GMAT: Data sufficiency 34

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:04.755
We're on problem 135.

00:00:04.755 --> 00:00:08.640
I just did some pilates, so I'm
slightly out of breath.

00:00:08.640 --> 00:00:12.190
If x is less than 0, is
y greater than 0?

00:00:22.110 --> 00:00:29.460
All right, statement number
1, x/y is less than 0.

00:00:34.690 --> 00:00:36.620
So this is actually
pretty useful.

00:00:36.620 --> 00:00:40.050
They're telling us that x
is a negative number.

00:00:40.050 --> 00:00:43.960
So we have a negative here, and
we're dividing it by some

00:00:43.960 --> 00:00:45.220
other number.

00:00:45.220 --> 00:00:47.990
They're telling us that
the resulting

00:00:47.990 --> 00:00:49.630
number is still negative.

00:00:49.630 --> 00:00:54.340
So we have a negative
divided by something

00:00:54.340 --> 00:00:56.770
is equal to a negative.

00:00:56.770 --> 00:00:58.660
So I think you learned when you
learned negative numbers,

00:00:58.660 --> 00:01:01.740
if y was negative-- well,
if y was 0 this would be

00:01:01.740 --> 00:01:04.800
undefined-- and if y is
negative, if this was a

00:01:04.800 --> 00:01:09.400
negative, then when you divide
a negative by a negative, you

00:01:09.400 --> 00:01:10.440
would have a positive.

00:01:10.440 --> 00:01:14.530
So y has to be a positive
number, and it can't be 0.

00:01:14.530 --> 00:01:15.660
So this actually tells us.

00:01:15.660 --> 00:01:20.550
Statement 1 alone tells us that
y is greater than 0 just

00:01:20.550 --> 00:01:23.780
based on the fact that
you have a negative.

00:01:23.780 --> 00:01:25.620
It's just telling you that
a negative divided

00:01:25.620 --> 00:01:27.780
by y is still negative.

00:01:27.780 --> 00:01:29.020
This whole thing is
still negative.

00:01:29.020 --> 00:01:31.470
So y has to be a positive.

00:01:31.470 --> 00:01:39.750
Statement number 2 tells us y
minus x is greater than 0.

00:01:39.750 --> 00:01:42.030
Let's see, add x
to both sides.

00:01:42.030 --> 00:01:44.880
That tells us that y
is greater than x.

00:01:44.880 --> 00:01:45.910
Well that's not enough.

00:01:45.910 --> 00:01:49.460
It says if x is 0, is
y greater than 0?

00:01:49.460 --> 00:01:55.610
Maybe x is equal to minus 10
and y is equal to minus 9.

00:01:55.610 --> 00:01:57.440
So in this case, y wouldn't
be greater than 0.

00:01:57.440 --> 00:02:01.400
Or maybe x is equal to minus 10
and y is equal to plus 9.

00:02:01.400 --> 00:02:03.460
Both of these would satisfy
this condition.

00:02:03.460 --> 00:02:04.570
So we don't know.

00:02:04.570 --> 00:02:07.280
This doesn't tell us any
information on whether y is

00:02:07.280 --> 00:02:09.630
greater than 0.

00:02:09.630 --> 00:02:12.060
So statement 1 alone us
sufficient, and statement 2

00:02:12.060 --> 00:02:15.280
doesn't do much for us.

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Problem 135, no, I already
did 135, 136.

00:02:23.732 --> 00:02:26.190
They've drawn a circle there,
so I guess I'll

00:02:26.190 --> 00:02:29.570
draw a circle as well.

00:02:29.570 --> 00:02:30.820
So that's the circle.

00:02:33.170 --> 00:02:45.290
They have a center like that
and then it's like that.

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There's a right angle
right here.

00:02:48.630 --> 00:02:50.240
They call this O.

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They call this X.

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They call this Z.

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Then they actually connect
those two points.

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Then they call this, this point
right here, that's Y.

00:03:05.290 --> 00:03:06.130
Let's see what they're asking.

00:03:06.130 --> 00:03:07.670
What is the circumference
of the circle

00:03:07.670 --> 00:03:08.900
above with center O?

00:03:08.900 --> 00:03:10.530
So they just want to know
the circumference.

00:03:10.530 --> 00:03:12.630
So if we could figure out
a radius, then we know

00:03:12.630 --> 00:03:14.750
circumference is 2 pi
times the radius,

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and we'd be all set.

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So let's see what information
they give us.

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Statement number 1, the
perimeter of triangle OXZ is

00:03:25.250 --> 00:03:35.290
20 plus 10 square roots of 2.

00:03:35.290 --> 00:03:35.960
Now this is interesting.

00:03:35.960 --> 00:03:37.410
What do we know about this?

00:03:37.410 --> 00:03:39.970
Well we know that it's going to
be an isosceles triangle.

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We know that if this length
right here is X, that this

00:03:43.480 --> 00:03:47.910
length is also going to be X,
because this is a circle.

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The radius is constant, and each
of these sides are the

00:03:51.570 --> 00:03:52.930
radius of the circle.

00:03:52.930 --> 00:03:56.140
So those are both
going to be x.

00:03:56.140 --> 00:03:58.910
What is this going to be?

00:03:58.910 --> 00:04:02.350
We just know from Pythagorean
theorem, x squared plus x

00:04:02.350 --> 00:04:05.980
squared is going to be equal
to that side squared.

00:04:05.980 --> 00:04:07.740
Let's call that c squared.

00:04:07.740 --> 00:04:12.500
So you get 2x squared is equal
to c squared or c-- I'm

00:04:12.500 --> 00:04:15.240
calling this side right here
c-- is equal to the square

00:04:15.240 --> 00:04:18.279
root of 2, x times the
square root of 2.

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Whatever x is.

00:04:20.709 --> 00:04:25.450
So the perimeter of this entire
triangle is going to be

00:04:25.450 --> 00:04:32.620
two x's, x plus x, so it's going
to be 2x plus x square

00:04:32.620 --> 00:04:33.870
roots of 2.

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Statement 1 told us that this
perimeter is equal to this

00:04:39.280 --> 00:04:39.860
right here.

00:04:39.860 --> 00:04:45.390
It's going to be equal to 20
plus 10 square roots of 2.

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Well I think you could do a
little pattern matching here

00:04:47.300 --> 00:04:49.200
and solve for x.

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This is actually, even though
it might not look completely

00:04:51.830 --> 00:04:53.740
like it, this is a
linear equation.

00:04:53.740 --> 00:04:55.790
You can solve for it.

00:04:55.790 --> 00:04:58.340
Immediately that should be, oh,
statement 1 is sufficient.

00:04:58.340 --> 00:04:59.780
But you can just do a little
pattern matching and say,

00:04:59.780 --> 00:05:02.910
well, this works when
x is equal to 10.

00:05:02.910 --> 00:05:06.840
So actually, we are able to
figure out what x is equal to.

00:05:06.840 --> 00:05:09.960
Then as I said before, x is
the radius of this circle.

00:05:09.960 --> 00:05:11.500
Then the circumference of the
circle, which is what the

00:05:11.500 --> 00:05:14.720
whole question is about, that's
2 pi times the radius.

00:05:14.720 --> 00:05:17.200
Circumference is 2 pi
times the radius.

00:05:17.200 --> 00:05:19.360
We just figured out
the radius is 10.

00:05:19.360 --> 00:05:20.980
So it equals 20 pi.

00:05:20.980 --> 00:05:23.310
So statement 1 is sufficient.

00:05:23.310 --> 00:05:25.650
What does statement 2 do?

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Statement 2, the length
of arc XYZ is 5 pi.

00:05:31.160 --> 00:05:32.910
XYZ is 5 pi.

00:05:32.910 --> 00:05:35.860
So they're telling us that
this length is 5 pi.

00:05:40.610 --> 00:05:43.030
Well do we know what proportion
that is of the

00:05:43.030 --> 00:05:44.790
entire circumference?

00:05:44.790 --> 00:05:48.140
Well sure, because they tell
us that this is 90 degrees

00:05:48.140 --> 00:05:49.440
right here.

00:05:49.440 --> 00:05:53.020
This is 90 degrees of 360
degrees of a circle.

00:05:53.020 --> 00:05:57.090
So this, the arc length right
here, is exactly going to be

00:05:57.090 --> 00:05:58.165
1/4 of the circle.

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We know that because 90
degrees is 1/4 of 360.

00:06:02.010 --> 00:06:07.250
So 5 pi is going to be equal
to 1/4 times the

00:06:07.250 --> 00:06:08.300
circumference.

00:06:08.300 --> 00:06:11.990
Or that the circumference is
going to be equal to 20 pi.

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So each statement independently,
statement 1 or

00:06:14.890 --> 00:06:17.920
statement 2, independently, is
sufficient to answer this

00:06:17.920 --> 00:06:21.950
question, the circumference
of the circle.

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137, what is the value
of a to the fourth

00:06:27.950 --> 00:06:29.200
minus b to the fourth?

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Immediately just even looking,
one thing I'm tempted to do is

00:06:38.350 --> 00:06:43.050
rewrite this as a
squared squared

00:06:43.050 --> 00:06:45.420
minus b squared squared.

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Because I just glanced at
statement 1 and they had

00:06:47.070 --> 00:06:47.810
squares there.

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So I'm going to do that.

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Let me do different letters.

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You know that x squared minus y
squared is equal to x plus y

00:06:58.290 --> 00:07:01.540
times x minus y.

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So here, x is a squared
and y is b squared.

00:07:05.160 --> 00:07:07.300
So this is going to be equal
to-- this is just the

00:07:07.300 --> 00:07:10.600
original, this is going to be
equal to a to the squared-- a

00:07:10.600 --> 00:07:17.620
squared plus b squared times
a squared minus b squared.

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Then we could simplified
this again.

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So this is going to be equal to
a squared plus b squared.

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We could simplify this as a
plus b times a minus b.

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I think that's going to be
helpful just glancing at the

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two statements that
they gave us.

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So statement 1 tells us that a
squared minus b squared is

00:07:46.270 --> 00:07:51.280
equal to 16.

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So that tells us that just
this is equal to 16.

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That is equal to 16.

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That doesn't help me much
because I don't know what a

00:08:00.920 --> 00:08:02.930
squared plus b squared
is going to be.

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It just tells me the difference

00:08:03.860 --> 00:08:06.440
between these two squares.

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I don't know whether they're
integers or anything.

00:08:08.620 --> 00:08:16.590
Statement 2 tells me that
a plus b is equal to 8.

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Once again, that doesn't help
me tremendously by itself.

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If I just know that a plus b
is equal to 8, and I don't

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know what a minus b
is, I don't know

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what all of this is.

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But when I use them together,
this is interesting.

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Because we know that a plus b
times a minus b is equal to a

00:08:39.539 --> 00:08:42.070
squared minus b squared.

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Well they just told us that
a plus b is equal to 8.

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So a plus b is equal to 8.

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So 8 times a minus b is going to
be equal to a squared minus

00:08:52.890 --> 00:08:55.640
b squared, which statement
one told us was 16.

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So that actually tells us that
a minus b is equal to 2.

00:09:00.180 --> 00:09:04.410
So now we have two equations
with two unknowns, and we

00:09:04.410 --> 00:09:05.680
could solve for a and b.

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If we can solve for a and b,
then we can definitely tell

00:09:07.530 --> 00:09:10.360
you what a to the fourth minus
b to the fourth is.

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Just to show you that,
let me do it.

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So let's see, a minus
b is equal to 2.

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Add these equations.

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2a is equal to 10.

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a is equal to 5.

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Then if a is equal to
5, b is equal to 3.

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So our original statement,
a to the

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fourth is 5 to the fourth.

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5 to the fourth minus
3 to the fourth.

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So whatever-- that's what?

00:09:33.690 --> 00:09:35.610
625 minus 81, I think.

00:09:35.610 --> 00:09:36.120
It doesn't matter.

00:09:36.120 --> 00:09:37.170
You can figure it out.

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So both statements combined
are sufficient

00:09:40.150 --> 00:09:42.050
to answer this question.

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See you in the next video.


GMAT: Data sufficiency 33

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:05.660
We're on problem 132.

00:00:05.660 --> 00:00:09.650
If the integer n is greater
than 1, is n equal to 2?

00:00:09.650 --> 00:00:13.490
So they tell us that the integer
n is greater than 1,

00:00:13.490 --> 00:00:18.350
and they ask us, is
n equal to 2?

00:00:18.350 --> 00:00:23.925
Statement 1, n has exactly
two positive factors.

00:00:31.400 --> 00:00:34.620
Well that's certainly true
of the number 2.

00:00:34.620 --> 00:00:36.220
But it's also true of
any prime number.

00:00:36.220 --> 00:00:38.770
I mean n could be 7.

00:00:38.770 --> 00:00:41.210
7 only has two positive
factors, 1 and 7.

00:00:41.210 --> 00:00:42.970
So it's true of any
prime number.

00:00:42.970 --> 00:00:45.885
So this, by itself, isn't enough
information to say that

00:00:45.885 --> 00:00:47.550
it's definitely 2.

00:00:47.550 --> 00:00:51.300
Statement 2 tells us the
difference of any two distinct

00:00:51.300 --> 00:00:56.180
positive factors of n is odd.

00:00:56.180 --> 00:00:57.430
This is interesting.

00:01:05.700 --> 00:01:07.355
We're just dealing in kind
of the positive world.

00:01:16.370 --> 00:01:18.030
So let's think about it.

00:01:18.030 --> 00:01:20.360
Just thinking about statement
1, we said, OK, what numbers

00:01:20.360 --> 00:01:22.120
have only two factors?

00:01:22.120 --> 00:01:24.860
Well prime numbers only
have two factors.

00:01:24.860 --> 00:01:28.890
This only tells us
that n is prime.

00:01:28.890 --> 00:01:32.670
2 is a prime number, but it's
not the only prime number.

00:01:32.670 --> 00:01:34.310
But for most other prime
numbers, what do

00:01:34.310 --> 00:01:36.360
we know about them?

00:01:36.360 --> 00:01:38.950
Well they're odd, right?

00:01:38.950 --> 00:01:40.670
Most prime numbers are odd.

00:01:40.670 --> 00:01:42.240
Other than 2, what are the
other prime numbers?

00:01:42.240 --> 00:01:47.480
You go to 3, you go to
7, you go to 11, 5.

00:01:47.480 --> 00:01:49.000
Let me ask you, why
are they odd?

00:01:49.000 --> 00:01:51.200
Because if they were even, they
would be divisible by 2

00:01:51.200 --> 00:01:52.580
and they wouldn't be prime.

00:01:52.580 --> 00:01:58.080
So by definition really, every
prime number that's not 2 is

00:01:58.080 --> 00:02:01.480
not divisible by 2 because
it's prime.

00:02:01.480 --> 00:02:03.220
So they have to be
odd numbers.

00:02:03.220 --> 00:02:08.020
So for any other prime number,
the number is going to be 1

00:02:08.020 --> 00:02:08.880
and itself.

00:02:08.880 --> 00:02:10.940
So let's call that p.

00:02:10.940 --> 00:02:13.060
So the difference between the
two numbers, and p is going to

00:02:13.060 --> 00:02:14.160
be an odd number.

00:02:14.160 --> 00:02:17.280
Every other prime number
other than 2 is odd.

00:02:17.280 --> 00:02:19.840
So the difference between
the two, p minus 1.

00:02:19.840 --> 00:02:22.910
If I take an odd number and I
subtract one from it, I get an

00:02:22.910 --> 00:02:25.750
even number.

00:02:25.750 --> 00:02:27.960
That's true for any
odd number.

00:02:27.960 --> 00:02:30.650
So every other prime
number is odd.

00:02:30.650 --> 00:02:31.830
And this happens.

00:02:31.830 --> 00:02:33.555
But they're saying that when I
subtract the difference I get

00:02:33.555 --> 00:02:34.960
an odd, I get an odd number.

00:02:34.960 --> 00:02:37.720
Well the only number that's
true for is 2.

00:02:37.720 --> 00:02:40.170
Because factors of
2 are 1 and 2.

00:02:40.170 --> 00:02:43.600
If I subtract 1 from 2, 2
minus 1 is equal to 1.

00:02:43.600 --> 00:02:48.070
That's because 2 is the only
even prime number.

00:02:48.070 --> 00:02:51.810
So this first statement says we
have to be dealing with a

00:02:51.810 --> 00:02:52.840
prime number.

00:02:52.840 --> 00:02:55.940
The second statement says, well,
the number really has to

00:02:55.940 --> 00:02:59.780
be an even number if it's
going to be prime.

00:02:59.780 --> 00:03:03.430
Let me think about the
other two-- if

00:03:03.430 --> 00:03:05.200
statement 2 alone is enough.

00:03:05.200 --> 00:03:11.790
The difference of any two
positive factors of n is odd,

00:03:11.790 --> 00:03:13.860
any two positive factors of n.

00:03:13.860 --> 00:03:17.690
Now this alone doesn't help us
because n could be-- I don't

00:03:17.690 --> 00:03:19.440
know-- it could be 1 and 6.

00:03:19.440 --> 00:03:21.260
The difference between
1 and 6 is 5, so it

00:03:21.260 --> 00:03:22.720
would satisfy this.

00:03:22.720 --> 00:03:25.880
So n could be 6 if we just
took statement 2.

00:03:25.880 --> 00:03:26.890
So we really need both.

00:03:26.890 --> 00:03:29.540
Statement 1 tells us
that n is prime.

00:03:29.540 --> 00:03:33.340
Statement 2 tells us that n has
to be even, that it has to

00:03:33.340 --> 00:03:36.420
be an even number.

00:03:36.420 --> 00:03:40.060
So frankly, there's only
one prime even

00:03:40.060 --> 00:03:41.310
number, and that's 2.

00:03:41.310 --> 00:03:43.740
So both statements together
are necessary

00:03:43.740 --> 00:03:46.260
to answer this question.

00:03:46.260 --> 00:03:55.490
133, every member of a certain
club volunteers to contribute

00:03:55.490 --> 00:04:00.600
equally to the purchase price
of a $60 gift certificate.

00:04:00.600 --> 00:04:02.350
How many members of
does a club have?

00:04:02.350 --> 00:04:05.750
So we want to know-- let's call
m for members-- how many

00:04:05.750 --> 00:04:07.530
members does a club have?

00:04:07.530 --> 00:04:11.900
Question 1, each member's
contribution is to be $4.

00:04:14.880 --> 00:04:18.060
Let's see, it says they
contribute equally to a

00:04:18.060 --> 00:04:19.170
purchase price.

00:04:19.170 --> 00:04:22.480
OK, so the number of members
times-- and they say every

00:04:22.480 --> 00:04:23.440
member of a certain club.

00:04:23.440 --> 00:04:25.150
So now they don't say
some members.

00:04:25.150 --> 00:04:27.660
So m is the number of
members, and they

00:04:27.660 --> 00:04:29.510
contribute equally $4.

00:04:29.510 --> 00:04:31.580
That's going to be
equal to $60.

00:04:31.580 --> 00:04:33.450
So then you immediately know
that you can solve for the

00:04:33.450 --> 00:04:34.080
number of members.

00:04:34.080 --> 00:04:36.780
There are 15 members
of this club.

00:04:36.780 --> 00:04:37.570
Maybe I'm missing something.

00:04:37.570 --> 00:04:43.200
Statement 2, if 5 club members
fail to contribute, the share

00:04:43.200 --> 00:04:47.185
of each contributing member
will increase by $2.

00:04:50.320 --> 00:04:56.090
So that means that if I were
to take the amount that 5

00:04:56.090 --> 00:05:00.710
members were to contribute--
so let's say that the

00:05:00.710 --> 00:05:03.610
contribution amount is c.

00:05:03.610 --> 00:05:09.630
So if we take the amount that 5
members would contribute, so

00:05:09.630 --> 00:05:17.240
5 times c, and divide it by
the remaining members, so

00:05:17.240 --> 00:05:23.570
divide by m minus 5, that
would be that each

00:05:23.570 --> 00:05:27.730
contributing member will
increase by $2.

00:05:27.730 --> 00:05:30.190
So this is the amount that would
have been contributed by

00:05:30.190 --> 00:05:32.640
those 5 people.

00:05:32.640 --> 00:05:34.215
Let me make sure that I'm
not missing anything.

00:05:40.490 --> 00:05:43.080
That is going to
be equal to $2.

00:05:43.080 --> 00:05:45.500
So this is the amount that those
5 members would have

00:05:45.500 --> 00:05:46.510
contributed.

00:05:46.510 --> 00:05:48.580
If they don't contribute it,
it's going to have to be

00:05:48.580 --> 00:05:50.180
divided by the other members.

00:05:50.180 --> 00:05:53.490
So however many members
there are minus 5.

00:05:53.490 --> 00:05:56.570
That is going to be-- when you
divide the amount divided by

00:05:56.570 --> 00:06:00.320
who has to pay for it-- is
$2 per leftover member.

00:06:00.320 --> 00:06:04.660
We also know that the members
times the contribution for

00:06:04.660 --> 00:06:08.180
member if everyone pays, is
going to be equal to $60.

00:06:08.180 --> 00:06:12.180
That they give us in the problem
statement, that m

00:06:12.180 --> 00:06:14.610
members are going to contribute
equally, and

00:06:14.610 --> 00:06:16.170
they're going to end
up with $60.

00:06:16.170 --> 00:06:17.450
So actually, we have two linear

00:06:17.450 --> 00:06:19.710
equations and two unknowns.

00:06:19.710 --> 00:06:22.510
So statement 2 alone is actually
sufficient to solve

00:06:22.510 --> 00:06:23.330
this problem.

00:06:23.330 --> 00:06:25.120
If this doesn't look like a
linear question, you can just

00:06:25.120 --> 00:06:30.320
multiply both sides by m minus
5 and you get 5c is equal to

00:06:30.320 --> 00:06:32.630
2m minus 10.

00:06:32.630 --> 00:06:36.150
Now this looks a lot more
like a linear equation.

00:06:36.150 --> 00:06:38.410
Well, actually, this isn't a
complete linear equation.

00:06:38.410 --> 00:06:42.720
But let me solve it just to make
the point clear for you.

00:06:42.720 --> 00:06:47.525
So if I were to say that c--
this isn't a linear equation,

00:06:47.525 --> 00:06:52.400
so I shouldn't have said that--
c is equal to 60/m.

00:06:52.400 --> 00:06:58.930
If c is equal to 60/m, so then
this turns to 5 times 60.

00:06:58.930 --> 00:07:06.830
So 300/m is equal
to 2m minus 10.

00:07:06.830 --> 00:07:08.170
Then you are left with what?

00:07:08.170 --> 00:07:12.610
Multiply both sides by m, you
get 300 is equal to 2m

00:07:12.610 --> 00:07:15.180
squared minus 10m.

00:07:15.180 --> 00:07:18.500
Divide both sides by 2, you
get 150 is equal to m

00:07:18.500 --> 00:07:21.530
squared minus 10m.

00:07:21.530 --> 00:07:22.860
Then subtract 150
from both sides.

00:07:22.860 --> 00:07:33.170
You get m squared minus 10m
minus 150 is equal to 0.

00:07:33.170 --> 00:07:38.360
Then let me see if I can just do
this just by factoring it.

00:07:38.360 --> 00:07:48.700
Minus 150, 30 times 5,
no, that's not good.

00:07:48.700 --> 00:07:50.510
15 times 10.

00:07:50.510 --> 00:07:57.850
So if I do m minus 15 times--
no, that doesn't work.

00:07:57.850 --> 00:08:00.910
15, 25, and 6.

00:08:00.910 --> 00:08:02.550
I know one of the
answers already.

00:08:02.550 --> 00:08:05.050
I know it's 15.

00:08:05.050 --> 00:08:09.550
m minus 15 times m plus 10.

00:08:09.550 --> 00:08:11.630
Oh, actually, I just realized
what my mistake was.

00:08:11.630 --> 00:08:13.140
I went from this step
to this step.

00:08:13.140 --> 00:08:16.120
So I divided both sides by 2.

00:08:16.120 --> 00:08:17.290
So 300 went into 150.

00:08:17.290 --> 00:08:21.130
2m squared-- this
had to be 5m.

00:08:21.130 --> 00:08:22.730
That's my mistake.

00:08:22.730 --> 00:08:23.030
5m.

00:08:23.030 --> 00:08:24.920
M squared minus 5m minus 150.

00:08:24.920 --> 00:08:28.780
So that's m minus 15 times
m plus 10 is equal to 0.

00:08:28.780 --> 00:08:34.120
So that tells us that m is
equal to 15 or minus 10.

00:08:34.120 --> 00:08:36.780
This is actually a quadratic
equation.

00:08:36.780 --> 00:08:39.289
We know that the members
can't be negative.

00:08:39.289 --> 00:08:41.630
There's a positive number
of members.

00:08:41.630 --> 00:08:44.830
So statement 2 alone is enough
information to know that there

00:08:44.830 --> 00:08:47.900
are exactly 15 members
in this club.

00:08:47.900 --> 00:08:48.650
Next problem.

00:08:48.650 --> 00:08:51.270
It had me stumped there because
of my careless mistake

00:08:51.270 --> 00:08:52.910
there for a second.

00:08:52.910 --> 00:08:54.160
Next problem.

00:08:58.380 --> 00:09:03.570
OK, problem 134.

00:09:03.570 --> 00:09:05.230
So that last one, I don't
know if I just said it.

00:09:05.230 --> 00:09:07.570
Each statement alone
is sufficient.

00:09:07.570 --> 00:09:08.540
So problem 134.

00:09:08.540 --> 00:09:14.700
If m and n are positive
integers, is the square root

00:09:14.700 --> 00:09:20.910
of n minus m an integer?

00:09:20.910 --> 00:09:22.730
So m and n are positive
integers.

00:09:22.730 --> 00:09:23.840
OK, so they tell us.

00:09:23.840 --> 00:09:34.920
Statement 1, they say n is
greater than m plus 15.

00:09:34.920 --> 00:09:39.190
Well you know, n is greater
than m plus 15.

00:09:39.190 --> 00:09:45.480
So if n is exactly 16 greater
than m, this is another way of

00:09:45.480 --> 00:09:48.270
saying that n minus m
is greater than 15.

00:09:48.270 --> 00:09:50.420
That tells that what's under
the denominator is

00:09:50.420 --> 00:09:51.470
greater than 15.

00:09:51.470 --> 00:09:54.510
Well, if what's under the
denominator is 16, then it is

00:09:54.510 --> 00:09:54.750
an integer.

00:09:54.750 --> 00:09:57.890
But if it's 17, which also meets
this requirement, it's

00:09:57.890 --> 00:09:58.580
not an integer.

00:09:58.580 --> 00:10:01.800
So this isn't enough information
by itself.

00:10:01.800 --> 00:10:10.730
Statement 2 says n is equal to m
times m plus 1, which is the

00:10:10.730 --> 00:10:14.830
same thing as m squared
plus m.

00:10:14.830 --> 00:10:18.420
So if we substitute that into
this equation, we get the

00:10:18.420 --> 00:10:29.200
square root of m squared
plus m minus m.

00:10:32.060 --> 00:10:33.410
That's m times m plus 1.

00:10:33.410 --> 00:10:34.820
It's m squared plus m.

00:10:34.820 --> 00:10:35.810
This minus m is right there.

00:10:35.810 --> 00:10:37.680
So that cancels out and we're
just left with square root of

00:10:37.680 --> 00:10:40.580
m squared, which is going
to be equal to

00:10:40.580 --> 00:10:42.400
m, which is an integer.

00:10:42.400 --> 00:10:46.780
So statement 2 alone is
sufficient to say that this

00:10:46.780 --> 00:10:49.500
would be an integer as long
as n is equal to m

00:10:49.500 --> 00:10:51.680
times m plus 1.

00:10:51.680 --> 00:10:52.410
I've run out of time.

00:10:52.410 --> 00:10:54.140
See you in the next video.


GMAT: Data sufficiency 32

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.360
We're on problem 129.

00:00:05.360 --> 00:00:09.330
If n is a positive integer, is
the value of b minus a at

00:00:09.330 --> 00:00:11.120
least-- OK, so let's see.

00:00:11.120 --> 00:00:12.320
So n is positive.

00:00:12.320 --> 00:00:15.610
n positive integer.

00:00:15.610 --> 00:00:21.080
And they're saying, is the value
of b minus a at least

00:00:21.080 --> 00:00:22.020
twice the value?

00:00:22.020 --> 00:00:27.520
So that's greater than or equal
to twice the value of 3

00:00:27.520 --> 00:00:32.670
to the n minus 2 to the n.

00:00:32.670 --> 00:00:35.020
That's what we need to test.

00:00:35.020 --> 00:00:45.000
Statement 1 tells us, a is equal
to 2 to the n plus 1.

00:00:45.000 --> 00:00:48.940
And b-- this is still statement
1-- is equal to 3 to

00:00:48.940 --> 00:00:51.390
the n plus 1.

00:00:51.390 --> 00:00:54.910
So let's think about
this a little bit.

00:00:54.910 --> 00:00:57.310
So let's substitute this
into this equation.

00:00:57.310 --> 00:01:07.370
So you get 3 to the n plus 1
minus 2 to the n plus 1 is

00:01:07.370 --> 00:01:09.770
greater than or equal to-- and
let's multiply this out.

00:01:09.770 --> 00:01:16.990
2 times 3 to the n minus
2 times 2 to the n.

00:01:16.990 --> 00:01:19.860
So what's 2 times 2 to the n?

00:01:19.860 --> 00:01:21.680
Well, that's just 2
to the n plus 1.

00:01:25.180 --> 00:01:27.240
2 times 2 to the n is 2
to the n plus 1, and

00:01:27.240 --> 00:01:28.140
there's a minus sign.

00:01:28.140 --> 00:01:29.530
That's where I got it from.

00:01:29.530 --> 00:01:30.170
So interesting.

00:01:30.170 --> 00:01:32.210
Just like that, we have a minus
2 to the n plus 1 on

00:01:32.210 --> 00:01:32.640
both sides.

00:01:32.640 --> 00:01:36.260
So we can just add 2 to the n
plus 1 from both sides to

00:01:36.260 --> 00:01:37.690
cancel them out.

00:01:37.690 --> 00:01:40.080
So we get rid of that,
get rid of that.

00:01:40.080 --> 00:01:45.900
And then our statement boils
down to, is 3 to the n plus 1

00:01:45.900 --> 00:01:52.440
greater than or equal to
2 times 3 to the n?

00:01:52.440 --> 00:01:58.500
We could rewrite 3 to the n plus
1 as 3 times 3 to the n.

00:01:58.500 --> 00:02:00.390
And we're testing whether
that's greater than 2

00:02:00.390 --> 00:02:02.750
times 3 to the n.

00:02:02.750 --> 00:02:05.240
And they tell us that n
is a positive number.

00:02:05.240 --> 00:02:06.450
But that actually
doesn't matter.

00:02:06.450 --> 00:02:09.310
Because we have 3 to the
n on both sides.

00:02:09.310 --> 00:02:11.520
Even if you rose this to a
negative exponent, you would

00:02:11.520 --> 00:02:13.310
still have a positive number.

00:02:13.310 --> 00:02:15.290
So you could divide both sides
by 3 to the n and the

00:02:15.290 --> 00:02:17.820
statement boils down to 3 is
greater than 2, which is

00:02:17.820 --> 00:02:18.880
absolutely correct.

00:02:18.880 --> 00:02:23.890
So statement 1 is enough
information to say whether

00:02:23.890 --> 00:02:24.926
this statement is true or not.

00:02:24.926 --> 00:02:27.140
And it actually proves
that it's true.

00:02:27.140 --> 00:02:29.300
Let's look at statement
number two.

00:02:29.300 --> 00:02:33.610
Statement number two tells
us that n is equal to 3.

00:02:37.450 --> 00:02:38.090
So let's see.

00:02:38.090 --> 00:02:41.230
If we look at the original
statement, if n is equal to 3,

00:02:41.230 --> 00:02:45.140
then we have to say-- This is
what the statement reduces to.

00:02:45.140 --> 00:02:51.200
It reduces to, b minus a is
greater than or equal to 2

00:02:51.200 --> 00:02:53.300
times 3 to the third power.

00:02:53.300 --> 00:02:55.390
So that's 27.

00:02:55.390 --> 00:02:57.720
Minus 2 to the third power.

00:02:57.720 --> 00:02:59.760
Well, 2 to the third is 8.

00:02:59.760 --> 00:03:03.550
So it reduces to, b minus a is
greater than or equal to-- 27

00:03:03.550 --> 00:03:04.590
minus 8 is 19.

00:03:04.590 --> 00:03:06.260
Times 2 is 38.

00:03:06.260 --> 00:03:07.690
Well, I don't know.

00:03:07.690 --> 00:03:08.640
They don't tell us
anything else

00:03:08.640 --> 00:03:09.920
about b, or a, or anything.

00:03:09.920 --> 00:03:12.110
So this is a useless
statement.

00:03:12.110 --> 00:03:13.690
And actually, going through
this was a little bit of a

00:03:13.690 --> 00:03:14.270
waste of time.

00:03:14.270 --> 00:03:16.180
You could have just said, hey,
they told me what n is.

00:03:16.180 --> 00:03:18.310
I can come up with a number
here, but that still doesn't

00:03:18.310 --> 00:03:20.640
give me any information
over what b or a is.

00:03:20.640 --> 00:03:23.020
So statement 1, alone,
is sufficient

00:03:23.020 --> 00:03:23.920
to answer this question.

00:03:23.920 --> 00:03:25.425
And 2 is not.

00:03:28.570 --> 00:03:30.640
Let me go to sky blue.

00:03:30.640 --> 00:03:34.710
Problem 130.

00:03:34.710 --> 00:03:38.590
The inflation index for the
year 1989, relative to the

00:03:38.590 --> 00:03:42.540
year 1970, was 3.56.

00:03:42.540 --> 00:03:46.740
Indicating that, on average, for
each dollar spent in 1970

00:03:46.740 --> 00:03:49.490
for goods, 3.56 had to
be spent for the

00:03:49.490 --> 00:03:51.710
same goods in 1989.

00:03:51.710 --> 00:03:52.540
Let me write that down.

00:03:52.540 --> 00:04:03.070
1970, $1.00.

00:04:03.070 --> 00:04:08.420
In 1989, that same thing
would cost $3.59.

00:04:08.420 --> 00:04:10.550
Oh, $3.56.

00:04:10.550 --> 00:04:12.560
I'm getting my 6's
and 9's confused.

00:04:12.560 --> 00:04:14.900
$3.56 had to be spent.

00:04:14.900 --> 00:04:18.209
If the price of a model K mixer
increased precisely

00:04:18.209 --> 00:04:21.240
according to the inflation
index, what was the price of

00:04:21.240 --> 00:04:22.970
the mixer in 1970?

00:04:22.970 --> 00:04:28.730
So we want to know the
mixer K in 1970.

00:04:28.730 --> 00:04:30.400
So statement number one.

00:04:30.400 --> 00:04:34.990
The price of the model K
mixer was $102.4 more

00:04:34.990 --> 00:04:38.860
in 1989 than 1970.

00:04:38.860 --> 00:04:40.840
So let's call this, K.

00:04:40.840 --> 00:04:44.630
We're trying to figure
out K 70.

00:04:44.630 --> 00:04:45.620
That's what we're trying
to figure out.

00:04:45.620 --> 00:04:47.420
Mixer K in 1970.

00:04:47.420 --> 00:04:52.620
So they say the price of the
model K mixer was $102.40 more

00:04:52.620 --> 00:04:54.450
in 1989 than 1970.

00:04:54.450 --> 00:05:03.920
So this, essentially, tells us
that mixer K in 1989, minus

00:05:03.920 --> 00:05:15.370
mixer K in 1970, is
equal to $102.40.

00:05:15.370 --> 00:05:16.840
But we also have other
information.

00:05:16.840 --> 00:05:21.730
We know that mixer K, it went
exactly with inflation, the

00:05:21.730 --> 00:05:25.080
increase in inflation price.

00:05:25.080 --> 00:05:36.580
So mixer K in 1989, is equal to
3.56 times mixer K in 1970.

00:05:36.580 --> 00:05:37.590
Well, how do we know that?

00:05:37.590 --> 00:05:39.970
Because they told us that mixer
K, the price of it,

00:05:39.970 --> 00:05:43.820
increased directly with
this inflation.

00:05:43.820 --> 00:05:47.700
Pretty much anything that went
with inflation was worth 3.56

00:05:47.700 --> 00:05:50.610
times more in 1989.

00:05:50.610 --> 00:05:54.680
So mixer K in '89 is
3.56 times more

00:05:54.680 --> 00:05:55.750
than it was in 1970.

00:05:55.750 --> 00:05:58.595
So now, we have this linear
equation, which was given in

00:05:58.595 --> 00:05:59.970
the problem statement.

00:05:59.970 --> 00:06:01.110
And we have this linear
equation.

00:06:01.110 --> 00:06:03.600
So we have two equations
and two unknowns.

00:06:03.600 --> 00:06:06.010
So we actually have enough
information just with

00:06:06.010 --> 00:06:08.810
statement 1 to solve
the problem.

00:06:08.810 --> 00:06:10.350
And I'm actually not even going
to go into it because it

00:06:10.350 --> 00:06:11.740
would be a waste of time.

00:06:11.740 --> 00:06:13.230
Hopefully, we're getting into
the rhythm of this.

00:06:13.230 --> 00:06:16.000
Two linear equations, two
unknowns, enough information.

00:06:16.000 --> 00:06:18.520
If they're quadratic or
something, then you'd have to

00:06:18.520 --> 00:06:21.290
dig a little bit deeper.

00:06:21.290 --> 00:06:23.270
Or if they're somehow parallel
equations, but I wouldn't

00:06:23.270 --> 00:06:25.590
worry about that too
much on the GMAT.

00:06:25.590 --> 00:06:27.760
Statement number two.

00:06:27.760 --> 00:06:31.840
The price of the model K mixer
was $142.40 in 1989.

00:06:31.840 --> 00:06:40.760
So they're telling us that
K in 1989 was 142.40

00:06:40.760 --> 00:06:43.880
Well, this is definitely enough
information, because we

00:06:43.880 --> 00:06:45.870
already have this equation.

00:06:45.870 --> 00:06:52.370
The '89 price, which is 142.40,
is equal to 3.56 times

00:06:52.370 --> 00:06:54.200
the 1970 price.

00:06:54.200 --> 00:06:57.120
So you just divide this by 3.56
and you have your answer.

00:06:57.120 --> 00:07:00.660
So each statement,
independently, is enough to

00:07:00.660 --> 00:07:01.910
solve this problem.

00:07:05.460 --> 00:07:09.260
Problem 131.

00:07:09.260 --> 00:07:11.306
Is 5 to the k less
than a thousand?

00:07:19.060 --> 00:07:25.620
Statement number one tells us
that 5 to the k plus 1 is

00:07:25.620 --> 00:07:27.430
greater than 3,000.

00:07:27.430 --> 00:07:28.910
I already have a sense this
isn't going to help us,

00:07:28.910 --> 00:07:31.980
because it's essentially setting
a lower bound on 5.

00:07:31.980 --> 00:07:34.230
So let me just explain it.

00:07:34.230 --> 00:07:35.340
So 5 to the k plus 1.

00:07:35.340 --> 00:07:38.430
That's the same thing as
5 to the k times 5.

00:07:38.430 --> 00:07:42.150
So that's equal to 5
times 5 to the k.

00:07:42.150 --> 00:07:43.670
You could view this as
5 to the first power.

00:07:43.670 --> 00:07:46.070
Then, you would add the
exponents to get here.

00:07:46.070 --> 00:07:49.000
And that, they say, is
greater than 3,000.

00:07:49.000 --> 00:07:53.630
You divide both sides by 5 and
you get 5 to the k-- that's a

00:07:53.630 --> 00:07:56.960
k-- is greater than 600.

00:07:56.960 --> 00:07:58.080
That doesn't help us.

00:07:58.080 --> 00:08:00.070
5 to the k could be 6,000.

00:08:00.070 --> 00:08:02.560
Well, 5 to the k could
be 700, in which case

00:08:02.560 --> 00:08:03.850
this would be true.

00:08:03.850 --> 00:08:05.460
5 to the k could be 7,000.

00:08:05.460 --> 00:08:06.600
In which case, this
wouldn't be true.

00:08:06.600 --> 00:08:10.930
So statement 1, by itself,
isn't enough information.

00:08:10.930 --> 00:08:20.230
Statement number two tells us, 5
to the k minus 1 is equal to

00:08:20.230 --> 00:08:26.820
5 to the k minus 500.

00:08:26.820 --> 00:08:28.940
This is interesting.

00:08:28.940 --> 00:08:31.620
Let's see what we can do here.

00:08:31.620 --> 00:08:32.990
Let's get all the
k's on one side.

00:08:32.990 --> 00:08:36.830
So this is the same thing as 5
to the k minus 1 minus 5 to

00:08:36.830 --> 00:08:45.430
the k is equal to minus 500.

00:08:45.430 --> 00:08:49.100
Well, this is the same
thing as 1/5, or 5 to

00:08:49.100 --> 00:08:50.030
the negative 1.

00:08:50.030 --> 00:08:51.030
Let me write that.

00:08:51.030 --> 00:08:55.290
5 to the negative 1
times 5 to the k.

00:08:55.290 --> 00:09:00.220
Then, minus 5 to the k is
equal to minus 500.

00:09:00.220 --> 00:09:03.180
Let's factor 5 to the
k out of this.

00:09:03.180 --> 00:09:08.240
So you get 5 to the
k times-- 5 to the

00:09:08.240 --> 00:09:11.720
negative 1, that's 1/5.

00:09:11.720 --> 00:09:15.220
Minus 1-- we factored a 5
to the k out there-- is

00:09:15.220 --> 00:09:19.300
equal to minus 500.

00:09:19.300 --> 00:09:20.540
And let's see what we can do.

00:09:20.540 --> 00:09:22.860
5 to the k.

00:09:22.860 --> 00:09:24.980
1/5 minus 1.

00:09:24.980 --> 00:09:27.020
That's minus 4/5, right?

00:09:27.020 --> 00:09:30.910
Times minus 4/5 is equal
to minus 500.

00:09:30.910 --> 00:09:32.610
We can get rid of the minus
sign on both sides.

00:09:32.610 --> 00:09:34.480
Multiply both sides
by negative 1.

00:09:34.480 --> 00:09:36.670
And then, we can multiply
both sides by 5/4.

00:09:36.670 --> 00:09:42.630
So we get 5 to the k is equal
to 500 times what?

00:09:42.630 --> 00:09:43.880
5/4.

00:09:46.570 --> 00:09:48.170
And that is what?

00:09:48.170 --> 00:09:52.180
That's 125.

00:09:52.180 --> 00:09:57.040
2,500 divided by 4.

00:09:57.040 --> 00:09:59.480
2,500 divided by 4.

00:09:59.480 --> 00:10:01.060
I want to say it's 475.

00:10:01.060 --> 00:10:03.040
Is that-- 5/4.

00:10:03.040 --> 00:10:03.740
No.

00:10:03.740 --> 00:10:07.170
So each fourth is 125 and
you have another.

00:10:07.170 --> 00:10:10.010
So it's equal to 625.

00:10:10.010 --> 00:10:13.310
Even if I just got my math
wrong, that doesn't matter.

00:10:13.310 --> 00:10:15.200
Because they just want to know
whether 5 to the k is less

00:10:15.200 --> 00:10:16.270
than a thousand.

00:10:16.270 --> 00:10:17.820
And it definitely is.

00:10:17.820 --> 00:10:24.220
Even if my math-- 500 times
5/4-- 1/4 of 500 is 125.

00:10:24.220 --> 00:10:30.130
125 times 5.

00:10:30.130 --> 00:10:32.090
So we'll have one
more of those.

00:10:32.090 --> 00:10:32.880
625.

00:10:32.880 --> 00:10:38.460
So statement 2, alone,
is sufficient

00:10:38.460 --> 00:10:39.390
to answer the question.

00:10:39.390 --> 00:10:42.310
Statement 1 was kind
of useless.

00:10:42.310 --> 00:10:43.460
I've run out of time.

00:10:43.460 --> 00:10:45.330
See you in the next video.


GMAT: Data sufficiency 31

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:09.250
All right, we're on problem
125 on page 288.

00:00:09.250 --> 00:00:14.290
If r and s are positive
integers, is r/s an integer?

00:00:14.290 --> 00:00:17.590
Is r/s an integer?

00:00:17.590 --> 00:00:19.640
So that's really just another
way of saying, is s

00:00:19.640 --> 00:00:21.010
divisible into r?

00:00:21.010 --> 00:00:24.130
So let's see what the
statements are.

00:00:24.130 --> 00:00:24.990
Statement 1.

00:00:24.990 --> 00:00:30.290
Every factor of s is
also a factor of r.

00:00:30.290 --> 00:00:31.700
That answers our question.

00:00:31.700 --> 00:00:41.990
Every factor of s
is factor of r.

00:00:41.990 --> 00:00:42.780
Well, let me ask
you a question.

00:00:42.780 --> 00:00:44.700
What is the largest
factor of s?

00:00:44.700 --> 00:00:48.270
The largest factor of s is s.

00:00:48.270 --> 00:00:52.860
So this statement tells us that
since s is a factor of s,

00:00:52.860 --> 00:00:55.260
that s is also going to
be a factor of r.

00:00:55.260 --> 00:00:57.060
And something being a factor of
something means that it's

00:00:57.060 --> 00:00:57.950
divisible into it.

00:00:57.950 --> 00:01:00.290
So that means that s is
divisible into r.

00:01:00.290 --> 00:01:05.459
So this means that r/s
is an integer.

00:01:05.459 --> 00:01:07.550
So it answers our question.

00:01:07.550 --> 00:01:10.480
So statement 1, alone,
is sufficient.

00:01:10.480 --> 00:01:16.410
Statement 2 tells us, every
prime factor of s is also a

00:01:16.410 --> 00:01:19.460
prime factor of r.

00:01:19.460 --> 00:01:23.540
Every prime factor-- so let me
think of-- Immediately, I can

00:01:23.540 --> 00:01:27.170
think of a case where that
doesn't hold up.

00:01:27.170 --> 00:01:30.520
Where I could have every prime
factor being a factor of r.

00:01:30.520 --> 00:01:34.132
So let's say s is equal to 4.

00:01:34.132 --> 00:01:38.300
And its prime factorization
is 2 times 2.

00:01:38.300 --> 00:01:42.570
And let's say that
R is equal to 6.

00:01:42.570 --> 00:01:47.030
And its prime factorization
is 2 times 3.

00:01:47.030 --> 00:01:49.820
So we have a case here where
every factor, every prime

00:01:49.820 --> 00:01:53.790
factor of s, is a prime
factor of R.

00:01:53.790 --> 00:01:55.360
The only prime factor
of s is 2.

00:01:55.360 --> 00:01:57.170
And that's a prime
factor of R.

00:01:57.170 --> 00:02:01.770
But if we were to say R/s,
R/s would be 6/4,

00:02:01.770 --> 00:02:03.860
which is not an integer.

00:02:03.860 --> 00:02:06.180
So even though these satisfy
the second condition, this

00:02:06.180 --> 00:02:07.080
isn't an integer.

00:02:07.080 --> 00:02:08.949
But then I could have-- instead
of making R equal to

00:02:08.949 --> 00:02:11.820
6, I could've made
R equal to 4.

00:02:11.820 --> 00:02:15.140
Sorry, I could have made R is
equal to 8, which is equal to

00:02:15.140 --> 00:02:17.010
2 times 2 times 2.

00:02:17.010 --> 00:02:19.040
And in this case, it would
have been an integer.

00:02:19.040 --> 00:02:24.340
R/s would be equal to 8/4,
which is equal to 2.

00:02:24.340 --> 00:02:26.950
So statement 2, really doesn't
give us information as to

00:02:26.950 --> 00:02:29.180
whether r/s is an integer.

00:02:29.180 --> 00:02:32.860
So statement 1, alone,
is sufficient.

00:02:32.860 --> 00:02:35.090
Next problem.

00:02:35.090 --> 00:02:37.410
Switch colors.

00:02:37.410 --> 00:02:41.750
126.

00:02:41.750 --> 00:02:50.090
If z to the n is equal to 1,
what is the value of z?

00:02:50.090 --> 00:02:52.850
z equals what?

00:02:52.850 --> 00:02:57.270
So statement 1 tells us,
n is a nonzero integer.

00:02:57.270 --> 00:02:59.642
n does not equal to 0.

00:02:59.642 --> 00:03:01.300
So let's think about this.

00:03:01.300 --> 00:03:05.160
For something to some power
to be equal to 1, what do

00:03:05.160 --> 00:03:06.660
we know about it?

00:03:06.660 --> 00:03:13.460
Well, if anything to the 0th
power is equal to 1, but they

00:03:13.460 --> 00:03:14.750
just told us that
it's nonzero.

00:03:14.750 --> 00:03:16.240
So we can't use this
condition.

00:03:16.240 --> 00:03:18.630
So that actually does restrict
z a good bit.

00:03:18.630 --> 00:03:21.860
So if n is a nonzero number,
what can I raise to the power

00:03:21.860 --> 00:03:22.390
to equal 1?

00:03:22.390 --> 00:03:23.760
Well, clearly 1, right?

00:03:23.760 --> 00:03:28.290
1 to anything is going
to be equal to 1.

00:03:28.290 --> 00:03:30.240
But what other?

00:03:30.240 --> 00:03:31.990
Well, there's also the
possibility of negative

00:03:31.990 --> 00:03:33.200
numbers, right?

00:03:33.200 --> 00:03:38.440
Negative 1 to the nth power is
equal to 1, if n is even.

00:03:42.760 --> 00:03:45.430
And I think these are the only
two, if we take imaginary

00:03:45.430 --> 00:03:46.370
numbers out of the picture.

00:03:46.370 --> 00:03:48.420
And I think, on the GMAT, we
assume that we don't have any

00:03:48.420 --> 00:03:49.800
imaginary numbers.

00:03:49.800 --> 00:03:54.760
So assuming that, the only--
If we know that n does not

00:03:54.760 --> 00:03:58.190
equal 0 and z to the n is
equal to 1, the only

00:03:58.190 --> 00:04:03.480
possibilities that this allows
for is that z is equal to 1 or

00:04:03.480 --> 00:04:04.470
negative 1.

00:04:04.470 --> 00:04:06.330
And if it's negative 1,
then n would have

00:04:06.330 --> 00:04:06.990
to be an even number.

00:04:06.990 --> 00:04:09.220
But they didn't restrict
that yet.

00:04:09.220 --> 00:04:11.230
So statement 1, by itself,
it helps us.

00:04:11.230 --> 00:04:14.330
But it doesn't actually give
us enough information.

00:04:14.330 --> 00:04:18.910
We can just narrow it down to
z being 1 or negative 1.

00:04:18.910 --> 00:04:21.019
Let's see what statement
2 tells us.

00:04:21.019 --> 00:04:25.210
Statement 2 tells us,
z is greater than 0.

00:04:25.210 --> 00:04:28.650
So that, by itself, is
useless information.

00:04:28.650 --> 00:04:34.140
Because if z is greater than
0 and n can be 0, right?

00:04:34.140 --> 00:04:36.550
We're assuming we don't
have statement 1 yet.

00:04:36.550 --> 00:04:40.300
So if z is greater than 0 and
n could be 0, z could be a

00:04:40.300 --> 00:04:42.090
hundred to the 0th power.

00:04:42.090 --> 00:04:44.000
z could be a hundred
and that equals 1.

00:04:44.000 --> 00:04:47.110
z could be 99 to the
0th power, and that

00:04:47.110 --> 00:04:47.810
could be equal to 1.

00:04:47.810 --> 00:04:49.800
So z could be anything to the
0th power as long as it's

00:04:49.800 --> 00:04:50.500
greater than 0.

00:04:50.500 --> 00:04:53.000
So this, by itself,
doesn't help us.

00:04:53.000 --> 00:04:55.300
But if we use statement 2 and
statement 1 in conjunction,

00:04:55.300 --> 00:04:56.570
then it's interesting.

00:04:56.570 --> 00:04:59.010
Because statement 1 essentially
told us that z has

00:04:59.010 --> 00:05:01.670
to be 1 or negative 1.

00:05:01.670 --> 00:05:04.830
Statement 2 tells us, z has
to be greater than 0.

00:05:04.830 --> 00:05:07.450
So if you use both the
conditions combined, it forces

00:05:07.450 --> 00:05:10.610
us to say, well, then z has
to be equal to positive 1.

00:05:10.610 --> 00:05:12.990
Because negative 1 is
not greater than 0.

00:05:12.990 --> 00:05:15.670
So both statements combined
are sufficient

00:05:15.670 --> 00:05:18.080
to answer this question.

00:05:18.080 --> 00:05:19.690
Next problem.

00:05:19.690 --> 00:05:24.040
127.

00:05:24.040 --> 00:05:25.970
OK, so they've written
this thing.

00:05:25.970 --> 00:05:33.100
They write, s is equal
2/n, all of that

00:05:33.100 --> 00:05:39.140
over 1/x plus 2/3x.

00:05:39.140 --> 00:05:41.930
And they say in the expression
above, if xn does not equal

00:05:41.930 --> 00:05:48.170
0-- so essentially saying that
neither x nor n is 0.

00:05:48.170 --> 00:05:49.770
What is the value of s?

00:05:49.770 --> 00:05:52.120
s is equal to what?

00:05:52.120 --> 00:05:55.400
So even before looking at the
statements, I just want to

00:05:55.400 --> 00:05:56.160
simplify this.

00:05:56.160 --> 00:05:57.770
Just because it's too
complex right now.

00:05:57.770 --> 00:05:58.210
So let's see.

00:05:58.210 --> 00:06:02.580
This is equal to 2/n
over-- let's see.

00:06:02.580 --> 00:06:05.650
You have a common
denominator 3x.

00:06:05.650 --> 00:06:05.950
Let's see.

00:06:05.950 --> 00:06:09.050
This is 3 over 3x.

00:06:09.050 --> 00:06:10.040
That's the same thing as 1/x.

00:06:10.040 --> 00:06:11.370
Plus 2.

00:06:11.370 --> 00:06:18.130
So that equals 2/n over
5/3x, which is equal

00:06:18.130 --> 00:06:22.900
to 2/n times 3x/5.

00:06:22.900 --> 00:06:25.190
Which is equal to 6x/5n.

00:06:27.950 --> 00:06:30.250
That's a much more pleasing
thing to look at and try to

00:06:30.250 --> 00:06:32.440
get your brain around.

00:06:32.440 --> 00:06:37.680
So statement 1 tells us,
x is equal to 2n.

00:06:40.470 --> 00:06:43.580
So if x is equal to 2n, let's
just substitute that into this

00:06:43.580 --> 00:06:44.330
statement for s.

00:06:44.330 --> 00:06:48.190
Then, s would be equal to 6
times x, which we know is

00:06:48.190 --> 00:06:50.700
equal to 2n.

00:06:50.700 --> 00:06:52.940
Divided by 5n.

00:06:52.940 --> 00:06:54.710
n's cancel out.

00:06:54.710 --> 00:06:56.365
So it equals 12/5.

00:06:56.365 --> 00:06:58.180
So statement 1, alone,
is sufficient

00:06:58.180 --> 00:06:59.850
to answer this question.

00:06:59.850 --> 00:07:06.700
Statement 2 tells us that
n is equal to 1/2.

00:07:06.700 --> 00:07:07.670
Well, that's fairly useless.

00:07:07.670 --> 00:07:10.180
Because this is what we
simplified s down to.

00:07:10.180 --> 00:07:14.090
If we put 1/2 here, then
we get s is equal to 6x

00:07:14.090 --> 00:07:16.150
over 5 times 1/2.

00:07:16.150 --> 00:07:18.930
so that's 5/2.

00:07:18.930 --> 00:07:21.050
I can simplify this more, but we
still don't know what x is.

00:07:21.050 --> 00:07:23.620
You can't cancel the x's
out or anything.

00:07:23.620 --> 00:07:25.050
This, alone, doesn't
help you at all.

00:07:25.050 --> 00:07:27.400
So the answer is statement
1, alone, is sufficient.

00:07:27.400 --> 00:07:29.850
And statement 2, by itself,
is fairly useless.

00:07:33.790 --> 00:07:39.670
Problem 128.

00:07:39.670 --> 00:07:43.640
If x is an integer, is x times--
I think that says the

00:07:43.640 --> 00:07:45.150
absolute-- it's kind
of strange to

00:07:45.150 --> 00:07:45.700
look at it like that.

00:07:45.700 --> 00:07:49.420
But they're saying, is x times
the absolute value

00:07:49.420 --> 00:07:53.810
of x less than 2x?

00:07:53.810 --> 00:07:56.770
And they tell us that
x is an integer.

00:07:56.770 --> 00:08:01.500
OK, statement 1 tells us
that x is less than 0.

00:08:01.500 --> 00:08:02.550
So how does that help us?

00:08:02.550 --> 00:08:08.620
If x is less than 0, then this
on the right-hand side is

00:08:08.620 --> 00:08:11.220
going to be less than 0.

00:08:11.220 --> 00:08:12.490
And this will also
be less than 0.

00:08:12.490 --> 00:08:13.720
Because you'll have
a negative number

00:08:13.720 --> 00:08:14.620
times a positive number.

00:08:14.620 --> 00:08:15.870
They'll both be less than 0.

00:08:18.660 --> 00:08:22.450
If we make x is equal to
negative 1, then this would

00:08:22.450 --> 00:08:25.620
not-- then this will
not be true.

00:08:25.620 --> 00:08:27.690
Because you'll have negative
1 times 1.

00:08:27.690 --> 00:08:32.659
So you'll have 1 is less
than negative 2.

00:08:32.659 --> 00:08:36.140
Sorry, you'd have negative 1
times 1, which is negative 1.

00:08:36.140 --> 00:08:38.179
Which is less than 2,
which is not true.

00:08:38.179 --> 00:08:40.140
It's not less than 2.

00:08:40.140 --> 00:08:45.830
Or, if x was less than negative
2, if x was negative

00:08:45.830 --> 00:08:50.310
3, then you would have
negative 3 times 3.

00:08:50.310 --> 00:08:52.330
Because the absolute value
of negative 3.

00:08:52.330 --> 00:08:56.980
So you would have minus 9 is
less than minus 3 times 2.

00:08:56.980 --> 00:08:58.080
Minus 6.

00:08:58.080 --> 00:08:59.220
This is true.

00:08:59.220 --> 00:09:02.510
So just this condition, alone,
doesn't get us there.

00:09:02.510 --> 00:09:06.430
Because I can still pick an x
that meets this condition.

00:09:06.430 --> 00:09:09.380
And depending on whether I make
that x greater than or

00:09:09.380 --> 00:09:12.100
less than negative 2, I can
make this true or not.

00:09:12.100 --> 00:09:15.520
So statement 1, by itself,
isn't enough.

00:09:15.520 --> 00:09:19.510
Statement 2 tells us, x
is equal to minus 10.

00:09:19.510 --> 00:09:20.400
Well, this is an easy one.

00:09:20.400 --> 00:09:21.540
We can just test it.

00:09:21.540 --> 00:09:25.240
So if we have minus 10 times
the absolute value of minus

00:09:25.240 --> 00:09:27.550
10, and we're going to test
whether that's less than 2

00:09:27.550 --> 00:09:29.910
times minus 10.

00:09:29.910 --> 00:09:32.960
So the absolute value of--
So this is minus 10 times

00:09:32.960 --> 00:09:34.330
positive 10.

00:09:34.330 --> 00:09:38.120
So it's minus 100, which
is less than minus 20.

00:09:38.120 --> 00:09:39.380
Which is completely true.

00:09:39.380 --> 00:09:42.530
So statement 2, alone,
is sufficient

00:09:42.530 --> 00:09:45.110
to answer this question.

00:09:45.110 --> 00:09:46.710
And I'm almost out of
time, so I'll see

00:09:46.710 --> 00:09:48.530
you in the next video.


GMAT: Data sufficiency 27

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:04.300
We're on problem 116.

00:00:04.300 --> 00:00:11.270
What is the value of
a plus b squared?

00:00:11.270 --> 00:00:13.300
Statement 1.

00:00:13.300 --> 00:00:20.650
Statement 1 tells us that
a times b is equal to 0.

00:00:20.650 --> 00:00:24.285
That, to me, seems
fairly useless.

00:00:24.285 --> 00:00:26.660
Let me see, what if we
expand that out.

00:00:26.660 --> 00:00:33.920
That's equal to to a squared
plus 2ab plus b squared.

00:00:33.920 --> 00:00:35.000
This is the same thing.

00:00:35.000 --> 00:00:37.310
I just expanded out
a plus b squared.

00:00:37.310 --> 00:00:38.410
And so they're giving
us a piece.

00:00:38.410 --> 00:00:41.530
They're telling us that this
term right here is going to be

00:00:41.530 --> 00:00:42.280
equal to 0.

00:00:42.280 --> 00:00:44.590
If ab is equal to 0, this term
right here is going to be

00:00:44.590 --> 00:00:45.140
equal to 0.

00:00:45.140 --> 00:00:47.690
But we still don't know what a
squared and b squared are.

00:00:47.690 --> 00:00:51.630
So statement 1 by itself:
not so useful.

00:00:51.630 --> 00:00:54.200
Statement 2.

00:00:54.200 --> 00:01:00.790
a minus b squared
is equal to 36.

00:01:00.790 --> 00:01:02.030
This is interesting.

00:01:02.030 --> 00:01:05.860
So let's expand this
out a little bit.

00:01:05.860 --> 00:01:08.870
So this tells us-- so you might
want to say, oh that

00:01:08.870 --> 00:01:12.325
means that a minus b
is plus or minus 6.

00:01:12.325 --> 00:01:13.900
But actually I don't
think that is going

00:01:13.900 --> 00:01:15.210
to help you as much.

00:01:15.210 --> 00:01:17.870
Well actually, that could
help you as well.

00:01:17.870 --> 00:01:20.460
But the way I'm thinking about
it is, let's expand this out.

00:01:20.460 --> 00:01:36.800
We get a squared minus 2ab plus
b squared is equal to 36.

00:01:36.800 --> 00:01:39.890
So, by itself the statement
isn't that useful.

00:01:39.890 --> 00:01:42.010
I mean, even here the statement
says that if you

00:01:42.010 --> 00:01:43.700
were to take the square root
of both sides, it tells us

00:01:43.700 --> 00:01:47.240
that the difference between a
and b is either positive 6 or

00:01:47.240 --> 00:01:47.746
negative 6.

00:01:47.746 --> 00:01:49.070
If you took the square
root of both sides.

00:01:49.070 --> 00:01:50.700
So statement 2 by itself
is kind of useless.

00:01:50.700 --> 00:01:53.500
But if you use them together,
you expand out

00:01:53.500 --> 00:01:55.150
statement 2 you get this.

00:01:55.150 --> 00:01:58.040
Now if we also take in statement
1 and say, oh, ab is

00:01:58.040 --> 00:01:58.740
equal to 0.

00:01:58.740 --> 00:02:00.610
So then that is equal to 0.

00:02:00.610 --> 00:02:06.630
And we're left with a squared
plus b squared is equal to 36.

00:02:06.630 --> 00:02:09.280
And if a squared plus b squared
is equal to 36, then

00:02:09.280 --> 00:02:12.920
we know that this a squared
plus b squared is

00:02:12.920 --> 00:02:14.950
also equal to 36.

00:02:14.950 --> 00:02:21.800
So that tells us that
a plus b squared is

00:02:21.800 --> 00:02:25.850
equal to 36 as well.

00:02:25.850 --> 00:02:28.220
So both statements together
are sufficient.

00:02:28.220 --> 00:02:29.460
And this is interesting.

00:02:29.460 --> 00:02:31.350
Because if you think about it,
it makes a lot of sense.

00:02:31.350 --> 00:02:33.920
This statement tells us that the
difference between a and

00:02:33.920 --> 00:02:37.130
b-- and this is all a waste of
time, but I just want to give

00:02:37.130 --> 00:02:40.320
you some intuition-- this tells
us that a minus b is

00:02:40.320 --> 00:02:43.320
equal to plus or minus 6.

00:02:43.320 --> 00:02:45.500
So it tells you that the
difference between a and b is

00:02:45.500 --> 00:02:47.060
either positive 6 or minus 6.

00:02:47.060 --> 00:02:49.350
So one is 6 bigger than the
other but we don't know which

00:02:49.350 --> 00:02:50.890
way it goes.

00:02:50.890 --> 00:02:55.020
This statement tells us that
one of them is equal to 0.

00:02:55.020 --> 00:02:56.720
Both of them aren't going to be
equal to 0 because there's

00:02:56.720 --> 00:02:58.340
a 6 difference.

00:02:58.340 --> 00:03:02.340
So if one of them is equal to
0, then the other one is

00:03:02.340 --> 00:03:04.060
either going to be
plus or minus 6.

00:03:07.110 --> 00:03:10.900
If we look at this statement,
one of these numbers is 0.

00:03:10.900 --> 00:03:12.540
The other one is plus
or minus 6.

00:03:12.540 --> 00:03:13.340
But it doesn't matter.

00:03:13.340 --> 00:03:17.030
If one is 0 and one is plus 6,
and you square it, you get 36.

00:03:17.030 --> 00:03:18.920
If one is 0 and the other
is minus 6, you

00:03:18.920 --> 00:03:20.520
square it, you get 36.

00:03:20.520 --> 00:03:22.020
So anyway, just want to give
you that little intuition.

00:03:22.020 --> 00:03:25.600
But the correct answer is that
both statements together are

00:03:25.600 --> 00:03:27.190
necessary to solve
this problem.

00:03:30.930 --> 00:03:33.740
117.

00:03:33.740 --> 00:03:36.050
The more connections you can
make while doing these math

00:03:36.050 --> 00:03:38.540
problems, the better you'll
be, really the better off

00:03:38.540 --> 00:03:41.180
you'll be in life, I think.

00:03:41.180 --> 00:03:43.190
It will all start
making sense.

00:03:43.190 --> 00:03:46.590
All right, they've
drawn this thing.

00:03:52.450 --> 00:03:55.450
And then they draw this
down like that.

00:03:58.810 --> 00:03:59.260
Let's see.

00:03:59.260 --> 00:04:09.170
This is L, M, N and K.

00:04:09.170 --> 00:04:13.060
In the figure above, what is
the ratio of KN to MN?

00:04:13.060 --> 00:04:17.870
So they want to know the ratio
of this side, KM to MN.

00:04:25.110 --> 00:04:26.830
Fair enough.

00:04:26.830 --> 00:04:27.370
OK.

00:04:27.370 --> 00:04:31.770
Statement number 1 tells us
the perimeter of rectangle

00:04:31.770 --> 00:04:42.790
KLMN is 30 meters.

00:04:42.790 --> 00:04:44.280
So this is a rectangle, right?

00:04:44.280 --> 00:04:48.200
So that tells us that-- let's
put it in terms of the things

00:04:48.200 --> 00:04:52.350
that we care about-- that 2
times MN-- because MN and KL

00:04:52.350 --> 00:04:53.320
are going to be the
same length,

00:04:53.320 --> 00:04:54.160
since it's a rectangle.

00:04:54.160 --> 00:05:01.720
So this tells us that 2 times
MN plus 2 times KN-- because

00:05:01.720 --> 00:05:04.230
KN and LM are the same length.

00:05:04.230 --> 00:05:05.540
So let's just put it
in the terms of the

00:05:05.540 --> 00:05:06.770
things we care about.

00:05:06.770 --> 00:05:13.100
Plus 2 times KN is
equal to 30.

00:05:13.100 --> 00:05:13.920
That's the perimeter.

00:05:13.920 --> 00:05:15.520
Twice this plus twice this.

00:05:15.520 --> 00:05:16.430
That's the perimeter.

00:05:16.430 --> 00:05:18.790
You could say that plus that
plus that plus that.

00:05:18.790 --> 00:05:21.570
But this keeps it in
terms of that.

00:05:21.570 --> 00:05:23.910
So I think we can solve.

00:05:23.910 --> 00:05:30.000
Let's divide both sides by 2.

00:05:30.000 --> 00:05:40.590
You get MN plus KN
is equal to 30.

00:05:40.590 --> 00:05:43.540
And actually, yeah, you still
can't figure out the ratio

00:05:43.540 --> 00:05:44.190
between the two.

00:05:44.190 --> 00:05:49.410
You just know that the sum of
the two are equal to 30.

00:05:49.410 --> 00:05:52.050
Let's look at statement 2.

00:05:52.050 --> 00:05:54.150
You can't figure out the
ratio from this.

00:05:54.150 --> 00:06:01.370
Statement 2 tells us the three
small rectangles have the same

00:06:01.370 --> 00:06:02.620
dimensions.

00:06:06.950 --> 00:06:08.950
This is interesting.

00:06:08.950 --> 00:06:10.980
So that's telling us.

00:06:10.980 --> 00:06:12.760
Well that's interesting.

00:06:12.760 --> 00:06:18.790
Because that tells us that
this distance and this

00:06:18.790 --> 00:06:21.280
distance have to be the same.

00:06:21.280 --> 00:06:27.060
That tells us that-- and if
this distance and this

00:06:27.060 --> 00:06:32.580
distance are the same, then this
distance right there is

00:06:32.580 --> 00:06:35.780
twice this distance.

00:06:35.780 --> 00:06:36.640
How did I get that?

00:06:36.640 --> 00:06:40.880
Well this distance is twice each
of this these distances.

00:06:40.880 --> 00:06:43.610
This longer side of
this horizontal

00:06:43.610 --> 00:06:46.120
rectangle is twice this.

00:06:46.120 --> 00:06:48.620
And this is the same
thing as this.

00:06:48.620 --> 00:06:51.130
So for all of these rectangles,
their ratio of

00:06:51.130 --> 00:06:55.170
their sides is 1:2.

00:06:55.170 --> 00:06:57.690
So let's create a unit
called the short

00:06:57.690 --> 00:07:00.420
side of this rectangle.

00:07:00.420 --> 00:07:03.010
So how many units long is MN?

00:07:03.010 --> 00:07:06.780
Well MN is going to be, this
side right here is going to be

00:07:06.780 --> 00:07:08.170
two of those units long.

00:07:08.170 --> 00:07:12.360
So, 1, 2, and then 3.

00:07:14.870 --> 00:07:18.060
We figured that out because the
long side of this is twice

00:07:18.060 --> 00:07:19.050
as long as the short side.

00:07:19.050 --> 00:07:21.120
So if we were measuring these in
terms of the short sides of

00:07:21.120 --> 00:07:26.070
the rectangle, MN is 3 of them
and then KN is 2 of them.

00:07:26.070 --> 00:07:29.430
1, 2.

00:07:29.430 --> 00:07:34.720
So the ratio of KN to MN, it
doesn't matter what units you

00:07:34.720 --> 00:07:37.020
measure in, you just want
to get the ratio.

00:07:37.020 --> 00:07:41.910
So the ratio is going to
be equal to 2 to 3.

00:07:41.910 --> 00:07:44.680
So statement 2 alone
is sufficient

00:07:44.680 --> 00:07:47.270
to solve this problem.

00:07:47.270 --> 00:07:48.520
Next question.

00:07:50.850 --> 00:07:52.840
118.

00:07:52.840 --> 00:08:00.620
If n is a positive integer,
is 150 over n an integer?

00:08:00.620 --> 00:08:04.330
So it's essentially asking us,
is n divisible into 150?

00:08:04.330 --> 00:08:05.900
That's the only way
that 150 over n is

00:08:05.900 --> 00:08:08.610
going to be an integer.

00:08:08.610 --> 00:08:14.270
Statement number 1 tells us
that n is less than 7.

00:08:14.270 --> 00:08:19.270
Now if every number less than 7
is divisible into 150, then

00:08:19.270 --> 00:08:19.770
we're all set.

00:08:19.770 --> 00:08:23.840
Because we know it's
positive as well.

00:08:23.840 --> 00:08:26.310
And 1 works.

00:08:26.310 --> 00:08:28.960
2 works, 75.

00:08:28.960 --> 00:08:31.370
3 works, 50.

00:08:31.370 --> 00:08:37.440
4-- let's see 4 goes into 100
and then does it go into 50?

00:08:37.440 --> 00:08:40.390
4 goes into 50-- no it
doesn't go into 50.

00:08:40.390 --> 00:08:42.490
So 4 does not work.

00:08:42.490 --> 00:08:45.900
So just by telling us that n is
less than 7 does not solve

00:08:45.900 --> 00:08:46.280
the problem.

00:08:46.280 --> 00:08:49.915
Because if n was 4, 4 does
not go into 150 evenly.

00:08:49.915 --> 00:08:52.920
It goes into it-- no, I'm not
going to figure it out.

00:08:52.920 --> 00:09:00.840
4 goes into it-- it goes in 25,
25 plus 12, 37.5 times.

00:09:00.840 --> 00:09:02.430
So that doesn't do
us any good.

00:09:02.430 --> 00:09:05.780
Statement 1 by itself
isn't that useful.

00:09:05.780 --> 00:09:07.180
Statement 2.

00:09:07.180 --> 00:09:08.960
n is a prime number.

00:09:08.960 --> 00:09:12.490
n is prime.

00:09:12.490 --> 00:09:14.950
So this statement by itself
is not useful.

00:09:14.950 --> 00:09:18.970
Because, I mean n could be a
prime number larger than 150.

00:09:18.970 --> 00:09:22.670
So that by itself doesn't tell
us that we definitely can get

00:09:22.670 --> 00:09:23.410
an integer.

00:09:23.410 --> 00:09:26.660
And n could easily be 3,
which would allow it.

00:09:26.660 --> 00:09:28.350
So n doesn't tell you
one way or another

00:09:28.350 --> 00:09:29.870
whether this is an integer.

00:09:29.870 --> 00:09:32.810
But if you combine the two
statements, if you say that n

00:09:32.810 --> 00:09:36.520
is a prime number that has to
be less than 7, then we can

00:09:36.520 --> 00:09:38.360
take 4 out of the picture.

00:09:38.360 --> 00:09:40.150
Because 4 isn't prime.

00:09:40.150 --> 00:09:42.260
And then you put 5 there.

00:09:42.260 --> 00:09:44.910
5 definitely does go into 150.

00:09:44.910 --> 00:09:48.030
And then 6, 6 does
go into 150.

00:09:48.030 --> 00:09:49.640
It goes into 120.

00:09:49.640 --> 00:09:52.060
And then 30 leftover,
so 6 goes into 120.

00:09:52.060 --> 00:09:55.930
So all of the prime numbers
less than 7 are

00:09:55.930 --> 00:09:57.490
divisible into 150.

00:09:57.490 --> 00:10:00.490
So both statements combined
are sufficient

00:10:00.490 --> 00:10:02.010
to answer this question.

00:10:02.010 --> 00:10:03.860
See you in the next video.


GMAT: Data sufficiency 30

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.900
We're on 121.

00:00:04.900 --> 00:00:19.520
It says, if x/2 is equal to
3/y, is x less than y?

00:00:19.520 --> 00:00:24.280
So this is essentially saying
that x times y is equal to 6.

00:00:24.280 --> 00:00:26.000
I just cross multiplied.

00:00:26.000 --> 00:00:27.110
Or you could do that
step by step.

00:00:27.110 --> 00:00:28.380
Multiply both sides by 2.

00:00:28.380 --> 00:00:29.740
Then, multiply both
sides by y.

00:00:29.740 --> 00:00:31.610
And you get xy is equal to 6.

00:00:31.610 --> 00:00:34.920
And they want to know whether
x is less than y.

00:00:34.920 --> 00:00:45.545
Statement 1 says, y is greater
than or equal to 3.

00:00:45.545 --> 00:00:49.880
If y is greater than or equal
to 3, what does that mean?

00:00:49.880 --> 00:00:51.630
So let's think about this.

00:00:51.630 --> 00:00:57.300
x is equal to 6 divided by y.

00:00:57.300 --> 00:01:03.710
If y is greater than or equal
to 3-- so if it's 3, that's

00:01:03.710 --> 00:01:06.610
the largest that x
can be if y is 3.

00:01:06.610 --> 00:01:10.320
If y is any larger number, then
x will only be smaller.

00:01:10.320 --> 00:01:15.780
So x will be less than
or equal to 6/3.

00:01:15.780 --> 00:01:19.090
Or x will be less than
or equal to 2.

00:01:19.090 --> 00:01:21.220
And just, I want you to
understand why I said that.

00:01:21.220 --> 00:01:22.220
Think about this.

00:01:22.220 --> 00:01:23.980
It says, y is greater
than or equal to 3.

00:01:23.980 --> 00:01:27.170
If y is 3, x is going to be 2.

00:01:27.170 --> 00:01:31.370
If y is 4, x is going
to be 1 and 1/2.

00:01:31.370 --> 00:01:33.990
If y is 6, x is going to be 1.

00:01:33.990 --> 00:01:37.760
So the highest value for x
happens when y is equal to 3.

00:01:37.760 --> 00:01:40.920
And then, as y gets larger and
larger, x just gets smaller.

00:01:40.920 --> 00:01:46.350
So y being greater than or equal
to 3 says that x is less

00:01:46.350 --> 00:01:48.310
than or equal to 2.

00:01:48.310 --> 00:01:52.060
So statement 1, alone, is
sufficient to say that x is

00:01:52.060 --> 00:01:53.480
less than y.

00:01:53.480 --> 00:01:55.950
If x is less than 2 and y is
greater than 3, then x is

00:01:55.950 --> 00:01:57.720
definitely less than y.

00:01:57.720 --> 00:02:01.885
Statement 2 tells us, y is
less than or equal to 4.

00:02:05.630 --> 00:02:07.990
So this is interesting.

00:02:07.990 --> 00:02:12.190
It's really just the inverse of
this statement right here.

00:02:12.190 --> 00:02:14.080
Now, you're saying, instead of y
being greater than a certain

00:02:14.080 --> 00:02:16.700
number, y is less than
a certain number.

00:02:16.700 --> 00:02:19.210
So if you think about it
this way-- so let's go

00:02:19.210 --> 00:02:20.840
back to this equation.

00:02:23.640 --> 00:02:32.480
If y is 4, x is going to be--
let's do a little table.

00:02:32.480 --> 00:02:34.400
y, x.

00:02:34.400 --> 00:02:39.570
If y is equal to 4,
x is equal to 1.5.

00:02:39.570 --> 00:02:43.070
If y is equal to 3,
x is equal to 2.

00:02:43.070 --> 00:02:47.780
If y is equal to 2, then
x is equal to 3.

00:02:47.780 --> 00:02:50.790
So actually, this constraint,
I can pick a bunch of

00:02:50.790 --> 00:02:53.420
different y's that meet
this constraint.

00:02:53.420 --> 00:02:56.220
But it still doesn't tell me
about anything whether x is

00:02:56.220 --> 00:02:57.470
greater than or less than y.

00:02:57.470 --> 00:02:59.540
Here, x is less than y.

00:02:59.540 --> 00:03:01.300
And here, x is greater than y.

00:03:01.300 --> 00:03:03.270
So this statement is useless.

00:03:03.270 --> 00:03:06.060
So statement 1, alone,
is sufficient

00:03:06.060 --> 00:03:07.270
to answer this question.

00:03:07.270 --> 00:03:10.660
Statement 2 doesn't
help us much.

00:03:10.660 --> 00:03:14.300
122.

00:03:14.300 --> 00:03:18.140
If v and w are different
integers, does v equal 0?

00:03:22.040 --> 00:03:23.420
Different integers.

00:03:23.420 --> 00:03:25.090
Remember, it says, different.

00:03:25.090 --> 00:03:33.433
They say that vw is equal
to v squared.

00:03:39.110 --> 00:03:41.420
Now, let's think about this.

00:03:41.420 --> 00:03:46.280
This doesn't tell me for sure
that v is equal to 0.

00:03:46.280 --> 00:03:49.420
And, in fact, as long as-- well,
let's think about this.

00:03:49.420 --> 00:03:55.670
The only way that this could be
true is if v is equal to 0,

00:03:55.670 --> 00:03:56.490
that's one case.

00:03:56.490 --> 00:04:01.770
Now, if v does not equal to 0--
So if v does not equal to

00:04:01.770 --> 00:04:05.760
0, then this case
implies what?

00:04:05.760 --> 00:04:07.110
That soon. w equals v.

00:04:07.110 --> 00:04:07.680
Think about it.

00:04:07.680 --> 00:04:10.430
If v doesn't equal 0, we can
divide both sides by v.

00:04:10.430 --> 00:04:12.440
Because we know we're
not dividing by 0.

00:04:12.440 --> 00:04:15.270
You divide both sides by
v, you get, on the

00:04:15.270 --> 00:04:16.589
left-hand side, w.

00:04:16.589 --> 00:04:18.149
Because this v cancels out.

00:04:18.149 --> 00:04:21.089
And then v squared divided
by v, you get v.

00:04:24.070 --> 00:04:26.330
And so you are left with--

00:04:26.330 --> 00:04:28.710
Well, I want to be careful
here, though.

00:04:28.710 --> 00:04:34.770
Because w would have
to equal v.

00:04:34.770 --> 00:04:36.690
And they already told
us that w and v are

00:04:36.690 --> 00:04:38.060
two different numbers.

00:04:38.060 --> 00:04:43.370
So we know that this situation--
if w equals v,

00:04:43.370 --> 00:04:45.340
then v does not equal 0.

00:04:45.340 --> 00:04:47.870
But they tell us in the problem
statement that w and v

00:04:47.870 --> 00:04:49.520
are two different integers.

00:04:49.520 --> 00:04:52.280
They tell us that w
does not equal v.

00:04:52.280 --> 00:04:55.420
So we know that the only way
that this could be satisfied,

00:04:55.420 --> 00:05:01.120
if w does not equal v, is
if v is equal to 0.

00:05:01.120 --> 00:05:02.780
So statement 1, alone,
is sufficient

00:05:02.780 --> 00:05:05.140
to answer this question.

00:05:05.140 --> 00:05:07.620
Statement 2.

00:05:07.620 --> 00:05:09.170
w is equal to 2.

00:05:12.240 --> 00:05:14.830
Well, that doesn't do
anything for me.

00:05:14.830 --> 00:05:18.030
I mean w could be 2
and v could be 0.

00:05:18.030 --> 00:05:19.990
Or v could be 3.

00:05:19.990 --> 00:05:22.150
All they told us was that v and
w are different integers

00:05:22.150 --> 00:05:24.080
and that they don't
equal each other.

00:05:24.080 --> 00:05:26.220
So this doesn't help me at all
in determining whether v is

00:05:26.220 --> 00:05:26.810
equal to 0.

00:05:26.810 --> 00:05:29.100
So statement 1, alone,
is sufficient.

00:05:32.320 --> 00:05:33.910
I just want to make sure I got
that first part right.

00:05:33.910 --> 00:05:37.290
I mean the only way that this is
true, this is true if v is

00:05:37.290 --> 00:05:38.330
equal to 0.

00:05:38.330 --> 00:05:42.220
Or it's true if w
is equal to v.

00:05:42.220 --> 00:05:45.940
Either of those cases
are true.

00:05:45.940 --> 00:05:47.950
That's the only way it works.

00:05:47.950 --> 00:05:49.640
And we know w does
not equal v.

00:05:49.640 --> 00:05:51.215
So v has to be equal to 0.

00:05:54.770 --> 00:05:55.510
What are we on?

00:05:55.510 --> 00:06:00.680
123.

00:06:00.680 --> 00:06:15.430
What is the value of 36,500
times 1.05 to the nth power?

00:06:15.430 --> 00:06:22.510
All right, statement number 1
tells us, n squared minus 5n

00:06:22.510 --> 00:06:25.730
plus 6 is equal to 0.

00:06:25.730 --> 00:06:26.930
So we could solve for n.

00:06:26.930 --> 00:06:27.940
There will be two solutions.

00:06:27.940 --> 00:06:33.500
So this is n minus 2
times n minus 3.

00:06:33.500 --> 00:06:34.220
I just factored it.

00:06:34.220 --> 00:06:36.920
I said, what two numbers, when
I multiply them, equal

00:06:36.920 --> 00:06:37.630
positive 6?

00:06:37.630 --> 00:06:39.130
So I said, minus 2, minus 3.

00:06:39.130 --> 00:06:40.970
And when you add them
become minus 5.

00:06:40.970 --> 00:06:43.065
That's why I picked the minuses,
as opposed to plus 2

00:06:43.065 --> 00:06:45.320
and plus 3 or some other
combination.

00:06:45.320 --> 00:06:46.440
Equals 0.

00:06:46.440 --> 00:06:50.140
So statement 1 tells us that
n is equal to 2 or

00:06:50.140 --> 00:06:52.020
n is equal to 3.

00:06:52.020 --> 00:06:53.890
So statement 1, by itself,
isn't enough.

00:06:53.890 --> 00:06:55.530
Because I don't know whether
I should put a

00:06:55.530 --> 00:06:56.680
2 here or a 3 here.

00:06:56.680 --> 00:06:59.690
So I can't determine this
value just yet.

00:06:59.690 --> 00:07:07.630
Statement 2 tells us, n minus
2 does not equal 0.

00:07:07.630 --> 00:07:10.730
This statement, by itself,
is fairly useless.

00:07:10.730 --> 00:07:14.430
Because it just tells me that
n does not equal 2.

00:07:14.430 --> 00:07:15.680
So n could be any
other number.

00:07:15.680 --> 00:07:17.150
So I don't know what
number to put here.

00:07:17.150 --> 00:07:20.250
But used in conjunction, these
two statements-- statement 1

00:07:20.250 --> 00:07:22.430
says that it is either 2 or 3.

00:07:22.430 --> 00:07:24.230
Statement 2 says, well,
it's not 2.

00:07:24.230 --> 00:07:26.860
So if you use both of these
together, we know that n is

00:07:26.860 --> 00:07:27.750
equal to 3.

00:07:27.750 --> 00:07:30.010
And then we can, obviously,
solve this problem.

00:07:30.010 --> 00:07:34.190
Taking 1.05 to the third power
and multiplying by 36,500,

00:07:34.190 --> 00:07:37.220
which I won't do because it
would be a waste of time.

00:07:37.220 --> 00:07:43.220
124.

00:07:43.220 --> 00:07:45.200
In the rectangular
coordinate system

00:07:45.200 --> 00:07:48.212
above-- let me draw that.

00:07:48.212 --> 00:07:51.320
They have the y-axis.

00:07:51.320 --> 00:07:54.870
They have the x-axis.

00:07:54.870 --> 00:07:56.020
Then, they draw on a triangle.

00:07:56.020 --> 00:07:58.540
I'll do that in a
different color.

00:07:58.540 --> 00:08:01.140
That and like that.

00:08:04.270 --> 00:08:14.210
And the letters are
O, P, you and Q.

00:08:14.210 --> 00:08:18.000
In the rectangular coordinate
system above, if OP is less

00:08:18.000 --> 00:08:21.260
than PQ-- so OP is
less than PQ.

00:08:21.260 --> 00:08:22.130
Let me write that down.

00:08:22.130 --> 00:08:26.850
OP is less than PQ.

00:08:26.850 --> 00:08:31.100
Is the area of OPQ
greater than 48?

00:08:31.100 --> 00:08:36.039
So we want to know, is
area greater than 48?

00:08:36.039 --> 00:08:37.690
OP is less than PQ.

00:08:37.690 --> 00:08:40.190
So the way I drew this
actually isn't right.

00:08:40.190 --> 00:08:42.080
It's probably a little
bit shifted to the

00:08:42.080 --> 00:08:43.179
left a little bit.

00:08:43.179 --> 00:08:44.620
But fair enough.

00:08:44.620 --> 00:08:45.540
We can visualize.

00:08:45.540 --> 00:08:47.070
We can imagine that.

00:08:47.070 --> 00:08:47.770
So let's see.

00:08:47.770 --> 00:08:48.610
Statement number 1.

00:08:48.610 --> 00:08:53.090
The coordinates of point
P are 6 comma 8.

00:08:53.090 --> 00:08:57.430
So this is 6 comma 8.

00:08:57.430 --> 00:08:57.720
Let's see.

00:08:57.720 --> 00:09:00.230
That tells us the height.

00:09:00.230 --> 00:09:02.246
When we figure out the area of
a triangle, we need to know

00:09:02.246 --> 00:09:04.100
the base times the height.

00:09:04.100 --> 00:09:07.700
And then multiply
that times 1/2.

00:09:07.700 --> 00:09:12.400
The height of this triangle is
the y-coordinate, which is 8.

00:09:12.400 --> 00:09:14.720
So the altitude of this
triangle is 8.

00:09:14.720 --> 00:09:16.440
So we have the height, but we
still don't know the distance

00:09:16.440 --> 00:09:18.210
from O to Q.

00:09:18.210 --> 00:09:21.810
So statement 1, by itself,
isn't much information.

00:09:21.810 --> 00:09:24.380
So that tells that the
point was 6, 8.

00:09:24.380 --> 00:09:26.130
That P was 6, 8.

00:09:26.130 --> 00:09:28.790
Statement 2 says, the
coordinates of point Q

00:09:28.790 --> 00:09:32.860
are 13 comma 0.

00:09:32.860 --> 00:09:33.590
Well, this is helpful.

00:09:33.590 --> 00:09:35.020
13 comma 0.

00:09:35.020 --> 00:09:38.830
So this distance right
here is 13.

00:09:38.830 --> 00:09:40.020
So by itself, that's useless.

00:09:40.020 --> 00:09:42.600
Because if I didn't have the
height, I still wouldn't be

00:09:42.600 --> 00:09:45.190
able to figure out the area
of this triangle.

00:09:45.190 --> 00:09:48.200
But now, if I use both
statements, I know the base

00:09:48.200 --> 00:09:48.940
times the height.

00:09:48.940 --> 00:09:50.650
So the area is going to
be the base times the

00:09:50.650 --> 00:09:51.740
height times 1/2.

00:09:51.740 --> 00:09:53.410
So 8 times 1/2 is 4.

00:09:53.410 --> 00:09:56.170
Times 13 is 52.

00:09:56.170 --> 00:10:00.015
And so 52 is clearly
greater than 48.

00:10:00.015 --> 00:10:01.520
Let me make sure.

00:10:01.520 --> 00:10:03.135
Because I didn't use
this information.

00:10:05.780 --> 00:10:08.410
I didn't use that information
at all.

00:10:11.680 --> 00:10:12.090
Let's see.

00:10:12.090 --> 00:10:14.290
Let me think about it.

00:10:14.290 --> 00:10:14.970
Actually, no.

00:10:14.970 --> 00:10:16.140
I think I'm wrong.

00:10:16.140 --> 00:10:18.250
I think statement 1, alone,
might be enough.

00:10:18.250 --> 00:10:21.860
Let's assume that we don't
have-- these are tricky.

00:10:21.860 --> 00:10:24.750
Let's assume that we don't
have statement 2 there.

00:10:24.750 --> 00:10:27.420
Let me see if I can figure out
whether the area is greater

00:10:27.420 --> 00:10:30.860
than 48 without it.

00:10:30.860 --> 00:10:32.230
Let me see.

00:10:32.230 --> 00:10:34.630
Yeah, I think I'm wrong.

00:10:34.630 --> 00:10:35.920
I caught myself.

00:10:35.920 --> 00:10:41.800
OK, so if this is 6 comma 8,
the area of this triangle

00:10:41.800 --> 00:10:43.800
right here, well, this
distance right here

00:10:43.800 --> 00:10:45.050
is going to be 6.

00:10:47.380 --> 00:10:49.570
This distance right here is 6.

00:10:49.570 --> 00:10:50.960
6 comma 8.

00:10:50.960 --> 00:10:51.330
Right.

00:10:51.330 --> 00:10:55.920
OK, so the area of this
triangle is 6 times 8.

00:10:55.920 --> 00:10:56.930
48 times 1/2.

00:10:56.930 --> 00:11:00.580
So the area of this
triangle is 24.

00:11:00.580 --> 00:11:03.580
But they told us that
OP is less than PQ.

00:11:03.580 --> 00:11:08.020
That this distance is less
than this distance.

00:11:08.020 --> 00:11:09.450
Maybe I should draw it
like that, so you

00:11:09.450 --> 00:11:10.940
can kind of see it.

00:11:10.940 --> 00:11:13.780
And the only way that this
distance is less than this

00:11:13.780 --> 00:11:20.150
distance is if this distance
is less than this distance.

00:11:20.150 --> 00:11:21.840
This is a tricky problem.

00:11:21.840 --> 00:11:25.380
This has to be less than this.

00:11:25.380 --> 00:11:26.620
Q has to be further away.

00:11:26.620 --> 00:11:28.310
Because that's the only way
you're going to get this

00:11:28.310 --> 00:11:30.390
altitude to be longer
than this altitude.

00:11:30.390 --> 00:11:32.700
Because you're assuming that
Q is along the axes.

00:11:32.700 --> 00:11:35.890
So that's the only thing that
you can shift back and forth.

00:11:35.890 --> 00:11:39.860
So if this distance is larger
than this distance, then the

00:11:39.860 --> 00:11:42.550
area of this triangle is going
to be at least 24.

00:11:42.550 --> 00:11:46.250
So the area is going to
be greater than 48.

00:11:46.250 --> 00:11:50.400
So actually, all you needed
was statement 1

00:11:50.400 --> 00:11:52.130
to solve this problem.

00:11:52.130 --> 00:11:53.620
That's all you needed.

00:11:53.620 --> 00:11:56.170
Statement 1 gave you
enough information.

00:11:56.170 --> 00:11:59.200
Statement 2 does not, by
itself, give you enough

00:11:59.200 --> 00:12:01.670
information because you don't
know how high P is.

00:12:01.670 --> 00:12:03.947
It could be really low down
here, in which case the area

00:12:03.947 --> 00:12:04.860
is going to be less than 48.

00:12:04.860 --> 00:12:06.920
It could be really high, in
which case the area would be

00:12:06.920 --> 00:12:08.150
greater than 48.

00:12:08.150 --> 00:12:11.090
But anyway, see, I almost
got that one wrong.

00:12:11.090 --> 00:12:11.810
But I'll leave you there.

00:12:11.810 --> 00:12:12.830
I'll see you in the
next video.

00:12:12.830 --> 00:12:15.790
Statement 1, alone, was
sufficient for this one.

00:12:15.790 --> 00:12:17.040
See


GMAT: Data sufficiency 28

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:07.310
All right we're on problem
113 on page 287.

00:00:07.310 --> 00:00:09.340
So they've drawn this thing.

00:00:09.340 --> 00:00:11.570
Let me see if I could draw it.

00:00:11.570 --> 00:00:14.872
So we have a right triangle
in the center.

00:00:14.872 --> 00:00:16.552
So that's the side.

00:00:16.552 --> 00:00:20.140
That's the other side,
and then the base.

00:00:22.900 --> 00:00:24.380
Right triangle.

00:00:24.380 --> 00:00:26.880
And then they drew some
squares on every

00:00:26.880 --> 00:00:28.180
side of that triangle.

00:00:28.180 --> 00:00:33.730
Let me see if I can do that.

00:00:33.730 --> 00:00:34.980
That's one square.

00:00:38.500 --> 00:00:39.950
Well, I don't know how I'm going
to draw the other square

00:00:39.950 --> 00:00:41.490
because I don't know how
to draw an angled one.

00:00:41.490 --> 00:00:44.604
I'll just draw the lines.

00:00:44.604 --> 00:00:47.110
That's the other square.

00:00:47.110 --> 00:00:48.480
And the last one.

00:00:48.480 --> 00:00:52.330
I'll just draw in some lines
something like this.

00:00:52.330 --> 00:01:00.472
It comes up like that, like
that and like that.

00:01:00.472 --> 00:01:02.130
So that's what they have.

00:01:02.130 --> 00:01:08.930
And they call this A, this B,
C, D, and they tell us that

00:01:08.930 --> 00:01:12.180
this is a right angle.

00:01:12.180 --> 00:01:15.880
They say, in the figure above,
if the area of triangular

00:01:15.880 --> 00:01:25.070
region D is 4, what is the
length of a side of

00:01:25.070 --> 00:01:27.090
square region A?

00:01:31.790 --> 00:01:36.830
So all of these are-- well, they
don't tell us that this

00:01:36.830 --> 00:01:39.130
and this are a square region
just yet, but they tell us

00:01:39.130 --> 00:01:41.230
this is a square region.

00:01:41.230 --> 00:01:44.120
So the length of this side-- if
we know the length of this

00:01:44.120 --> 00:01:48.670
side and that side, then we can
figure out the length of

00:01:48.670 --> 00:01:50.520
the hypotenuse of this
right triangle, so

00:01:50.520 --> 00:01:51.530
that's all we need.

00:01:51.530 --> 00:01:57.570
And let's see, let's call this
length right here little b.

00:01:57.570 --> 00:01:59.870
And let's call this
length little c.

00:01:59.870 --> 00:02:03.905
So we already know that, if we
know the area of this triangle

00:02:03.905 --> 00:02:07.800
is 4, we know that little b
times little c, so height

00:02:07.800 --> 00:02:12.600
times base times 1/2,
is equal to 4.

00:02:12.600 --> 00:02:14.600
But that also tells us that
b be times little

00:02:14.600 --> 00:02:16.890
c is equal to 8.

00:02:16.890 --> 00:02:17.910
I don't know that does for us.

00:02:17.910 --> 00:02:19.410
Let's read the statements.

00:02:19.410 --> 00:02:21.470
Statement number 1.

00:02:21.470 --> 00:02:25.190
The area of square
region B is 9.

00:02:25.190 --> 00:02:28.150
So B is equal to 9.

00:02:28.150 --> 00:02:29.460
Well, that's useful.

00:02:29.460 --> 00:02:33.250
If the area of this region,
and it's a square, is 9,

00:02:33.250 --> 00:02:35.220
what's each of the sides
going to be?

00:02:35.220 --> 00:02:37.800
The sides are going to be
the square root of that.

00:02:37.800 --> 00:02:40.550
Little b is going to
be equal to 3.

00:02:40.550 --> 00:02:43.160
If little b is equal to
3, then the area of

00:02:43.160 --> 00:02:45.090
this is 3 times 3.

00:02:45.090 --> 00:02:47.760
We can do that because it's a
square, which would be 9.

00:02:47.760 --> 00:02:51.740
So if little b is 3,
then what is C?

00:02:51.740 --> 00:02:57.630
Well, then we get that C is
equal to 8/3 using this

00:02:57.630 --> 00:02:59.260
information.

00:02:59.260 --> 00:03:01.520
And we got that, remember, just
based on the fact that

00:03:01.520 --> 00:03:06.550
the area of this triangle
right here is 4.

00:03:06.550 --> 00:03:10.910
So little b times little c times
1/2 would equal to 4.

00:03:10.910 --> 00:03:13.700
So, if we know what little b and
we know what little c is,

00:03:13.700 --> 00:03:15.880
then we can solve for-- let's
call that little a.

00:03:15.880 --> 00:03:18.410
And little a is what the
whole point was about.

00:03:18.410 --> 00:03:21.520
What is the length of a side
of the square region A?

00:03:21.520 --> 00:03:26.220
So we could just solve for b
squared plus c squared is

00:03:26.220 --> 00:03:28.550
equal to a squared, right?

00:03:28.550 --> 00:03:30.590
This squared soon. plus this
squared is equal to the

00:03:30.590 --> 00:03:32.330
hypotenuse squared.

00:03:32.330 --> 00:03:33.880
And we just solve for
a and we'd be done.

00:03:33.880 --> 00:03:36.430
So statement one alone
is sufficient.

00:03:36.430 --> 00:03:38.570
And this is really messy,
especially taking an 8/3

00:03:38.570 --> 00:03:40.110
squared and all that, so
I'm not going to do it.

00:03:40.110 --> 00:03:43.030
And it would be a waste of
time on the GMAT anyway.

00:03:43.030 --> 00:03:44.380
Let's look at statement
number two.

00:03:48.280 --> 00:03:53.020
The area of square region
C is 64/9, so C is

00:03:53.020 --> 00:03:55.240
equal to 64 over 9.

00:03:55.240 --> 00:03:57.760
So this is kind of very similar
to the information

00:03:57.760 --> 00:03:58.860
they gave us in statement one.

00:03:58.860 --> 00:04:02.530
If the area of this is 64/9 and
this is a square region,

00:04:02.530 --> 00:04:05.480
then the length of each of its
sides, or in this case, little

00:04:05.480 --> 00:04:08.430
c, the length of little c is
just going to be the square

00:04:08.430 --> 00:04:08.940
root of that.

00:04:08.940 --> 00:04:12.690
So little c is going to be 8/3,
which is exactly what we

00:04:12.690 --> 00:04:13.790
figured out here.

00:04:13.790 --> 00:04:15.680
And then we could use this
information that we got from

00:04:15.680 --> 00:04:19.390
the problem statement that b
times c is going to be equal

00:04:19.390 --> 00:04:21.800
to 8, and we got that from the
fact that the area of this

00:04:21.800 --> 00:04:23.560
triangle is 4.

00:04:23.560 --> 00:04:29.410
So we use that and we can figure
out that b times 8/3 is

00:04:29.410 --> 00:04:31.220
equal to 8.

00:04:31.220 --> 00:04:34.400
And then you get b is equal to
3, and then you apply this

00:04:34.400 --> 00:04:36.570
again to solve for a.

00:04:36.570 --> 00:04:39.930
So each of the statements one
and two are each individually

00:04:39.930 --> 00:04:43.730
sufficient to answer
this question.

00:04:43.730 --> 00:04:46.110
Next problem.

00:04:46.110 --> 00:04:50.500
114.

00:04:50.500 --> 00:04:54.210
If x is to be selected at random
from set T, what is the

00:04:54.210 --> 00:05:04.490
probability that 1/4x minus 5
is less than or equal to 0?

00:05:04.490 --> 00:05:07.210
Let's simplify this real fast.
That means that what's the

00:05:07.210 --> 00:05:11.310
probably that 1/4x
is less than 5?

00:05:11.310 --> 00:05:12.850
And that means what's
the probability?

00:05:12.850 --> 00:05:14.290
Let's multiply both
sides by 4.

00:05:14.290 --> 00:05:16.590
I just added 5 to
both sides here.

00:05:16.590 --> 00:05:18.230
And let's multiply
both sides by 4.

00:05:18.230 --> 00:05:20.540
And we don't have to change the
inequality since 4 is a

00:05:20.540 --> 00:05:21.670
positive number.

00:05:21.670 --> 00:05:23.480
So it's the same thing as what's
the probability that x

00:05:23.480 --> 00:05:25.630
is less than 20?

00:05:25.630 --> 00:05:27.770
And I like this a lot more
because it's just simple and

00:05:27.770 --> 00:05:31.920
it lets my brain grasp
the problem better.

00:05:31.920 --> 00:05:34.990
Now, what are the statements
they give us?

00:05:34.990 --> 00:05:39.020
Statement one: T is a
set of 8 integers.

00:05:39.020 --> 00:05:40.870
T has 8 integers.

00:05:45.730 --> 00:05:48.430
That by itself doesn't help me
because I don't know what

00:05:48.430 --> 00:05:49.580
those eight integers
look like.

00:05:49.580 --> 00:05:55.740
I mean, maybe all eight could
be larger than 20, in which

00:05:55.740 --> 00:05:57.990
case, it would be impossible to
find one that's less than

00:05:57.990 --> 00:06:00.320
or equal to 20, or maybe all 8
are less than 20, in which

00:06:00.320 --> 00:06:02.160
case, you would have
100% chance.

00:06:02.160 --> 00:06:05.480
So statement one by itself
is fairly useless.

00:06:05.480 --> 00:06:06.910
Statement two.

00:06:06.910 --> 00:06:21.890
T is contained in the set of
integers from 1 to 25.

00:06:21.890 --> 00:06:23.690
So that also doesn't help me.

00:06:23.690 --> 00:06:29.910
So T within 1 to 25.

00:06:29.910 --> 00:06:32.500
So they're essentially telling
us that every number in T is

00:06:32.500 --> 00:06:33.930
between this range.

00:06:33.930 --> 00:06:36.720
But if we just take statement
two alone, they're not telling

00:06:36.720 --> 00:06:39.360
us, first of all, how many
numbers are in T.

00:06:39.360 --> 00:06:40.750
Statement one tells us that,
but statement two

00:06:40.750 --> 00:06:41.740
doesn't tell us.

00:06:41.740 --> 00:06:44.900
And then even if we knew how
many numbers were in T, we

00:06:44.900 --> 00:06:46.200
don't know where they're
distributed.

00:06:46.200 --> 00:06:50.010
Maybe all of the numbers in T
are less than 20, in which

00:06:50.010 --> 00:06:52.144
case, there is 100% chance that
you would select one that

00:06:52.144 --> 00:06:53.150
was less than 20.

00:06:53.150 --> 00:06:57.660
But maybe 4 of them are
greater than 20.

00:06:57.660 --> 00:07:00.470
you Maybe you have
21, 22, 23, 24.

00:07:00.470 --> 00:07:03.400
And four of them are less
than 20, or maybe

00:07:03.400 --> 00:07:04.280
something in between.

00:07:04.280 --> 00:07:07.670
So given that, you still don't
know where T is distributed

00:07:07.670 --> 00:07:08.900
relative to 20.

00:07:08.900 --> 00:07:12.740
So I say that we still don't
have enough information, even

00:07:12.740 --> 00:07:16.300
with both of these statements,
to solve this problem.

00:07:16.300 --> 00:07:19.200
Next question.

00:07:19.200 --> 00:07:19.810
115.

00:07:19.810 --> 00:07:23.470
It takes the most confidence to
choose that E, to say that

00:07:23.470 --> 00:07:26.040
one cannot solve this problem.

00:07:26.040 --> 00:07:28.720
115.

00:07:28.720 --> 00:07:29.950
I'm always second-guessing
myself.

00:07:29.950 --> 00:07:31.440
I'm like, well, let me make
sure I haven't missed some

00:07:31.440 --> 00:07:32.260
piece of information.

00:07:32.260 --> 00:07:33.370
No, but I'm pretty
confident here.

00:07:33.370 --> 00:07:33.850
All right.

00:07:33.850 --> 00:07:35.020
115.

00:07:35.020 --> 00:07:44.370
If Sarah's age is exactly twice
Bill's age, what is

00:07:44.370 --> 00:07:45.160
Sarah's age?

00:07:45.160 --> 00:07:47.600
They want to know Sarah's age.

00:07:47.600 --> 00:07:48.845
S for Sarah's age.

00:07:48.845 --> 00:07:52.490
Let me scroll down
a little bit.

00:07:52.490 --> 00:07:56.750
Problem statement number one
says four years ago, Sarah's

00:07:56.750 --> 00:07:59.740
age was exactly three
times Bill's age.

00:07:59.740 --> 00:08:03.380
So Sarah's age four years
ago, so Sarah minus 4.

00:08:03.380 --> 00:08:04.640
S is Sarah's age today.

00:08:04.640 --> 00:08:13.090
So Sarah minus 4 was exactly
three times Bill's age, so was

00:08:13.090 --> 00:08:18.230
three times Bill's age then,
so B minus 4, right?

00:08:18.230 --> 00:08:20.565
Sarah's age four years ago
was three times Bill's

00:08:20.565 --> 00:08:22.060
age four years ago.

00:08:22.060 --> 00:08:23.010
Fair enough.

00:08:23.010 --> 00:08:25.070
So just with this and this,
can I solve for S?

00:08:25.070 --> 00:08:25.850
Well, sure.

00:08:25.850 --> 00:08:29.850
I have two linear equations
and two unknowns, so I can

00:08:29.850 --> 00:08:31.530
solve for either variable.

00:08:31.530 --> 00:08:35.880
And to prove it to you, I'll
do it, although this, once

00:08:35.880 --> 00:08:41.169
again, is a waste of time from
the point of view of the GMAT.

00:08:41.169 --> 00:08:46.660
S is equal to 3B minus 8.

00:08:46.660 --> 00:08:51.680
And then B-- from there we get
B is equal to S/2, so S is

00:08:51.680 --> 00:08:53.000
equal to 3, right?

00:08:53.000 --> 00:08:56.140
I just divide 2 into
both sides.

00:08:56.140 --> 00:09:02.400
So 3S/2 minus 8.

00:09:02.400 --> 00:09:03.960
What can I do here?

00:09:03.960 --> 00:09:05.740
So I have 1 and 1/2 S.

00:09:05.740 --> 00:09:07.140
I could subtract it
from both sides.

00:09:07.140 --> 00:09:11.580
I'll have minus-- I'm doing it
fast just to-- minus 1/2S is

00:09:11.580 --> 00:09:13.230
equal to minus 8.

00:09:13.230 --> 00:09:16.730
And then multiply both sides by
minus 2, and you get Sarah

00:09:16.730 --> 00:09:18.480
is 16 years old.

00:09:18.480 --> 00:09:19.300
And you're done.

00:09:19.300 --> 00:09:20.960
But you didn't have
to do any of this.

00:09:20.960 --> 00:09:22.260
That was all a waste of time.

00:09:22.260 --> 00:09:25.155
You just had to see that you had
two linear equations and

00:09:25.155 --> 00:09:27.230
two unknowns, which is
enough to solve for

00:09:27.230 --> 00:09:30.120
either of the variables.

00:09:30.120 --> 00:09:35.590
Statement two tells us, eight
years from now, Sarah's age

00:09:35.590 --> 00:09:38.690
will be exactly 1.5
times Bill's age.

00:09:38.690 --> 00:09:39.770
So, same thing.

00:09:39.770 --> 00:09:46.860
Sarah's age eight years from now
is going to be equal to 1

00:09:46.860 --> 00:09:53.050
and 1/2 times Bill's age
eight years from now.

00:09:53.050 --> 00:09:53.730
Same thing.

00:09:53.730 --> 00:09:57.060
We have two linear equations,
one that they gave in the

00:09:57.060 --> 00:09:58.810
problem statement and one
they gave in statement

00:09:58.810 --> 00:10:00.330
two, and two unknowns.

00:10:00.330 --> 00:10:03.300
No reason why you can't solve
for both variables.

00:10:03.300 --> 00:10:04.290
And you might want to do it for

00:10:04.290 --> 00:10:05.710
yourself just as an exercise.

00:10:05.710 --> 00:10:08.100
But you'd do it the same way
you did it right here.

00:10:08.100 --> 00:10:10.210
I always find substitution to
be the easiest, especially

00:10:10.210 --> 00:10:14.870
when one of the equations is
really easy to manipulate.

00:10:14.870 --> 00:10:17.260
So each statement individually
is sufficient

00:10:17.260 --> 00:10:19.140
to solve this problem.

00:10:19.140 --> 00:10:20.750
Next problem.

00:10:20.750 --> 00:10:23.220
116.

00:10:23.220 --> 00:10:24.560
Oh, I just realized
I'm out of time.

00:10:24.560 --> 00:10:26.040
I'll do 116 in the next video.

00:10:26.040 --> 00:10:27.290
See