GMAT: Math 4

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.970
We're on problem 20.

00:00:03.970 --> 00:00:08.640
Of the 5 coordinates associated
with points A, B,

00:00:08.640 --> 00:00:11.350
C, D and E on the number line
above, which has the greatest

00:00:11.350 --> 00:00:13.740
absolute value?

00:00:13.740 --> 00:00:15.520
OK.

00:00:15.520 --> 00:00:17.420
Let me just draw the number
line that they drew.

00:00:22.050 --> 00:00:25.980
And they say from 0 to 1.

00:00:25.980 --> 00:00:30.350
0, 1, 2.

00:00:30.350 --> 00:00:34.550
And then they say minus
1, minus 2.

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And then they say these are
choices A, B, C, D, and E.

00:00:43.090 --> 00:00:46.440
Of the 5 coordinates associated
with the points A,

00:00:46.440 --> 00:00:51.060
B, C, D, and E on the number
line above, which has the

00:00:51.060 --> 00:00:54.070
greatest absolute value?

00:00:54.070 --> 00:00:57.000
So you can view absolute value
as the distance from zero So

00:00:57.000 --> 00:00:58.460
the greatest distance from
zero is going to

00:00:58.460 --> 00:01:00.250
happen in two places.

00:01:00.250 --> 00:01:02.930
It's going to happen
here at A.

00:01:02.930 --> 00:01:05.480
It's also going to
happen here at E.

00:01:05.480 --> 00:01:06.960
Can you have more than
one right answer?

00:01:06.960 --> 00:01:09.210
Because both of those
are choices.

00:01:09.210 --> 00:01:10.460
Let's see.

00:01:15.240 --> 00:01:18.000
Indicate the best of the
answer choices given.

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I'm looking up the thing.

00:01:22.930 --> 00:01:24.820
Is it possible to have more
than one right answer?

00:01:24.820 --> 00:01:31.160
Because this doesn't seem quite
right that I can have

00:01:31.160 --> 00:01:35.320
two possible answers
for this one.

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The answer is A.

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The coordinate A is the
farthest from zero.

00:01:38.190 --> 00:01:40.340
This has the greatest
absolute value.

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I disagree with that.

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It could be choice E as well.

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The absolute value of the
number x is the distance

00:01:48.120 --> 00:01:49.710
between x and 0 on
the number line.

00:01:49.710 --> 00:01:51.865
The coordinate of point A is the
farthest from 0, and thus

00:01:51.865 --> 00:01:54.110
has the greatest
absolute value.

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That's false.

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I mean, that's true, but
E is just as far.

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I don't know, I think
they made a mistake.

00:01:59.930 --> 00:02:03.430
I think this could be choices
A or E, because the absolute

00:02:03.430 --> 00:02:04.260
value of either of those is 2.

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I think they made a mistake.

00:02:05.590 --> 00:02:08.199
And they say the choice is A,
which I agree with, but it

00:02:08.199 --> 00:02:10.319
could just as easily
be choice E.

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So I don't know.

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Write a letter to GMAT.

00:02:12.640 --> 00:02:15.040
I mean, this seems pretty
straightforward.

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Of the 5 coordinates associated
with choices A, B,

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C, D, and E on the number line,
which has the greatest

00:02:18.820 --> 00:02:19.590
absolute value?

00:02:19.590 --> 00:02:21.540
The absolute value of
this point is 2.

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The absolute value of
this point is 2.

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They're equal.

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Anyway.

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Next problem.

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I think it should be A or E.

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It's surprising that they would
let a question like that

00:02:31.750 --> 00:02:32.730
get through their screen.

00:02:32.730 --> 00:02:34.680
Or at least they should
get rid of choice E.

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If x and y are prime numbers,
which of the following cannot

00:02:43.030 --> 00:02:44.976
be the sum of x and y?

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x plus y cannot equal what?

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So statement 1: 5.

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Well, I can immediately say
2 plus 3 is equal to 5.

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So that doesn't work.

00:03:00.460 --> 00:03:02.995
Cannot equal, so B.

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9, I could do 2 plus
7 is equal to 9.

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This was A.

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This is B.

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Let's see what C is.

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So it's not B.

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C is 13.

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So let me think.

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What could it be?

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11 plus 2 is equal to 13,
so it's not choice C.

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D.

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16.

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The sum.

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So let me see, 16.

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9 and 7.

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No.

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11 and 5.

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11 plus 5 is equal to 16.

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So without even looking at E,
it's probably going to be E.

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E , 23.

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Let me see if I can come up with
two prime numbers that

00:03:42.370 --> 00:03:44.360
add up to 23.

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19 and 4.

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No.

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Let's see.

00:03:48.160 --> 00:03:49.250
2 and 21.

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No.

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First of all, we found examples
for all of these, so

00:03:52.100 --> 00:03:53.630
if you don't want to waste time
on the GMAT, you should

00:03:53.630 --> 00:03:56.350
already have chosen E and moved
on by this point, but we

00:03:56.350 --> 00:04:00.030
could just look through a bunch
of prime numbers and

00:04:00.030 --> 00:04:03.380
think if they can add up to 23,
but I can't think of any

00:04:03.380 --> 00:04:04.010
right at the moment.

00:04:04.010 --> 00:04:05.780
And I feel pretty good that E
is the answer, because I was

00:04:05.780 --> 00:04:08.080
able to find prime numbers that
added up to all of the

00:04:08.080 --> 00:04:10.280
other four choices.

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Next question.

00:04:11.930 --> 00:04:14.060
22.

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I'll switch colors just
to ease the monotony.

00:04:16.820 --> 00:04:19.200
If each of the following
fractions were written as a

00:04:19.200 --> 00:04:24.530
repeating decimal, which would
have the longest sequence of

00:04:24.530 --> 00:04:26.073
different digits?

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Are they going to sit here and
make us-- so let me write all

00:04:31.900 --> 00:04:33.270
the choices down.

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Maybe we can do some
cancellation from the get-go.

00:04:35.840 --> 00:04:37.670
2/11.

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B is 1/3.

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And we already know
that that's just

00:04:41.440 --> 00:04:43.600
going to be 0.33 repeating.

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Longest sequence of
different digits.

00:04:45.440 --> 00:04:48.820
So it's not going to be choice
B because choice B doesn't

00:04:48.820 --> 00:04:49.760
have different digits.

00:04:49.760 --> 00:04:52.330
It just keeps being a 3, so it's
not going to be choice B,

00:04:52.330 --> 00:04:53.650
so we don't have to
worry about that.

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Choice C.

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41/99.

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Choice D is 2/3.

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We know that that's 0.666.

00:05:02.370 --> 00:05:03.300
It just keeps going.

00:05:03.300 --> 00:05:04.250
So it's not going to be that.

00:05:04.250 --> 00:05:06.720
It's the same digit that
just keeps repeating.

00:05:06.720 --> 00:05:13.870
And choice E is 23
divided by 37.

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So 2 divided by 11 repeats.

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Let's just try it, though.

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11 goes into 2.000 one time.

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1 times 11 is 11.

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You get a 90.

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11 goes into 90 eight times.

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88.

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It's 20.

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I already see the pattern.

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11 goes into 20 two times.

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Sorry, it goes into
20 one time.

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1 times 11 is 11.

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Then you get a 90.

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It goes into it eight times,
so it's just 0.181818.

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It's just going to keep
repeating 0.18.

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So that's 2/11.

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41/99.

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Let me do it in a different
color just so I

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don't get too messy.

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99 goes into 41.0000.

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So 99 goes into 410
four times.

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4 times 99 is what?

00:06:04.020 --> 00:06:05.890
396.

00:06:05.890 --> 00:06:07.900
4 times 9 is 6.

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3.

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4 times 9 is 36.

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Yep, 396.

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This is equal to 140.

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99 goes into 140 one time.

00:06:17.300 --> 00:06:20.610
1 times 99 is 99.

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This becomes 41.

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Already started repeating,
right?

00:06:24.470 --> 00:06:25.620
99 goes into 410.

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That's just like that.

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So it's going to be 0.41
repeating over and over again

00:06:30.370 --> 00:06:32.060
because we got the same
number again.

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We're going 99 into
41 four times.

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We're going to get
396 and it's just

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going to keep happening.

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Then we're going to get 396.

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And then we're going
to get 140 again.

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It's 41, 41.

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So it's 41 repeating.

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So if I had to guess, it's
already going to be choice E,

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without having to do any work
because these just have the

00:06:52.590 --> 00:06:55.310
same numbers repeating
from the get-go.

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Choices A and C have the
same two numbers

00:06:57.430 --> 00:06:59.400
repeating from the get-go.

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So I'm guessing that this
one has a lot more.

00:07:01.600 --> 00:07:04.180
So if you just wanted to worry
about time, you could just

00:07:04.180 --> 00:07:05.240
pick E and move on.

00:07:05.240 --> 00:07:08.180
But let's do it just to
prove to ourselves.

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So 37 goes into 23.

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I don't remember when I took
the GMAT having to do this

00:07:13.170 --> 00:07:18.780
much decimal division or
decimal multiplication.

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So 37 goes into 230.

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I don't know.

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Does it go into it six times?

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No, probably five times.

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5 times 7 is 35.

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5 times 3 is 15, so
it's 18, plus 3.

00:07:36.110 --> 00:07:36.590
OK.

00:07:36.590 --> 00:07:37.510
30.

00:07:37.510 --> 00:07:38.610
85.

00:07:38.610 --> 00:07:40.310
Actually, this would have
gone six times.

00:07:40.310 --> 00:07:41.560
Let me change that to 6.

00:07:44.610 --> 00:07:46.130
Let me to it in a
different color.

00:07:46.130 --> 00:07:47.530
I don't want to make
it too messy.

00:07:47.530 --> 00:07:53.580
6 times 7 is 42.

00:07:53.580 --> 00:07:57.620
6 times 3 is 18, plus 4 is 22.

00:07:57.620 --> 00:07:58.490
80.

00:07:58.490 --> 00:08:00.820
37 goes into 80 two times.

00:08:00.820 --> 00:08:03.410
2 times 37 is what?

00:08:03.410 --> 00:08:07.900
74, right?

00:08:07.900 --> 00:08:09.060
60 plus 14.

00:08:09.060 --> 00:08:11.240
74.

00:08:11.240 --> 00:08:12.152
You get a 60.

00:08:12.152 --> 00:08:13.460
30 is going to go into 60.

00:08:13.460 --> 00:08:16.360
So already we haven't even
started repeating and we're

00:08:16.360 --> 00:08:17.720
already three digits into it.

00:08:17.720 --> 00:08:21.090
So this is going to have
the longest sequence

00:08:21.090 --> 00:08:22.000
of different digits.

00:08:22.000 --> 00:08:22.950
We don't have to keep going.

00:08:22.950 --> 00:08:23.940
We've answered our question.

00:08:23.940 --> 00:08:27.560
The choice is definitely E.

00:08:27.560 --> 00:08:29.710
Next question.

00:08:29.710 --> 00:08:31.550
They've drawn something there.

00:08:31.550 --> 00:08:35.780
In the figure above the
coordinates of point V are--

00:08:35.780 --> 00:08:37.179
well, this is just kind
of a reading the

00:08:37.179 --> 00:08:38.140
graph type of problem.

00:08:38.140 --> 00:08:44.750
I'm just going to draw this 4th
quadrant just so that I

00:08:44.750 --> 00:08:48.180
can show you how I
think about it.

00:08:48.180 --> 00:08:52.340
OK, so let me just count: one,
two, three, four, five; one,

00:08:52.340 --> 00:08:54.850
two, three, four, five.

00:08:54.850 --> 00:08:59.350
They tell us that each of those
slashes are 1, because

00:08:59.350 --> 00:09:00.060
they do it in a couple
of places.

00:09:00.060 --> 00:09:03.700
They go one, two, three, four,
five, and then they say this

00:09:03.700 --> 00:09:04.940
is negative 5.

00:09:04.940 --> 00:09:06.870
So that's information that
you actually need.

00:09:06.870 --> 00:09:08.470
You have to know that each
of those slashes

00:09:08.470 --> 00:09:09.740
are definitely 1.

00:09:09.740 --> 00:09:11.860
But then on this, they
go one, two, three,

00:09:11.860 --> 00:09:14.120
four, five, six, seven.

00:09:14.120 --> 00:09:19.210
So they go one, two, three,
four, five, six, seven, so

00:09:19.210 --> 00:09:20.540
this is point 7.

00:09:20.540 --> 00:09:22.160
And you have to just count
it on your paper.

00:09:22.160 --> 00:09:24.960
So this is one, two,
three, four, five.

00:09:24.960 --> 00:09:28.500
And this is point one, two,
three, four, five.

00:09:28.500 --> 00:09:29.910
This is minus 5.

00:09:29.910 --> 00:09:32.380
And this is where V is.

00:09:32.380 --> 00:09:34.960
So they're just testing
to see if you know

00:09:34.960 --> 00:09:37.330
how to graph a point.

00:09:37.330 --> 00:09:39.310
So this is x is 7.

00:09:39.310 --> 00:09:41.880
And y is minus 5.

00:09:41.880 --> 00:09:44.370
So it's 7 minus 5.

00:09:47.680 --> 00:09:50.430
And actually, that's
interesting.

00:09:50.430 --> 00:09:52.290
You don't even have to count the
points, because you say,

00:09:52.290 --> 00:09:53.120
you know what?

00:09:53.120 --> 00:09:54.970
This is in the 4th quadrant.

00:09:54.970 --> 00:09:56.110
x is positive.

00:09:56.110 --> 00:09:57.190
y is negative.

00:09:57.190 --> 00:09:59.380
And if you look at all the
choices, there's only one

00:09:59.380 --> 00:10:01.540
choice where x is positive
and y is negative, and

00:10:01.540 --> 00:10:02.730
that's choice E.

00:10:02.730 --> 00:10:04.770
So you really didn't even have
to count it, although I think

00:10:04.770 --> 00:10:06.610
this wouldn't have taken much
time to count it, so that's

00:10:06.610 --> 00:10:08.070
not like a huge shortcut.

00:10:08.070 --> 00:10:10.510
Anyway, see you in
the next video.


GMAT: Data sufficiency 40

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:05.860
We're on problem 151.

00:00:05.860 --> 00:00:09.940
If x and y are positive,
is the ratio of x to y

00:00:09.940 --> 00:00:11.190
greater than 3?

00:00:14.810 --> 00:00:18.430
And they're telling us that
x and y are positive.

00:00:18.430 --> 00:00:19.180
Fair enough.

00:00:19.180 --> 00:00:22.040
Statement one.

00:00:22.040 --> 00:00:25.210
x is 2 more than 3 times y.

00:00:25.210 --> 00:00:30.830
So x is equal to 3y plus 2.

00:00:30.830 --> 00:00:33.770
Let's rewrite the original
question because I think it's

00:00:33.770 --> 00:00:34.950
going to look a lot like this.

00:00:34.950 --> 00:00:38.570
If we multiply y times both
sides of the equation-- and we

00:00:38.570 --> 00:00:40.270
don't have to change the
inequality because they told

00:00:40.270 --> 00:00:41.470
us that y is positive.

00:00:41.470 --> 00:00:43.750
They're both positive.

00:00:43.750 --> 00:00:45.830
So if you multiply both sides
of the equation times y, you

00:00:45.830 --> 00:00:49.600
get x is greater than 3y.

00:00:49.600 --> 00:00:51.960
This is the same question
as this.

00:00:51.960 --> 00:00:54.870
Well, if we look at statement
number one, clearly x is

00:00:54.870 --> 00:00:56.060
greater than 3y.

00:00:56.060 --> 00:00:59.700
Because x is 2 more than 3y
based on statement number one.

00:00:59.700 --> 00:01:01.880
So statement number
one is sufficient

00:01:01.880 --> 00:01:03.190
to answer this question.

00:01:03.190 --> 00:01:04.730
x is definitely greater
than 3y.

00:01:04.730 --> 00:01:07.360
x actually equals 3y plus 2.

00:01:07.360 --> 00:01:09.860
This plus 2 tells you that
x is that much greater.

00:01:09.860 --> 00:01:11.790
2 greater.

00:01:11.790 --> 00:01:13.430
Second statement.

00:01:13.430 --> 00:01:21.520
The ratio of 2x to 3y
is greater than 2.

00:01:21.520 --> 00:01:23.580
Let's just do some algebraic
manipulation.

00:01:23.580 --> 00:01:25.520
3y, once again, we know
that's positive.

00:01:25.520 --> 00:01:27.750
So when we multiply both sides
of the equation by it, we

00:01:27.750 --> 00:01:29.360
don't have to switch the sign.

00:01:29.360 --> 00:01:36.010
So you get 2x is equal to 6y.

00:01:36.010 --> 00:01:37.990
Now we can divide
both sides by 2.

00:01:37.990 --> 00:01:40.270
And you get-- oh, sorry,
why did I put equals?

00:01:40.270 --> 00:01:42.170
It should be greater than.

00:01:42.170 --> 00:01:46.240
Divide both sides by 2 and you
get x is greater than 3y.

00:01:46.240 --> 00:01:48.610
So this statement and this
statement are the same thing,

00:01:48.610 --> 00:01:50.770
so 2 alone is also sufficient.

00:01:50.770 --> 00:01:54.990
So either statement alone is
sufficient to answer this or

00:01:54.990 --> 00:01:56.240
this question.

00:01:58.090 --> 00:02:03.680
152.

00:02:03.680 --> 00:02:06.470
OK, if arc PQ-- OK, let me
see if I can draw this.

00:02:06.470 --> 00:02:09.270
So they've drawn a semicircle.

00:02:09.270 --> 00:02:11.740
So let me draw a circle.

00:02:11.740 --> 00:02:15.800
That's a circle, and I can
make it a semicircle by

00:02:15.800 --> 00:02:18.955
drawing a line from
there to there.

00:02:21.630 --> 00:02:23.920
That's about as good
as I can do.

00:02:23.920 --> 00:02:26.175
And they have it connecting
to some point.

00:02:26.175 --> 00:02:28.850
Let me see if I could do this.

00:02:28.850 --> 00:02:30.710
Looks like that.

00:02:30.710 --> 00:02:35.260
And then it looks like that.

00:02:35.260 --> 00:02:37.220
And then they actually have
drawn an altitude

00:02:37.220 --> 00:02:38.275
coming down like that.

00:02:38.275 --> 00:02:41.840
And I think I've done
all my line drawing.

00:02:41.840 --> 00:02:45.080
And then, let's see, they
labeled this P.

00:02:45.080 --> 00:02:47.950
They labeled this Q.

00:02:47.950 --> 00:02:50.970
They labeled this R.

00:02:50.970 --> 00:02:54.000
And this is 2.

00:02:54.000 --> 00:02:56.540
And they say that this distance
right here is a.

00:02:56.540 --> 00:02:58.470
This distance right here.

00:02:58.470 --> 00:03:02.020
And then they say that this
distance right here is b.

00:03:02.020 --> 00:03:02.900
Fair enough.

00:03:02.900 --> 00:03:07.110
So they say if arc PQR-- so
that's that arc right there--

00:03:07.110 --> 00:03:10.410
is a semicircle, what is the
length of diameter PR?

00:03:10.410 --> 00:03:13.120
So they essentially want to know
what a plus b is, right?

00:03:17.140 --> 00:03:18.810
That's PR.

00:03:18.810 --> 00:03:21.240
So that's their question.

00:03:21.240 --> 00:03:23.840
So immediately-- well let's just
look at their statements.

00:03:23.840 --> 00:03:25.260
What did they give us?

00:03:25.260 --> 00:03:28.800
Statement one, they tell us
that a is equal to 4.

00:03:28.800 --> 00:03:31.080
So from knowing that a is equal
to 4, if we can figure

00:03:31.080 --> 00:03:34.535
out what b is, then we would
know what a plus b is and we

00:03:34.535 --> 00:03:35.600
would be done.

00:03:35.600 --> 00:03:37.150
Well, let's see if
we can do that.

00:03:37.150 --> 00:03:41.410
Statement two is b
is equal to 1.

00:03:41.410 --> 00:03:45.320
So if we know what b is, and
then we can figure out a from

00:03:45.320 --> 00:03:47.480
it, then we know what
a plus b is.

00:03:47.480 --> 00:03:49.970
So actually, I think if either
of these is sufficient, the

00:03:49.970 --> 00:03:51.020
other one probably is.

00:03:51.020 --> 00:03:55.240
Because if statement one is
sufficient-- if knowing a we

00:03:55.240 --> 00:03:57.640
can figure out b, then probably
knowing b we could

00:03:57.640 --> 00:03:58.230
figure out a.

00:03:58.230 --> 00:04:00.220
But let's see if that holds.

00:04:00.220 --> 00:04:02.990
So one thing that immediately
should pop out of your head,

00:04:02.990 --> 00:04:05.700
although I don't know if it
will-- and this pops out of my

00:04:05.700 --> 00:04:07.320
head, really, from going
to math competitions

00:04:07.320 --> 00:04:08.630
when I was a kid.

00:04:08.630 --> 00:04:12.420
But if you have a triangle
inscribed in a semicircle,

00:04:12.420 --> 00:04:15.410
that triangle-- and one of its
sides is a diameter, right?

00:04:15.410 --> 00:04:16.910
And that's the case
with this one.

00:04:16.910 --> 00:04:18.540
Because we know it's a
semicircle, so we know that

00:04:18.540 --> 00:04:21.730
this is the diameter--
then that

00:04:21.730 --> 00:04:24.120
triangle is a right triangle.

00:04:24.120 --> 00:04:25.370
This is a right triangle.

00:04:25.370 --> 00:04:26.920
Even the way they drew it, it
kind of looks like that.

00:04:26.920 --> 00:04:28.320
But you that's something
you should know.

00:04:28.320 --> 00:04:30.610
Even if this triangle was drawn
like that, it's always

00:04:30.610 --> 00:04:32.780
going to be a right triangle.

00:04:32.780 --> 00:04:35.150
And I'll maybe do another
YouTube video proving that.

00:04:35.150 --> 00:04:36.510
But that's just a good thing
to know for the gmat.

00:04:36.510 --> 00:04:38.950
So let's see if we could use
that information and the

00:04:38.950 --> 00:04:42.550
Pythagorean theorem to come
up with a formula that

00:04:42.550 --> 00:04:44.340
relates a and b.

00:04:44.340 --> 00:04:51.040
So if we look at just
this triangle right

00:04:51.040 --> 00:04:53.570
there, we know what?

00:04:53.570 --> 00:05:00.920
We know that a squared plus 4 is
equal to-- let's call this

00:05:00.920 --> 00:05:06.730
side c squared-- is equal
to c squared.

00:05:06.730 --> 00:05:07.880
Fair enough.

00:05:07.880 --> 00:05:12.670
If we look at this smaller
triangle right here, we know

00:05:12.670 --> 00:05:19.050
that b squared plus 4 is equal
to-- let's call this d

00:05:19.050 --> 00:05:22.500
squared, that's the hypotenuse
of the smaller triangle-- is

00:05:22.500 --> 00:05:25.570
equal to d squared.

00:05:25.570 --> 00:05:28.340
Now, if we look at the larger
triangle, which we also know

00:05:28.340 --> 00:05:32.790
is a right triangle, now c
squared and d squared are the

00:05:32.790 --> 00:05:36.930
non-hypotenuse sides, and a plus
b is a hypotenuse side.

00:05:36.930 --> 00:05:42.180
So we know that c squared plus
d squared is equal to this

00:05:42.180 --> 00:05:48.520
entire side squared, which
is a plus b squared.

00:05:48.520 --> 00:05:50.160
Well, we know what c squared
and d squared are.

00:05:50.160 --> 00:05:51.100
We can substitute that.

00:05:51.100 --> 00:05:53.840
So c squared is a squared plus
4-- and I'll switch colors

00:05:53.840 --> 00:05:55.530
just to avoid the monotnony.

00:05:55.530 --> 00:05:57.660
So c squared is a
squared plus 4.

00:06:02.990 --> 00:06:05.690
And d squared is b
squared plus 4.

00:06:10.350 --> 00:06:12.370
And all of that's equal
to a plus b squared.

00:06:12.370 --> 00:06:14.050
Let's just multiply that
out because I think

00:06:14.050 --> 00:06:15.210
we can do some canceling.

00:06:15.210 --> 00:06:22.000
So we get a squared plus
2ab plus b squared.

00:06:22.000 --> 00:06:23.560
OK.

00:06:23.560 --> 00:06:26.250
So, we could subtract a squared
from both sides.

00:06:26.250 --> 00:06:28.310
a squared, a squared.

00:06:28.310 --> 00:06:30.600
Subtract b squared
from both sides.

00:06:30.600 --> 00:06:32.710
b squared and b squared.

00:06:32.710 --> 00:06:35.760
And this simplifies
out quite nicely.

00:06:35.760 --> 00:06:36.020
Let's see.

00:06:36.020 --> 00:06:38.625
4 plus 4 is 8, is
equal to 2ab.

00:06:41.500 --> 00:06:48.900
Divide both sides by 2, and
you get ab is equal to 4.

00:06:48.900 --> 00:06:52.660
So all of this crazy
information, that this is

00:06:52.660 --> 00:06:55.600
inscribed in the semicircle and
this is a, this is b, all

00:06:55.600 --> 00:06:59.320
of that reduces down to
ab is equal to 4.

00:06:59.320 --> 00:07:00.080
So now we know.

00:07:00.080 --> 00:07:03.580
If we know a, if a is 4,
then we know that b has

00:07:03.580 --> 00:07:06.420
to be equal to 1.

00:07:06.420 --> 00:07:08.000
And then we can figure
out what a plus b is.

00:07:08.000 --> 00:07:08.510
It's 5.

00:07:08.510 --> 00:07:10.560
So statement one alone
is sufficient.

00:07:10.560 --> 00:07:12.200
And similarly, if we know
what b is, we know

00:07:12.200 --> 00:07:13.810
that a has to be 4.

00:07:13.810 --> 00:07:16.360
And so a plus b is equal to 5.

00:07:16.360 --> 00:07:18.230
So that was actually
less hairy than I

00:07:18.230 --> 00:07:20.300
thought going into it.

00:07:20.300 --> 00:07:21.970
Next problem.

00:07:21.970 --> 00:07:24.410
And this really is
the home stretch.

00:07:24.410 --> 00:07:28.390
153.

00:07:28.390 --> 00:07:34.970
Does the integer k have a factor
p such that 1 is less

00:07:34.970 --> 00:07:36.540
than p is less than k.

00:07:41.240 --> 00:07:44.540
OK, so they're essentially
saying, is k prime?

00:07:44.540 --> 00:07:47.180
This is another way of saying
is k-- or you could

00:07:47.180 --> 00:07:49.280
say, is k not prime?

00:07:49.280 --> 00:07:50.730
Or k is composite.

00:07:50.730 --> 00:07:52.890
I think that's the
word for it.

00:07:52.890 --> 00:07:53.860
A non-prime number.

00:07:53.860 --> 00:07:55.590
So that's all they're asking.

00:07:55.590 --> 00:07:59.400
Does k have some other factor
other than 1 and itself.

00:07:59.400 --> 00:08:01.310
So let's see, statement
number one.

00:08:01.310 --> 00:08:04.330
k is greater than 4 factorial.

00:08:04.330 --> 00:08:07.780
Well, 4 factorial is 4 times
3 times 2 times 1.

00:08:07.780 --> 00:08:08.770
So that's 12 times 2.

00:08:08.770 --> 00:08:09.650
That's 24.

00:08:09.650 --> 00:08:13.040
So this boils down to k
is greater than 24.

00:08:13.040 --> 00:08:16.430
So I can give you both prime
and non-prime numbers

00:08:16.430 --> 00:08:17.800
greater than 24.

00:08:17.800 --> 00:08:21.430
29 is prime.

00:08:21.430 --> 00:08:24.460
30 is not prime.

00:08:24.460 --> 00:08:30.230
So statement one, at least by
itself, is pretty useless.

00:08:30.230 --> 00:08:33.110
k can be greater than 24 and
be prime or not prime.

00:08:33.110 --> 00:08:35.400
Statement two.

00:08:35.400 --> 00:08:37.140
This looks interesting.

00:08:37.140 --> 00:08:41.340
13 factorial-- which is a large
number, so it's hard to

00:08:41.340 --> 00:08:48.490
multiply it out-- plus 2 is less
than or equal to k, which

00:08:48.490 --> 00:08:57.220
is less than or equal to
13 factorial plus 13.

00:08:57.220 --> 00:09:01.700
So, essentially, what we need
to prove, somehow, is that

00:09:01.700 --> 00:09:06.690
every number between this
and this is not prime.

00:09:06.690 --> 00:09:09.400
And let's think about
how we can do it.

00:09:09.400 --> 00:09:12.660
Well, let's take the initial
example of 13

00:09:12.660 --> 00:09:14.810
factorial plus 2.

00:09:14.810 --> 00:09:20.550
So 13 factorial, just so you
know, is 13 times 12, times

00:09:20.550 --> 00:09:24.750
11-- you keep going-- times
3 times 2 times 1.

00:09:24.750 --> 00:09:27.640
That's 13 factorial.

00:09:27.640 --> 00:09:31.030
So 13 factorial is definitely
divisible by 2.

00:09:31.030 --> 00:09:33.160
So we could rewrite
13 factorial.

00:09:33.160 --> 00:09:41.010
We could write that as-- 13
factorial divided by 2

00:09:41.010 --> 00:09:44.180
times 2 plus 2.

00:09:44.180 --> 00:09:46.670
I just rewrote 13 factorial
plus 2.

00:09:46.670 --> 00:09:48.580
And I want you to see that this
is the same thing, right?

00:09:48.580 --> 00:09:50.800
I just divided by 2 and
multiplied by 2.

00:09:50.800 --> 00:09:53.870
And you'll see in a second
why I did that.

00:09:53.870 --> 00:09:55.000
And this is a tricky problem.

00:09:55.000 --> 00:09:57.840
When I looked at this at first,
which I actually did

00:09:57.840 --> 00:10:02.080
glance at before the video,
because-- I don't know if this

00:10:02.080 --> 00:10:03.740
is something I would have gotten
right if I was under

00:10:03.740 --> 00:10:04.650
time pressure on the gmat.

00:10:04.650 --> 00:10:07.470
But when I thought about it a
little bit more it made sense.

00:10:07.470 --> 00:10:11.540
So if I divide by 2 and times 2,
I could-- and this is going

00:10:11.540 --> 00:10:12.520
to be an integer.

00:10:12.520 --> 00:10:14.670
Because we know that 13
factorial is divisible by 2.

00:10:14.670 --> 00:10:17.570
2 is actually one
of its factors.

00:10:17.570 --> 00:10:20.655
So as long as you understand
where I got-- you can divide

00:10:20.655 --> 00:10:24.590
anything by 2 and multiply by 2
and not change it So this is

00:10:24.590 --> 00:10:28.170
the same thing as this.

00:10:28.170 --> 00:10:36.060
And then if we factor the 2 out,
this becomes 2 times 13

00:10:36.060 --> 00:10:42.470
factorial over 2 plus 1.

00:10:42.470 --> 00:10:43.870
I just factored the 2 out.

00:10:43.870 --> 00:10:45.180
This isn't a fraction.

00:10:45.180 --> 00:10:48.830
That's just a line I did just to
not make it look too messy.

00:10:48.830 --> 00:10:54.530
So what we show is that 13
factorial plus 2 is definitely

00:10:54.530 --> 00:10:57.430
divisible by 2.

00:10:57.430 --> 00:10:59.450
Whatever that number is, it's
definitely divisible by 2.

00:10:59.450 --> 00:11:00.500
And how do I know that?

00:11:00.500 --> 00:11:02.800
Because this is just another
way of writing 13 factorial

00:11:02.800 --> 00:11:04.070
divided by 2.

00:11:04.070 --> 00:11:06.420
I'm sorry, this is another way--
let me write this down--

00:11:06.420 --> 00:11:09.660
this is the same thing as
13 factorial plus 2.

00:11:09.660 --> 00:11:11.350
It's just another way
of writing it.

00:11:11.350 --> 00:11:14.650
I took 13 factorial divided by
2 and multiplying by 2, and

00:11:14.650 --> 00:11:16.410
then taking out the 2.

00:11:16.410 --> 00:11:19.640
And the reason why I know that
2 is divisible into this is

00:11:19.640 --> 00:11:21.400
because I know that this
is an integer.

00:11:21.400 --> 00:11:24.280
We know that 13 factorial is
divided by 2 because 2 is one

00:11:24.280 --> 00:11:25.570
of the things that
goes into it.

00:11:25.570 --> 00:11:27.110
And so this is an integer.

00:11:27.110 --> 00:11:27.900
You add 1.

00:11:27.900 --> 00:11:28.970
So this is an integer.

00:11:28.970 --> 00:11:31.060
And so you're multiplying
2 times some integer.

00:11:31.060 --> 00:11:32.690
So this has to be
divisible by 2.

00:11:32.690 --> 00:11:36.990
So 13 factorial plus 2
is divisible by 2.

00:11:36.990 --> 00:11:40.460
Now let's just extend
that generally.

00:11:40.460 --> 00:11:43.930
13 factorial plus x.

00:11:43.930 --> 00:11:48.010
Let's prove that as long as x
is less than or equal to 13

00:11:48.010 --> 00:11:51.100
that this is divisible by x.

00:11:51.100 --> 00:11:52.980
So we can do the same trick.

00:11:52.980 --> 00:11:56.950
You could say this is equal to
13 factorial divided by x,

00:11:56.950 --> 00:12:00.160
times x plus x.

00:12:00.160 --> 00:12:05.050
That's equal to-- you can factor
the x out-- x times 13

00:12:05.050 --> 00:12:09.500
factorial over x plus 1.

00:12:09.500 --> 00:12:12.010
And how do we know that this
number, which is the same

00:12:12.010 --> 00:12:14.060
thing as this number, how
do we know that this is

00:12:14.060 --> 00:12:15.710
divisible by x?

00:12:15.710 --> 00:12:17.050
Well, just a little thought
experiment.

00:12:19.730 --> 00:12:27.780
We know as long as x is greater
than or equal to 1 and

00:12:27.780 --> 00:12:30.270
less than or equal to 13,
and x is an integer.

00:12:30.270 --> 00:12:32.330
As long as x is an integer
that's greater than or equal

00:12:32.330 --> 00:12:36.610
to 1 and less than or equal to
13, it is one of the factors

00:12:36.610 --> 00:12:38.170
of 13 factorial.

00:12:38.170 --> 00:12:40.470
13 factorial is 13 times
12 times 11.

00:12:40.470 --> 00:12:42.380
It's all of those numbers
multiplied together.

00:12:42.380 --> 00:12:46.150
So we know that as long as x is
less than or equal to 13,

00:12:46.150 --> 00:12:49.570
we know that this right here,
this is an integer.

00:12:49.570 --> 00:12:51.510
If this is an integer,
then the integer plus

00:12:51.510 --> 00:12:52.990
1 is also an integer.

00:12:52.990 --> 00:12:56.836
So 13 factorial plus x
can be rewritten as

00:12:56.836 --> 00:12:59.930
x times some integer.

00:12:59.930 --> 00:13:02.160
I'll call it I.

00:13:02.160 --> 00:13:04.680
But the bottom line is, the
reason why I wanted to do

00:13:04.680 --> 00:13:09.760
that, it shows you that every
number between 13 factorial

00:13:09.760 --> 00:13:17.850
plus 2 and 13 factorial plus
13 is actually divisible by

00:13:17.850 --> 00:13:20.940
whatever number you're adding
to the factorial.

00:13:20.940 --> 00:13:24.370
So since it's divisible by those
numbers, the numbers

00:13:24.370 --> 00:13:26.120
cannot be prime.

00:13:26.120 --> 00:13:28.200
So hopefully-- I've actually
run way over time-- but

00:13:28.200 --> 00:13:31.440
hopefully I've shown you that
that information alone,

00:13:31.440 --> 00:13:37.540
statement two alone, is
sufficient to show that k is

00:13:37.540 --> 00:13:41.500
not prime or that k does
have a factor p that's

00:13:41.500 --> 00:13:43.000
between 1 and k.

00:13:43.000 --> 00:13:44.660
Anyway, I hope didn't
confuse you.

00:13:44.660 --> 00:13:46.600
I think that was one of the
hardest problems we've done,

00:13:46.600 --> 00:13:48.730
so don't feel bad if you found
that a little daunting.

00:13:48.730 --> 00:13:50.830
See you in the next video.


GMAT: Data sufficiency 41

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:05.930
We're on problem 154.

00:00:05.930 --> 00:00:07.250
They ask us, is x negative?

00:00:07.250 --> 00:00:11.150
So the question is
x less than 0.

00:00:11.150 --> 00:00:14.840
Statement number 1 tells us--
let's see, they say x to the

00:00:14.840 --> 00:00:20.630
third times 1 minus x squared
is less than 0.

00:00:20.630 --> 00:00:21.480
Now let's simplify it.

00:00:21.480 --> 00:00:25.030
Let's see, x to the third
minus x to the fifth

00:00:25.030 --> 00:00:28.140
is less than 0.

00:00:28.140 --> 00:00:31.240
We could add x to the fifth to
both sides. x to the third is

00:00:31.240 --> 00:00:33.820
less than x to the fifth.

00:00:33.820 --> 00:00:35.700
Let's divide both sides
by x squared.

00:00:35.700 --> 00:00:37.740
That won't change the inequality
since x squared is

00:00:37.740 --> 00:00:38.980
definitely going
to be positive.

00:00:38.980 --> 00:00:41.860
Any integer squared, even if
it's negative, is a positive.

00:00:41.860 --> 00:00:45.390
So you divide both sides by x
squared, you get x is less

00:00:45.390 --> 00:00:48.290
than x to the third.

00:00:48.290 --> 00:00:50.830
Now let's see, does this
answer that question?

00:00:50.830 --> 00:00:53.185
Let's see what x's would
satisfy this condition.

00:00:57.050 --> 00:01:02.820
So we can already say that x
cannot be equal to 0, right?

00:01:02.820 --> 00:01:03.760
Anything where this
would have been an

00:01:03.760 --> 00:01:04.400
inequality doesn't work.

00:01:04.400 --> 00:01:08.160
So x can't be equal 0, it can't
be able to 1, because 1

00:01:08.160 --> 00:01:10.010
is equal 1 to the third,
and it can't be

00:01:10.010 --> 00:01:11.290
equal to negative 1.

00:01:11.290 --> 00:01:14.100
So we know that immediately.

00:01:14.100 --> 00:01:17.200
We know that for regular
positive numbers, for numbers

00:01:17.200 --> 00:01:22.340
greater than 1, taking it to
the third power is always

00:01:22.340 --> 00:01:24.000
going to be greater than
the number itself.

00:01:24.000 --> 00:01:26.170
If you took 2 to the third
power, that's greater than 2.

00:01:26.170 --> 00:01:28.520
3 to the third power
is greater than 3.

00:01:28.520 --> 00:01:31.620
1.5 to the third power
is greater than 1.5.

00:01:31.620 --> 00:01:36.200
So we know that this statement
right here implies that x is

00:01:36.200 --> 00:01:38.070
greater than 1.

00:01:38.070 --> 00:01:41.490
Now let's see, is there
a situation

00:01:41.490 --> 00:01:43.260
where x can be negative?

00:01:43.260 --> 00:01:44.670
Well, let's see.

00:01:44.670 --> 00:01:47.202
First off, x is between
0 and 1.

00:01:52.610 --> 00:01:54.100
1/2 is not less than 1/8.

00:01:54.100 --> 00:01:55.480
So it doesn't work
between 0 and 1.

00:01:55.480 --> 00:02:04.000
But what if x were between
negative 1 and 0?

00:02:04.000 --> 00:02:06.400
Let's try an example.

00:02:06.400 --> 00:02:09.889
Negative 1/2 is less
than negative 1/8.

00:02:09.889 --> 00:02:11.940
It's more negative.

00:02:11.940 --> 00:02:17.700
Negative 1/2 is less than
negative 1/8, right?

00:02:17.700 --> 00:02:19.060
You could try that with
any fraction.

00:02:19.060 --> 00:02:20.870
Because when you take it to the
third power, it becomes a

00:02:20.870 --> 00:02:22.390
smaller fraction,
but it becomes a

00:02:22.390 --> 00:02:24.220
smaller negative fraction.

00:02:24.220 --> 00:02:28.840
So x will be less than it,
because it's more negative.

00:02:28.840 --> 00:02:31.490
So this is true.

00:02:31.490 --> 00:02:35.780
If this is true, then x is
either greater than 1, or it's

00:02:35.780 --> 00:02:38.400
between negative 1 and 0.

00:02:38.400 --> 00:02:41.760
Let's see, does it work if x
is less than negative 1.

00:02:41.760 --> 00:02:45.360
Negative 2, is negative 2
less than negative 8?

00:02:45.360 --> 00:02:46.550
Nope.

00:02:46.550 --> 00:02:46.850
Right.

00:02:46.850 --> 00:02:50.730
So these are the only ranges
where it works.

00:02:50.730 --> 00:02:52.830
This statement implies this.

00:02:52.830 --> 00:02:55.260
But this still isn't enough
information to tell us whether

00:02:55.260 --> 00:02:56.870
x is less than 0.

00:02:56.870 --> 00:02:59.530
x could be greater than 1, which
is definitely greater

00:02:59.530 --> 00:03:01.280
than 0, or x could
be less than 0.

00:03:01.280 --> 00:03:03.980
So we don't know yet just
from statement 1.

00:03:03.980 --> 00:03:12.440
Statement 2 tells us x squared
minus 1 is less than 0.

00:03:12.440 --> 00:03:15.660
So add 1 to both sides, that
tells us that x squared is

00:03:15.660 --> 00:03:17.850
less than 1.

00:03:17.850 --> 00:03:22.020
So when could something squared
be less than 1?

00:03:22.020 --> 00:03:23.150
Well, it's going to
have to be between

00:03:23.150 --> 00:03:28.270
negative 1 and 1, right?

00:03:28.270 --> 00:03:30.900
It's essentially going to have
be a positive or a negative

00:03:30.900 --> 00:03:32.140
fraction less than 1.

00:03:32.140 --> 00:03:33.820
That's the only way that when
you square it, you get a

00:03:33.820 --> 00:03:36.340
number less than 1.

00:03:36.340 --> 00:03:38.030
If it was greater than 1, you're
definitely going to get

00:03:38.030 --> 00:03:40.600
a number greater than 1 and
it's going to be positive

00:03:40.600 --> 00:03:42.390
either way.

00:03:42.390 --> 00:03:45.640
If it's 1, it's going to be 1,
and then 0 is in our range.

00:03:45.640 --> 00:03:48.570
So statement 2 tells us this,
but it includes both positive

00:03:48.570 --> 00:03:49.700
and negative numbers.

00:03:49.700 --> 00:03:51.360
So it doesn't answer our
question whether x

00:03:51.360 --> 00:03:53.530
is less than 0.

00:03:53.530 --> 00:03:57.120
But if we use both statements
combined, what is the

00:03:57.120 --> 00:03:57.970
intersection?

00:03:57.970 --> 00:04:00.630
What is the overlap
of this and this?

00:04:00.630 --> 00:04:05.550
Well, if x has to be between
negative 1 and 1 and it has to

00:04:05.550 --> 00:04:09.620
be one of these, the only
one that it overlaps

00:04:09.620 --> 00:04:11.800
with is this one.

00:04:11.800 --> 00:04:16.010
You cannot have x being greater
than 1 and x being in

00:04:16.010 --> 00:04:17.050
this range.

00:04:17.050 --> 00:04:19.930
You can have x being in this
range and x being in this

00:04:19.930 --> 00:04:22.460
range, since this is
a subset of this.

00:04:22.460 --> 00:04:25.470
So this is actually the
more restrictive.

00:04:25.470 --> 00:04:28.370
And so since we go to that, that
actually tells us that x

00:04:28.370 --> 00:04:30.200
has to be negative.

00:04:30.200 --> 00:04:33.230
So both statements combined are
sufficient to answer this

00:04:33.230 --> 00:04:36.690
question, but individually
they're not.

00:04:36.690 --> 00:04:41.330
The last question, 155.

00:04:41.330 --> 00:04:43.740
Marsha's bucket can hold
a maximum of how

00:04:43.740 --> 00:04:44.970
many liters of water?

00:04:44.970 --> 00:04:46.855
So we want to know the capacity
of her bucket.

00:04:50.740 --> 00:04:53.560
Statement number 1 says, the
bucket currently contains 9

00:04:53.560 --> 00:04:54.660
liters of water.

00:04:54.660 --> 00:04:55.430
That's useless.

00:04:55.430 --> 00:04:57.750
We don't know how-- that
could be the bucket, 9

00:04:57.750 --> 00:04:58.830
liters could be that.

00:04:58.830 --> 00:05:01.940
Or maybe 9 liters is
the full capacity.

00:05:01.940 --> 00:05:04.380
That tells us nothing about
what the capacity

00:05:04.380 --> 00:05:06.660
of the bucket is.

00:05:06.660 --> 00:05:08.810
If I told you I had a bite of
the sandwich, that tells you

00:05:08.810 --> 00:05:11.760
nothing as far as how large
of a sandwich I can eat.

00:05:11.760 --> 00:05:14.880
So statement 1 doesn't
seem that useful.

00:05:14.880 --> 00:05:20.090
Statement 2, if 3 liters of
water are added to the bucket

00:05:20.090 --> 00:05:22.760
when it is 1/2 full of water.

00:05:22.760 --> 00:05:26.630
So 1/2 times the capacity,
right?

00:05:26.630 --> 00:05:30.220
When it's 1/2 full of water,
you add 3 liters.

00:05:30.220 --> 00:05:32.910
So if 3 liters of water are
added to the bucket when it is

00:05:32.910 --> 00:05:39.480
1/2 full of water, the amount
of water in the bucket will

00:05:39.480 --> 00:05:40.730
increase by 1/3.

00:05:46.930 --> 00:05:52.210
To increase something by 1/3,
that is like multiplying it by

00:05:52.210 --> 00:05:57.970
1 1/3 That's 1 1/3 times
the capacity.

00:05:57.970 --> 00:06:00.870
That's what they're saying.

00:06:00.870 --> 00:06:03.340
The amount of water in the
bucket will increase by 1.

00:06:06.950 --> 00:06:07.700
Sorry.

00:06:07.700 --> 00:06:10.270
You're going to have 1 1/3 times
what you started off

00:06:10.270 --> 00:06:11.565
with, which was half
your capacity.

00:06:15.220 --> 00:06:15.770
That make sense?

00:06:15.770 --> 00:06:19.110
We say we're starting with 1/2
of our capacity and we're

00:06:19.110 --> 00:06:20.360
adding 3 liters.

00:06:22.930 --> 00:06:25.340
They're saying that that will
increase the amount

00:06:25.340 --> 00:06:28.100
we have by 1 1/3.

00:06:28.100 --> 00:06:28.680
Let me write that.

00:06:28.680 --> 00:06:35.500
Let's say we start with x, and
we're adding 3 liters.

00:06:35.500 --> 00:06:40.150
They're saying that will equal
1 1/3 or 4/3 times x, right?

00:06:40.150 --> 00:06:43.740
This shows that x is
increasing by 1/3.

00:06:43.740 --> 00:06:47.490
You could view this as
1 plus 1/3 times x.

00:06:47.490 --> 00:06:50.120
This is an increase of 1/3.

00:06:50.120 --> 00:06:53.210
They tell us that our starting
point is 1/2 of capacity.

00:06:53.210 --> 00:06:56.630
So that's where I get the 1/2
c-- that's our starting

00:06:56.630 --> 00:07:01.270
point-- plus 3 is equal to 4/3
times the starting point, or

00:07:01.270 --> 00:07:04.530
where we're adding water
to, 1/2 times c.

00:07:04.530 --> 00:07:06.550
We could easily solve
for c here.

00:07:06.550 --> 00:07:09.450
So you get a 2, this
cancels out.

00:07:09.450 --> 00:07:17.470
Then you get 1/2 c plus
3 is equal to 2/3 c.

00:07:17.470 --> 00:07:20.640
Let's see, we could subtract 1/2
c from both sides, and you

00:07:20.640 --> 00:07:26.950
get 3 is equal to
2/3 minus 1/2 c.

00:07:26.950 --> 00:07:30.180
Then you can just do that
fraction, and you get c is

00:07:30.180 --> 00:07:33.620
equal to 3 over-- well,
what is this?

00:07:33.620 --> 00:07:41.230
1/2-- so if you do it over a 6,
4 minus 3, so this is 1/6.

00:07:41.230 --> 00:07:43.360
So this turns into 1/6.

00:07:43.360 --> 00:07:47.572
So c is equal to 18.

00:07:47.572 --> 00:07:48.190
Wait a minute.

00:07:48.190 --> 00:07:49.150
You didn't have to
do all that.

00:07:49.150 --> 00:07:50.970
You should have just realized
that as soon as you can write

00:07:50.970 --> 00:07:53.540
this algebraic equation down,
it's a linear equation with

00:07:53.540 --> 00:07:55.980
one unknown, and that one
unknown is what we're trying

00:07:55.980 --> 00:07:58.060
to solve for, you have
enough information

00:07:58.060 --> 00:07:59.220
to solve the problem.

00:07:59.220 --> 00:08:03.090
So statement 2, alone,
is sufficient

00:08:03.090 --> 00:08:04.810
to solve this problem.

00:08:04.810 --> 00:08:06.580
We're all done with
data sufficiency.

00:08:06.580 --> 00:08:08.840
See you all in the
next section.


GMAT: Data sufficiency 39

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:06.250
We're on problem 148.

00:00:06.250 --> 00:00:09.310
What is the tens digit of
positive integer x.

00:00:09.310 --> 00:00:10.560
We want to know the
tens digit.

00:00:13.230 --> 00:00:16.680
Statement one tells us x
divided by 100 has a

00:00:16.680 --> 00:00:19.280
remainder of 30.

00:00:19.280 --> 00:00:23.140
Well, then we know the tens
digit is 3, right?

00:00:23.140 --> 00:00:27.890
Because, let's say 30 divided by
100 has remainder of 30 and

00:00:27.890 --> 00:00:30.340
its tens digit is 3.

00:00:30.340 --> 00:00:33.300
130 divided by 100 has a
remainder of 30 and its

00:00:33.300 --> 00:00:34.950
remainder is 3.

00:00:34.950 --> 00:00:37.730
And I can make an argument
that any number, when you

00:00:37.730 --> 00:00:41.280
divide it by 100 that has a
remainder of 30, its tens

00:00:41.280 --> 00:00:42.650
digit is going to be 3.

00:00:42.650 --> 00:00:46.550
In fact, that number is
going to be 3, 0.

00:00:46.550 --> 00:00:48.140
And its going to have a
bunch of other digits

00:00:48.140 --> 00:00:48.680
here, there, there.

00:00:48.680 --> 00:00:50.730
And if you just divide this by
100, you're going to be left

00:00:50.730 --> 00:00:52.120
over with 30.

00:00:52.120 --> 00:00:54.560
So statement one is sufficient
to figure out what the tens

00:00:54.560 --> 00:00:57.020
digit of the number is.

00:00:57.020 --> 00:00:59.545
Statement two.

00:00:59.545 --> 00:01:06.880
So we could write x divided
by 110 has a

00:01:06.880 --> 00:01:09.470
remainder equal to 30.

00:01:09.470 --> 00:01:12.140
Well, this is a little
different case.

00:01:12.140 --> 00:01:15.045
So, definitely, when 30 is
divided by 110 the remainder

00:01:15.045 --> 00:01:17.700
is 30, because it goes
into it 0 times.

00:01:17.700 --> 00:01:19.500
So the remainder is 30.

00:01:19.500 --> 00:01:21.850
And then the tens digit is 3.

00:01:21.850 --> 00:01:24.490
So let's see, I'm going to try
to find other numbers when

00:01:24.490 --> 00:01:27.160
they're divided by 110 its
remainder is 30 but its tens

00:01:27.160 --> 00:01:28.740
digit is something else.

00:01:28.740 --> 00:01:29.770
So let's see.

00:01:29.770 --> 00:01:31.400
140.

00:01:31.400 --> 00:01:38.320
If I divide that by 110, I'm
left with a remainder of 30.

00:01:38.320 --> 00:01:40.200
But its tens digit is what?

00:01:40.200 --> 00:01:42.940
Its tens digit is 4, right?

00:01:42.940 --> 00:01:44.990
I shouldn't draw
an arrow here.

00:01:44.990 --> 00:01:48.830
If I did-- let's see, I
go up one more-- 250.

00:01:48.830 --> 00:01:52.270
250, you divide it by 110,
its remainder is 30.

00:01:52.270 --> 00:01:54.600
But its tens digit is 5.

00:01:54.600 --> 00:01:56.940
So this statement gives
me no information.

00:01:56.940 --> 00:01:58.760
Saying that the remainder
is 30 still gives me no

00:01:58.760 --> 00:01:59.450
information.

00:01:59.450 --> 00:02:01.650
The tens digit can be any of
these numbers, or a bunch more

00:02:01.650 --> 00:02:03.100
if I just kept going.

00:02:03.100 --> 00:02:07.520
So statement one is sufficient
to answer this question.

00:02:07.520 --> 00:02:13.070
Question 149.

00:02:13.070 --> 00:02:20.740
If x, y, and z are positive
integers, is x minus y odd?

00:02:20.740 --> 00:02:23.070
In order for this to happen,
one of these have to be odd

00:02:23.070 --> 00:02:24.080
and one have to be even.

00:02:24.080 --> 00:02:25.560
And you can just think
about that.

00:02:25.560 --> 00:02:29.480
If you take the difference,
or really the sum, of two

00:02:29.480 --> 00:02:31.310
numbers, the only way that
that difference or sum is

00:02:31.310 --> 00:02:35.060
going to be odd is if one
is odd and one is even.

00:02:35.060 --> 00:02:36.150
So let's think about that.

00:02:36.150 --> 00:02:37.640
Let's look at the statements.

00:02:37.640 --> 00:02:39.660
Just going to see if they
give us any information.

00:02:39.660 --> 00:02:42.775
They say that x is equal
to z squared.

00:02:45.360 --> 00:02:47.950
Well, that gives me no
information about y.

00:02:47.950 --> 00:02:49.795
And it gives me, actually, very
little information about

00:02:49.795 --> 00:02:50.780
x just yet.

00:02:50.780 --> 00:02:53.100
I mean, it tells me that x is
a perfect square, but a

00:02:53.100 --> 00:02:55.190
perfect square could be
odd or even, right?

00:02:55.190 --> 00:02:57.070
It could be 16, it could be 9.

00:02:57.070 --> 00:03:00.830
16's an even perfect square,
9 is an odd one.

00:03:00.830 --> 00:03:03.780
So this doesn't give me much
information by itself.

00:03:03.780 --> 00:03:05.700
Let's see what statement
two tells us.

00:03:05.700 --> 00:03:14.280
Statement two says that y is
equal to z minus 1 squared.

00:03:14.280 --> 00:03:16.620
So this statement, by itself, is
kind of like statement one.

00:03:16.620 --> 00:03:18.780
It just tells me that y is
a perfect square of some

00:03:18.780 --> 00:03:20.440
integer, right?

00:03:20.440 --> 00:03:22.310
Because z is an integer, so
z minus 1 is an integer.

00:03:22.310 --> 00:03:23.840
So it just tells you y
is a perfect square.

00:03:23.840 --> 00:03:26.410
And a perfect square could
be even or odd.

00:03:26.410 --> 00:03:29.370
But if we take both of these
together, then something

00:03:29.370 --> 00:03:30.690
interesting happens.

00:03:30.690 --> 00:03:35.130
For example, if we assume that z
is odd, then its square will

00:03:35.130 --> 00:03:36.170
also be odd, right?

00:03:36.170 --> 00:03:39.510
3 squared is 9, 7
squared is 49.

00:03:39.510 --> 00:03:42.300
So if z is odd, then x is odd.

00:03:42.300 --> 00:03:45.930
And then z minus 1 would be even
and then y would be even.

00:03:45.930 --> 00:03:48.110
So one would be odd,
one would be even.

00:03:48.110 --> 00:03:48.870
And you could do
the other way.

00:03:48.870 --> 00:03:51.540
You could say if z is
even, x is even.

00:03:51.540 --> 00:03:54.190
And if z is even, then
z minus one is odd.

00:03:54.190 --> 00:03:55.250
And y is odd.

00:03:55.250 --> 00:03:57.110
And I can prove it to
you mathematically.

00:03:57.110 --> 00:03:59.570
Let's write this and
substitute for z.

00:03:59.570 --> 00:04:06.660
So x minus y becomes z squared
minus z minus 1 squared.

00:04:06.660 --> 00:04:10.900
And so this becomes z squared
minus z squared

00:04:10.900 --> 00:04:13.770
minus 2z plus 1.

00:04:13.770 --> 00:04:17.959
And that equals z
squared minus z

00:04:17.959 --> 00:04:21.620
squared plus 2z, right?

00:04:21.620 --> 00:04:22.940
Distribute the negative sign.

00:04:22.940 --> 00:04:25.000
Plus 2z minus 1.

00:04:25.000 --> 00:04:31.550
The z squares cancel out and
we're left with 2z minus 1.

00:04:31.550 --> 00:04:33.870
If we use both statements,
x minus y

00:04:33.870 --> 00:04:36.160
simplifies to 2z minus 1.

00:04:36.160 --> 00:04:38.970
They told us that z is
a positive integer.

00:04:38.970 --> 00:04:41.420
So this-- this part of
the statement right

00:04:41.420 --> 00:04:42.490
here-- has to be even.

00:04:42.490 --> 00:04:43.870
It's a multiple of 2.

00:04:43.870 --> 00:04:45.600
So this is even.

00:04:45.600 --> 00:04:48.090
And if you subtract 1 from an
even number, this whole

00:04:48.090 --> 00:04:50.800
expression has to be odd.

00:04:50.800 --> 00:04:54.400
So both statements together are
sufficient to say that x

00:04:54.400 --> 00:04:58.050
minus y is odd.

00:04:58.050 --> 00:04:58.620
Next problem.

00:04:58.620 --> 00:05:01.950
This one looks hairy.

00:05:01.950 --> 00:05:05.070
150.

00:05:05.070 --> 00:05:07.930
Henry purchased three
items during a sale.

00:05:07.930 --> 00:05:11.450
He received a 20% discount off
the regular price off the most

00:05:11.450 --> 00:05:15.860
expensive item and a 10%
discount off the regular price

00:05:15.860 --> 00:05:17.840
of the other two.

00:05:17.840 --> 00:05:19.940
Was the total amount of the
three discounts greater than

00:05:19.940 --> 00:05:22.410
15% of the sum of the
regular price?

00:05:22.410 --> 00:05:23.170
OK.

00:05:23.170 --> 00:05:25.610
So what was the total amount
of the three discounts?

00:05:25.610 --> 00:05:34.450
It was 20% times-- let's just
call it item one-- 20% times

00:05:34.450 --> 00:05:41.570
item one plus 10% times
item too, plus

00:05:41.570 --> 00:05:47.200
10% times item three.

00:05:47.200 --> 00:05:48.970
And I'm assuming this is
the most expensive.

00:05:48.970 --> 00:05:50.780
Let's just say this is the
second most, I guess.

00:05:50.780 --> 00:05:51.950
This is third most expensive.

00:05:51.950 --> 00:05:54.570
Maybe they're the same
price, I don't know.

00:05:54.570 --> 00:05:55.700
This is the total discount.

00:05:55.700 --> 00:05:56.730
This isn't what he paid.

00:05:56.730 --> 00:05:58.240
This is the discount
on item one.

00:05:58.240 --> 00:05:59.170
The discount on item two.

00:05:59.170 --> 00:06:01.490
I'm not saying what they
actually paid for item two.

00:06:01.490 --> 00:06:03.800
So what the question is, was
a total amount of the three

00:06:03.800 --> 00:06:10.090
discounts, that's this number,
greater than 15% of the sum of

00:06:10.090 --> 00:06:11.030
the regular prices.

00:06:11.030 --> 00:06:16.050
So they're saying, was that
greater than 15% of I1, plus

00:06:16.050 --> 00:06:18.895
I2, plus I3.

00:06:18.895 --> 00:06:21.240
And I think we can
simplify this.

00:06:21.240 --> 00:06:25.200
Because if we distribute this
right-hand side, you get 0.15

00:06:25.200 --> 00:06:35.110
times I1 plus 0.15 times
I2 plus 0.15 times I3.

00:06:35.110 --> 00:06:40.990
And let's see, if we subtract
out 0.15 I1 from both sides,

00:06:40.990 --> 00:06:48.600
you get point-- let's see, 0.2
minus 0.15 is 0.05 I1.

00:06:48.600 --> 00:06:51.840
And I want to keep everything
positive, so let me subtract

00:06:51.840 --> 00:06:53.410
these from the right-hand
side.

00:06:53.410 --> 00:06:57.180
So I'm going to subtract 0.1
I2 from-- I'm going to

00:06:57.180 --> 00:06:59.610
subtract these from both
sides of the equation.

00:06:59.610 --> 00:07:06.040
So that's going to be greater
than 0.15 I2 minus 0.1 I2.

00:07:06.040 --> 00:07:11.570
So that's 0.05 I2.

00:07:11.570 --> 00:07:13.690
Plus, and now to do the
same thing for I3.

00:07:13.690 --> 00:07:15.550
0.15, 0.1.

00:07:15.550 --> 00:07:18.140
So I'm going to subtract
0.1 from 0.15.

00:07:18.140 --> 00:07:22.170
So 0.05 I3.

00:07:22.170 --> 00:07:25.420
All I did, going from this to
this, is I just distributed

00:07:25.420 --> 00:07:28.390
the 0.15 and I subtracted
and added to

00:07:28.390 --> 00:07:29.590
simplify it a little bit.

00:07:29.590 --> 00:07:31.400
And, actually, this is
interesting, too, because I

00:07:31.400 --> 00:07:32.970
just have these 0.05's
everywhere.

00:07:32.970 --> 00:07:34.150
That's a positive number.

00:07:34.150 --> 00:07:35.640
And I was able to do that
because I just added and

00:07:35.640 --> 00:07:37.060
subtracted from both sides.

00:07:37.060 --> 00:07:39.820
But if I multiply or divide by
a positive number, then I

00:07:39.820 --> 00:07:41.750
don't have to change the
inequality, and you could say,

00:07:41.750 --> 00:07:44.620
let's just divide both sides
of this equation by 0.05.

00:07:44.620 --> 00:07:47.460
Or the equivalent is to multiply
both sides by 20.

00:07:47.460 --> 00:07:53.720
But then we're left with I1 is
greater than I2 plus I3.

00:07:53.720 --> 00:07:55.550
And I think this simplification,
without having

00:07:55.550 --> 00:07:57.080
read the statements,
was worth it.

00:07:57.080 --> 00:07:59.320
Because we went from something
very convoluted in a very

00:07:59.320 --> 00:08:02.630
convoluted problem statement
to something very simple.

00:08:02.630 --> 00:08:07.470
So, essentially, they're asking
us was the price of I1,

00:08:07.470 --> 00:08:10.830
was the price of item one,
greater than the price of item

00:08:10.830 --> 00:08:13.000
2 plus item 3.

00:08:13.000 --> 00:08:14.850
If we can answer this question,
we can answer the

00:08:14.850 --> 00:08:16.090
harder question.

00:08:16.090 --> 00:08:18.260
So statement number one.

00:08:18.260 --> 00:08:21.710
The regular price of the most
expensive item was $50.

00:08:21.710 --> 00:08:25.230
I1 was equal to $50.

00:08:25.230 --> 00:08:26.560
And the regular price
for the next most

00:08:26.560 --> 00:08:27.910
expensive item was $20.

00:08:27.910 --> 00:08:31.120
So I2 was equal to $20.

00:08:31.120 --> 00:08:35.409
So now the question boils down
to, was $50 greater than $20

00:08:35.409 --> 00:08:37.880
plus the third most
expensive item.

00:08:37.880 --> 00:08:39.390
Well, I don't know, depends.

00:08:39.390 --> 00:08:42.710
Well, actually, this answers
our question, right?

00:08:42.710 --> 00:08:44.830
Because I was about to say, well
the third most expensive

00:08:44.830 --> 00:08:46.330
item, maybe it's $30.

00:08:46.330 --> 00:08:49.010
But by definition, we know
that it's not $30.

00:08:49.010 --> 00:08:49.840
Why?

00:08:49.840 --> 00:08:51.970
Because it was the third
most expensive item.

00:08:51.970 --> 00:08:54.560
The second most expensive
item is $20.

00:08:54.560 --> 00:08:58.780
So this thing has to be less
than $20 if we are to consider

00:08:58.780 --> 00:09:01.430
it the third most
expensive item.

00:09:01.430 --> 00:09:05.220
So if this thing is less than
$20, then the right-hand side

00:09:05.220 --> 00:09:07.615
of the equation is definitely--
it's going to

00:09:07.615 --> 00:09:08.820
actually be less than $40.

00:09:08.820 --> 00:09:10.940
So this is definitely going to
be less than $50, so this is

00:09:10.940 --> 00:09:12.010
going to be true.

00:09:12.010 --> 00:09:14.790
So it turns out that statement
one by itself is sufficient.

00:09:14.790 --> 00:09:17.050
Because you just have to realize
that I3 has to be less

00:09:17.050 --> 00:09:20.390
than $20 I2 is equal to $20.

00:09:20.390 --> 00:09:22.270
Statement two.

00:09:22.270 --> 00:09:26.770
The regular price of the least
expensive item was $15.

00:09:26.770 --> 00:09:27.180
So let's see.

00:09:27.180 --> 00:09:33.570
That statement by itself, we get
I1 is greater than I2 plus

00:09:33.570 --> 00:09:36.840
$15, where this is the
least expensive item.

00:09:36.840 --> 00:09:40.342
So just looking at this, we know
this is going to be $15,

00:09:40.342 --> 00:09:43.630
this is going to be more than
$15-- yeah, this is hard.

00:09:43.630 --> 00:09:51.140
You can't say anything about
this, because maybe I1 is $17.

00:09:51.140 --> 00:09:53.400
Maybe I2 is $16 plus $15.

00:09:53.400 --> 00:09:55.590
And in this case, it would
not be the case.

00:09:55.590 --> 00:10:01.230
Or maybe I1 is $170 and I2 is
$16 plus $15, in which case it

00:10:01.230 --> 00:10:02.170
would be the case.

00:10:02.170 --> 00:10:04.600
So statement two, by itself,
isn't sufficient.

00:10:04.600 --> 00:10:06.010
So the answer to this is a.

00:10:06.010 --> 00:10:08.010
Statement one alone
is sufficient

00:10:08.010 --> 00:10:09.760
to answer this question.

00:10:09.760 --> 00:10:11.720
See you in the next video.


GMAT: Data sufficiency 38

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:07.200
We're on problem 145.

00:00:07.200 --> 00:00:21.130
It says, is 1/p greater than
r/r squared plus 2.

00:00:21.130 --> 00:00:22.580
That's their question.

00:00:22.580 --> 00:00:24.900
Statement number one-- I realize
I go on these tangents

00:00:24.900 --> 00:00:26.410
before even looking at the
statement; I should look at

00:00:26.410 --> 00:00:30.410
the statements first--
p is equal to r.

00:00:30.410 --> 00:00:32.560
So let's see if we can
simplify this.

00:00:32.560 --> 00:00:37.770
So if p is equal to r, then we
get 1/r-- instead of a p,

00:00:37.770 --> 00:00:44.340
right?-- is greater than
r/r squared plus 2.

00:00:44.340 --> 00:00:46.460
Let's multiply both sides
of this equation by r.

00:00:48.980 --> 00:00:54.150
Well, we don't know that r is
necessarily greater than 0.

00:00:54.150 --> 00:00:55.400
That's the problem.

00:00:58.720 --> 00:01:02.890
So if r is greater than 0--
well, it's going to change the

00:01:02.890 --> 00:01:03.990
inequality one way
or the other.

00:01:03.990 --> 00:01:07.620
So let's just assume r
is greater than 0.

00:01:07.620 --> 00:01:10.040
So if we multiply both sides of
the equation by r, then we

00:01:10.040 --> 00:01:11.595
don't have to switch the
inequality, because we're

00:01:11.595 --> 00:01:13.760
assuming r greater than 0.

00:01:13.760 --> 00:01:15.220
That's an assumption.

00:01:15.220 --> 00:01:19.390
Then you get 1 is greater
than r squared over r

00:01:19.390 --> 00:01:21.310
squared plus 2.

00:01:21.310 --> 00:01:23.330
And then if r is greater than
0, then r squared plus 2--

00:01:23.330 --> 00:01:25.080
actually, r squared plus 2 is
always going to be greater

00:01:25.080 --> 00:01:26.480
than 0, because r squared
[INAUDIBLE]

00:01:26.480 --> 00:01:26.920
positive.

00:01:26.920 --> 00:01:28.170
Let's multiply both sides
of this equation by r

00:01:28.170 --> 00:01:29.180
squared plus 2.

00:01:29.180 --> 00:01:32.880
And you get r squared plus 2
is greater than r squared.

00:01:32.880 --> 00:01:35.410
You could subtract r squared
from both sides, and you get 2

00:01:35.410 --> 00:01:37.040
is greater than 0,
which is true.

00:01:37.040 --> 00:01:39.425
But remember, I had to make
this assumption that r is

00:01:39.425 --> 00:01:40.580
greater than 0.

00:01:40.580 --> 00:01:45.360
If you assume that r less than
0, then all of this is going

00:01:45.360 --> 00:01:47.080
to break down.

00:01:47.080 --> 00:01:49.610
Because when you multiply both
sides by r, you have to switch

00:01:49.610 --> 00:01:53.150
the inequality to get 1 is less
than r squared over r

00:01:53.150 --> 00:01:54.620
squared plus 2.

00:01:54.620 --> 00:02:00.600
And then, eventually, you're
going to end up with r squared

00:02:00.600 --> 00:02:03.010
plus 2 is less than r squared.

00:02:03.010 --> 00:02:06.920
Or that 2 his less than
0, which is false.

00:02:06.920 --> 00:02:10.090
So this statement alone
isn't enough.

00:02:10.090 --> 00:02:13.640
We have to know whether
or not r is greater 0.

00:02:13.640 --> 00:02:17.153
Statement two-- well, there you
go; I really should look

00:02:17.153 --> 00:02:20.480
at the statements first--
r is greater 0.

00:02:20.480 --> 00:02:22.320
And just so you know, statement
two by itself is

00:02:22.320 --> 00:02:24.420
insufficient, because if you
know that r is greater than 0,

00:02:24.420 --> 00:02:25.950
you have no idea what p is.

00:02:25.950 --> 00:02:27.420
So you can't answer it yet.

00:02:27.420 --> 00:02:29.710
But both statements combined
are sufficient

00:02:29.710 --> 00:02:30.960
to answer this question.

00:02:33.350 --> 00:02:36.400
Next question.

00:02:36.400 --> 00:02:39.680
146.

00:02:39.680 --> 00:02:41.065
Is n an integer?

00:02:48.430 --> 00:02:51.700
Statement one says n squared
is an integer.

00:02:57.230 --> 00:02:58.290
Well, that doesn't help us.

00:02:58.290 --> 00:03:03.070
I mean, what if n was equal
to the square root of 2?

00:03:03.070 --> 00:03:06.290
And then n squared
would be equal to

00:03:06.290 --> 00:03:08.050
2, which is an integer.

00:03:08.050 --> 00:03:10.260
But this is clearly
not an integer.

00:03:10.260 --> 00:03:13.460
But on the other hand, n could
be equal to 2, in which case n

00:03:13.460 --> 00:03:15.740
squared would be an integer.

00:03:15.740 --> 00:03:18.120
So whether or not n squared is
an integer, both of these are

00:03:18.120 --> 00:03:20.910
cases where n squared are
integers, but one ends up

00:03:20.910 --> 00:03:23.490
where n is an integer, one is
where n isn't an integer.

00:03:23.490 --> 00:03:26.630
So this, by itself, is not
enough to tell me whether n is

00:03:26.630 --> 00:03:28.460
an integer.

00:03:28.460 --> 00:03:34.700
Statement two tells
us, the square

00:03:34.700 --> 00:03:38.450
root of n is an integer.

00:03:41.120 --> 00:03:41.550
OK.

00:03:41.550 --> 00:03:48.360
So that essentially tells us
that n is equal to some

00:03:48.360 --> 00:03:51.640
integer squared, right?

00:03:51.640 --> 00:03:53.110
You could take the square
root of both sides.

00:03:53.110 --> 00:03:55.380
You'd get the square root of
n is equal to some integer.

00:03:55.380 --> 00:03:58.160
So you take any integer, you
square it, you're going to get

00:03:58.160 --> 00:03:59.050
an integer.

00:03:59.050 --> 00:04:01.320
So statement two alone
is sufficient

00:04:01.320 --> 00:04:02.590
to answer this question.

00:04:02.590 --> 00:04:04.350
That was a strangely
easy question.

00:04:04.350 --> 00:04:06.650
They they were getting a
little bit hairy and

00:04:06.650 --> 00:04:13.780
confusing, but that was a nice
little rest question, I think.

00:04:13.780 --> 00:04:17.850
147.

00:04:17.850 --> 00:04:20.930
If n is a positive integer--
so n is a positive integer.

00:04:20.930 --> 00:04:22.300
Well, n is greater than 0.

00:04:22.300 --> 00:04:22.870
It's an integer.

00:04:22.870 --> 00:04:24.130
Well, I didn't write
that down.

00:04:24.130 --> 00:04:32.040
Is n to the third minus
n divisible by 4?

00:04:32.040 --> 00:04:34.080
Fascinating.

00:04:34.080 --> 00:04:35.290
Let's see.

00:04:35.290 --> 00:04:39.110
So my initial reaction is to see
if I can simplify this as

00:04:39.110 --> 00:04:42.150
a product of simpler
expressions.

00:04:42.150 --> 00:04:44.030
I look already at statement
number one.

00:04:44.030 --> 00:04:45.150
Let me look at statement
number one.

00:04:45.150 --> 00:04:50.486
Statement number one says, n is
equal to 2k plus 1, where k

00:04:50.486 --> 00:04:51.330
is an integer.

00:04:51.330 --> 00:04:55.150
I don't feel like taking 2k
plus 1 and cubing it.

00:04:55.150 --> 00:04:56.230
That's not an easy
thing to do.

00:04:56.230 --> 00:04:57.360
It'll take some time.

00:04:57.360 --> 00:05:00.550
So my intuition is that maybe
we should simplify this a

00:05:00.550 --> 00:05:01.410
little bit.

00:05:01.410 --> 00:05:04.020
So if we factor out an n, that's
equal to n times n

00:05:04.020 --> 00:05:06.670
squared, minus 1.

00:05:06.670 --> 00:05:09.560
And that's equal to n times--
what's n squared minus 1?

00:05:09.560 --> 00:05:13.870
That's n plus 1, times
n minus 1.

00:05:13.870 --> 00:05:16.490
And now this is something
that's much easier to

00:05:16.490 --> 00:05:18.006
substitute this into.

00:05:18.006 --> 00:05:19.010
So let's do that.

00:05:19.010 --> 00:05:21.730
Let's substitute n equals
2k plus 1 into this.

00:05:21.730 --> 00:05:28.830
So if n is equal to 2k plus 1,
you get 2k plus 1, times 2k

00:05:28.830 --> 00:05:29.960
plus 1, plus 1.

00:05:29.960 --> 00:05:32.560
So that's 2k plus 2.

00:05:32.560 --> 00:05:33.660
That's this one.

00:05:33.660 --> 00:05:35.990
And then you have 2k
plus 1, minus 1.

00:05:35.990 --> 00:05:38.440
So that's just 2k.

00:05:38.440 --> 00:05:40.070
And then what does
that simplify to?

00:05:40.070 --> 00:05:43.540
You get 2k plus 1.

00:05:43.540 --> 00:05:46.990
Remember, in the back my mind, I
want a 4 to show up, because

00:05:46.990 --> 00:05:49.620
I want this thing to
be divisible by 4.

00:05:49.620 --> 00:05:56.850
2k plus 1, times 4k
squared plus 4k.

00:05:56.850 --> 00:05:57.730
Interesting.

00:05:57.730 --> 00:05:59.860
So now I can factor out a 4.

00:05:59.860 --> 00:06:07.680
So it's 2k plus 1, times k
squared plus k, times 4.

00:06:07.680 --> 00:06:11.530
And we know that this is an
integer, because they told us

00:06:11.530 --> 00:06:13.600
that k is an integer.

00:06:13.600 --> 00:06:14.600
We know that this
is an integer,

00:06:14.600 --> 00:06:15.850
because k is an integer.

00:06:15.850 --> 00:06:17.010
And we know 4 is an integer.

00:06:17.010 --> 00:06:21.420
So we've essentially kind of
factored n cubed minus n.

00:06:21.420 --> 00:06:23.950
And 4 is one of it's factors.

00:06:23.950 --> 00:06:25.360
So it's definitely
divisible by 4.

00:06:25.360 --> 00:06:27.320
I mean, we can divide
it by 4 right now.

00:06:27.320 --> 00:06:30.890
If we divide it by 4, we'd be
left with this, which is

00:06:30.890 --> 00:06:33.890
clearly an integer, because
k is an integer.

00:06:33.890 --> 00:06:37.020
So n to the third minus n is
divisible by 4 if we can

00:06:37.020 --> 00:06:39.910
assume that n is equal
to 2k plus 1,

00:06:39.910 --> 00:06:42.170
where k is some integer.

00:06:42.170 --> 00:06:44.240
Now, what do they tell
us in statement two?

00:06:47.430 --> 00:06:53.420
Statement two tells us,
n squared plus n is

00:06:53.420 --> 00:06:59.010
divisible by 6.

00:06:59.010 --> 00:07:03.400
Well, I don't see how that's
helpful at all.

00:07:03.400 --> 00:07:06.180
n squared plus n is
divisible by 6.

00:07:09.110 --> 00:07:10.980
You can't even relate
that to that.

00:07:10.980 --> 00:07:14.270
I mean, n squared--
I could try to do

00:07:14.270 --> 00:07:16.070
something fancy here.

00:07:16.070 --> 00:07:18.420
No.

00:07:18.420 --> 00:07:21.286
This is so different than this.
n squared plus n is so

00:07:21.286 --> 00:07:23.660
different from n to the third.

00:07:23.660 --> 00:07:24.750
I could factor out an n.

00:07:24.750 --> 00:07:29.650
I could say n times n plus
1 is divisible by 6.

00:07:29.650 --> 00:07:31.180
Well, I guess that's
interesting.

00:07:31.180 --> 00:07:37.450
That tells us that this part
is divisible by 6.

00:07:37.450 --> 00:07:41.550
n times n plus 1 is
divisible by 6.

00:07:48.150 --> 00:07:50.480
Do they tell us that n is
a positive integer?

00:07:50.480 --> 00:07:53.420
Yeah, they do tell us that
n is a positive integer.

00:07:53.420 --> 00:07:56.440
So this could be 2 and 3.

00:07:56.440 --> 00:08:01.550
If this is 2 and 3, then
this would be 1.

00:08:01.550 --> 00:08:04.180
This could be 2 and 3.

00:08:04.180 --> 00:08:06.790
In fact-- well, this could be
2 and 3, but it could also

00:08:06.790 --> 00:08:09.580
be-- well, actually, that's the
only numbers it could be.

00:08:09.580 --> 00:08:13.890
It could be 2 and 3, in which
case this is 1, in which case

00:08:13.890 --> 00:08:19.620
this whole thing-- if you say
this, then n could be 2.

00:08:19.620 --> 00:08:20.870
n is equal to 2.

00:08:23.400 --> 00:08:25.355
n plus 1 is equal to 3.

00:08:28.040 --> 00:08:31.722
And then n minus 1 would
be equal to 1.

00:08:31.722 --> 00:08:32.400
And let me see.

00:08:32.400 --> 00:08:34.386
Is that the only case?

00:08:34.386 --> 00:08:36.167
I'm looking at the answer right
now, just because I want

00:08:36.167 --> 00:08:37.049
to make sure.

00:08:37.049 --> 00:08:38.419
They say that this
isn't sufficient.

00:08:38.419 --> 00:08:42.210
But, once again, kind of like
problem 142, actually--

00:08:42.210 --> 00:08:48.210
because if you say that n is a
positive integer and that n

00:08:48.210 --> 00:08:55.990
times n plus 1 is divisible by
6, I can't think of any other

00:08:55.990 --> 00:08:59.000
numbers where you multiply one
number times the next number,

00:08:59.000 --> 00:09:04.030
and their integers, where
you get a multiple of 6.

00:09:04.030 --> 00:09:05.190
No, I take that back.

00:09:05.190 --> 00:09:07.500
It could be 3 and 4.

00:09:07.500 --> 00:09:08.920
There could be 3 and 4.

00:09:08.920 --> 00:09:12.590
It could be-- yeah, there's
actually a bunch of them.

00:09:12.590 --> 00:09:17.310
This could be 3, this could be
4, because it's a multiple 6,

00:09:17.310 --> 00:09:18.470
and then this would be 2.

00:09:18.470 --> 00:09:19.020
So that's right.

00:09:19.020 --> 00:09:20.420
This is not enough
information.

00:09:20.420 --> 00:09:21.030
So that's good.

00:09:21.030 --> 00:09:25.590
Because by this information, n
could be 2, n plus 1 could be

00:09:25.590 --> 00:09:29.370
3, in which case, this number
would be actually 6.

00:09:29.370 --> 00:09:30.930
But that still doesn't help
us, because 6 isn't

00:09:30.930 --> 00:09:31.860
divisible by 4.

00:09:31.860 --> 00:09:34.280
But then I could come up with
the situation where this

00:09:34.280 --> 00:09:39.800
number right here, where n is 3,
n plus 1 is 4, and n minus

00:09:39.800 --> 00:09:43.270
1 is 2, where this does
become divisible by 4.

00:09:43.270 --> 00:09:46.260
So this doesn't give us
enough information.

00:09:46.260 --> 00:09:47.680
Not useful.

00:09:47.680 --> 00:09:48.770
And I'm out of time.

00:09:48.770 --> 00:09:50.700
I'll see you in the
next video.


GMAT: Data sufficiency 35

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.760
We're on problem 141.

00:00:03.760 --> 00:00:07.665
On Jane's credit card account,
the average daily balance for

00:00:07.665 --> 00:00:12.292
a 30 day billing cycle-- when I
see these long paragraphs it

00:00:12.292 --> 00:00:14.430
starts to give you a headache--
the average daily

00:00:14.430 --> 00:00:18.560
balance for a 30 day billing
cycle is the average of the

00:00:18.560 --> 00:00:22.660
daily balances at the end
of each of the 30 days.

00:00:22.660 --> 00:00:25.330
OK, so they take the average
balance of each of the days

00:00:25.330 --> 00:00:26.730
and then they average it.

00:00:26.730 --> 00:00:27.400
OK.

00:00:27.400 --> 00:00:30.940
At the beginning of a certain
30 day billing cycle, Jane's

00:00:30.940 --> 00:00:33.850
credit card account had
a balance of $600.

00:00:33.850 --> 00:00:35.780
At the beginning.

00:00:35.780 --> 00:00:36.630
What problem is this?

00:00:36.630 --> 00:00:38.320
141.

00:00:38.320 --> 00:00:41.230
So at the beginning, she
had a balance of $600.

00:00:41.230 --> 00:00:42.530
Fair enough.

00:00:42.530 --> 00:00:46.040
Jane made a payment of
$300 on the account

00:00:46.040 --> 00:00:47.200
during the billing cycle.

00:00:47.200 --> 00:00:49.900
So this is the beginning of the
month, and then at some

00:00:49.900 --> 00:00:51.960
point-- this is 30 days later--
at some point she made

00:00:51.960 --> 00:00:54.230
a $300 payment.

00:00:54.230 --> 00:00:56.040
Fair enough.

00:00:56.040 --> 00:00:56.520
Continuing.

00:00:56.520 --> 00:00:59.370
If no other amounts were added
or subtracted from the account

00:00:59.370 --> 00:01:02.590
during the billing cycle, what
was the average daily balance

00:01:02.590 --> 00:01:04.930
on Jane's account for
the billing cycle?

00:01:04.930 --> 00:01:07.590
OK, so I think I'm visualizing
this right.

00:01:07.590 --> 00:01:13.670
If we say that this is days,
where this is day 0

00:01:13.670 --> 00:01:16.020
and this is day 30.

00:01:16.020 --> 00:01:20.300
So essentially, at the end of
every day, so at the end of

00:01:20.300 --> 00:01:22.960
day 1-- so at the beginning of a
certain billing cycle Jane's

00:01:22.960 --> 00:01:25.050
credit card account had
a balance of $600.

00:01:25.050 --> 00:01:25.480
So we don't know.

00:01:25.480 --> 00:01:26.910
She might have paid
it on day 1.

00:01:26.910 --> 00:01:28.170
So I'm not going to put
any labels here.

00:01:28.170 --> 00:01:30.350
But she starts at 600.

00:01:30.350 --> 00:01:36.580
And that's her balance until on
some day she pays off half

00:01:36.580 --> 00:01:40.190
of it, and her balance
goes down like that.

00:01:40.190 --> 00:01:41.950
So the question is, what
is the average?

00:01:41.950 --> 00:01:46.680
So it's going to be $600 times a
certain number of days, plus

00:01:46.680 --> 00:01:50.270
$300 times a certain number
of days, divided by 30.

00:01:50.270 --> 00:01:51.190
So let me write that down.

00:01:51.190 --> 00:01:54.940
The average is going to be equal
to $600 times however

00:01:54.940 --> 00:01:59.610
many days she carried the $600
balance plus $300 times

00:01:59.610 --> 00:02:02.530
however many days she kept
the $300 balance.

00:02:02.530 --> 00:02:07.580
And that's going to
be 30 minus x.

00:02:07.580 --> 00:02:10.064
If she paid after 15 days, she
would have had a $600 balance

00:02:10.064 --> 00:02:13.850
for 15 days and then she would
have had the $300 for the

00:02:13.850 --> 00:02:14.730
remainder of the days.

00:02:14.730 --> 00:02:17.180
If she paid after one day, then
you would have the $300

00:02:17.180 --> 00:02:18.850
balance for 29 days.

00:02:18.850 --> 00:02:20.860
And all of that divided by 30.

00:02:20.860 --> 00:02:22.170
That's the average.

00:02:22.170 --> 00:02:23.780
And I try to do that from the
get-go, because I just want to

00:02:23.780 --> 00:02:26.690
get my algebraic brain around
the problem so it

00:02:26.690 --> 00:02:28.540
becomes less abstract.

00:02:28.540 --> 00:02:31.705
So statement number one.

00:02:31.705 --> 00:02:33.610
Let me keep that
in the screen.

00:02:33.610 --> 00:02:37.000
Statement number one tells us
Jane's payment cycle was

00:02:37.000 --> 00:02:39.960
credited on the 21st day
of the billing cycle.

00:02:46.180 --> 00:02:49.960
So that means she had a $600--
that x in this example-- that

00:02:49.960 --> 00:02:52.880
she had a $600 balance
for 20 days.

00:02:52.880 --> 00:02:54.735
And then on the 21st day
her balance would

00:02:54.735 --> 00:02:55.790
have gone to $300.

00:02:55.790 --> 00:02:59.500
So the average is going to be
equal to $600-- she had a $600

00:02:59.500 --> 00:03:02.200
balance for the first 20 days.

00:03:02.200 --> 00:03:07.610
Times 20 plus 300 times
the remainder days.

00:03:07.610 --> 00:03:09.315
So 300 for 10 days.

00:03:09.315 --> 00:03:12.310
And that's all divided by
the number of days.

00:03:12.310 --> 00:03:14.750
So statement one, alone, is
enough to figure out her

00:03:14.750 --> 00:03:17.130
average balance for the month.

00:03:17.130 --> 00:03:19.060
Statement two.

00:03:19.060 --> 00:03:22.740
The average daily balance
through the 25th day of the

00:03:22.740 --> 00:03:27.010
billing cycle was $540.

00:03:27.010 --> 00:03:28.550
This is interesting.

00:03:28.550 --> 00:03:34.440
So $540 is the average
through the 25th day.

00:03:34.440 --> 00:03:42.300
So if we average the first 25
days-- so her balance was $600

00:03:42.300 --> 00:03:47.390
for x days, plus $300
for the remainder.

00:03:47.390 --> 00:03:48.910
Only the first 25 days.

00:03:48.910 --> 00:03:53.040
So it's x minus 25.

00:03:53.040 --> 00:03:55.980
So you can actually take this
equation and solve for x.

00:03:55.980 --> 00:03:57.680
It's actually a linear
equation.

00:03:57.680 --> 00:03:58.460
And I'm sure this time.

00:03:58.460 --> 00:04:00.750
Multiply both sides by
25, you get a number.

00:04:00.750 --> 00:04:02.450
And then you can distribute
this out.

00:04:02.450 --> 00:04:04.180
Add all the x terms.
Solve for x.

00:04:04.180 --> 00:04:07.190
And then once you solve for x,
you can just use this equation

00:04:07.190 --> 00:04:08.860
up here to figure out
the average daily

00:04:08.860 --> 00:04:11.460
balance for 30 days.

00:04:11.460 --> 00:04:14.680
So actually, each of these
statements independently are

00:04:14.680 --> 00:04:16.630
sufficient to figure
out this problem.

00:04:16.630 --> 00:04:19.140
Actually, it was very critical
that we kind of thought about

00:04:19.140 --> 00:04:20.350
in this term from the get-go.

00:04:20.350 --> 00:04:22.880
Otherwise this would have been
a very hard problem to get

00:04:22.880 --> 00:04:23.900
your hand around.

00:04:23.900 --> 00:04:25.180
But it's interesting.

00:04:25.180 --> 00:04:27.900
I, strangely, really
like that problem.

00:04:27.900 --> 00:04:28.900
Anyway, next problem.

00:04:28.900 --> 00:04:35.730
142.

00:04:35.730 --> 00:04:43.070
If x is an integer, they're
asking is 9 to the x plus 9 to

00:04:43.070 --> 00:04:48.150
the minus x equal to b?

00:04:48.150 --> 00:04:49.760
Who knows?

00:04:49.760 --> 00:04:58.800
Problem one says 3 to the x plus
3 to the minus x is equal

00:04:58.800 --> 00:05:02.450
to the square root
of b plus 2.

00:05:02.450 --> 00:05:04.560
So right from the get-go I don't
see-- I mean there's 3's

00:05:04.560 --> 00:05:07.170
there's 9's, there seems to
be some relationship.

00:05:07.170 --> 00:05:09.890
Let's square both sides of this
equation, see if it can

00:05:09.890 --> 00:05:12.180
reduce to something that's
useful here.

00:05:12.180 --> 00:05:13.510
So the right hand
side is easy.

00:05:13.510 --> 00:05:19.020
The left hand side, if you
square it you get 3 to the 2x.

00:05:19.020 --> 00:05:22.910
And then you get plus
2 times these

00:05:22.910 --> 00:05:24.210
multiplied by each other.

00:05:24.210 --> 00:05:32.650
3 to the x, 3 to the minus x
plus 3 to the minus 2x is

00:05:32.650 --> 00:05:34.940
equal to b plus 2.

00:05:34.940 --> 00:05:36.730
I just squared both sides.

00:05:36.730 --> 00:05:39.190
3 to the x times 3 to
the minus x, that

00:05:39.190 --> 00:05:41.890
just equals 1, right?

00:05:41.890 --> 00:05:43.970
You could add the exponents,
that's equal to 3 to the 0.

00:05:43.970 --> 00:05:44.630
That equals 1.

00:05:44.630 --> 00:05:46.750
Or you could view that as 3 to
the x divided by 3 to the x.

00:05:46.750 --> 00:05:49.060
Either way, that equals 1.

00:05:49.060 --> 00:05:51.335
Then you're left with 2 on both
sides of the equation.

00:05:51.335 --> 00:05:53.970
So you can get rid of that.

00:05:53.970 --> 00:06:00.870
So then we're left with 3 to the
2x plus 3 to the minus 2x

00:06:00.870 --> 00:06:02.340
is equal to b.

00:06:02.340 --> 00:06:03.390
But then we could
rewrite this.

00:06:03.390 --> 00:06:03.950
Think about this.

00:06:03.950 --> 00:06:08.060
3 to the 2x, that's the same
thing as 3 squared to the x.

00:06:08.060 --> 00:06:12.105
And this is the same thing as
3 to the minus 2 to the x is

00:06:12.105 --> 00:06:13.250
equal to b.

00:06:13.250 --> 00:06:16.960
And I think now bells are
ringing in your head.

00:06:16.960 --> 00:06:20.720
3 squared, that's the same thing
as 9 to the x plus-- I

00:06:20.720 --> 00:06:22.570
should do 3 squared
to the minus x.

00:06:22.570 --> 00:06:23.290
That's easier.

00:06:23.290 --> 00:06:26.450
Plus 9 to the minus
x is equal to b.

00:06:26.450 --> 00:06:28.870
So statement number one actually
reduced to what we

00:06:28.870 --> 00:06:30.170
were trying to prove.

00:06:30.170 --> 00:06:33.060
So statement number one, at
least alone, is sufficient.

00:06:33.060 --> 00:06:35.500
Let's see what statement
number two gets us.

00:06:35.500 --> 00:06:43.970
Statement number two says x is
greater than 0 is equal to b.

00:06:49.860 --> 00:06:50.560
Let's see.

00:06:50.560 --> 00:06:53.890
The original was 9 to the
x plus 9 to the minus

00:06:53.890 --> 00:06:57.590
x is equal to b.

00:06:57.590 --> 00:07:00.200
So they're now saying that
b is equal to 0.

00:07:00.200 --> 00:07:01.440
So how does that help us?

00:07:01.440 --> 00:07:07.130
If b is equal to 0-- let's
subtract 9 to the minus x from

00:07:07.130 --> 00:07:07.550
both sides.

00:07:07.550 --> 00:07:16.740
You get 9 to the x is equal
to minus 9 to the minus x.

00:07:16.740 --> 00:07:23.290
Now let me multiply both sides
by negative 9 to the x.

00:07:26.080 --> 00:07:30.810
I just want to make this--
minus 9 to the x times

00:07:30.810 --> 00:07:32.130
minus 9 to the x.

00:07:32.130 --> 00:07:34.140
I'm just trying to
simplify it.

00:07:34.140 --> 00:07:35.380
So what do you have here?

00:07:35.380 --> 00:07:40.070
The minus sign, minus
9 to the 2x, right?

00:07:40.070 --> 00:07:42.350
Just add the exponents.

00:07:42.350 --> 00:07:46.750
Is equal to-- a minus times a
minus is a positive-- and then

00:07:46.750 --> 00:07:49.110
a minus x and a positive x.

00:07:49.110 --> 00:07:50.350
Add them together, you get 0.

00:07:50.350 --> 00:07:52.370
So 9 and 0 is 1.

00:07:52.370 --> 00:07:58.670
That simplifies to minus 9
to the 2x is equal to 1.

00:08:05.790 --> 00:08:09.013
So can this ever be true?

00:08:12.040 --> 00:08:13.670
Let me think about that.

00:08:13.670 --> 00:08:18.160
The only way something that's
non 1 or non negative 1, the

00:08:18.160 --> 00:08:21.960
only way that when you raise it
to a power you can get a 1

00:08:21.960 --> 00:08:24.280
is if you raise it
to a 0th power.

00:08:24.280 --> 00:08:30.900
So this is only true
if x is equal to 0.

00:08:30.900 --> 00:08:34.960
But they tell us that
x is greater than 0.

00:08:34.960 --> 00:08:36.780
So, actually, this is
an interesting case.

00:08:36.780 --> 00:08:39.570
Using the information they
gave us in problem number

00:08:39.570 --> 00:08:47.940
two-- I just want to make sure
I'm not missing something.

00:08:47.940 --> 00:08:50.570
Statement one was sufficient.

00:08:50.570 --> 00:08:52.950
Can we prove--

00:08:52.950 --> 00:08:54.900
well, we don't have
enough information

00:08:54.900 --> 00:08:56.090
to prove the statement.

00:08:56.090 --> 00:08:59.120
We have enough information to
disprove the statement but not

00:08:59.120 --> 00:09:00.060
to prove it.

00:09:00.060 --> 00:09:01.630
So I'm going to stick with a.

00:09:01.630 --> 00:09:04.940
I was going to say that we can
say whether the statement is

00:09:04.940 --> 00:09:06.840
true or false, but we definitely
don't have enough

00:09:06.840 --> 00:09:11.300
information with just statement
b to prove-- so they

00:09:11.300 --> 00:09:12.800
want to know, right.

00:09:12.800 --> 00:09:16.190
You have to be able to prove
the statement is true.

00:09:16.190 --> 00:09:18.020
Statement two actually proves
that it is false.

00:09:18.020 --> 00:09:18.970
It doesn't prove
that it's true.

00:09:18.970 --> 00:09:20.840
It answers the question
but says no.

00:09:20.840 --> 00:09:25.520
So statement one, alone, is
sufficient to say that that

00:09:25.520 --> 00:09:27.360
statement is true.

00:09:27.360 --> 00:09:28.660
And I'll continue in
the next video.