:05:09.300 --> 00:05:11.280
So the range of the difference
between the highest and the

00:05:11.280 --> 00:05:18.030
lowest. No matter what, the
range of S has to be at least

00:05:18.030 --> 00:05:19.960
as large as the range of T.

00:05:19.960 --> 00:05:20.930
Think about it.

00:05:20.930 --> 00:05:26.280
In this case, the range of T
is 9 minus 1, which is 8.

00:05:29.220 --> 00:05:32.005
You could make the range of
S larger if you use 50.

00:05:32.005 --> 00:05:35.500
If 50 was the extra number, then
the range is 50 minus 1.

00:05:35.500 --> 00:05:38.110
But considering that all of
these numbers in T are also in

00:05:38.110 --> 00:05:42.900
S, the difference between the
highest number in T and the

00:05:42.900 --> 00:05:47.910
lowest number in T has to be
less than or equal to the

00:05:47.910 --> 00:05:50.240
range of S.

00:05:50.240 --> 00:05:53.590
So they're saying the opposite,
the range of S is

00:05:53.590 --> 00:05:55.520
less than the range of T.

00:05:55.520 --> 00:06:00.680
We know this cannot be true
because T is a subset of S.

00:06:00.680 --> 00:06:03.920
So the choice is E.

00:06:03.920 --> 00:06:10.340
Next problem-- it took me more
time that I wanted to-- 209,

00:06:10.340 --> 00:06:17.520
How many different positive
integers are factors of 441?

00:06:17.520 --> 00:06:20.380
Well, essentially it's
just factoring it.

00:06:20.380 --> 00:06:23.800
Well let's see, this should be
divisible by-- let's see, 4

00:06:23.800 --> 00:06:29.640
plus 4 is 8 plus 1 is 9-- so
this should be divisible by 3.

00:06:29.640 --> 00:06:32.140
So 3 goes into 441.

00:06:32.140 --> 00:06:37.020
1 times 3 is 1.

00:06:37.020 --> 00:06:40.540
Bring down a 4, it goes
in it 4 times, 12, and

00:06:40.540 --> 00:06:45.560
then you get 21, 27.

00:06:45.560 --> 00:06:51.140
So, you get 3 times 147.

00:06:51.140 --> 00:06:54.550
3 should also go into this,
because 1 plus 4 plus 7 is 12,

00:06:54.550 --> 00:06:56.310
which is divisible by 3.

00:06:56.310 --> 00:06:59.630
So 3 goes into 147.

00:06:59.630 --> 00:07:08.430
147, you go 4 times
12, 27, 49 times.

00:07:08.430 --> 00:07:11.510
So let's just do a prime
factorization of 441.

00:07:11.510 --> 00:07:15.210
You get 3 and 147.

00:07:15.210 --> 00:07:24.950
147 factors into 3 and 27, which
then factors into 3 and

00:07:24.950 --> 00:07:30.450
9, which then factors
into 3 and 3.

00:07:30.450 --> 00:07:30.870
OK.

00:07:30.870 --> 00:07:32.890
So I didn't realize, but
this is a [? link ?].

00:07:32.890 --> 00:07:35.520
This is the same thing
as 3 to the fifth.

00:07:35.520 --> 00:07:36.770
Is that right?

00:07:41.390 --> 00:07:42.060
Oh no, sorry.

00:07:42.060 --> 00:07:44.500
I made a mistake.

00:07:44.500 --> 00:07:45.810
Let me redo this.

00:07:45.810 --> 00:07:50.200
3 goes into 147 not 27
times, but 49 times.

00:07:50.200 --> 00:07:53.550
So then this is, if
you you prime

00:07:53.550 --> 00:07:57.410
factorization, a 7 and a 7.

00:07:57.410 --> 00:08:00.370
I was about to say, 441 isn't
3 to the fifth power.

00:08:00.370 --> 00:08:02.520
So let's write that down here.

00:08:02.520 --> 00:08:03.380
Let me do it in a
different color.

00:08:03.380 --> 00:08:04.220
I don't want to be confusing.

00:08:04.220 --> 00:08:13.330
So 441 is equal to 3 times
3 times 7 times 7.

00:08:13.330 --> 00:08:15.110
So if they want to know how
many different positive

00:08:15.110 --> 00:08:18.310
integers are factors, so all you
have to think about is how

00:08:18.310 --> 00:08:21.680
many positive integers
can I construct with

00:08:21.680 --> 00:08:23.540
two 3's and two 7's?

00:08:23.540 --> 00:08:24.790
So let's just list them.

00:08:24.790 --> 00:08:26.450
3 definitely works.

00:08:26.450 --> 00:08:28.140
3 times 3 will definitely
work.

00:08:28.140 --> 00:08:28.670
Well, let's go in order.

00:08:28.670 --> 00:08:30.460
7 will definitely work.

00:08:30.460 --> 00:08:31.690
3 times 3 will work.

00:08:31.690 --> 00:08:33.610
That's 9.

00:08:33.610 --> 00:08:36.000
3 times 7 would work.

00:08:36.000 --> 00:08:37.990
21.

00:08:37.990 --> 00:08:42.700
3 times 49 will work.

00:08:42.700 --> 00:08:46.630
3 times 7 times 7, so 3 times
49 will work, which was

00:08:46.630 --> 00:08:49.610
actually 147.

00:08:49.610 --> 00:08:52.620
147 works.

00:08:52.620 --> 00:08:57.680
Then 9 times 7 would
work, which is 63.

00:08:57.680 --> 00:08:59.320
9 times 7.

00:08:59.320 --> 00:09:03.200
Then you have 9 times
7 times 7.

00:09:03.200 --> 00:09:05.850
That's essentially 441.

00:09:05.850 --> 00:09:08.260
Let me make sure I haven't
missed any combinations.

00:09:08.260 --> 00:09:10.270
You have one 3 and one 7.

00:09:10.270 --> 00:09:12.160
So you have a 3 and a 7.

00:09:12.160 --> 00:09:16.140
You have a 3 and two 7's.

00:09:16.140 --> 00:09:18.545
You have two 3's and a 7.

00:09:21.910 --> 00:09:28.600
Then you could have two
3's and two 7's.

00:09:28.600 --> 00:09:30.430
Then, of course, you have
the 3 and the 7.

00:09:30.430 --> 00:09:33.400
So I have 1, 2, 3,
4, 5, 6 numbers.

00:09:33.400 --> 00:09:39.460
Here I have 1, 2, 3-- what
am I missing here?

00:09:39.460 --> 00:09:45.170
So, I have a 3, 7--
oh, of course.

00:09:45.170 --> 00:09:49.620
I'm actually missing the 3 times
the 3, and I'm missing

00:09:49.620 --> 00:09:51.840
the 7 times the 7.

00:09:51.840 --> 00:09:59.190
So, it's 1, 2, 3, 4, 5, 6, 7,
8 factors of which 441-- and

00:09:59.190 --> 00:10:00.485
that's not one of the choices.

00:10:04.170 --> 00:10:05.600
How many different positives
integers?

00:10:05.600 --> 00:10:08.870
Oh, and of course,
the number 1.

00:10:08.870 --> 00:10:10.000
So, how many does
that come to?

00:10:10.000 --> 00:10:14.710
1, 2, 3, 4, 5, 6,
7, 8, 9 numbers.

00:10:14.710 --> 00:10:17.010
So, that is D.

00:10:17.010 --> 00:10:18.580
There must be a faster
way of doing that.

00:10:18.580 --> 00:10:20.610
I'll let you think
about that one.

00:10:20.610 --> 00:10:20.950
Anyway.

00:10:20.950 --> 00:10:22.780
See you in the next video.


GMAT: Math 42

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:06.220
We're on problem 205.

00:00:06.220 --> 00:00:08.640
In the table above, what
is the number of green

00:00:08.640 --> 00:00:10.410
marbles in jar R?

00:00:10.410 --> 00:00:11.430
OK.

00:00:11.430 --> 00:00:13.950
Let's see, they give us
a table, they have

00:00:13.950 --> 00:00:21.680
jars P, Q, and R.

00:00:21.680 --> 00:00:31.860
They say number of red,
is x, y, and x.

00:00:31.860 --> 00:00:41.460
And they tell us the number
of green is y, z, and z.

00:00:41.460 --> 00:00:52.520
And then they tell us the total
is 80, 120, and 160.

00:00:52.520 --> 00:00:54.540
And they say, in the table
above, what is the number of

00:00:54.540 --> 00:00:57.610
green marbles in jars?

00:00:57.610 --> 00:00:59.875
So we need to solve for z.

00:01:02.600 --> 00:01:07.140
So what we have here is
three variables and

00:01:07.140 --> 00:01:08.220
we have three unknowns.

00:01:08.220 --> 00:01:10.220
We actually have three equations
because we know that

00:01:10.220 --> 00:01:15.100
the red plus the green is equal
to the total for any one

00:01:15.100 --> 00:01:16.090
of the jars.

00:01:16.090 --> 00:01:23.310
y plus z is equal to 120 and
x plus z is equal to 160.

00:01:23.310 --> 00:01:26.170
So let's just try to solve
this equation.

00:01:26.170 --> 00:01:29.720
So let's try to get everything--
I don't know,

00:01:29.720 --> 00:01:32.290
what do we want to
cancel out first?

00:01:32.290 --> 00:01:34.980
What happens if we subtract that
equation from this one?

00:01:34.980 --> 00:01:36.490
Let me write that one over.

00:01:36.490 --> 00:01:42.630
So that's x plus y
is equal to 80.

00:01:42.630 --> 00:01:45.220
I just rewrote that
column down here.

00:01:45.220 --> 00:01:49.390
Now, if we were to subtract this
from that-- let's just

00:01:49.390 --> 00:01:50.560
well multiply it
by negative 1.

00:01:50.560 --> 00:01:53.300
So, minus x, minus
y, minus 80.

00:01:53.300 --> 00:01:54.710
And now let's just add them.

00:01:54.710 --> 00:02:01.190
We get 0 plus z minus y, so we
get z minus y is equal to--

00:02:01.190 --> 00:02:02.960
what's 160 minus 80?

00:02:02.960 --> 00:02:06.220
Well that's just 80.

00:02:06.220 --> 00:02:08.834
And now, this is interesting,
we have x minus y equals 80,

00:02:08.834 --> 00:02:10.669
and z plus y is equal to 120.

00:02:10.669 --> 00:02:11.850
So I think we're pretty close.

00:02:11.850 --> 00:02:12.780
Let's write that.

00:02:12.780 --> 00:02:14.390
So we could write
this equation--

00:02:14.390 --> 00:02:16.140
I'll write it in yellow.

00:02:16.140 --> 00:02:22.710
z plus y-- I just switched the
order-- is equal to 120.

00:02:22.710 --> 00:02:24.340
But, in the back of my mind, I'm
like, well, all I have to

00:02:24.340 --> 00:02:25.110
do is figure out these.

00:02:25.110 --> 00:02:26.570
If I can figure out
these as fast as

00:02:26.570 --> 00:02:28.920
possible, all the better.

00:02:28.920 --> 00:02:30.560
Add these two equations.

00:02:30.560 --> 00:02:35.980
2z plus 0 is equal to 240.

00:02:35.980 --> 00:02:39.080
z is equal to 120.

00:02:39.080 --> 00:02:43.400
And that's the number of
green marbles in jar R.

00:02:43.400 --> 00:02:45.070
120.

00:02:45.070 --> 00:02:49.420
And that is not one
of the choices.

00:02:49.420 --> 00:02:51.500
Where did I make a mistake?

00:02:51.500 --> 00:02:53.920
I often make a mistake
right near the end.

00:02:53.920 --> 00:02:55.160
Let's see.

00:02:55.160 --> 00:02:57.270
x plus z.

00:02:57.270 --> 00:02:57.940
Let me see.

00:02:57.940 --> 00:03:00.270
X plus y is equal to 80.

00:03:00.270 --> 00:03:01.780
y plus z is equal to 120.

00:03:01.780 --> 00:03:07.240
x plus z is equal to 160.

00:03:07.240 --> 00:03:11.350
Now, I took this equation--
no, I took this one and I

00:03:11.350 --> 00:03:12.640
multiplied it by negative
1 and put it here.

00:03:12.640 --> 00:03:15.550
Minus x, minus y, is
equal to minus 80.

00:03:15.550 --> 00:03:17.150
Then I added these two.

00:03:17.150 --> 00:03:21.000
So the x's cancel out and you're
left with z minus y is

00:03:21.000 --> 00:03:26.700
equal to 80.

00:03:26.700 --> 00:03:31.310
And then, this equation says y
plus z is equal to 120-- oh, I

00:03:31.310 --> 00:03:33.760
see my mistake.

00:03:33.760 --> 00:03:35.320
You see, I made it
right at the end.

00:03:35.320 --> 00:03:36.930
z minus y is equal to 80.

00:03:36.930 --> 00:03:39.540
z plus y is equal to 120.

00:03:39.540 --> 00:03:42.840
80 plus 120 is not 240.

00:03:42.840 --> 00:03:45.820
80 plus 120 is 200.

00:03:45.820 --> 00:03:47.980
So 2z is equal to 200.

00:03:47.980 --> 00:03:49.520
z is equal to 100.

00:03:49.520 --> 00:03:51.320
Boneheaded mistake.

00:03:51.320 --> 00:03:53.730
So the answer is 100.

00:03:53.730 --> 00:03:55.800
D.

00:03:55.800 --> 00:03:57.690
Next question.

00:03:57.690 --> 00:04:01.200
Question 206.

00:04:01.200 --> 00:04:07.060
In the circle above, PQ-- let
me draw the circle above.

00:04:07.060 --> 00:04:08.950
This is the circle above.

00:04:08.950 --> 00:04:10.890
Let's see, in the circle
above, PQ-- OK

00:04:10.890 --> 00:04:12.180
they then draw diameter.

00:04:12.180 --> 00:04:14.382
Let me draw diameter as well.

00:04:14.382 --> 00:04:17.130
Because I want to be accurate.

00:04:17.130 --> 00:04:24.020
So that's diameter and they
labelled this as O and R.

00:04:24.020 --> 00:04:26.940
And they say PQ-- so they have
this other line up here that

00:04:26.940 --> 00:04:29.960
this is parallel to
the diameter.

00:04:29.960 --> 00:04:32.590
So let me draw it up here.

00:04:32.590 --> 00:04:35.160
Maybe it looks something
like this.

00:04:35.160 --> 00:04:37.160
That's parallel.

00:04:37.160 --> 00:04:39.970
And they call this point
P and this point Q.

00:04:39.970 --> 00:04:41.845
And then they have another
line they say

00:04:41.845 --> 00:04:43.790
is a 35 degree angle.

00:04:43.790 --> 00:04:46.600
Let me see if I can draw that.

00:04:46.600 --> 00:04:49.600
You have a line that goes
from here to here.

00:04:49.600 --> 00:04:51.720
And they tell us
that this angle

00:04:51.720 --> 00:04:56.140
right here is 35 degrees.

00:04:56.140 --> 00:04:59.880
In the circle above, PQ is
parallel to diameter OR.

00:05:02.820 --> 00:05:07.620
And OR has length 18.

00:05:07.620 --> 00:05:12.090
What is the length
of minor arc PQ?

00:05:12.090 --> 00:05:13.750
Fascinating.

00:05:13.750 --> 00:05:21.080
So, they want to know
what this is.

00:05:21.080 --> 00:05:23.930
So, we can figure out the
circumference of this circle.

00:05:23.930 --> 00:05:25.330
The circumference, we
know its diameter.

00:05:25.330 --> 00:05:28.540
It's just going to be the
diameter times pi.

00:05:28.540 --> 00:05:30.800
So, we already know that the
circumference of the circle is

00:05:30.800 --> 00:05:33.470
18 pi or 2 times the
radius times pi.

00:05:33.470 --> 00:05:34.920
But same thing.

00:05:34.920 --> 00:05:38.810
Now, in order to figure out this
arc length, what would be

00:05:38.810 --> 00:05:41.540
useful-- and I don't know yet if
I can figure it out, is if

00:05:41.540 --> 00:05:45.380
we could figure out this
angle right here.

00:05:45.380 --> 00:05:46.970
If we could figure out
this angle right

00:05:46.970 --> 00:05:49.740
here, we'd be done.

00:05:49.740 --> 00:05:54.495
We could easily figure it out
because that angle as a

00:05:54.495 --> 00:05:57.690
proportion of 360 would be the
same relative proportion of

00:05:57.690 --> 00:05:59.140
the total radius.

00:05:59.140 --> 00:06:02.600
So let's see if we can figure
out what that angle is.

00:06:02.600 --> 00:06:05.100
Just given the information
they've given us.

00:06:05.100 --> 00:06:10.130
This angle here's is going
to be 35 degrees.

00:06:10.130 --> 00:06:11.080
And how did I know that?

00:06:11.080 --> 00:06:13.080
Well, those are those are
opposite inside angles.

00:06:16.030 --> 00:06:16.620
Let me see.

00:06:16.620 --> 00:06:20.540
What matters is-- let me think
about it a little bit.

00:06:25.740 --> 00:06:27.250
And this is on the circle.

00:06:27.250 --> 00:06:28.500
Oh, well this is interesting.

00:06:32.320 --> 00:06:33.580
We should use the information.

00:06:33.580 --> 00:06:34.220
This is a circle.

00:06:34.220 --> 00:06:37.930
So, what is the length
of that right there?

00:06:37.930 --> 00:06:43.530
Well, if the diameter is 18 then
the radius is 9, right?

00:06:43.530 --> 00:06:48.800
If that length is 9, than this
length is also going to be 9.

00:06:48.800 --> 00:06:52.060
And if that length is 9--
oh this is interesting.

00:06:52.060 --> 00:06:54.690
This length is 9, this
length is also 9.

00:06:57.610 --> 00:06:58.730
That length also has to be 9.

00:06:58.730 --> 00:07:01.490
They're both radiuses of the
circle which tells us that

00:07:01.490 --> 00:07:03.380
this, right here, is kind
of an isosceles

00:07:03.380 --> 00:07:05.300
triangle on its side.

00:07:05.300 --> 00:07:06.550
It looks like that.

00:07:08.950 --> 00:07:10.050
Where these sides are equal.

00:07:10.050 --> 00:07:11.250
They're both equal to 9.

00:07:11.250 --> 00:07:15.250
So, if this is a 35 degree
angle, then this, right here,

00:07:15.250 --> 00:07:17.350
has to be a 35 degree
angle as well.

00:07:17.350 --> 00:07:18.530
I know you can't read it.

00:07:18.530 --> 00:07:24.030
Because the base angles of an
isosceles triangle are equal.

00:07:24.030 --> 00:07:25.260
I could redraw it like this.

00:07:25.260 --> 00:07:26.510
This is an interesting
problem.

00:07:29.160 --> 00:07:31.510
So, if we know that length is
equal to that length, we know

00:07:31.510 --> 00:07:34.190
that angle is equal
to that angle.

00:07:34.190 --> 00:07:38.230
So, then we know that this angle
right here has also got

00:07:38.230 --> 00:07:43.800
to be equal to 35.

00:07:43.800 --> 00:07:47.550
This is useful because if this
angle is 35 and this angle is

00:07:47.550 --> 00:07:50.060
35, what is this entire angle?

00:07:50.060 --> 00:07:52.170
What is that entire angle?

00:07:52.170 --> 00:07:54.840
Well that entire angle--
let me draw it up here.

00:07:54.840 --> 00:07:57.300
So that's the PQ side.

00:07:57.300 --> 00:07:59.100
So, this is also 35.

00:07:59.100 --> 00:08:02.180
So this entire angle
right here is 70.

00:08:02.180 --> 00:08:04.850
35 plus 35.

00:08:04.850 --> 00:08:06.670
I should have drawn
this a lot bigger.

00:08:06.670 --> 00:08:08.330
So, this right here
is 70 degrees.

00:08:08.330 --> 00:08:08.980
And how do I figure it out?

00:08:08.980 --> 00:08:12.640
I figure out this 35 because
this is an isosceles triangle.

00:08:12.640 --> 00:08:15.460
And I figured out
that this is 35.

00:08:15.460 --> 00:08:19.960
I figured out that this is right
here is 35 because it's

00:08:19.960 --> 00:08:22.120
an opposite angle from a
transversal between two

00:08:22.120 --> 00:08:22.930
parallel lines.

00:08:22.930 --> 00:08:28.210
Because that line is parallel
to that line.

00:08:28.210 --> 00:08:32.070
So, that was all
pretty useful.

00:08:32.070 --> 00:08:36.520
Now, what if there was only
a way to figure out

00:08:36.520 --> 00:08:37.620
what this angle was?

00:08:37.620 --> 00:08:40.419
Because if we know this angle
and this angle, then we know

00:08:40.419 --> 00:08:42.870
for sure this angle.

00:08:42.870 --> 00:08:45.440
Well, I'm going to use
the same argument.

00:08:45.440 --> 00:08:50.350
This line right here, this is
a radius of the triangle.

00:08:50.350 --> 00:08:52.900
So this side right
here is also 9.

00:08:52.900 --> 00:08:54.030
That side.

00:08:54.030 --> 00:08:58.470
so, if we know that this side is
equal to this side-- let me

00:08:58.470 --> 00:09:00.560
draw the triangle that I'm
talking about now, because I

00:09:00.560 --> 00:09:02.450
think this is an important
problem.

00:09:02.450 --> 00:09:04.465
So, I'm trying to figure
out the value of

00:09:04.465 --> 00:09:06.800
this upside down triangle.

00:09:06.800 --> 00:09:09.230
And I'll label it so you know
what I'm talking about.

00:09:09.230 --> 00:09:10.710
So, this is P.

00:09:10.710 --> 00:09:11.830
This is Q.

00:09:11.830 --> 00:09:15.180
And this is the center
of the circle.

00:09:15.180 --> 00:09:18.360
And we know that the side
is 9, this side is 9.

00:09:18.360 --> 00:09:21.960
And we just figured out that
this angle right here is 70.

00:09:21.960 --> 00:09:25.820
This was 35 plus another 35.

00:09:25.820 --> 00:09:27.840
Once again, another isosceles
triangle.

00:09:27.840 --> 00:09:30.116
If this side is equal to that
side, that this angle is equal

00:09:30.116 --> 00:09:31.210
to this angle.

00:09:31.210 --> 00:09:34.460
So, this angle also has
to be equal to 70.

00:09:34.460 --> 00:09:35.800
And now, if we know
two angles of a

00:09:35.800 --> 00:09:37.570
triangle, we know the third.

00:09:37.570 --> 00:09:38.430
This has to be--

00:09:38.430 --> 00:09:38.580
[PHONE RINGING]

00:09:38.580 --> 00:09:39.090
I'm on a roll.

00:09:39.090 --> 00:09:41.760
I'm not going to answer
the phone.

00:09:41.760 --> 00:09:44.180
Oh I think it's my wife.

00:09:44.180 --> 00:09:46.880
I'm going to call her back
right after this.

00:09:46.880 --> 00:09:48.060
So what was I doing?

00:09:48.060 --> 00:09:49.390
I lost my train of thought.

00:09:49.390 --> 00:09:50.080
I was on a roll.

00:09:50.080 --> 00:09:53.900
If this angle is 70 degrees,
this angle is 70 degrees, then

00:09:53.900 --> 00:09:54.930
this angle has to be what?

00:09:54.930 --> 00:10:01.740
70 plus 70 plus x is equal
to 180 degrees.

00:10:01.740 --> 00:10:06.920
140 plus x is equal
to 40 degrees.

00:10:06.920 --> 00:10:08.540
Sorry, is equal to 180.

00:10:08.540 --> 00:10:10.150
My brain was skipping ahead.

00:10:10.150 --> 00:10:12.580
x is equal to 40 degrees.

00:10:12.580 --> 00:10:13.040
OK.

00:10:13.040 --> 00:10:16.080
So that angle is 40 degrees
which is the same thing as

00:10:16.080 --> 00:10:17.350
that angle.

00:10:17.350 --> 00:10:19.410
So the length of-- so let's
go back to the problem.

00:10:19.410 --> 00:10:22.580
We have to figure out what
this arc length is.

00:10:22.580 --> 00:10:29.750
So, this angle, which is 40 over
all the degrees in the

00:10:29.750 --> 00:10:34.300
circle, over 360, is going to be
equal to this arc length--

00:10:34.300 --> 00:10:37.370
let's just call it x-- over
the circumference

00:10:37.370 --> 00:10:38.550
of the whole circle.

00:10:38.550 --> 00:10:40.770
And we know that the
circumference of the full

00:10:40.770 --> 00:10:45.300
circle is 18 pi.

00:10:45.300 --> 00:10:46.620
Let's see if we can
simplify this.

00:10:46.620 --> 00:10:49.820
40 goes into 360-- that's 1/9.

00:10:49.820 --> 00:10:51.825
So to this becomes 1 over 9.

00:10:54.530 --> 00:10:59.646
Then we could cross multiply and
get 9x is equal to 18 pi.

00:10:59.646 --> 00:11:06.090
x is equal to, 18 pi divided
by 9, is equal to 2 pi.

00:11:06.090 --> 00:11:08.120
And, luckily, that's
one of the choices.

00:11:08.120 --> 00:11:09.700
It's choice A.

00:11:09.700 --> 00:11:10.720
See you in the next video.

00:11:10.720 --> 00:11:12.920
I now have to call my wife.


GMAT: Math 41

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.270
We're on problem 201 which I
had continued in the last

00:00:04.270 --> 00:00:07.020
video, but, frankly, I had bad
reading comprehension.

00:00:07.020 --> 00:00:09.770
I had read the first statement
wrong, and I went down an

00:00:09.770 --> 00:00:11.050
incorrect path.

00:00:11.050 --> 00:00:12.230
So let me read it correctly.

00:00:12.230 --> 00:00:15.700
If the sum of n consecutive
integers is 0, which of the

00:00:15.700 --> 00:00:16.730
following must be true?

00:00:16.730 --> 00:00:18.730
So I have n numbers in
the row, and their

00:00:18.730 --> 00:00:20.760
sum has to be 0.

00:00:20.760 --> 00:00:22.470
So, first of all, that tells us
that some of them have to

00:00:22.470 --> 00:00:24.990
be positive and some of them
have to be negative.

00:00:24.990 --> 00:00:26.710
And they're consecutive,
right?

00:00:26.710 --> 00:00:30.430
So if we put this as 0-- let's
think about how we can even

00:00:30.430 --> 00:00:33.090
construct this, how this
could ever happen.

00:00:33.090 --> 00:00:36.705
n consecutive numbers,
the sum being 0.

00:00:41.480 --> 00:00:44.730
Really in, order to do that--
and they're consecutive, which

00:00:44.730 --> 00:00:48.280
is a trick.

00:00:48.280 --> 00:00:55.770
If a is one of the numbers,
then, really, negative a has

00:00:55.770 --> 00:00:58.390
to be one of the numbers
as well.

00:00:58.390 --> 00:00:59.190
And why is that?

00:00:59.190 --> 00:01:03.270
You might say, oh well why can't
I have some other set of

00:01:03.270 --> 00:01:05.370
negative numbers
that offset a?

00:01:05.370 --> 00:01:07.080
Well, the key here is
that we're dealing

00:01:07.080 --> 00:01:08.030
with consecutive numbers.

00:01:08.030 --> 00:01:09.910
Because those other negative
numbers are going to have to

00:01:09.910 --> 00:01:12.760
offset all of the other positive
numbers in between.

00:01:12.760 --> 00:01:15.920
Maybe I should do this
with more abstract.

00:01:15.920 --> 00:01:19.550
Let's say that n
is equal to 1.

00:01:19.550 --> 00:01:22.720
The sum of n consecutive
integers is 0.

00:01:22.720 --> 00:01:24.760
So, first of all,
can n equal 0?

00:01:24.760 --> 00:01:25.880
Well sure.

00:01:25.880 --> 00:01:29.760
Only if that one integer is 0.

00:01:29.760 --> 00:01:31.910
Now what if n is equal to 2?

00:01:31.910 --> 00:01:33.580
Can n equal 2?

00:01:33.580 --> 00:01:35.200
Well let's think about it.

00:01:35.200 --> 00:01:38.160
We have to add one integer
to this, and

00:01:38.160 --> 00:01:40.000
they're integers right?

00:01:40.000 --> 00:01:43.030
So if we add one integer-- if
we add any positive integer

00:01:43.030 --> 00:01:49.140
here, a, then the sum is
going to be 0 plus a.

00:01:49.140 --> 00:01:52.544
If you're going to cross the
negative boundary, if you're

00:01:52.544 --> 00:01:56.700
going to cross 0, 0 has to
be one of your integers.

00:01:56.700 --> 00:01:58.270
So that's always going
to be counted.

00:01:58.270 --> 00:02:01.050
So, in order to get into
negative a, you need to have

00:02:01.050 --> 00:02:04.360
at least three integers.

00:02:04.360 --> 00:02:08.630
So, if I said that n is equal
to 3, you could do 1,

00:02:08.630 --> 00:02:11.500
0, and minus 1.

00:02:11.500 --> 00:02:15.060
Now, if n is equal to 4, can
I just add one number?

00:02:15.060 --> 00:02:16.320
Can I add either 2
or negative 2?

00:02:16.320 --> 00:02:18.960
No, that'll change
the sum again.

00:02:18.960 --> 00:02:24.050
So I have to add a
2 and a minus 2.

00:02:24.050 --> 00:02:26.250
So I think you could see here
that there's a pattern because

00:02:26.250 --> 00:02:29.130
0 is always going to be one
of the integers and it's

00:02:29.130 --> 00:02:30.860
symmetric on the right-
or left-hand side.

00:02:30.860 --> 00:02:35.470
We're always going to have an
odd number of integers we're

00:02:35.470 --> 00:02:36.040
summing up.

00:02:36.040 --> 00:02:37.800
So we know that n is odd.

00:02:42.310 --> 00:02:43.310
So, we know statement one.

00:02:43.310 --> 00:02:45.480
They say statement
one, n even.

00:02:45.480 --> 00:02:46.230
That's not true.

00:02:46.230 --> 00:02:48.830
We actually can't construct
it with n as even.

00:02:48.830 --> 00:02:53.660
Statement two says, n is odd
which we've just shown.

00:02:53.660 --> 00:02:57.020
And then statement three
is, the average of the

00:02:57.020 --> 00:02:59.380
n integers is 0.

00:02:59.380 --> 00:03:00.840
Well sure.

00:03:00.840 --> 00:03:03.430
What's the average?

00:03:03.430 --> 00:03:08.750
It's the sum of the integers
divided by n.

00:03:08.750 --> 00:03:12.340
Well, they're telling us that
the sum of the integers is 0.

00:03:12.340 --> 00:03:14.490
So 0 divided by anything, as
long as we don't have zero

00:03:14.490 --> 00:03:17.280
integers, is 0.

00:03:17.280 --> 00:03:21.150
So two and three are
both correct.

00:03:21.150 --> 00:03:24.250
So the choice is e.

00:03:24.250 --> 00:03:25.500
Next question.

00:03:28.120 --> 00:03:38.390
202.

00:03:38.390 --> 00:03:46.690
In the formula, v is equal
to 1 over 2r cubed.

00:03:46.690 --> 00:03:51.800
If r is halved, then v is
multiplied by what?

00:03:51.800 --> 00:03:53.400
The easiest thing I always
find to do in

00:03:53.400 --> 00:03:55.100
this is take numbers.

00:03:55.100 --> 00:03:57.490
So let's say, if r is
2 what happens?

00:03:57.490 --> 00:03:59.600
And let's halve it when
r equals 1 and see

00:03:59.600 --> 00:04:00.550
what happens to v.

00:04:00.550 --> 00:04:03.350
We can do it analytically, but
this is, frankly, the easiest

00:04:03.350 --> 00:04:04.400
way to do it.

00:04:04.400 --> 00:04:07.510
So you can say when r is equal
to 2, v is equal to 1 over 2

00:04:07.510 --> 00:04:11.870
times 2, is 4, to the
third power is 64.

00:04:11.870 --> 00:04:16.740
And then when r is 1, you get 1
over-- let's see, 2 times 1

00:04:16.740 --> 00:04:18.130
is 2 to the third power.

00:04:18.130 --> 00:04:20.209
So you get 1/8.

00:04:20.209 --> 00:04:22.180
So what happened to v?

00:04:22.180 --> 00:04:26.830
We, essentially, multiplied
that by 8.

00:04:26.830 --> 00:04:30.260
So that is choice b.

00:04:30.260 --> 00:04:33.760
To go from 1/64 to 1/8 you
have to multiply by 8.

00:04:33.760 --> 00:04:37.070
You can view 1/8 as the same
thing as 8/64 if that

00:04:37.070 --> 00:04:38.380
clarifies it at all.

00:04:38.380 --> 00:04:40.700
I always find it easier to
deal with actual physical

00:04:40.700 --> 00:04:43.050
numbers where they say, if you
change this variable by a

00:04:43.050 --> 00:04:46.190
factor of 5 what happened
to that variable?

00:04:46.190 --> 00:04:47.830
So easy to just-- well anyway.

00:04:47.830 --> 00:04:51.060
It could get very confusing if
you try to do an r1 and an r2,

00:04:51.060 --> 00:04:54.800
and r1 is 2 times
r2 and all that.

00:04:54.800 --> 00:04:57.350
Next question.

00:04:57.350 --> 00:05:01.840
203.

00:05:01.840 --> 00:05:05.040
A certain bakery has
six employees.

00:05:05.040 --> 00:05:08.220
It pays annual salaries
of $14,000

00:05:08.220 --> 00:05:09.785
to each of two employees.

00:05:13.330 --> 00:05:17.300
$16,000 to one employee.

00:05:17.300 --> 00:05:20.050
And $17,000 to each of the
remaining three employees.

00:05:22.690 --> 00:05:28.240
The average annual salary of
these employees is closest to

00:05:28.240 --> 00:05:29.620
which of the following?

00:05:29.620 --> 00:05:31.730
So, we have to take the sum
of all their salaries

00:05:31.730 --> 00:05:33.460
and divide by 6.

00:05:33.460 --> 00:05:39.490
So it'll be 2 times 14, plus 1
times 16-- which is 16, plus 3

00:05:39.490 --> 00:05:44.090
times 17, and we
divide it by 6.

00:05:44.090 --> 00:05:51.290
2 times 14 is 28, plus 16, plus
51, all of that over 6.

00:05:51.290 --> 00:05:52.450
Let's see, what is
that equal to?

00:05:52.450 --> 00:06:02.790
28, 16, 51, 8 plus 6 is 14, 14
plus 1 is 15, 1 plus 2 is 3, 3

00:06:02.790 --> 00:06:08.070
plus 1 is 4, 4 plus 5 is 9.

00:06:08.070 --> 00:06:09.360
This was 8 plus 6.

00:06:09.360 --> 00:06:12.410
So, it's 95 divided by 6.

00:06:12.410 --> 00:06:15.090
Sorry, 95 divided by 6.

00:06:15.090 --> 00:06:17.850
I think my brain was computing
ahead of me because 96 divided

00:06:17.850 --> 00:06:23.710
by 6 is equal to 16.

00:06:23.710 --> 00:06:26.230
So this number is going
to be just under 16.

00:06:26.230 --> 00:06:28.790
It's 15 and 5/6.

00:06:28.790 --> 00:06:32.210
So they want to get an
approximate, as close as-- oh,

00:06:32.210 --> 00:06:34.210
actually they have a lot of
numbers that are really close.

00:06:34.210 --> 00:06:39.490
Let me make sure that
I get it right.

00:06:39.490 --> 00:06:42.650
So, 6 goes into 95-- if they
didn't have numbers that

00:06:42.650 --> 00:06:44.070
weren't close, I wouldn't
do this.

00:06:44.070 --> 00:06:46.830
6 goes into 95, 6 into 9, 6.

00:06:46.830 --> 00:06:52.570
35, goes 5 times, 5 times
6 is 30, remainder of 5.

00:06:52.570 --> 00:06:55.600
Actually let's keep going
because we want a decimal.

00:06:55.600 --> 00:06:58.530
We're doing it in thousands
So, bring down a 0.

00:06:58.530 --> 00:07:02.670
6 goes into 50 8 times.

00:07:02.670 --> 00:07:04.150
8 times 6 is 48.

00:07:04.150 --> 00:07:07.140
Remember, we were dealing in
thousands when I said 16, that

00:07:07.140 --> 00:07:07.730
was thousands.

00:07:07.730 --> 00:07:10.840
So, now we're at
15.8 thousand.

00:07:10.840 --> 00:07:16.190
So 8 times 48, 20, 6 goes
into 20 3 times.

00:07:16.190 --> 00:07:18.420
I think that's close enough
to what we need.

00:07:18.420 --> 00:07:21.750
So it's going to be
roughly $15,800.

00:07:21.750 --> 00:07:23.920
Or 15.8 thousands.

00:07:23.920 --> 00:07:27.850
And that is choice C.

00:07:27.850 --> 00:07:29.650
At first I said maybe we can
approximate, and they had

00:07:29.650 --> 00:07:33.510
16,000 there, but then I saw the
15,800 so we should do a

00:07:33.510 --> 00:07:34.300
little bit more math.

00:07:34.300 --> 00:07:37.230
We don't want to get
too careless.

00:07:37.230 --> 00:07:39.010
Next problem.

00:07:39.010 --> 00:07:42.810
204.

00:07:42.810 --> 00:07:48.260
If x is equal to the sum of the
even integers from 40 to

00:07:48.260 --> 00:07:58.480
60 inclusive.

00:07:58.480 --> 00:08:00.990
This is my ad hoc notation.

00:08:00.990 --> 00:08:03.510
And y is the number
of even integers.

00:08:08.660 --> 00:08:11.900
Let's see, if x is equal to the
sum of the even integers

00:08:11.900 --> 00:08:18.750
from 40 to 60, inclusive, so it
includes 40 and 60, and y

00:08:18.750 --> 00:08:22.640
is the number of even integers
from 40 to 60 inclusive, what

00:08:22.640 --> 00:08:25.915
is the value of x plus y?

00:08:30.060 --> 00:08:32.120
The number-- y is
an easy thing.

00:08:32.120 --> 00:08:34.890
How many integers are there
between 40 and 60, if you

00:08:34.890 --> 00:08:37.520
include both of them?

00:08:37.520 --> 00:08:39.580
This is always a tricky
thing to think about.

00:08:39.580 --> 00:08:44.260
But, if you think about it, if
I said the numbers 1, 2, and

00:08:44.260 --> 00:08:46.740
3, how many integers
are there here?

00:08:46.740 --> 00:08:47.860
Well, clearly, you
can count them.

00:08:47.860 --> 00:08:49.270
1, 2, and 3.

00:08:49.270 --> 00:08:53.150
But if you wanted a quick way
you could say 3 minus-- if I

00:08:53.150 --> 00:08:58.360
just took the endpoint minus
the starting point, I would

00:08:58.360 --> 00:08:59.960
get these integers.

00:08:59.960 --> 00:09:02.300
Then I want to add 1.

00:09:02.300 --> 00:09:05.150
If I want to know how integers
are there between 60 and 40, I

00:09:05.150 --> 00:09:07.960
would say 60 minus 40.

00:09:07.960 --> 00:09:11.180
And that'll give me 41,
42, 43, 44, all the

00:09:11.180 --> 00:09:12.840
way to 59, and 60.

00:09:12.840 --> 00:09:14.210
And then I have to add
1 because we're

00:09:14.210 --> 00:09:15.920
including 40 as well.

00:09:15.920 --> 00:09:20.870
So y is equal to 21.

00:09:20.870 --> 00:09:26.490
Now let's think about, is there
an easy way to add up

00:09:26.490 --> 00:09:30.200
all of the integers between
40 and-- all of the

00:09:30.200 --> 00:09:32.360
even, oh no, no sorry.

00:09:32.360 --> 00:09:36.600
y is the number of even integers
from 40 to 60.

00:09:36.600 --> 00:09:38.430
So that's not right.

00:09:38.430 --> 00:09:39.700
So, even integers.

00:09:39.700 --> 00:09:41.920
Frankly, it might just
be easier to-- let's

00:09:41.920 --> 00:09:43.110
just write them out.

00:09:43.110 --> 00:09:56.740
40, 42, 44, 46, 48, 50, 52,
54, 56, 58, and 60.

00:09:56.740 --> 00:10:02.460
1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11.

00:10:02.460 --> 00:10:03.760
11 intergers.

00:10:03.760 --> 00:10:05.300
So that's y.

00:10:05.300 --> 00:10:09.140
And we need to take the
sum of all of these.

00:10:09.140 --> 00:10:11.390
So there's a couple of ways
we could think about it.

00:10:11.390 --> 00:10:18.170
You could view this as 40
plus 0, 40 plus 2, 40

00:10:18.170 --> 00:10:20.620
plus 3, and so forth.

00:10:20.620 --> 00:10:23.540
All the way to 40 plus 8.

00:10:23.540 --> 00:10:27.580
And so if you wanted to add up
at least up to this point,

00:10:27.580 --> 00:10:29.650
it's a pretty straightforward
way of doing it.

00:10:29.650 --> 00:10:34.300
You could just say, there's
1, 2, 3, 4, 5 of them.

00:10:34.300 --> 00:10:38.140
So it's equal to 40 times 5.

00:10:38.140 --> 00:10:39.460
That adds up all the 40's.

00:10:39.460 --> 00:10:42.330
Plus 2, plus 4, plus
6, plus 8.

00:10:42.330 --> 00:10:47.540
So 2 plus 4 is 6, plus 6
is 12, plus 8 is 20.

00:10:47.540 --> 00:10:48.880
So, plus 20.

00:10:48.880 --> 00:10:52.290
So, that's 40 times 5 is what?

00:10:52.290 --> 00:10:55.663
200 plus 20 is equal to 220.

00:10:55.663 --> 00:10:58.020
And you use the same
logic here.

00:10:58.020 --> 00:11:06.070
This is 50 plus 0, 50 plus 2,
all the way to 50 plus 10.

00:11:06.070 --> 00:11:07.010
And how many are there?

00:11:07.010 --> 00:11:10.590
There's 1, 2, 3, 4, 5, 6.

00:11:10.590 --> 00:11:21.170
So, it's equal to 6 times 50,
plus 2 plus 4 is 6, plus 6 is

00:11:21.170 --> 00:11:24.970
12, plus 8 is 20.

00:11:24.970 --> 00:11:25.960
Which we actually
already knew.

00:11:25.960 --> 00:11:26.880
We already calculated it there.

00:11:26.880 --> 00:11:28.050
It was 20.

00:11:28.050 --> 00:11:32.130
Because these don't contribute
to the-- oh no, no plus 10.

00:11:32.130 --> 00:11:33.120
Sorry, plus 10.

00:11:33.120 --> 00:11:34.010
Don't want to get careless.

00:11:34.010 --> 00:11:36.070
Plus 10.

00:11:36.070 --> 00:11:38.300
Because there's six of these.

00:11:38.300 --> 00:11:42.310
There's plus 0, plus 2, plus
4, plus 6, plus 8, plus 10.

00:11:42.310 --> 00:11:44.940
So, it's 20 plus 10 is 30.

00:11:44.940 --> 00:11:48.510
2 plus 4 plus 6 plus
8 plus 10 is 30.

00:11:48.510 --> 00:11:54.710
So, 6 times 6 is 300, plus
30 is equal to 330.

00:11:54.710 --> 00:11:57.320
So, 330 plus 220.

00:11:57.320 --> 00:12:00.970
These numbers have a strange
pattern that-- well there is a

00:12:00.970 --> 00:12:04.470
formula if you want to add the
first n numbers, but they've

00:12:04.470 --> 00:12:06.510
changed it a little bit by
talking about even numbers so

00:12:06.510 --> 00:12:08.820
I wanted to do it manually.

00:12:08.820 --> 00:12:11.750
So you go 0, 5, 5.

00:12:11.750 --> 00:12:15.670
So 550 is the sum.

00:12:15.670 --> 00:12:18.030
And then we know that
y-- this is x.

00:12:18.030 --> 00:12:19.280
x is equal to 550.

00:12:19.280 --> 00:12:20.640
And y is equal to 11.

00:12:20.640 --> 00:12:22.300
We figured that out before.

00:12:22.300 --> 00:12:30.320
So x plus y is equal to 550,
plus 11 which is equal to 561,

00:12:30.320 --> 00:12:33.820
which is, thank goodness,
actually one of the choices.

00:12:33.820 --> 00:12:35.470
That's choice D.

00:12:35.470 --> 00:12:37.500
See you in the next video.


GMAT: Math 40

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:05.140
We're on problem 198.

00:00:05.140 --> 00:00:17.190
If 1/2 plus 1/3, plus 1/4 is
equal to 13/x, which of the

00:00:17.190 --> 00:00:19.840
following must be an integer?

00:00:19.840 --> 00:00:21.420
Fascinating.

00:00:21.420 --> 00:00:22.590
So let's see what
they're saying.

00:00:22.590 --> 00:00:22.940
I don't know.

00:00:22.940 --> 00:00:24.510
I'm tempted to just
work through this.

00:00:24.510 --> 00:00:27.150
So if I get a common denominator
here, the common

00:00:27.150 --> 00:00:28.540
denominator is going to be 12.

00:00:28.540 --> 00:00:33.330
1/2 is 6/12 plus 4/12,
plus 3/12.

00:00:33.330 --> 00:00:36.110
6 plus 4, plus 3, is 13.

00:00:36.110 --> 00:00:38.470
So, that equals 13/12.

00:00:38.470 --> 00:00:41.080
So that tell us that
x is equal to 13.

00:00:41.080 --> 00:00:41.950
I'm sorry.

00:00:41.950 --> 00:00:44.950
That tells us that
x is equal to 12.

00:00:44.950 --> 00:00:46.510
It's the denominator.

00:00:46.510 --> 00:00:48.260
OK, so they give us a
couple of choices.

00:00:48.260 --> 00:00:51.140
Choice one, x/8.

00:00:51.140 --> 00:00:51.820
Is that an integer?

00:00:51.820 --> 00:00:54.190
No, 12 divided by 8
is not an integer.

00:00:54.190 --> 00:00:55.400
It's 1 and 1/2.

00:00:55.400 --> 00:00:59.050
Choice two, x divided by 12.

00:00:59.050 --> 00:01:00.130
Is that an integer?

00:01:00.130 --> 00:01:00.900
Yeah, sure.

00:01:00.900 --> 00:01:02.480
I mean 12 divided by
12 is equal to 1.

00:01:02.480 --> 00:01:03.980
So, that works.

00:01:03.980 --> 00:01:07.220
Choice three, x divided by 24.

00:01:07.220 --> 00:01:08.580
No, x is 12.

00:01:08.580 --> 00:01:10.700
So, x divided by 24 is 1/2.

00:01:10.700 --> 00:01:15.660
So, two only is the answer,
and that is choice B.

00:01:15.660 --> 00:01:21.430
Problem 199.

00:01:21.430 --> 00:01:25.510
And they have drawn a graph,
which I will draw.

00:01:25.510 --> 00:01:28.390
So that's a y-axis.

00:01:28.390 --> 00:01:31.260
And there's an x-axis.

00:01:31.260 --> 00:01:33.790
And that's y.

00:01:33.790 --> 00:01:36.670
That is x.

00:01:36.670 --> 00:01:38.320
See, in the rectangular
coordinate system above, the

00:01:38.320 --> 00:01:41.440
line, y equals x is the
perpendicular bisector of

00:01:41.440 --> 00:01:45.510
segment AB, not shown.

00:01:45.510 --> 00:01:49.860
OK, so let me draw this
y equals x graph.

00:01:49.860 --> 00:01:51.420
Start there.

00:01:51.420 --> 00:01:51.720
Nope.

00:01:51.720 --> 00:01:54.820
I thought I was using the
line tool, but I'm not.

00:01:54.820 --> 00:01:56.970
Let me use the line tool.

00:01:56.970 --> 00:02:01.000
That's the y equals x graph.

00:02:01.000 --> 00:02:02.890
y is equal to x.

00:02:02.890 --> 00:02:05.160
And they draw point A.

00:02:05.160 --> 00:02:07.190
The place they drew it, it looks
like it's roughly at the

00:02:07.190 --> 00:02:09.460
point 2, 3.

00:02:09.460 --> 00:02:11.500
I don't know if that's going
to become relevant.

00:02:11.500 --> 00:02:15.530
2, 1, 2, 3.

00:02:15.530 --> 00:02:18.810
So A looks like it's
at the point 2,3.

00:02:18.810 --> 00:02:22.870
I don't know if we
can assume that.

00:02:22.870 --> 00:02:26.170
The line, y equals x, is a
perpendicular bisector of AB,

00:02:26.170 --> 00:02:27.720
and they don't show us B.

00:02:27.720 --> 00:02:33.140
And the x-axis is the
perpendicular bisector of BC.

00:02:33.140 --> 00:02:33.730
All right.

00:02:33.730 --> 00:02:34.720
Not shown.

00:02:34.720 --> 00:02:37.440
If the coordinates of point A
are 2 comma 3-- OK, we were

00:02:37.440 --> 00:02:42.050
right; A is 2 comma
3-- what are the

00:02:42.050 --> 00:02:45.330
coordinates of point C?

00:02:45.330 --> 00:02:48.270
OK, perpendicular bisector means
there's some line that's

00:02:48.270 --> 00:02:52.100
exactly perpendicular to
this line, where it

00:02:52.100 --> 00:02:53.850
splits it in half.

00:02:53.850 --> 00:02:57.410
So if you think about it,
point A is at 2, 3.

00:02:57.410 --> 00:03:01.070
So where would B have to be in
order for the line between A

00:03:01.070 --> 00:03:03.950
and B to be perpendicular to
this line, and to be split

00:03:03.950 --> 00:03:05.560
directly in half?

00:03:05.560 --> 00:03:07.650
Well, it would be at the point--
I mean, you could just

00:03:07.650 --> 00:03:12.140
eyeball it-- it'd be at
the point 3 comma 2.

00:03:12.140 --> 00:03:15.980
If we're at the point 3 comma 2,
there's a lot of, I guess,

00:03:15.980 --> 00:03:18.090
intuitive reasons why you could
think that this would be

00:03:18.090 --> 00:03:19.340
the perpendicular bisector.

00:03:21.600 --> 00:03:24.170
We know y equals x is a
perpendicular bisector.

00:03:24.170 --> 00:03:26.960
The first is, you can see that
the slope of this line-- you

00:03:26.960 --> 00:03:28.970
could calculate it if you want--
the slope of this line

00:03:28.970 --> 00:03:30.340
is negative 1.

00:03:30.340 --> 00:03:32.960
And I don't know if you remember
in calculus class,

00:03:32.960 --> 00:03:35.025
but if you have the slope of
one line-- so the slope of

00:03:35.025 --> 00:03:38.900
this line is 1-- the slope of
a perpendicular line will be

00:03:38.900 --> 00:03:40.390
the negative inverse
of this slope.

00:03:40.390 --> 00:03:41.830
That's something that,
hopefully, you learned in

00:03:41.830 --> 00:03:44.470
calculus-- oh, in algebra
a long time ago.

00:03:44.470 --> 00:03:47.110
So, the negative inverse
of 1 is negative 1.

00:03:47.110 --> 00:03:50.270
So this line has a slope
of negative 1.

00:03:50.270 --> 00:03:52.670
And you could make a bunch
of arguments for why it's

00:03:52.670 --> 00:03:54.690
equidistant.

00:03:54.690 --> 00:03:56.720
Both of these points are on
the same line, and they're

00:03:56.720 --> 00:03:59.490
equidistant from the
point 2 comma 2.

00:03:59.490 --> 00:04:01.960
So that's why, you know, that
it's the bisector, that this

00:04:01.960 --> 00:04:03.860
distance is equal to
this distance.

00:04:03.860 --> 00:04:05.800
And, frankly, on the GMAT,
you're doing it fast. You

00:04:05.800 --> 00:04:07.370
don't have to do a
rigorous proof.

00:04:07.370 --> 00:04:08.210
You can eyeball it.

00:04:08.210 --> 00:04:09.640
You could say, oh, you know,
those are going to be equal.

00:04:09.640 --> 00:04:11.030
I'm just going to take the
mirror image and go on the

00:04:11.030 --> 00:04:11.790
other side.

00:04:11.790 --> 00:04:14.020
So this is point B.

00:04:14.020 --> 00:04:19.130
And they tell us that the x-axis
is a perpendicular

00:04:19.130 --> 00:04:21.930
bisector of segment BC.

00:04:21.930 --> 00:04:25.260
OK, so x is going to be

00:04:25.260 --> 00:04:27.290
perpendicular to this new line.

00:04:27.290 --> 00:04:29.610
So this new line has to be
parallel to the y-axis.

00:04:29.610 --> 00:04:32.120
So this new line is going
to go like that.

00:04:32.120 --> 00:04:35.100
And whatever the distance is
between B and the x-axis has

00:04:35.100 --> 00:04:38.800
to be the distance between
x-axis and C.

00:04:38.800 --> 00:04:43.200
So B is two above the x-axis, so
C is going to be two below

00:04:43.200 --> 00:04:44.570
the x-axis.

00:04:44.570 --> 00:04:46.360
So that's where C is.

00:04:46.360 --> 00:04:49.450
And x, at that point,
is equal to 3.

00:04:49.450 --> 00:04:53.920
And so this y would be
equal to negative 2.

00:04:53.920 --> 00:04:55.220
So what are the coordinates
of point C?

00:04:55.220 --> 00:04:56.680
3, negative 2.

00:04:56.680 --> 00:04:59.450
And that is choice D.

00:04:59.450 --> 00:05:00.870
Interesting problem.

00:05:00.870 --> 00:05:07.200
Problem 200.

00:05:07.200 --> 00:05:10.200
A store currently charges
the same price for each

00:05:10.200 --> 00:05:11.990
towel that it sells.

00:05:11.990 --> 00:05:17.530
If the current price of each
towel were to increase by $1,

00:05:17.530 --> 00:05:23.160
10 fewer of the towels could
be bought for $120,

00:05:23.160 --> 00:05:24.810
excluding sales tax.

00:05:35.630 --> 00:05:40.560
What is the current price
of each towel?

00:05:40.560 --> 00:05:44.410
So let's say the current
price of each towel.

00:05:44.410 --> 00:05:48.570
So if we take 120 and we divide
it by the current price

00:05:48.570 --> 00:05:53.990
of each towel, we could buy,
I don't know, n towels.

00:05:53.990 --> 00:05:58.050
Now, they're saying that if we
had $120, and if we were to

00:05:58.050 --> 00:06:03.740
increase the price by $1-- so
now the new price is p plus 1;

00:06:03.740 --> 00:06:06.860
now I can't buy n towels
anymore-- they're saying I can

00:06:06.860 --> 00:06:08.370
only buy 10 fewer towels.

00:06:08.370 --> 00:06:11.920
So I can only buy n
minus 10 towels.

00:06:11.920 --> 00:06:14.590
Now we have two equations
and two unknowns.

00:06:14.590 --> 00:06:16.420
Well, they're not linear
equations, but I think we

00:06:16.420 --> 00:06:19.130
should be able to solve them
one way or the other.

00:06:19.130 --> 00:06:25.300
So here we get p times
n is equal to 120.

00:06:25.300 --> 00:06:29.190
And here, let's see, we can
multiply-- we have 120 is

00:06:29.190 --> 00:06:34.620
equal to p plus 1,
times n minus 10.

00:06:34.620 --> 00:06:51.570
So that's equal to p times n,
minus 10p, plus n, minus 10.

00:06:51.570 --> 00:06:54.570
And so both of these are equal
to 120, so we could set them

00:06:54.570 --> 00:06:55.590
equal to each other.

00:06:55.590 --> 00:06:57.130
Both of these are
equal to 120.

00:06:57.130 --> 00:06:58.960
So we could use both
of these equations.

00:06:58.960 --> 00:07:09.800
Say pn is equal to pn minus
10p, plus n, minus 10.

00:07:09.800 --> 00:07:11.850
So let's see if we can make
any headway here.

00:07:16.110 --> 00:07:18.640
I could subtract pn
from both sides.

00:07:18.640 --> 00:07:26.116
So I get 0 is equal to minus
10p, plus n, minus 10.

00:07:26.116 --> 00:07:30.910
And let's see, I could add 10p
to both sides, and I get 10p

00:07:30.910 --> 00:07:38.050
is equal to n minus 10.

00:07:38.050 --> 00:07:41.290
Well, what's n?

00:07:41.290 --> 00:07:42.890
Actually, I shouldn't have
substituted like this.

00:07:42.890 --> 00:07:44.330
I'm kind of doing it
a little messily.

00:07:44.330 --> 00:07:45.810
But we can do another
substitution.

00:07:45.810 --> 00:07:51.870
n is equal to 120/p, so
that should help us.

00:07:51.870 --> 00:07:53.870
The real way I should have done
it is-- well, actually,

00:07:53.870 --> 00:07:55.970
no, this is I think the
easiest way to do it.

00:07:55.970 --> 00:07:57.180
10p.

00:07:57.180 --> 00:07:59.350
n is 120/p.

00:07:59.350 --> 00:08:01.620
A lot of times you just have to
go forth with the algebra

00:08:01.620 --> 00:08:02.670
and see what happens.

00:08:02.670 --> 00:08:05.100
Minus 10.

00:08:05.100 --> 00:08:07.520
And let's multiply both sides
of the equation by p.

00:08:07.520 --> 00:08:13.210
So you get 10 is equal-- sorry,
10p squared is equal to

00:08:13.210 --> 00:08:17.890
120 minus 10p.

00:08:17.890 --> 00:08:19.990
Let's divide both sides by 10
just to get rid of them.

00:08:19.990 --> 00:08:24.900
So you get p squared is
equal to 12, minus p.

00:08:24.900 --> 00:08:26.750
Let's take these on to
the left-hand side.

00:08:26.750 --> 00:08:32.320
So you get p squared plus p,
minus 12, is equal to 0.

00:08:32.320 --> 00:08:33.720
We have a quadratic.

00:08:33.720 --> 00:08:34.720
We can just factor it.

00:08:34.720 --> 00:08:41.260
That is p plus 4, times p
minus 3 is equal to 0.

00:08:41.260 --> 00:08:44.290
What two numbers, when you
add them, equal plus 1?

00:08:44.290 --> 00:08:45.930
The coefficient there.

00:08:45.930 --> 00:08:48.050
When you multiply them they
equal negative 12.

00:08:48.050 --> 00:08:50.130
That's just practice you got
in algebra doing that.

00:08:50.130 --> 00:08:51.530
So that's p plus 4,
times p, minus 3.

00:08:51.530 --> 00:08:57.030
So p could be minus 4, or
p could be equal to 3.

00:08:57.030 --> 00:08:59.220
And p can't be might minus 4
unless he was giving away the

00:08:59.220 --> 00:09:01.770
towels, he was paying people
to take the towels, which

00:09:01.770 --> 00:09:02.920
isn't the case.

00:09:02.920 --> 00:09:04.820
We're assuming that
price is positive.

00:09:04.820 --> 00:09:07.660
So the price had to be $3,
which, luckily enough, after

00:09:07.660 --> 00:09:09.810
doing all that algebra,
is one of the choices.

00:09:09.810 --> 00:09:12.360
Choice C.

00:09:12.360 --> 00:09:14.200
Next question.

00:09:14.200 --> 00:09:15.780
201.

00:09:15.780 --> 00:09:18.120
I think I have time
to do this one.

00:09:18.120 --> 00:09:20.520
201.

00:09:20.520 --> 00:09:34.270
If the sum of n consecutive
integers is 0, --this is my ad

00:09:34.270 --> 00:09:38.600
hoc notation-- which of the
following must be true?

00:09:38.600 --> 00:09:41.900
n consecutive integers is 0.

00:09:41.900 --> 00:09:44.330
n must be an even number.

00:09:50.360 --> 00:09:52.410
So let's think a lot about,
like, our even

00:09:52.410 --> 00:09:54.750
and odd type of talk.

00:09:54.750 --> 00:10:00.540
So we know that an even
plus an even is even.

00:10:00.540 --> 00:10:05.390
And we know that an odd plus
an odd is also even.

00:10:05.390 --> 00:10:06.730
And you could prove
it to yourself.

00:10:06.730 --> 00:10:07.650
You could try a bunch
of numbers.

00:10:07.650 --> 00:10:10.950
Or if you just write an odd as
2k plus 1, and you add it to

00:10:10.950 --> 00:10:15.240
another 2m plus 1, 2k-- that
becomes 2 times k,

00:10:15.240 --> 00:10:17.590
plus m, plus 2.

00:10:17.590 --> 00:10:18.980
So, that's an even number.

00:10:18.980 --> 00:10:20.430
It's divisible by 2.

00:10:20.430 --> 00:10:22.460
So an plus an odd is even.

00:10:22.460 --> 00:10:25.530
And so the only way you're going
to get an odd number is

00:10:25.530 --> 00:10:30.090
if you have an even
plus an odd.

00:10:30.090 --> 00:10:33.690
Now, the only way you're going
to have an even plus an odd--

00:10:33.690 --> 00:10:37.270
so we're summing n consecutive
numbers.

00:10:37.270 --> 00:10:41.950
So, I don't know, let's call
it number 1, plus number 2,

00:10:41.950 --> 00:10:42.830
plus number 3.

00:10:42.830 --> 00:10:46.360
Let's think about it a couple
of different ways.

00:10:46.360 --> 00:10:50.710
If this number is odd, then this
number is obviously even,

00:10:50.710 --> 00:10:53.310
and then this number is odd.

00:10:53.310 --> 00:10:59.990
Now, if we're taking the sum
of an even number, if n is

00:10:59.990 --> 00:11:03.880
even, we're going to get the
same number of evens and odds.

00:11:07.130 --> 00:11:10.010
So n could be 2.

00:11:10.010 --> 00:11:12.610
If n could be 2, we're going to
have one even and one odd

00:11:12.610 --> 00:11:13.560
no matter what, right?

00:11:13.560 --> 00:11:14.080
This could be odd.

00:11:14.080 --> 00:11:14.700
This could be even.

00:11:14.700 --> 00:11:16.710
Or this could be even and
this could be odd.

00:11:16.710 --> 00:11:20.210
And if n is 2, which is an
even number, we can very

00:11:20.210 --> 00:11:23.780
easily-- you know an odd plus
an even is an odd number.

00:11:23.780 --> 00:11:32.750
So, they say if the sum of n
consecutive integers is 0--

00:11:32.750 --> 00:11:33.670
oh, you know what?

00:11:33.670 --> 00:11:35.240
I completely missed
that point.

00:11:35.240 --> 00:11:37.810
Let me continue this
in the next video.