GMAT: Math 39

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:05.260
We're on problem 195 and they
have drawn a little

00:00:05.260 --> 00:00:06.230
bit of a grid here.

00:00:06.230 --> 00:00:08.510
Let me try to draw it as well.

00:00:08.510 --> 00:00:12.646
1, 2, 3.

00:00:12.646 --> 00:00:15.300
And then they do this in
the other direction.

00:00:15.300 --> 00:00:20.050
1, 2, 3.

00:00:20.050 --> 00:00:23.900
And they call this
First Street.

00:00:23.900 --> 00:00:25.660
And there's actually 4
going that direction.

00:00:25.660 --> 00:00:27.350
Let me draw that one.

00:00:27.350 --> 00:00:31.390
So, there's another
street like that.

00:00:31.390 --> 00:00:33.710
And they call this First Street,
Second Street, Third

00:00:33.710 --> 00:00:35.340
Street, Fourth Street.

00:00:35.340 --> 00:00:40.200
And Avenue A, B, and C.

00:00:40.200 --> 00:00:42.140
And they say that this point
right here-- I'll do that in a

00:00:42.140 --> 00:00:43.450
different color.

00:00:43.450 --> 00:00:46.690
That point right there
is x, and that point

00:00:46.690 --> 00:00:49.830
right there is y.

00:00:49.830 --> 00:00:52.590
Pat will walk from intersection
x to intersection

00:00:52.590 --> 00:00:55.970
y along a route that is confined
to the square grid of

00:00:55.970 --> 00:00:58.320
4 streets and 3 avenues
shown on the map.

00:00:58.320 --> 00:01:03.150
How many routes from x to y
can Pat take that have the

00:01:03.150 --> 00:01:04.450
minimum possible length?

00:01:04.450 --> 00:01:06.120
So essentially the minimum
possible length so he doesn't

00:01:06.120 --> 00:01:08.930
waste time.

00:01:08.930 --> 00:01:10.550
So let's think about
it this way.

00:01:10.550 --> 00:01:15.920
So he could go this way-- so we
could think about how many

00:01:15.920 --> 00:01:19.580
different ways can you get to
each point on this graph.

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So to get there, there's
one way.

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One possible way.

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To get there-- this seems to be
a very similar problem to

00:01:30.595 --> 00:01:32.390
what I saw on a Computer
Science exam a

00:01:32.390 --> 00:01:34.280
long, long time ago.

00:01:34.280 --> 00:01:37.170
But there's one way
to get there.

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That's obvious.

00:01:38.300 --> 00:01:41.500
So, how many ways are
there to get here?

00:01:41.500 --> 00:01:45.780
Well, you can view it as-- well,
to go from this path to

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this path and this path to this
path, you can sum the two

00:01:50.170 --> 00:01:51.050
ways to get there.

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So it's 2.

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Which makes sense.

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You could go like that or
you could go like that.

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How many ways are there
to go here?

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Well, there's only one
way to go here.

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Likewise only one
way to go here.

00:02:01.720 --> 00:02:04.400
Now this is where it
gets interesting.

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How many ways are there
to go here?

00:02:09.229 --> 00:02:12.770
I can either come from there,
and there's only one way to

00:02:12.770 --> 00:02:14.890
get there, so that's one path.

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Or I could come from here.

00:02:17.790 --> 00:02:19.800
But there's two ways
to get here.

00:02:19.800 --> 00:02:23.180
So there's two ways to get
here via this path, and

00:02:23.180 --> 00:02:25.020
there's one way to get
here via this path.

00:02:25.020 --> 00:02:27.110
So there are three
ways to get here.

00:02:27.110 --> 00:02:29.750
So essentially I just added
the one and the two.

00:02:29.750 --> 00:02:31.360
You think of the same
logic here.

00:02:31.360 --> 00:02:32.920
How many ways are there
to get there?

00:02:32.920 --> 00:02:37.620
Well, I can come via this path,
and that'll be-- there's

00:02:37.620 --> 00:02:39.720
only one way to get here so
there's only one way to get

00:02:39.720 --> 00:02:41.130
from this direction.

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Or I could come from this
direction, but there's two

00:02:43.130 --> 00:02:43.910
ways to get here.

00:02:43.910 --> 00:02:46.240
So there's two ways to come
from this direction.

00:02:46.240 --> 00:02:50.200
So, there's three ways to
get from that direction.

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If you use the same logic,
there's one way to get here.

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You just go straight up.

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How many ways are there
to get here?

00:02:57.470 --> 00:02:59.960
Well, one way to get there,
three ways to get there, so

00:02:59.960 --> 00:03:02.460
there are four ways to get
here because you can go

00:03:02.460 --> 00:03:05.200
through that one way or through
these three ways.

00:03:05.200 --> 00:03:07.130
So how many ways are
there to get here?

00:03:07.130 --> 00:03:09.890
You go three ways to there,
three ways to there, and I can

00:03:09.890 --> 00:03:11.420
come from either here or here.

00:03:11.420 --> 00:03:14.400
So there are six ways.

00:03:14.400 --> 00:03:15.960
So, how many ways to get to y?

00:03:15.960 --> 00:03:20.040
Four ways to get here, six ways
to get here, 6 plus 4.

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Ten ways to get here.

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I could either come from one
of the six ways from this

00:03:23.660 --> 00:03:26.510
direction, or one of the four
ways from this direction.

00:03:26.510 --> 00:03:27.980
So ten ways.

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And that is choice C.

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Next problem.

00:03:33.720 --> 00:03:38.220
196.

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The ratio, by volume, of soap
to alcohol to water in a

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certain solution is
equal to 2:50:100.

00:03:52.850 --> 00:03:55.940
The solution will be altered so
that the ratio of soap to

00:03:55.940 --> 00:03:59.750
alcohol is doubled,
while the ratio of

00:03:59.750 --> 00:04:02.243
soap to water is halved.

00:04:22.700 --> 00:04:30.010
If the altered solution will
contain 100 cubic centimeters

00:04:30.010 --> 00:04:36.950
of alcohol, how many cubic
centimeters of

00:04:36.950 --> 00:04:39.710
water will it contain?

00:04:39.710 --> 00:04:42.260
So, let's think about what
the new ratio is.

00:04:42.260 --> 00:04:56.100
The old ratio of soap
to alcohol was 2:50.

00:04:56.100 --> 00:04:58.390
Now they want to double
this ratio.

00:04:58.390 --> 00:05:03.360
So the new ratio of soap to
alcohol is going to be 4:50.

00:05:03.360 --> 00:05:04.650
Fair enough.

00:05:04.650 --> 00:05:12.110
Now the old ratio of soap
to water was 2:100.

00:05:12.110 --> 00:05:14.500
Now what's the new ratio
of soap to water?

00:05:14.500 --> 00:05:16.540
It's going to be halved.

00:05:16.540 --> 00:05:19.040
So, that equals soap to water.

00:05:19.040 --> 00:05:21.580
We want to half this ratio, so
now the new ratio of soap to

00:05:21.580 --> 00:05:25.770
water is 1:100.

00:05:25.770 --> 00:05:27.120
So let's see how we can
think about it.

00:05:30.220 --> 00:05:33.000
Let's try this as a ratio
of 4 to something.

00:05:33.000 --> 00:05:33.700
So that's the same.

00:05:33.700 --> 00:05:41.330
1:100 is the same
thing as 4:400.

00:05:41.330 --> 00:05:44.010
So, let's see if we can
rewrite these ratios.

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So the ratio of soap to alcohol
to water now becomes

00:05:51.050 --> 00:06:01.260
4:50 and the ratio of soap
to water is 4:400.

00:06:01.260 --> 00:06:05.160
So, they give us alcohol and
they want to figure out water.

00:06:05.160 --> 00:06:14.370
So the ratio of alcohol to
water is equal to 50:400.

00:06:14.370 --> 00:06:18.070
And that's another way
of saying 1:8.

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Alcohol to water is
equal to 1:8.

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And they're saying that
we have 100 cubic

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centimeters of alcohol.

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So, how much water?

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So 100 over water
is equal to 1:8.

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And you can even eyeball that.

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That's 100 over 800.

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So, water has to be equal to
800 cubic centimeters.

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And that's choice E.

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Next question.

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197.

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If 75% of a class answered the
first question on a test

00:06:57.850 --> 00:07:05.820
correctly-- so 75%, let's say
3/4, question one correctly.

00:07:08.370 --> 00:07:10.290
55% answered the second

00:07:10.290 --> 00:07:11.970
question on the test correctly.

00:07:11.970 --> 00:07:13.110
Let me write it this way.

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Question one, 75% true.

00:07:16.180 --> 00:07:21.070
Question two, 55% answered the
second question correctly, and

00:07:21.070 --> 00:07:24.290
20% answered neither of the
questions correctly.

00:07:24.290 --> 00:07:27.580
So three, 20%.

00:07:27.580 --> 00:07:28.500
So, that's not question three.

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20% neither.

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What percent answered
both correctly?

00:07:39.240 --> 00:07:45.970
So 3/4 answered question one
correctly and 1/4 answered it

00:07:45.970 --> 00:07:46.890
incorrectly.

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Q1, they got it wrong.

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They're telling us that 20%
got neither correct.

00:07:55.820 --> 00:07:59.300
So that means that they
got Q2 wrong, as well.

00:08:02.510 --> 00:08:04.900
And this is 20% of the
entire population.

00:08:04.900 --> 00:08:06.230
So this is 1/5.

00:08:08.830 --> 00:08:11.280
This is 1/5 of the entire
population.

00:08:11.280 --> 00:08:14.800
So my question is, if this is
1/5 of the entire population,

00:08:14.800 --> 00:08:17.900
what fraction of this
population was it?

00:08:17.900 --> 00:08:23.270
Well, of these people, the
fraction that got Q2

00:08:23.270 --> 00:08:25.320
incorrect, let's write
that as x.

00:08:25.320 --> 00:08:31.710
So, x times 1/4 is
equal to 1/5.

00:08:31.710 --> 00:08:33.150
Multiply both sides
by a fourth, you

00:08:33.150 --> 00:08:35.730
get x is equal 4/5.

00:08:35.730 --> 00:08:38.809
So, what we know is since this
is 1/5 of the entire

00:08:38.809 --> 00:08:42.600
population, that of the people
who got question one

00:08:42.600 --> 00:08:49.930
incorrect, 4/5 also got question
two incorrect.

00:08:49.930 --> 00:08:51.350
And if you want to know what
percentage that is of the

00:08:51.350 --> 00:08:53.670
whole population, multiply
1/4 times 4/5.

00:08:53.670 --> 00:08:56.890
And you get 1/5 which is that
data that they had given us.

00:08:56.890 --> 00:08:57.650
But this helps us.

00:08:57.650 --> 00:09:00.250
Because this tells us, of these
people who got question

00:09:00.250 --> 00:09:02.660
one incorrect, what fraction
got it right?

00:09:02.660 --> 00:09:04.440
Got question two right?

00:09:04.440 --> 00:09:09.280
Well, if 4/5 got it wrong, 1/5
got question two correct.

00:09:15.480 --> 00:09:17.350
So let me just take
a step back.

00:09:17.350 --> 00:09:18.940
I want to make sure I'm not
doing this the slowest

00:09:18.940 --> 00:09:19.450
possible way.

00:09:19.450 --> 00:09:25.060
They tell us that 75% got
question one right and 55% got

00:09:25.060 --> 00:09:27.350
question two right.

00:09:27.350 --> 00:09:28.730
So let's think about it.

00:09:28.730 --> 00:09:38.590
So of this, some percentage
got question two wrong.

00:09:38.590 --> 00:09:39.720
Actually, let's think
of it this way.

00:09:39.720 --> 00:09:43.770
Of this, some percentage
got question two right.

00:09:43.770 --> 00:09:45.850
Let's call this y.

00:09:49.440 --> 00:09:56.740
Let's call that y and let's
call this-- I'm

00:09:56.740 --> 00:09:58.010
going to use x again.

00:09:58.010 --> 00:10:00.220
Let's call this x.

00:10:00.220 --> 00:10:00.850
I'll use x.

00:10:00.850 --> 00:10:02.720
This is a different x.

00:10:02.720 --> 00:10:05.920
So, first of all, what is the
total proportion of people who

00:10:05.920 --> 00:10:09.460
got question one wrong and
then question two right?

00:10:09.460 --> 00:10:12.150
Well, it's 1/4 times 1/5.

00:10:12.150 --> 00:10:17.260
So, it's 1/20 of the people
got question one wrong and

00:10:17.260 --> 00:10:18.450
question two right.

00:10:18.450 --> 00:10:21.030
And that's equal to x.

00:10:21.030 --> 00:10:25.690
Now, what percentage of the
people got question one right

00:10:25.690 --> 00:10:26.470
and question two wrong?

00:10:26.470 --> 00:10:29.510
Well, that's going to be y.

00:10:29.510 --> 00:10:32.510
Now if you think about--
and y is actually what

00:10:32.510 --> 00:10:33.160
we're solving for.

00:10:33.160 --> 00:10:35.130
They want to know how
many students

00:10:35.130 --> 00:10:37.900
answered both correctly.

00:10:37.900 --> 00:10:43.850
So, if you think about it, y
plus 1/20, that represents all

00:10:43.850 --> 00:10:47.360
of the people who answered
question two correctly.

00:10:47.360 --> 00:10:49.140
All the people who answered
question two correctly-- I

00:10:49.140 --> 00:10:51.240
know it's a little confusing--
are those

00:10:51.240 --> 00:10:54.950
people and those people.

00:10:54.950 --> 00:10:56.720
That's all the question twos
answered correctly.

00:10:56.720 --> 00:10:59.660
We already know that this is
1/20 of the population.

00:10:59.660 --> 00:11:05.970
And we know that y plus 1/20 of
the population is equal to

00:11:05.970 --> 00:11:08.610
55% of the population.

00:11:08.610 --> 00:11:11.440
So, we can write that
as 55 over 100.

00:11:11.440 --> 00:11:15.430
Or, we could divide both by
5 and you get 11 over 20.

00:11:15.430 --> 00:11:19.490
This is 11/20 of
the population.

00:11:19.490 --> 00:11:20.930
So, what is y?

00:11:20.930 --> 00:11:24.580
Well, subtract 1/20 from both
sides. y is equal to-- 11/20

00:11:24.580 --> 00:11:31.720
minus 1/20 is equal to 10/20,
which is equal to 1/2.

00:11:31.720 --> 00:11:37.050
So, 1/2 of the population got
both the first and the second

00:11:37.050 --> 00:11:38.760
question right.

00:11:38.760 --> 00:11:41.120
Actually, we didn't even have to
use-- well, we did have to

00:11:41.120 --> 00:11:43.990
use this 3/4 information
to get this 1/4 here.

00:11:43.990 --> 00:11:44.770
But anyway.

00:11:44.770 --> 00:11:46.390
And let's see if that's
one of the choices.

00:11:46.390 --> 00:11:46.740
Right.

00:11:46.740 --> 00:11:47.170
D.

00:11:47.170 --> 00:11:49.100
50%.

00:11:49.100 --> 00:11:50.370
And I'm all out of time.

00:11:50.370 --> 00:11:52.380
See you in the next video.


GMAT: Math 38

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:05.550
We're on problem 190.

00:00:05.550 --> 00:00:10.810
If the operation star with a
circle around it is defined

00:00:10.810 --> 00:00:15.850
for all a and b by the equation,
a star with a circle

00:00:15.850 --> 00:00:24.120
around it b is equal to a
squared times b over 3, then

00:00:24.120 --> 00:00:31.800
what does 2 star with a circle
around it 3 star with the

00:00:31.800 --> 00:00:35.000
circle around it
minus 1 equal?

00:00:35.000 --> 00:00:37.270
So 3 star minus 1.

00:00:37.270 --> 00:00:41.800
That's equal to-- let's see,
I'll do this in magenta-- 3

00:00:41.800 --> 00:00:42.460
star minus 1.

00:00:42.460 --> 00:00:49.730
That's equal to 3 squared times
minus 1 which is minus 3

00:00:49.730 --> 00:00:52.670
squared, or minus 9.

00:00:52.670 --> 00:00:54.220
3 squared, 9.

00:00:54.220 --> 00:00:54.940
Times minus 1.

00:00:54.940 --> 00:00:59.750
That's minus 9 divided by 3
which is equal to minus 3.

00:00:59.750 --> 00:01:04.980
So this reduces to 2 star minus
3 which is equal to 2

00:01:04.980 --> 00:01:09.630
squared, which is 4,
times minus 3.

00:01:09.630 --> 00:01:12.050
All of that over 3.

00:01:12.050 --> 00:01:13.530
Minus 3 divided by 3.

00:01:13.530 --> 00:01:15.510
That's just minus 1.

00:01:15.510 --> 00:01:18.350
So you have minus 1 times
4 is equal to minus 4.

00:01:18.350 --> 00:01:22.280
And that's choice E.

00:01:22.280 --> 00:01:28.920
Problem 192.

00:01:28.920 --> 00:01:30.180
No, 191!

00:01:30.180 --> 00:01:33.870
Don't want to skip problems.
The inside dimensions of a

00:01:33.870 --> 00:01:44.990
rectangular wooden box
are 6 by 8 by 10.

00:01:44.990 --> 00:01:48.270
A cylindrical canister is to be
placed inside the box so it

00:01:48.270 --> 00:01:52.120
stands upright when the
closed box rests on

00:01:52.120 --> 00:01:53.480
one of its 6 bases.

00:02:02.480 --> 00:02:06.040
Of all such canisters that could
be used, what is the

00:02:06.040 --> 00:02:12.990
radius in inches of the one
that has maximum volume?

00:02:12.990 --> 00:02:17.865
OK, so they're going to put
one canister in this box.

00:02:22.370 --> 00:02:24.270
And how do you maximize
its volume?

00:02:28.390 --> 00:02:30.790
So let's think about
it this way.

00:02:30.790 --> 00:02:35.100
Let's think about what a volume
of a cylinder is.

00:02:35.100 --> 00:02:41.730
If this is my cylinder, its
volume is the area of the side

00:02:41.730 --> 00:02:43.980
times the height.

00:02:43.980 --> 00:02:47.480
If the area of the side is
equal to-- if this is the

00:02:47.480 --> 00:02:58.090
radius-- it would be pi r
squared times the height.

00:02:58.090 --> 00:03:00.960
Height times pi r squared.

00:03:00.960 --> 00:03:04.210
Now what's r going
to be equal to?

00:03:04.210 --> 00:03:06.185
r has to go all the
way around.

00:03:06.185 --> 00:03:09.090
So, r is going to be equal to
half-- well, it obviously

00:03:09.090 --> 00:03:10.975
equals half the diameter
of this circle.

00:03:14.070 --> 00:03:19.520
But if we look at the base
that's 6 and 8, if we use that

00:03:19.520 --> 00:03:22.890
as the base, and the 10 comes
out of the page, the radius

00:03:22.890 --> 00:03:23.670
can only be what?

00:03:23.670 --> 00:03:26.220
The radius can only be 3 because
the diameter of the

00:03:26.220 --> 00:03:27.470
circle can only be 6.

00:03:30.390 --> 00:03:32.415
We don't get to use the 8.

00:03:32.415 --> 00:03:36.600
So, this is the case if we
use the 6 and the 8 side.

00:03:36.600 --> 00:03:39.050
And so we'll kind of be-- I
guess the best way to think

00:03:39.050 --> 00:03:43.070
about it is we'll be wasting
this base over here.

00:03:43.070 --> 00:03:50.170
The other possibility, if we had
the 8 and the 10 side, if

00:03:50.170 --> 00:03:54.955
this was our base then we could
have a radius of 4.

00:04:01.540 --> 00:04:02.330
Let me think about it.

00:04:02.330 --> 00:04:04.040
Yeah, radius would be a 4.

00:04:04.040 --> 00:04:05.580
Because we would have
the height 8, we'd

00:04:05.580 --> 00:04:07.880
have a radius of 4.

00:04:07.880 --> 00:04:11.540
Now when we look at this
equation, if we had to

00:04:11.540 --> 00:04:14.330
maximize either the-- this is
going to be the volume of the

00:04:14.330 --> 00:04:17.209
cylinder-- if we had to maximize
either the height or

00:04:17.209 --> 00:04:19.570
the radius, because those are
the two things that we can

00:04:19.570 --> 00:04:23.560
deal with, which one, at least
to you, seems more important?

00:04:23.560 --> 00:04:26.190
Well, to me, the radius seems
more important because I'm

00:04:26.190 --> 00:04:27.770
squaring it.

00:04:27.770 --> 00:04:32.330
So, these, in my mind, are
the two contenders.

00:04:32.330 --> 00:04:35.270
The worst contender is where
we use the 6 and the

00:04:35.270 --> 00:04:37.140
10 side as a base.

00:04:37.140 --> 00:04:38.370
Because then we use
a lot of space.

00:04:38.370 --> 00:04:40.840
But we can figure that
one out, too.

00:04:40.840 --> 00:04:42.840
Because then we waste a lot of
space and we have a height of

00:04:42.840 --> 00:04:45.920
8 and the radius is still--.

00:04:45.920 --> 00:04:49.060
So, first of all, this one is
clearly better than this one.

00:04:49.060 --> 00:04:49.990
And why is that?

00:04:49.990 --> 00:04:53.240
They have the same area
of their base because

00:04:53.240 --> 00:04:54.980
the radius is 6.

00:04:54.980 --> 00:04:57.300
Sorry, the radius is 3,
the diameter is 6.

00:04:57.300 --> 00:05:00.080
But in here the height gets to
be 10, and here the height

00:05:00.080 --> 00:05:01.590
only gets to be 8.

00:05:01.590 --> 00:05:03.690
So this one is a non-option.

00:05:03.690 --> 00:05:06.590
So, really, we have to
compare these two.

00:05:06.590 --> 00:05:08.780
And I've drawn their bases.

00:05:08.780 --> 00:05:11.130
So, if you think about this one,
we have an area of the

00:05:11.130 --> 00:05:14.270
base of the radius is 4.

00:05:14.270 --> 00:05:20.450
So, 10 pi times the
height of six.

00:05:20.450 --> 00:05:22.010
That's the volume of
this cylinder.

00:05:22.010 --> 00:05:26.760
Here we have a radius--
we have a radius of 3.

00:05:26.760 --> 00:05:28.750
Half of the 6 diameter.

00:05:28.750 --> 00:05:31.200
So, the area is 9 pi.

00:05:31.200 --> 00:05:32.650
And we're going to multiply
that times the

00:05:32.650 --> 00:05:34.480
height, which is 10.

00:05:34.480 --> 00:05:36.580
So, here we have 90 pi.

00:05:36.580 --> 00:05:41.430
And here we have 6 times
16, which is 96 pi.

00:05:41.430 --> 00:05:45.570
So, this is the best
of our cases.

00:05:45.570 --> 00:05:47.100
And that was the correct
intuition.

00:05:47.100 --> 00:05:49.040
We're maximizing the radius.

00:05:49.040 --> 00:05:53.850
And when you maximize the
radius, we get a radius of 4.

00:05:53.850 --> 00:05:55.760
So what did they ask us
in the question again?

00:05:55.760 --> 00:05:58.630
What is the radius in inches
of the one that has the

00:05:58.630 --> 00:06:00.120
maximum volume?

00:06:00.120 --> 00:06:01.740
And so that radius is 4.

00:06:01.740 --> 00:06:03.910
Where one side is 8 and
one side is 10.

00:06:03.910 --> 00:06:05.390
And that's choice B.

00:06:09.080 --> 00:06:12.180
Next question.

00:06:12.180 --> 00:06:15.090
192.

00:06:15.090 --> 00:06:15.910
Square root.

00:06:15.910 --> 00:06:18.060
OK, they have all sorts
of interesting things.

00:06:18.060 --> 00:06:19.330
They essentially want
us to simplify.

00:06:19.330 --> 00:06:24.460
The square root of 2 plus 1
times the square root of 2

00:06:24.460 --> 00:06:30.490
minus 1 times the square root
of 3 plus 1 times the square

00:06:30.490 --> 00:06:33.010
root of 3 minus 1.

00:06:33.010 --> 00:06:35.890
So, this seems to be just an
application of-- I'm sure

00:06:35.890 --> 00:06:39.700
you've learned this in algebra--
that a plus b times

00:06:39.700 --> 00:06:43.930
a minus b is equal to a squared
minus b squared.

00:06:43.930 --> 00:06:44.870
You don't have to
memorize this.

00:06:44.870 --> 00:06:48.010
You could actually multiply it
all out and you'll get this.

00:06:48.010 --> 00:06:50.055
But it's a good thing
to know if you want

00:06:50.055 --> 00:06:51.480
to do things quickly.

00:06:51.480 --> 00:06:55.180
So, square root of 2 plus 1,
times square root of 2 minus

00:06:55.180 --> 00:07:03.710
one, is equal to square root of
2 squared, minus 1 squared.

00:07:03.710 --> 00:07:05.980
We're going to multiply that
times-- using the same

00:07:05.980 --> 00:07:12.380
principle-- square root of 3
squared minus one squared.

00:07:12.380 --> 00:07:13.110
Square root of 2 squared.

00:07:13.110 --> 00:07:20.030
That's 2 minus 1 times
3 minus 1.

00:07:20.030 --> 00:07:22.330
That's equal to 1 times 2.

00:07:22.330 --> 00:07:24.430
Which is equal to 2.

00:07:24.430 --> 00:07:27.650
And that is choice A.

00:07:27.650 --> 00:07:32.680
Problem 193.

00:07:32.680 --> 00:07:35.890
In a certain Calculus class,
the ratio of the number of

00:07:35.890 --> 00:07:39.060
math majors to the number of
students who are not math

00:07:39.060 --> 00:07:41.500
majors is 2:5.

00:07:41.500 --> 00:07:47.100
So, math to not math
is equal to 2:5.

00:07:47.100 --> 00:07:50.790
If 2 more math majors were to
enter the class the ratio

00:07:50.790 --> 00:07:52.670
would be 1:2.

00:07:52.670 --> 00:07:56.440
So if I have 2 more math majors,
then the ratio of them

00:07:56.440 --> 00:08:01.080
to the non-math majors
is equal to 1:2.

00:08:01.080 --> 00:08:02.920
How many students are
in the class?

00:08:05.830 --> 00:08:10.430
They want to know what m plus
nm, non-math majors, is.

00:08:10.430 --> 00:08:10.780
Let's see.

00:08:10.780 --> 00:08:12.390
Let's see if we can solve
for anything.

00:08:12.390 --> 00:08:14.520
We actually have two linear
equations and two unknowns.

00:08:14.520 --> 00:08:16.470
You might not see it yet,
but if we cross

00:08:16.470 --> 00:08:19.540
multiply we get 5m.

00:08:19.540 --> 00:08:21.560
I'll just call these
n, non-math majors.

00:08:21.560 --> 00:08:23.160
I don't want to keep
saying nm.

00:08:23.160 --> 00:08:25.110
5m is equal to 2n.

00:08:25.110 --> 00:08:29.050
And here you get 2 times
m plus 2 is 2m plus

00:08:29.050 --> 00:08:33.500
4 is equal to n.

00:08:33.500 --> 00:08:35.100
We could substitute that.

00:08:35.100 --> 00:08:39.030
If n is equal to that, then that
means that 5m is equal to

00:08:39.030 --> 00:08:41.890
2 times n. n is equal
to 2m plus 4.

00:08:44.932 --> 00:08:50.620
So, you get 5m is equal
to 4m plus 8.

00:08:50.620 --> 00:08:55.700
Subtract 4m from both sides,
you get m is equal to 8.

00:08:55.700 --> 00:08:59.070
If m is equal to 8, we could
say 5m is equal to 2n.

00:08:59.070 --> 00:09:03.280
So, 5 times 8 is 40.

00:09:03.280 --> 00:09:06.960
And so n is equal to 20.

00:09:06.960 --> 00:09:10.540
Let's see, are they asking for
the-- if 2 more students were

00:09:10.540 --> 00:09:11.280
to enter the class?

00:09:11.280 --> 00:09:11.460
OK.

00:09:11.460 --> 00:09:13.070
So they want to know how many
are in the class right now.

00:09:13.070 --> 00:09:15.430
So they want to know m plus n.

00:09:15.430 --> 00:09:19.230
Well, that's equal to
8 plus 20 which

00:09:19.230 --> 00:09:21.730
is equal to 28 students.

00:09:21.730 --> 00:09:25.600
And that is choice D.

00:09:25.600 --> 00:09:26.470
Next question.

00:09:26.470 --> 00:09:28.160
I think I have time to
fit this one in.

00:09:28.160 --> 00:09:30.150
I'll continue in the
next if I don't.

00:09:30.150 --> 00:09:31.850
194.

00:09:31.850 --> 00:09:43.080
What is the units digit of 13 to
the fourth times 17 squared

00:09:43.080 --> 00:09:47.020
times 29 to the third?

00:09:47.020 --> 00:09:49.110
So, here you just have to
realize that the units digit

00:09:49.110 --> 00:09:52.140
in any of these numbers, are
just the units digit of each

00:09:52.140 --> 00:09:53.790
of these numbers times
each other.

00:09:53.790 --> 00:09:55.540
And then you take the
units digit of that.

00:09:55.540 --> 00:09:58.100
So, we really just have to worry
about the units digits.

00:09:58.100 --> 00:10:00.210
The answer to this question
would be the exact same thing

00:10:00.210 --> 00:10:03.920
as if I said the units digit
of 3 to the fourth.

00:10:03.920 --> 00:10:05.700
We just have to worry about
the units digits.

00:10:05.700 --> 00:10:11.480
3 to the fourth times 7 squared
times 9 to the third.

00:10:11.480 --> 00:10:13.640
It would be the exact
same thing.

00:10:13.640 --> 00:10:15.560
So, 3 to the fourth, what's
the units digit?

00:10:15.560 --> 00:10:17.840
3 to the one is equal to 3.

00:10:17.840 --> 00:10:19.720
3 squared is equal to 9.

00:10:19.720 --> 00:10:23.730
3 to the third is 27, but
the units digit is 7.

00:10:23.730 --> 00:10:29.520
And then 3 to the fourth is 81
and the units digit is 1.

00:10:29.520 --> 00:10:31.980
And just so you can see this
principle that I just talked

00:10:31.980 --> 00:10:35.550
about-- and we could prove it if
I had more time-- is that I

00:10:35.550 --> 00:10:38.280
could just multiply 3 times 7
and the units digit is just 1.

00:10:38.280 --> 00:10:41.230
Or I could just multiply
3 times 9 and the

00:10:41.230 --> 00:10:42.600
units digit is 7.

00:10:42.600 --> 00:10:44.980
We could just ignore
everything else.

00:10:44.980 --> 00:10:48.860
So here the units digit is 1.

00:10:48.860 --> 00:10:51.510
7 squared, what's
the units digit?

00:10:51.510 --> 00:10:53.820
Well, 7 squared, the units
digit is-- well,

00:10:53.820 --> 00:10:55.110
7 squared is 49.

00:10:55.110 --> 00:10:57.720
The units digit is 9.

00:10:57.720 --> 00:11:00.670
And then 9 to the third,
what's the units digit?

00:11:00.670 --> 00:11:02.260
9 to the 1 is 9.

00:11:02.260 --> 00:11:03.510
9 squared.

00:11:03.510 --> 00:11:05.130
The units digit is 1.

00:11:05.130 --> 00:11:06.270
It's 81.

00:11:06.270 --> 00:11:08.090
And then 9 to the third, we
don't have to worry about

00:11:08.090 --> 00:11:08.660
anything else.

00:11:08.660 --> 00:11:11.350
When you multiply 81 times 9,
the units digit is going to be

00:11:11.350 --> 00:11:13.640
9 times that 1 in 81.

00:11:13.640 --> 00:11:15.770
So, that equals 9.

00:11:15.770 --> 00:11:17.980
So, it's 1 times 9 times 9.

00:11:17.980 --> 00:11:20.290
And then 9 times 9 is 81.

00:11:20.290 --> 00:11:23.170
We want the units digit,
which is 1.

00:11:23.170 --> 00:11:25.720
So the answer is E.

00:11:25.720 --> 00:11:27.470
And I'll let you think about
that a little bit.

00:11:27.470 --> 00:11:29.860
Maybe you want to play around
and see why we only have to

00:11:29.860 --> 00:11:32.540
worry about the units digit
and why I was able to make

00:11:32.540 --> 00:11:34.480
this statement right here
to avoid doing a

00:11:34.480 --> 00:11:35.840
lot of harder math.

00:11:35.840 --> 00:11:38.110
Anyway, see you in
the next video.


GMAT: Math 37

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:05.050
We're on problem 185.

00:00:05.050 --> 00:00:10.710
Car x and car y traveled
the same 80 mile route.

00:00:10.710 --> 00:00:21.660
If car x took 2 hours-- so, x
went 80 miles in 2 hours, and

00:00:21.660 --> 00:00:30.080
that's 40 miles per hour-- and
car y travelled at an average

00:00:30.080 --> 00:00:33.710
speed that was 50% percent
faster than the average

00:00:33.710 --> 00:00:35.420
speed of car x.

00:00:35.420 --> 00:00:36.500
So, 50% faster.

00:00:36.500 --> 00:00:40.670
So y went 50% faster than x.

00:00:40.670 --> 00:00:43.550
50% faster times 40
miles per hour.

00:00:43.550 --> 00:00:45.130
Well that's 40 plus 20.

00:00:45.130 --> 00:00:49.510
That's equal to 60
miles per hour.

00:00:52.210 --> 00:00:55.410
How many hours did it take car
y to travel the route?

00:00:55.410 --> 00:00:59.210
So, distance is equal
to rate times time.

00:00:59.210 --> 00:01:01.070
80 miles is the distance.

00:01:01.070 --> 00:01:02.760
The rate is 60 miles an hour.

00:01:02.760 --> 00:01:11.270
So the time is equal to 80/60,
which is equal to 1 and 20/60.

00:01:11.270 --> 00:01:15.270
Which is equal to
1 and 1/3 hours.

00:01:15.270 --> 00:01:16.410
And that is what they
have there.

00:01:16.410 --> 00:01:17.290
They have choice c.

00:01:17.290 --> 00:01:20.120
I thought they might have 1 hour
and 20 minutes written.

00:01:20.120 --> 00:01:23.160
Next problem.

00:01:23.160 --> 00:01:25.920
186.

00:01:25.920 --> 00:01:35.070
If the average of the four
numbers k, 2k plus 3, 3k minus

00:01:35.070 --> 00:01:42.810
5, and 5k plus 1 is 63, what
is the value of k?

00:01:42.810 --> 00:01:44.590
So let's just average
the numbers.

00:01:44.590 --> 00:01:55.400
k, plus 2k plus 3, plus 3k
minus 5, plus 5k plus 1.

00:01:55.400 --> 00:01:57.660
And that was four numbers.

00:01:57.660 --> 00:01:59.650
So, divide by 4.

00:01:59.650 --> 00:02:04.200
That is equal to 63.

00:02:04.200 --> 00:02:05.230
Let's add up all the k's.

00:02:05.230 --> 00:02:07.110
I have 1k plus 2k is 3k.

00:02:10.889 --> 00:02:12.360
So, 6k.

00:02:12.360 --> 00:02:13.450
8k.

00:02:13.450 --> 00:02:15.540
So, I get 8k.

00:02:15.540 --> 00:02:21.500
And then I get 3 minus 5 is
minus 2 plus 1 is minus 1.

00:02:21.500 --> 00:02:24.390
And if you multiply the 4
times the 63, if we just

00:02:24.390 --> 00:02:28.940
multiply both sides of this
equation by 4 you get

00:02:28.940 --> 00:02:35.710
4 times 63 is 252.

00:02:35.710 --> 00:02:36.720
Let me think about that.

00:02:36.720 --> 00:02:39.250
4 times 60 is 240 plus 12.

00:02:39.250 --> 00:02:44.020
Yeah, 252.

00:02:44.020 --> 00:02:49.770
And then 8k would be-- add
1 to both sides-- 253.

00:02:49.770 --> 00:02:55.370
And so k is equal to
253 divided by 8.

00:02:55.370 --> 00:02:56.190
Let's see what that is.

00:02:56.190 --> 00:03:01.630
8 goes into 253-- 8 goes
into 25 three times.

00:03:01.630 --> 00:03:03.320
24.

00:03:03.320 --> 00:03:04.800
Bring down-- equals one time.

00:03:04.800 --> 00:03:06.270
1 times 8 is 8.

00:03:06.270 --> 00:03:07.550
You have a remainder of 5.

00:03:07.550 --> 00:03:11.440
So, it's 31 and 5/8 is k.

00:03:11.440 --> 00:03:14.270
And that is not one
of the choices.

00:03:14.270 --> 00:03:15.890
Let me see where I
made a mistake.

00:03:15.890 --> 00:03:17.600
So the average of k.

00:03:17.600 --> 00:03:22.630
2k plus 3, 3k minus
5, and 5k plus 1.

00:03:22.630 --> 00:03:26.470
That's four numbers, right?

00:03:26.470 --> 00:03:29.070
What is the value of k?

00:03:29.070 --> 00:03:31.870
All right, so I add up the four
numbers, and I divide by

00:03:31.870 --> 00:03:34.130
4 and they should
be equal to 63.

00:03:34.130 --> 00:03:37.940
So, maybe I just made a mistake
in the simplification.

00:03:37.940 --> 00:03:46.070
So, I have k plus 2k is
3k, 3k plus 3k is 6k.

00:03:46.070 --> 00:03:48.350
Oh, 6k plus 5k is 11k.

00:03:48.350 --> 00:03:50.740
I don't know what I
was-- that's 11k.

00:03:50.740 --> 00:03:55.330
And then you have 3 minus 5
which is minus 2 plus 1.

00:03:55.330 --> 00:03:56.050
Minus 1.

00:03:56.050 --> 00:03:57.090
So, 11k minus 1.

00:03:57.090 --> 00:04:01.030
So, this is 11k.

00:04:01.030 --> 00:04:08.540
So, k is equal to 253 divided by
11, which is-- 11 goes into

00:04:08.540 --> 00:04:12.040
253 two times.

00:04:12.040 --> 00:04:14.860
22, 33.

00:04:14.860 --> 00:04:15.844
Oh, that's nicer.

00:04:15.844 --> 00:04:20.089
It goes in exactly 23 times
and that's choice D.

00:04:20.089 --> 00:04:23.630
Sorry for the error.

00:04:23.630 --> 00:04:24.490
Next question.

00:04:24.490 --> 00:04:27.930
187.

00:04:27.930 --> 00:04:32.930
If p is an integer, p is even.

00:04:32.930 --> 00:04:36.270
They're telling us q is odd.

00:04:36.270 --> 00:04:40.790
Which is the following must
be an odd integer?

00:04:40.790 --> 00:04:44.140
OK, so they give us
a bunch of things.

00:04:44.140 --> 00:04:46.670
So, an even times an odd.

00:04:46.670 --> 00:04:48.620
Well, by definition, that's
going to be even.

00:04:48.620 --> 00:04:49.290
Why is that?

00:04:49.290 --> 00:04:54.510
Well, you can view it as if you
multiply p times q, if p

00:04:54.510 --> 00:04:58.510
is even that means that p is
divisible by 2, right?

00:04:58.510 --> 00:05:01.770
So, if 2 is one of the factors
of p, it's definitely going to

00:05:01.770 --> 00:05:04.050
be one of the factors
of p times q.

00:05:04.050 --> 00:05:06.960
So, this one will not be odd.

00:05:06.960 --> 00:05:08.250
Let me think about it.

00:05:08.250 --> 00:05:15.400
If I add an even plus an odd,
that gives me an odd.

00:05:15.400 --> 00:05:18.340
So, let's see if any of
the choices are that.

00:05:18.340 --> 00:05:21.410
Let's see, p divided by q.

00:05:21.410 --> 00:05:27.680
An even divided by an odd does
not necessarily-- well, this

00:05:27.680 --> 00:05:28.560
is an interesting one.

00:05:28.560 --> 00:05:31.040
2p plus q.

00:05:31.040 --> 00:05:34.250
That's choice c.

00:05:34.250 --> 00:05:37.445
Now if p is even-- well, it
doesn't matter whether p is

00:05:37.445 --> 00:05:39.340
even or not, but 2p is actually

00:05:39.340 --> 00:05:42.290
definitely going to be even.

00:05:42.290 --> 00:05:43.770
This is going to be even.

00:05:43.770 --> 00:05:46.040
And q, we already
know, is odd.

00:05:46.040 --> 00:05:47.950
So, this is an even
plus an odd.

00:05:47.950 --> 00:05:50.310
So, it's going to be odd.

00:05:50.310 --> 00:05:51.260
So, the choice is c.

00:05:51.260 --> 00:05:52.700
And we could even prove
it a little bit.

00:05:52.700 --> 00:05:57.860
If you say that p is equal
to 2 times some number k.

00:05:57.860 --> 00:05:58.060
Right?

00:05:58.060 --> 00:05:59.320
We don't know what that is.

00:05:59.320 --> 00:06:03.140
We say that q is equal to 2
times some number, I don't

00:06:03.140 --> 00:06:06.910
know, m plus 1, then what
does this simplify to?

00:06:06.910 --> 00:06:10.860
This simplifies to 2 times p,
which is 2 times some number

00:06:10.860 --> 00:06:16.230
k-- we don't know what k is--
plus q which is 2 times some

00:06:16.230 --> 00:06:18.390
number m plus 1.

00:06:18.390 --> 00:06:19.490
Some integer m.

00:06:19.490 --> 00:06:25.450
So, this becomes 4 times some
number k, plus 2 times some

00:06:25.450 --> 00:06:27.390
number m, plus 1.

00:06:27.390 --> 00:06:32.360
Well, 4 times some number k,
that's the same thing as 2

00:06:32.360 --> 00:06:34.690
times some other number n.

00:06:34.690 --> 00:06:37.880
You know it's 2 times some other
integer where n is twice

00:06:37.880 --> 00:06:39.690
what k is, right?

00:06:39.690 --> 00:06:43.200
Plus 2m plus 1.

00:06:43.200 --> 00:06:45.510
And this is the same thing
as 2 times n plus m-- and

00:06:45.510 --> 00:06:50.640
remember, n, m k, these are all
just integers-- plus 1.

00:06:50.640 --> 00:06:52.750
And so, if n and m are integers,
then we could say,

00:06:52.750 --> 00:06:54.720
well, that's just equal to 2
times some other integer--

00:06:54.720 --> 00:06:56.780
let's call it l-- plus 1.

00:06:56.780 --> 00:06:58.880
And this would be
an odd number.

00:06:58.880 --> 00:06:59.710
Because this is even.

00:06:59.710 --> 00:07:01.925
And then you're adding
1 to an even number.

00:07:01.925 --> 00:07:03.120
But you didn't have
to do that.

00:07:03.120 --> 00:07:05.530
You just had to recognize
that an odd plus an even

00:07:05.530 --> 00:07:08.100
is equal to an odd.

00:07:08.100 --> 00:07:10.070
Choice c.

00:07:10.070 --> 00:07:11.940
Next problem.

00:07:11.940 --> 00:07:15.800
188.

00:07:15.800 --> 00:07:20.810
Drum x is 1/2 full of oil.

00:07:20.810 --> 00:07:26.020
So x is 1/2 full.

00:07:26.020 --> 00:07:28.590
And drum y, which has twice
the capacity of

00:07:28.590 --> 00:07:31.410
drum x, is 2/3 full.

00:07:31.410 --> 00:07:39.067
So y is 2/3 full and has
two times x capacity.

00:07:41.990 --> 00:07:47.270
If all of the oil in drum x is
poured into drum y, then drum

00:07:47.270 --> 00:07:51.580
y will be filled to what
fraction of its capacity?

00:07:51.580 --> 00:07:52.790
OK.

00:07:52.790 --> 00:07:56.250
So if x is 1/2 full.

00:07:56.250 --> 00:08:04.620
So x is equal to 1/2 of its--
well, let me think of the best

00:08:04.620 --> 00:08:05.810
way to write this
algebraically.

00:08:05.810 --> 00:08:12.600
x is 1/2 full and y
is twice as big.

00:08:12.600 --> 00:08:16.100
So, if you poured all of this
1/2 into y, it would

00:08:16.100 --> 00:08:19.580
contribute to 1/4
of y's capacity.

00:08:19.580 --> 00:08:26.690
I guess another way, we could
say capacity of x is equal to

00:08:26.690 --> 00:08:29.325
1/2 the capacity of y.

00:08:33.090 --> 00:08:36.909
So, if we were to say 1/2 the
capacity of x, if we were to

00:08:36.909 --> 00:08:41.100
multiply both sides of this by
1/2, that's equal to 1/4 the

00:08:41.100 --> 00:08:42.570
capacity of y.

00:08:42.570 --> 00:08:46.080
So, this 1/2, that's equal to
1/2 the capacity of x which is

00:08:46.080 --> 00:08:49.400
equal to 1/4 the
capacity of y.

00:08:49.400 --> 00:08:52.540
So, if we add 1/4 of the
capacity of y to 2/3 of the

00:08:52.540 --> 00:08:55.950
capacity of y-- that's 2/3 full,
that's another way of

00:08:55.950 --> 00:08:59.230
saying 2/3 times the capacity
of y-- what do we get?

00:08:59.230 --> 00:09:04.300
We get 1/4 plus 2/3 which is
equal to-- get a common

00:09:04.300 --> 00:09:12.570
denominator 12--
3/4 plus 8/12.

00:09:12.570 --> 00:09:15.170
3/12 plus 8/12, that's 2/3.

00:09:15.170 --> 00:09:17.300
You get 11/12.

00:09:17.300 --> 00:09:20.270
So, y would be 11/12 full
of its capacity.

00:09:20.270 --> 00:09:21.840
So, that is choice C.

00:09:26.430 --> 00:09:32.565
Problem 189.

00:09:32.565 --> 00:09:41.720
If x is greater than 0,
x/50 plus x/25 is what

00:09:41.720 --> 00:09:44.220
percentage of x?

00:09:44.220 --> 00:09:46.040
So, you get a common
denominator.

00:09:46.040 --> 00:09:48.525
This is equal to over 50.

00:09:48.525 --> 00:09:50.255
x/50 is x/50.

00:09:50.255 --> 00:09:53.770
x/25 is 2x over 50.

00:09:53.770 --> 00:10:00.150
So, that is equal to 3x over
50 which is the same

00:10:00.150 --> 00:10:03.520
thing as 3/50 x.

00:10:03.520 --> 00:10:06.170
And that's the same thing as--
since we want it as a fraction

00:10:06.170 --> 00:10:08.340
of 100, since we want to do a
percentage-- that's the same

00:10:08.340 --> 00:10:11.290
thing as if you multiply the
numerator and the denominator

00:10:11.290 --> 00:10:17.890
by 2, you get 6 over 100 x which
is equal to .06x which

00:10:17.890 --> 00:10:20.600
is equal to 6% of x.

00:10:20.600 --> 00:10:23.450
And that's choice A.

00:10:23.450 --> 00:10:23.740
Anyway.

00:10:23.740 --> 00:10:25.540
See you in the next video.


GMAT: Math 36

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.800
Before moving on, I want to
address problem 180 again.

00:00:03.800 --> 00:00:05.660
Because we got it right in the
last video, but then as soon

00:00:05.660 --> 00:00:07.260
as I finished the video, I
realized there was a much,

00:00:07.260 --> 00:00:08.840
much simpler way
of doing that.

00:00:08.840 --> 00:00:10.790
They said in a nationwide
poll, n people were

00:00:10.790 --> 00:00:11.490
interviewed.

00:00:11.490 --> 00:00:16.180
1/4 of them answered yes
to question one.

00:00:16.180 --> 00:00:19.820
And of those, 1/3 answered
yes to question two.

00:00:19.820 --> 00:00:20.980
So then you have another 1/3.

00:00:20.980 --> 00:00:24.880
So what fraction of the entire
population said yes to both?

00:00:24.880 --> 00:00:28.470
Well, it would be 1/4
of-- or 1/3 of 1/4

00:00:28.470 --> 00:00:33.840
which is equal to 1/12.

00:00:33.840 --> 00:00:36.440
1/12 of the population
said yes to both.

00:00:36.440 --> 00:00:39.380
Or you could say 1/12 of
n said yes to both.

00:00:39.380 --> 00:00:43.330
And they ask us which of the
following expressions

00:00:43.330 --> 00:00:45.270
represents the number of people
interviewed who did not

00:00:45.270 --> 00:00:46.720
answer yes to both?

00:00:46.720 --> 00:00:48.030
So it's everybody else.

00:00:48.030 --> 00:00:49.600
These are the people who
said yes to both.

00:00:49.600 --> 00:00:51.800
So essentially you just
subtract that from 1.

00:00:51.800 --> 00:00:58.170
So 1 minus 1/12 is equal to
12/12 minus 1/12, which is

00:00:58.170 --> 00:01:00.590
equal to 11/12 of the
population did

00:01:00.590 --> 00:01:02.030
not say yes to both.

00:01:02.030 --> 00:01:04.526
And so 11/12 and then the
population is n, so

00:01:04.526 --> 00:01:05.790
it's 11n over 12.

00:01:05.790 --> 00:01:09.220
And that's how we
got choice E.

00:01:09.220 --> 00:01:12.430
Problem 181.

00:01:12.430 --> 00:01:15.370
The ratio of two quantities
is 3:4.

00:01:15.370 --> 00:01:19.210
So, let's say one is x, one
is y, and it equals 3:4.

00:01:21.820 --> 00:01:24.650
If each of the quantities is
increased by 5, what is the

00:01:24.650 --> 00:01:25.900
ratio of these new quantities?

00:01:29.480 --> 00:01:33.640
x plus 5 to y plus 5.

00:01:33.640 --> 00:01:36.400
Well, it really depends
on what multiple these

00:01:36.400 --> 00:01:37.955
are of 3 and 4.

00:01:40.570 --> 00:01:42.800
Well, I already have a sense
that it's choice E.

00:01:42.800 --> 00:01:45.250
It cannot be determined from
the information given.

00:01:45.250 --> 00:01:46.540
And let's prove it.

00:01:46.540 --> 00:01:48.930
I'm going to show you two
different x's and y's, and

00:01:48.930 --> 00:01:50.730
when you add 5 to both you're
going to get a completely

00:01:50.730 --> 00:01:51.430
different answer.

00:01:51.430 --> 00:01:53.900
So you cannot determine it
from the information.

00:01:53.900 --> 00:01:56.620
So they snuck in a data
sufficiency question into the

00:01:56.620 --> 00:01:57.750
problem solving.

00:01:57.750 --> 00:02:00.170
So, x could be 3.

00:02:00.170 --> 00:02:04.120
If x is equal to 3 and y is
equal to 4, that definitely

00:02:04.120 --> 00:02:05.490
satisfies this condition.

00:02:05.490 --> 00:02:11.030
Or we could have x is equal
to 6 and y is equal to 8.

00:02:11.030 --> 00:02:11.610
Right?

00:02:11.610 --> 00:02:14.290
That also satisfies
the 3 to 4 ratio.

00:02:14.290 --> 00:02:16.890
What happens when we add
5 to both of these?

00:02:16.890 --> 00:02:22.130
x plus 5 would be equal to 8.

00:02:22.130 --> 00:02:26.840
And y plus 5 would
be equal to 9.

00:02:26.840 --> 00:02:29.290
So, the new ratio becomes 8:9.

00:02:29.290 --> 00:02:30.820
What about this case?

00:02:30.820 --> 00:02:34.290
You would have x plus
5 is equal to 6

00:02:34.290 --> 00:02:36.430
plus 5, which is 11.

00:02:36.430 --> 00:02:42.780
And then y plus 5 is equal
to 8 plus 5, which is 13.

00:02:42.780 --> 00:02:46.040
And 8/9 is a very different
fraction than 11/13.

00:02:46.040 --> 00:02:48.150
It's not like you can reduce
one into the other.

00:02:48.150 --> 00:02:51.810
So I can find an x and a y to
satisfy this, but when I add 5

00:02:51.810 --> 00:02:53.480
to the top and bottom I get
two different answers.

00:02:53.480 --> 00:02:56.460
So you cannot determine this
with the information given.

00:02:56.460 --> 00:02:57.820
So the choice is E.

00:02:57.820 --> 00:02:59.520
Sneaky data sufficiency
problem.

00:03:02.160 --> 00:03:07.180
182.

00:03:07.180 --> 00:03:13.540
If the average arithmetic mean
of x and y is 60-- so

00:03:13.540 --> 00:03:18.370
essentially you're saying x plus
y over 2 is equal to 60.

00:03:18.370 --> 00:03:20.020
The average of them is 60.

00:03:20.020 --> 00:03:22.270
And the average of
y and z is 80.

00:03:22.270 --> 00:03:28.120
So, y plus z over 2
is equal to 80.

00:03:28.120 --> 00:03:34.050
What is the value
of z minus x?

00:03:34.050 --> 00:03:35.340
So let's see if we
can do that.

00:03:35.340 --> 00:03:36.800
So let's rewrite this.

00:03:36.800 --> 00:03:41.120
Let's solve for x
in terms of y.

00:03:41.120 --> 00:03:43.250
I'll do it in a different
color.

00:03:43.250 --> 00:03:48.010
We get x plus y is
equal to 120.

00:03:48.010 --> 00:03:49.780
Just multiplied both
sides by 2.

00:03:49.780 --> 00:03:53.530
You get x is equal
to 120 minus y.

00:03:53.530 --> 00:03:55.730
Now let's solve for z
in terms of y here.

00:03:55.730 --> 00:04:01.680
Multiply both sides by 2, you
get y plus z is equal to 160.

00:04:01.680 --> 00:04:06.730
z is equal to 160 minus y.

00:04:06.730 --> 00:04:08.360
So what's z minus x?

00:04:08.360 --> 00:04:19.970
So z is 160 minus y, minus
x, 120 minus y.

00:04:19.970 --> 00:04:27.160
So that is equal to 160 minus
y, minus 120 plus y.

00:04:27.160 --> 00:04:29.180
And it's good that the y's
are canceling out.

00:04:29.180 --> 00:04:32.560
Whenever they give you something
like this, it's

00:04:32.560 --> 00:04:35.440
usually a safe guess that if
you just go forth with the

00:04:35.440 --> 00:04:37.700
algebra that nice things
will happen like this.

00:04:37.700 --> 00:04:41.950
So these y's cancel out and
we're left with 160 minus 120

00:04:41.950 --> 00:04:44.020
which is 40.

00:04:44.020 --> 00:04:46.970
And that is choice B.

00:04:50.720 --> 00:04:56.980
Problem 183.

00:04:56.980 --> 00:05:01.900
If 1/2 of the air in a tank is
removed with each stroke of a

00:05:01.900 --> 00:05:11.830
vacuum pump-- so we have
1/2 tank per stroke.

00:05:11.830 --> 00:05:13.480
I'm guessing this is going
to be a rate of change.

00:05:13.480 --> 00:05:17.030
What fraction of the original
amount of air has been removed

00:05:17.030 --> 00:05:18.010
after four strokes?

00:05:18.010 --> 00:05:18.260
Oh no.

00:05:18.260 --> 00:05:19.020
This is interesting.

00:05:19.020 --> 00:05:19.710
OK.

00:05:19.710 --> 00:05:23.290
So, after one stroke, what
fraction of the original

00:05:23.290 --> 00:05:24.390
amount of air has
been removed?

00:05:24.390 --> 00:05:28.160
Let's just figure out how
much has been left.

00:05:28.160 --> 00:05:29.030
We could do both actually.

00:05:29.030 --> 00:05:30.565
Left and then removed.

00:05:33.910 --> 00:05:38.340
So, after one stroke-- 1/2 of
the air in the tank is removed

00:05:38.340 --> 00:05:39.100
with each stroke.

00:05:39.100 --> 00:05:41.480
So after one stroke, you
have 1/2 left and

00:05:41.480 --> 00:05:43.510
you have 1/2 removed.

00:05:43.510 --> 00:05:46.890
But, after two strokes, it takes
out half of the air.

00:05:46.890 --> 00:05:47.810
Right.

00:05:47.810 --> 00:05:50.890
So it takes out 1/2
of this half.

00:05:50.890 --> 00:05:54.580
So if you take out 1/2 of this
half you have 1/4 left over.

00:05:54.580 --> 00:05:57.830
And you've also removed
another 1/4.

00:05:57.830 --> 00:05:58.430
Right?

00:05:58.430 --> 00:06:01.280
Because these add up to
what was left before.

00:06:01.280 --> 00:06:05.940
After three strokes, you
take out 1/2 of this.

00:06:05.940 --> 00:06:07.950
So you have 1/8 left over.

00:06:07.950 --> 00:06:11.170
And you took out another 1/8.

00:06:11.170 --> 00:06:15.210
And then after four strokes,
you have 1/16 of

00:06:15.210 --> 00:06:16.330
the air left over.

00:06:16.330 --> 00:06:17.660
Actually I didn't even have
to do this column.

00:06:17.660 --> 00:06:20.710
After four strokes you have
1/16 of the original air.

00:06:20.710 --> 00:06:22.680
But they want to know what
fraction of the original

00:06:22.680 --> 00:06:24.260
amount of air has
been removed.

00:06:24.260 --> 00:06:26.080
So this is what you have left.

00:06:26.080 --> 00:06:28.240
So what's been removed
is 1 minus this.

00:06:28.240 --> 00:06:30.220
So, 1 minus 1/16.

00:06:30.220 --> 00:06:34.070
Well, that's 16 over
16 minus 1 over 16.

00:06:34.070 --> 00:06:36.470
Which is equal to 15 over 16.

00:06:36.470 --> 00:06:37.110
Choice A.

00:06:37.110 --> 00:06:39.080
I didn't even have to worry
about that column.

00:06:41.960 --> 00:06:48.010
Problem 184.

00:06:48.010 --> 00:06:55.690
If the 2 digit integers M and
N are positive and have the

00:06:55.690 --> 00:07:00.120
same digits, but in reverse
order, which of the following

00:07:00.120 --> 00:07:04.000
cannot be the sum of M and N?

00:07:04.000 --> 00:07:05.680
All right, same digits
in reverse order.

00:07:05.680 --> 00:07:08.450
So M could be AB.

00:07:08.450 --> 00:07:12.390
And N could be BA.

00:07:12.390 --> 00:07:13.990
So which cannot be the sum?

00:07:13.990 --> 00:07:14.870
I'm just experimenting.

00:07:14.870 --> 00:07:16.600
Let's see you if you
add AB and BA.

00:07:21.770 --> 00:07:27.610
So, if B plus A-- I guess
there's two assumptions.

00:07:27.610 --> 00:07:30.000
If B plus A is less 10,
let's do this one.

00:07:30.000 --> 00:07:33.590
B plus A is less than 10.

00:07:33.590 --> 00:07:37.880
Then you would have-- this
digit would be B plus A.

00:07:37.880 --> 00:07:41.470
And then this digit over here
would also be B plus A.

00:07:44.250 --> 00:07:48.290
So, definitely choices
D and E work.

00:07:48.290 --> 00:07:50.500
So we're left with choices
A, B, and C.

00:07:50.500 --> 00:07:53.150
Because none of these fit this
paradigm where we have the

00:07:53.150 --> 00:07:54.460
same digit twice.

00:07:54.460 --> 00:07:59.080
So, all of those assume that B
plus A is greater than 10.

00:07:59.080 --> 00:08:03.330
So, if B plus A is greater than
10, then what happens?

00:08:03.330 --> 00:08:05.010
A plus B.

00:08:05.010 --> 00:08:07.920
B plus A.

00:08:07.920 --> 00:08:10.850
So let's think about
it a little bit.

00:08:10.850 --> 00:08:24.210
If B plus A, you'll have the
ones digit from B plus A.

00:08:24.210 --> 00:08:26.350
And then you'll have
a 1 up here.

00:08:26.350 --> 00:08:28.800
You add 1 plus A plus B.

00:08:28.800 --> 00:08:34.289
So, you'd have B
plus A plus 1.

00:08:34.289 --> 00:08:37.740
And so this will be a
two-digit number.

00:08:37.740 --> 00:08:42.850
And if you think about it-- if
we do the three digits of the

00:08:42.850 --> 00:08:48.110
number, this is going
to be the ones digit

00:08:48.110 --> 00:08:50.500
from B plus A right?

00:08:50.500 --> 00:08:54.730
This is going to be the ones
digit from B plus A plus 1.

00:08:54.730 --> 00:08:57.940
So this number is going to be
one more than this number.

00:08:57.940 --> 00:08:59.720
That's an interesting problem.

00:08:59.720 --> 00:09:01.940
This one has to be one
more than that one.

00:09:01.940 --> 00:09:05.150
And so both B and C satisfy
that, right.

00:09:05.150 --> 00:09:07.460
165.

00:09:07.460 --> 00:09:09.270
6 is one more than 5.

00:09:09.270 --> 00:09:10.570
And it should work out.

00:09:10.570 --> 00:09:12.640
We should be able to figure
out a B and an A

00:09:12.640 --> 00:09:14.040
that satisfy that.

00:09:14.040 --> 00:09:16.070
And then C also satisfies that.

00:09:16.070 --> 00:09:16.800
121.

00:09:16.800 --> 00:09:18.430
2 is one more than 1.

00:09:18.430 --> 00:09:19.630
I want to make sure you
understand that because it's

00:09:19.630 --> 00:09:21.980
actually pretty interesting.

00:09:21.980 --> 00:09:23.860
Deductive reasoning,
A is the answer.

00:09:23.860 --> 00:09:26.540
But if you think about it, we're
assuming B plus A is

00:09:26.540 --> 00:09:28.050
greater than 0.

00:09:28.050 --> 00:09:30.775
So, here, you'd have the ones
digit of B plus A and

00:09:30.775 --> 00:09:32.170
you'd carry a 1.

00:09:32.170 --> 00:09:35.880
Now when you add A plus B and
that 1, the ones digit is

00:09:35.880 --> 00:09:36.900
going to be 1 greater.

00:09:36.900 --> 00:09:39.630
This is going to be the ones
digit of B plus A plus 1.

00:09:39.630 --> 00:09:42.710
And that's why I'm saying that
the tens digit has to be 1

00:09:42.710 --> 00:09:44.330
greater than the ones.

00:09:44.330 --> 00:09:45.890
That was an interesting
problem.

00:09:45.890 --> 00:09:48.260
Anyway, see you in
the next video.


GMAT: Math 35

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.200
My wife is looking at me kind
of strange right now.

00:00:03.200 --> 00:00:05.350
So forgive me if I have a
problem doing some of these

00:00:05.350 --> 00:00:08.460
problems. But anyway,
problem 177.

00:00:08.460 --> 00:00:10.770
She said she wanted to observe
me making videos.

00:00:10.770 --> 00:00:13.310
Let's see how this goes.

00:00:13.310 --> 00:00:14.270
Do you want to say hi?

00:00:14.270 --> 00:00:15.100
No, no, OK.

00:00:15.100 --> 00:00:16.670
No, I'm not going
to record it.

00:00:16.670 --> 00:00:20.100
A company accountant estimates
that airfares next year for

00:00:20.100 --> 00:00:24.880
business trips of 1,000 miles or
less will increase by 20%.

00:00:24.880 --> 00:00:27.990
So less than 1,000 miles.

00:00:27.990 --> 00:00:32.509
Airfares next year going
are going to go up 20%.

00:00:32.509 --> 00:00:33.920
And for all other business
trips, it

00:00:33.920 --> 00:00:35.460
will increase by 10%.

00:00:35.460 --> 00:00:39.340
So essentially greater than or
equal to 1,000 miles, airfare

00:00:39.340 --> 00:00:42.690
is going to increase by 10%.

00:00:42.690 --> 00:00:45.630
This year total airfares for
business trips of 1,000 miles

00:00:45.630 --> 00:00:47.790
or less were $9,900.

00:00:47.790 --> 00:00:50.250
So that's the total for
1,000 miles or less.

00:00:50.250 --> 00:00:51.480
And airfares for all
other business

00:00:51.480 --> 00:00:55.790
trips were 13,000 miles.

00:00:55.790 --> 00:00:57.965
According to the accountant's
estimate, if the same number

00:00:57.965 --> 00:01:01.020
of business trips will be made
next year as this year, how

00:01:01.020 --> 00:01:04.739
much will be spent for
airfares next year?

00:01:04.739 --> 00:01:07.120
I have a problem with what they
say, because they say the

00:01:07.120 --> 00:01:08.580
same number of business trips.

00:01:08.580 --> 00:01:11.020
But we really, I'm thinking,
have to assume that not only

00:01:11.020 --> 00:01:13.120
are the same number of business
trips made, but the

00:01:13.120 --> 00:01:16.580
allocation between less than
1,000 and greater than 1,000

00:01:16.580 --> 00:01:18.420
is also going to have
to be the same.

00:01:18.420 --> 00:01:20.860
But anyway, let's just do what
I think they're assuming.

00:01:20.860 --> 00:01:23.620
So essentially you just have
to take the $9,900 and

00:01:23.620 --> 00:01:25.950
increase it by 20%.

00:01:25.950 --> 00:01:30.090
9,900 times 1.2.

00:01:30.090 --> 00:01:33.900
And then add that to the 13,000
increased by 10%.

00:01:33.900 --> 00:01:38.490
So 1.1 times 13,000.

00:01:38.490 --> 00:01:39.945
That's a 13, not a B.

00:01:39.945 --> 00:01:41.195
So let's see.

00:01:44.560 --> 00:01:50.000
Let's just write 9,900
times 1.2.

00:01:50.000 --> 00:01:51.010
2 times 0 is 0.

00:01:51.010 --> 00:01:52.110
2 times 0 is 0.

00:01:52.110 --> 00:01:54.730
2 times 9 is 18.

00:01:54.730 --> 00:01:58.730
2 times 9, 18, plus 1 is 19.

00:01:58.730 --> 00:02:00.290
Add a 0 and then
you have 9,900.

00:02:00.290 --> 00:02:09.560
0, 0, 9, 9-- scroll down--
0, 0, 8, 9 plus 9 is

00:02:09.560 --> 00:02:12.900
18, 2 plus 9 is 11.

00:02:12.900 --> 00:02:15.780
And we have one number
behind the decimal.

00:02:15.780 --> 00:02:18.660
So that's 11,880.

00:02:18.660 --> 00:02:21.760
And then 1.1 times 13,000.

00:02:21.760 --> 00:02:27.050
So it's 13,000 times 1.1.

00:02:27.050 --> 00:02:38.480
1 times 13,000 is 13,000, 0,
1 times 13,000 is 13,000.

00:02:38.480 --> 00:02:42.050
0, 0, 0, 3, 4, 1.

00:02:42.050 --> 00:02:44.210
And we have one number
behind the decimal.

00:02:44.210 --> 00:02:45.630
So it's 14,300.

00:02:45.630 --> 00:02:48.560
And if we were to add
these two together.

00:02:48.560 --> 00:02:54.680
Let's see, if I add 11,880
to this, I get 0, 8.

00:02:54.680 --> 00:02:58.620
3 plus 8 is 11.

00:02:58.620 --> 00:03:01.810
So, 26,180.

00:03:01.810 --> 00:03:04.210
And that's choice B.

00:03:04.210 --> 00:03:06.600
Next problem.

00:03:06.600 --> 00:03:11.610
Problem 178.

00:03:11.610 --> 00:03:24.910
If x star y is equal to xy minus
2 times x plus y, for

00:03:24.910 --> 00:03:28.616
all integers x and y,
then what does 2

00:03:28.616 --> 00:03:32.340
star minus 3 equal?

00:03:32.340 --> 00:03:34.130
So we essentially just have
to do pattern matching.

00:03:34.130 --> 00:03:37.570
Everywhere we see an x here,
we substitute with 2.

00:03:37.570 --> 00:03:40.165
And everywhere we see a y, we
substitute with minus 3.

00:03:40.165 --> 00:03:42.580
And just to be clear, this isn't
an equal, that's a minus

00:03:42.580 --> 00:03:43.420
right there.

00:03:43.420 --> 00:03:55.260
So it's 2 times minus 3, minus
2 times 2, plus minus 3.

00:03:55.260 --> 00:03:57.600
2 times minus 3 is minus 6.

00:03:57.600 --> 00:04:02.425
Minus 2 times 2 plus
minus 3 is minus 1.

00:04:02.425 --> 00:04:04.880
That's the same thing
as 2 minus 3.

00:04:04.880 --> 00:04:08.250
And let's see, you have a minus
and a minus, so we can

00:04:08.250 --> 00:04:09.640
make those both plus.

00:04:09.640 --> 00:04:11.750
So you have minus 6 plus 2.

00:04:11.750 --> 00:04:13.550
When you multiply it by 1,
that's the same thing.

00:04:13.550 --> 00:04:17.160
So minus 6 plus 2 is
equal to minus 4.

00:04:17.160 --> 00:04:18.435
And that is choice C.

00:04:21.490 --> 00:04:22.890
Problem 179.

00:04:22.890 --> 00:04:26.670
I'll switch colors to
ease the monotony.

00:04:26.670 --> 00:04:31.080
179, the table above shows the
number of students in the 3

00:04:31.080 --> 00:04:32.530
clubs at [? McCullough ?]

00:04:32.530 --> 00:04:32.975
School.

00:04:32.975 --> 00:04:34.720
So let me write the table.

00:04:34.720 --> 00:04:35.970
So they have the club.

00:04:38.600 --> 00:04:39.935
And they say the number
of students.

00:04:43.960 --> 00:04:46.060
And they have chess, drama,
and math clubs.

00:04:52.360 --> 00:04:58.510
The chess club has 40 students,
drama has 30, and

00:04:58.510 --> 00:05:00.880
math has 25.

00:05:00.880 --> 00:05:03.210
The table above shows the number
of students in 3 clubs

00:05:03.210 --> 00:05:03.850
at [? McCullough ?]

00:05:03.850 --> 00:05:04.660
School.

00:05:04.660 --> 00:05:08.610
Although no student is in all
3 clubs, 10 students are in

00:05:08.610 --> 00:05:10.095
both chess and drama.

00:05:14.660 --> 00:05:17.270
I feel a Venn diagram
coming on.

00:05:17.270 --> 00:05:18.680
10 students are in both
chess and drama.

00:05:18.680 --> 00:05:20.906
5 students are in both
chess and math.

00:05:25.990 --> 00:05:28.800
And 6 students are in
both drama and math.

00:05:32.710 --> 00:05:36.090
How many different students
are there in the 3 clubs?

00:05:36.090 --> 00:05:38.140
Yeah, I think we need to
do a Venn diagram.

00:05:38.140 --> 00:05:41.870
So let's draw the chess students
first in yellow.

00:05:44.420 --> 00:05:45.706
Actually I'm not going to do
circles, because I think I

00:05:45.706 --> 00:05:46.990
have to draw it a little
bit different.

00:05:46.990 --> 00:05:50.270
Because there's no students
in all 3 clubs.

00:05:50.270 --> 00:05:53.600
So let's say that this
is the chess club.

00:05:53.600 --> 00:05:55.160
There's 40 people in that.

00:05:55.160 --> 00:05:57.520
And let's do drama next.

00:05:57.520 --> 00:06:03.840
There are 30 students
in the drama club.

00:06:03.840 --> 00:06:05.590
And then how many people
are in both?

00:06:05.590 --> 00:06:06.810
Well they told us that
there are 10 people

00:06:06.810 --> 00:06:07.840
in chess and drama.

00:06:07.840 --> 00:06:10.240
So this is right here is 10.

00:06:10.240 --> 00:06:14.010
And then let's draw math.

00:06:14.010 --> 00:06:16.890
So math will intersect
with both of these.

00:06:16.890 --> 00:06:18.950
It looks like that.

00:06:18.950 --> 00:06:24.610
Math has a total
of 25 students.

00:06:24.610 --> 00:06:30.900
And their overlap with
math and drama is 6.

00:06:30.900 --> 00:06:37.150
And the overlap with math
and chest is 5.

00:06:37.150 --> 00:06:39.530
So the main question we have to
ask is, if we were to add

00:06:39.530 --> 00:06:42.820
up all of these-- if we were to
add up the 40 plus the 30

00:06:42.820 --> 00:06:45.550
plus the 25-- how many
times are we counting

00:06:45.550 --> 00:06:46.355
each of these sections?

00:06:46.355 --> 00:06:48.270
So let's just try that, because
we just want to know

00:06:48.270 --> 00:06:51.210
the total number of students
in 3 clubs.

00:06:51.210 --> 00:06:54.750
So let's take the 40.

00:06:54.750 --> 00:06:57.200
We're already counting this 10
once, right, because they're

00:06:57.200 --> 00:06:59.490
in that 40.

00:06:59.490 --> 00:07:02.890
So let's add the 30.

00:07:02.890 --> 00:07:03.990
So 40 plus 30.

00:07:03.990 --> 00:07:06.800
If we add the 40 plus the 30,
we're double counting this 10

00:07:06.800 --> 00:07:08.100
that's in both clubs.

00:07:08.100 --> 00:07:12.250
So if we were to subtract one
of them, this number right

00:07:12.250 --> 00:07:15.160
here is the number of students
in the chess and

00:07:15.160 --> 00:07:17.140
drama clubs, right?

00:07:17.140 --> 00:07:21.570
So that is-- what is that-- that
is 40 plus 30 minus 20.

00:07:21.570 --> 00:07:22.910
That's 60 students.

00:07:22.910 --> 00:07:25.750
So 60 students in
chess and drama.

00:07:25.750 --> 00:07:32.650
And now if we add 25 students--
so we're adding the

00:07:32.650 --> 00:07:35.870
math club to that-- who are
we double counting?

00:07:35.870 --> 00:07:38.340
Now we're double counting these
6 that we've already

00:07:38.340 --> 00:07:39.000
counted right?

00:07:39.000 --> 00:07:41.620
Because we added them in the
30 from the drama club.

00:07:41.620 --> 00:07:45.110
And we're also double counting
these 5, which we added when

00:07:45.110 --> 00:07:49.710
we first took into account the
40 from the chess club, right?

00:07:49.710 --> 00:07:52.760
So since we've already counted
those, we need to subtract

00:07:52.760 --> 00:07:53.870
those from the 25.

00:07:53.870 --> 00:07:56.970
So minus 5 and minus 6.

00:07:56.970 --> 00:07:59.480
This problem is all about
not double counting.

00:07:59.480 --> 00:08:04.430
So let's see, 60 plus
25 is 85, minus 5 is

00:08:04.430 --> 00:08:09.580
80, minus 6 is 74.

00:08:09.580 --> 00:08:11.010
So 74 total students.

00:08:11.010 --> 00:08:15.020
So that's choice C.

00:08:15.020 --> 00:08:21.170
Problem 180.

00:08:21.170 --> 00:08:24.010
In a nationwide poll, N people
were interviewed.

00:08:28.510 --> 00:08:30.480
So question number one.

00:08:30.480 --> 00:08:32.190
That's in the question,
question number 1.

00:08:32.190 --> 00:08:35.549
So it said for question number
1, 1/4 said yes.

00:08:38.370 --> 00:08:42.636
And of those, 1/3 answered
yes to question 2.

00:08:47.230 --> 00:08:49.610
Which of the following
expressions represents the

00:08:49.610 --> 00:08:51.850
number of people interviewed
who did not

00:08:51.850 --> 00:08:54.560
answer yes to both questions?

00:08:57.860 --> 00:09:01.520
OK, so first of all, you have
the people who answered no to

00:09:01.520 --> 00:09:02.960
question 1.

00:09:02.960 --> 00:09:06.590
So that was 3/4 of the
people, right?

00:09:06.590 --> 00:09:08.620
No on question 1.

00:09:08.620 --> 00:09:11.510
And who cares what they answered
on question 2, right?

00:09:11.510 --> 00:09:15.350
Because they definitely said
no to one of the questions.

00:09:15.350 --> 00:09:16.830
Because they want to
say who did not

00:09:16.830 --> 00:09:18.420
answer yes to both questions?

00:09:18.420 --> 00:09:20.010
So these guys fall into
that category.

00:09:20.010 --> 00:09:21.560
They said no to one
of the questions.

00:09:21.560 --> 00:09:24.270
And then you want to say, what
percentage of the total people

00:09:24.270 --> 00:09:28.420
said yes on the first question
and then no on the second?

00:09:28.420 --> 00:09:32.510
That combination would be if 1/3
of the people who said yes

00:09:32.510 --> 00:09:34.340
on the first question, said
yes on the second,

00:09:34.340 --> 00:09:37.560
then 2/3 said no.

00:09:37.560 --> 00:09:40.360
And what percentage, or what
fraction is this of the entire

00:09:40.360 --> 00:09:40.930
population?

00:09:40.930 --> 00:09:43.250
Well it's 2/3 of 1/4.

00:09:43.250 --> 00:09:50.460
So that is 2/3 times 1/4
is equal to, 1, 2.

00:09:50.460 --> 00:09:53.500
That's equal to 1/6.

00:09:53.500 --> 00:09:56.950
1/6 of the entire population
said yes on the first and no

00:09:56.950 --> 00:09:57.750
on the second.

00:09:57.750 --> 00:10:00.790
So if we were to add these two
up, these are people who said

00:10:00.790 --> 00:10:06.410
no in either or both questions,
you get 1/6 plus

00:10:06.410 --> 00:10:10.860
3/4 is equal to-- let's give a
common denominator of 12-- so

00:10:10.860 --> 00:10:18.570
that's 2/6 plus 3/4,
that is 9/12.

00:10:18.570 --> 00:10:22.850
So that is 11/12.

00:10:22.850 --> 00:10:25.350
Oh, there were N people
interviewed.

00:10:25.350 --> 00:10:30.890
So 11/12 of the people did not
answer yes to both questions.

00:10:30.890 --> 00:10:32.820
And so if they want to know
the number of people, you

00:10:32.820 --> 00:10:36.140
would say 11/12 times the total
population, which is N.

00:10:36.140 --> 00:10:38.240
So 11N over 12.

00:10:38.240 --> 00:10:40.630
And that's choice E.

00:10:40.630 --> 00:10:42.680
And my wife is still
staring at me.

00:10:42.680 --> 00:10:43.930
See you in the next video.


GMAT: Math 33

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:05.000
We're on problem 167.

00:00:05.000 --> 00:00:12.540
If 1 plus 1 over x is equal to
2 minus 2 over x, then what

00:00:12.540 --> 00:00:14.410
does x equal?

00:00:14.410 --> 00:00:16.830
The first thing I would do is--
I don't like these x's in

00:00:16.830 --> 00:00:17.580
the denominator.

00:00:17.580 --> 00:00:21.180
So let's just multiply both
sides of this equation by x.

00:00:21.180 --> 00:00:31.840
So I'll get x plus-- x over x is
just 1-- equals 2x minus 2.

00:00:31.840 --> 00:00:34.692
Now I can subtract 1
from both sides.

00:00:34.692 --> 00:00:38.710
Actually, let's add 2 to both
sides, so you get x plus 3 is

00:00:38.710 --> 00:00:40.570
equal to 2x.

00:00:40.570 --> 00:00:42.030
Let's subtract x from
both sides.

00:00:42.030 --> 00:00:44.900
You get 3 is equal to x.

00:00:44.900 --> 00:00:47.470
And that's choice E.

00:00:47.470 --> 00:00:50.095
I think I have the hiccups.

00:00:50.095 --> 00:00:53.330
I hope you don't notice, or if
you do, I hope it doesn't

00:00:53.330 --> 00:00:53.800
bother you.

00:00:53.800 --> 00:00:58.170
Problem 168.

00:00:58.170 --> 00:01:01.220
Last year, for every 100 million
vehicles that traveled

00:01:01.220 --> 00:01:03.680
on a certain highway,
96 vehicles

00:01:03.680 --> 00:01:04.879
were involved in accidents.

00:01:04.879 --> 00:01:09.970
So 96 vehicles for every
100 million.

00:01:09.970 --> 00:01:15.700
So 100 times a million is 10
to the sixth vehicles.

00:01:15.700 --> 00:01:20.060
If 3 billion vehicles traveled
on the highway last year, how

00:01:20.060 --> 00:01:24.350
many of those vehicles were
involved in accidents?

00:01:24.350 --> 00:01:26.370
You could view this as the
fraction that will be involved

00:01:26.370 --> 00:01:28.650
in accidents, so we just
multiply that times the total

00:01:28.650 --> 00:01:29.960
number of vehicles.

00:01:29.960 --> 00:01:34.130
That's 3 times a billion.

00:01:34.130 --> 00:01:37.000
And they even write a billion
as 1 with 9 0's, so it's 3

00:01:37.000 --> 00:01:40.150
times 10 to the ninth.

00:01:40.150 --> 00:01:41.940
So if we simplify this
a little bit, let's

00:01:41.940 --> 00:01:43.200
see what we can do.

00:01:43.200 --> 00:01:47.840
So 100 times 10 to the sixth,
this is the same thing as 1

00:01:47.840 --> 00:01:50.130
times 10 to the eighth.

00:01:50.130 --> 00:01:53.260
You could view this as 10 to
the second times 10 to the

00:01:53.260 --> 00:01:55.570
sixth is 10 to the eighth.

00:01:55.570 --> 00:01:58.180
If you have a 10 to the ninth in
the denominator divided by

00:01:58.180 --> 00:02:00.630
10 to the eighth, this cancels
out and you're

00:02:00.630 --> 00:02:02.230
just left with a 10.

00:02:02.230 --> 00:02:06.590
So the denominator becomes
1, and the numerator

00:02:06.590 --> 00:02:09.625
becomes 96 times 30.

00:02:09.625 --> 00:02:19.620
And 96 times 30 is-- let's put a
0 here and then 3 times 6 is

00:02:19.620 --> 00:02:31.660
18, 3 times 9 is 27
plus 1 is 28.

00:02:31.660 --> 00:02:34.340
How many vehicles are involved
in accidents?

00:02:34.340 --> 00:02:38.660
2,880, C.

00:02:38.660 --> 00:02:40.840
I think this number is low
compared to the reality.

00:02:40.840 --> 00:02:45.930
I think actually it's closer
to the 40,000 range.

00:02:45.930 --> 00:02:46.580
That's what want.

00:02:46.580 --> 00:02:48.330
How many of those vehicles are
involved in accidents?

00:02:48.330 --> 00:02:51.265
We multiply the fraction that
have an accident times the

00:02:51.265 --> 00:02:54.390
total number of vehicles, and
then you get your answer.

00:02:54.390 --> 00:02:55.880
2,880.

00:02:55.880 --> 00:03:01.100
Next question, 169.

00:03:01.100 --> 00:03:04.880
30% of the members of a swim
team have passed the

00:03:04.880 --> 00:03:09.310
lifesaving test.
So 30% passed.

00:03:09.310 --> 00:03:12.870
Among the members will have
passed the test, 12 have taken

00:03:12.870 --> 00:03:14.360
the preparatory course.

00:03:14.360 --> 00:03:17.950
So this the 70% not passed.

00:03:17.950 --> 00:03:24.100
And they say that 12 have taken
a course, so 12 course.

00:03:24.100 --> 00:03:25.730
And 30 have not taken
the course.

00:03:29.730 --> 00:03:31.900
How many members are
in the swim club?

00:03:31.900 --> 00:03:34.690
So many people have not
passed the test?

00:03:34.690 --> 00:03:37.560
They said that 12 have taken
a course, 30 haven't.

00:03:37.560 --> 00:03:38.730
I'm assuming that's
all of them.

00:03:38.730 --> 00:03:43.290
So the 70% that have not passed,
that is 12 plus 30 is

00:03:43.290 --> 00:03:44.710
equal to 42.

00:03:44.710 --> 00:03:48.340
And this is 70% of the total
people in the swim club.

00:03:48.340 --> 00:03:53.180
So 70% of the total
is equal to 42.

00:03:53.180 --> 00:03:59.810
So the total is equal to
42 divided by 0.7.

00:03:59.810 --> 00:04:03.060
42 divided by 7 would be 6, but
since we're dividing it by

00:04:03.060 --> 00:04:05.000
1/10 of that, it's going
to be equal to 60.

00:04:05.000 --> 00:04:05.920
And that make sense.

00:04:05.920 --> 00:04:07.910
70% of 60 is 42.

00:04:07.910 --> 00:04:09.780
So there's 60 people in the
swim club, and that's A.

00:04:13.450 --> 00:04:17.920
Problem 170.

00:04:17.920 --> 00:04:20.500
In a certain company, the
ratio of the number of

00:04:20.500 --> 00:04:22.300
managers to the number
of production

00:04:22.300 --> 00:04:30.750
line workers is 5:72.

00:04:30.750 --> 00:04:34.260
If 8 additional production line
workers were to be hired,

00:04:34.260 --> 00:04:36.460
the ratio of the number of
managers to the number of

00:04:36.460 --> 00:04:39.730
production line workers
would be 5:74.

00:04:39.730 --> 00:04:45.510
So if we hire 8 more
workers, then the

00:04:45.510 --> 00:04:54.900
ratio is equal to 5:74.

00:04:54.900 --> 00:05:00.720
How many managers does
the company have?

00:05:00.720 --> 00:05:03.360
So this is interesting.

00:05:03.360 --> 00:05:04.670
Let's think of it this way.

00:05:04.670 --> 00:05:07.550
There's a quick way to think
about it, and then there's an

00:05:07.550 --> 00:05:09.650
algebraic way, and we'll
do both of them.

00:05:09.650 --> 00:05:13.950
But when we add 8 to the bottom
of this-- when we add 8

00:05:13.950 --> 00:05:18.110
to the number of workers--
the ratio actually only

00:05:18.110 --> 00:05:19.360
increased by 2.

00:05:23.200 --> 00:05:26.910
So what's happening is we
probably have 4 times as many

00:05:26.910 --> 00:05:28.260
workers as this ratio
predicts.

00:05:28.260 --> 00:05:36.080
So the ratio is probably
20:288.

00:05:36.080 --> 00:05:37.720
And let's think about
that logic.

00:05:37.720 --> 00:05:41.540
Because 20/288-- this is 5/72 if
you were to divide the top

00:05:41.540 --> 00:05:46.980
and the bottom by 4-- and if you
were add 8 to the bottom,

00:05:46.980 --> 00:05:50.420
you would get 20/296.

00:05:50.420 --> 00:05:53.537
And then if you divide this by
4, if you just take this to

00:05:53.537 --> 00:05:56.070
the lowest reduced
form, you get 5.

00:05:56.070 --> 00:05:57.770
Divide that by 4.

00:05:57.770 --> 00:05:59.740
This divided by 4 is what?

00:05:59.740 --> 00:06:02.120
It's 74.

00:06:02.120 --> 00:06:05.740
So if you see that intuition
that I'm talking about, that

00:06:05.740 --> 00:06:10.550
when you add 8 workers it only
increased this reduced form by

00:06:10.550 --> 00:06:14.550
2, that means that this reduced
form is probably 1/4

00:06:14.550 --> 00:06:16.300
of the real number.

00:06:16.300 --> 00:06:18.190
Because 2 is 1/4 of 8.

00:06:18.190 --> 00:06:19.910
If that doesn't make sense to
you, don't worry about it.

00:06:19.910 --> 00:06:21.330
But if you did it this
way, you could say

00:06:21.330 --> 00:06:22.700
there must be 20 managers.

00:06:22.700 --> 00:06:25.240
And that would be a very fast
way of solving this problem.

00:06:25.240 --> 00:06:28.145
But we can also do it
alegebraically.

00:06:28.145 --> 00:06:31.070
So you could use this
information to say 5 times the

00:06:31.070 --> 00:06:38.050
workers is equal to 72
times the managers.

00:06:38.050 --> 00:06:43.120
And then you can use this
information to say 5 times the

00:06:43.120 --> 00:06:48.210
workers plus 40-- I'm just
cross-multiplying, so 5 times

00:06:48.210 --> 00:06:52.700
w8-- is equal to 72 times
the managers.

00:06:52.700 --> 00:06:55.960
And now we can do
some algebra.

00:06:55.960 --> 00:06:57.870
So let's subtract this from.

00:06:57.870 --> 00:07:01.520
Let me say minus 5w
is equal to minus

00:07:01.520 --> 00:07:04.630
72 times the managers.

00:07:04.630 --> 00:07:05.950
Subtract this from this.

00:07:05.950 --> 00:07:06.620
These cancel out.

00:07:06.620 --> 00:07:10.550
You get 40 is equal
to 74 minus 72

00:07:10.550 --> 00:07:12.320
is 2 times the managers.

00:07:12.320 --> 00:07:15.265
So then you get the managers are
equal to 20, which is what

00:07:15.265 --> 00:07:16.090
we figured out the first time.

00:07:16.090 --> 00:07:17.790
Maybe this is simpler for you.

00:07:17.790 --> 00:07:19.510
And I would probably do that
on the exam to make sure I

00:07:19.510 --> 00:07:22.400
didn't make a careless
mistake.

00:07:22.400 --> 00:07:31.200
Next question, 171.

00:07:31.200 --> 00:07:38.350
If x minus 1 squared is equal to
400, which of the following

00:07:38.350 --> 00:07:40.750
could be the value
of x minus 1?

00:07:40.750 --> 00:07:44.110
x minus 1 squared is equal to
400, so that means x minus 1

00:07:44.110 --> 00:07:48.430
is equal to positive 20, the
positive square root of 400.

00:07:48.430 --> 00:07:49.340
And don't forget this.

00:07:49.340 --> 00:07:53.580
This also means that x minus 1
could equal minus 20, because

00:07:53.580 --> 00:07:56.660
minus 20 squared is also 400,
the negative square root.

00:07:56.660 --> 00:07:58.720
In this case, if you add 1
to both sides, you get

00:07:58.720 --> 00:08:00.900
x is equal to 21.

00:08:00.900 --> 00:08:04.520
In this case, if you add 1 to
both sides, you get x is equal

00:08:04.520 --> 00:08:09.720
to minus 19.

00:08:09.720 --> 00:08:11.050
So that's what the
values of x are.

00:08:11.050 --> 00:08:15.000
So what can be a value
of x minus 5?

00:08:15.000 --> 00:08:16.330
That's what they ask for,
they're not asking for the

00:08:16.330 --> 00:08:17.390
value of x.

00:08:17.390 --> 00:08:21.400
x minus 5 in this case would
be-- what's 21 minus 5?

00:08:21.400 --> 00:08:29.040
That's 16, minus 19 minus 5.

00:08:29.040 --> 00:08:30.270
That's minus 24.

00:08:30.270 --> 00:08:35.240
So x minus 5 could be 16,
or it could be minus 24.

00:08:35.240 --> 00:08:39.059
And they only have this one
there, so it's choice C.

00:08:39.059 --> 00:08:41.990
So they were just testing to see
if you remembered that x

00:08:41.990 --> 00:08:45.200
minus 1 could be the minus
square root of 400, the

00:08:45.200 --> 00:08:46.450
negative square root.

00:08:49.400 --> 00:08:53.550
Problem 172.

00:08:53.550 --> 00:08:57.530
Which of the following describes
all values of x for

00:08:57.530 --> 00:09:05.880
which 1 minus x squared is
greater than or equal to 0?

00:09:05.880 --> 00:09:09.640
So let's add x squared to both
sides of this equation.

00:09:09.640 --> 00:09:13.360
So then you get 1-- x squared
plus minus x squared, that

00:09:13.360 --> 00:09:16.990
zero-- is greater than or
equal to x squared.

00:09:16.990 --> 00:09:18.410
I'm just going to switch it
because that's just how I

00:09:18.410 --> 00:09:21.630
think. x squared is less
than or equal to 1.

00:09:24.950 --> 00:09:27.030
We're dealing with real numbers,
so x squared is going

00:09:27.030 --> 00:09:28.310
to be a positive number.

00:09:28.310 --> 00:09:32.520
So the only way that I can
square a positive number and

00:09:32.520 --> 00:09:35.720
have it be less than or equal 1,
is if x itself is less than

00:09:35.720 --> 00:09:36.600
or equal to 1.

00:09:36.600 --> 00:09:40.290
Or it could be greater than or
equal to negative 1, because

00:09:40.290 --> 00:09:43.820
those numbers will square
to being positive.

00:09:43.820 --> 00:09:48.100
So x would be less than
or equal to 1.

00:09:48.100 --> 00:09:50.620
But it can also be greater than
or equal to negative 1.

00:09:50.620 --> 00:09:51.220
And you could try it.

00:09:51.220 --> 00:09:52.470
Try negative 1/2.

00:09:52.470 --> 00:09:56.090
Negative 1/2 squared
is positive 1/4.

00:09:56.090 --> 00:09:58.270
No matter what, whether it's
positive or negative, this is

00:09:58.270 --> 00:09:59.750
going to be a positive number.

00:09:59.750 --> 00:10:02.490
And the magnitude can
be at most 1.

00:10:02.490 --> 00:10:04.940
So that takes us from
minus 1 to 1.

00:10:04.940 --> 00:10:05.965
And even 0 works.

00:10:05.965 --> 00:10:07.870
So let's see if that's
one of the choices.

00:10:07.870 --> 00:10:09.850
That's choice E.

00:10:09.850 --> 00:10:12.180
And I'll see you in
the next video.


GMAT: Math 32

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.986
I was interrupted in the last
video by a phone call.

00:00:03.986 --> 00:00:06.740
I was actually expecting a phone
call from someone whose

00:00:06.740 --> 00:00:08.550
call I wanted to take, but that
ended up being one of

00:00:08.550 --> 00:00:09.730
those robocallers.

00:00:09.730 --> 00:00:12.210
So, anyway, I was
on problem 163.

00:00:12.210 --> 00:00:15.090
Let me just start it over so
that I don't interrupt your

00:00:15.090 --> 00:00:16.090
train of thought.

00:00:16.090 --> 00:00:19.820
A fruit salad mixture consists
of apples, peaches, and grapes

00:00:19.820 --> 00:00:22.290
in the ratio of 6:5:2.

00:00:22.290 --> 00:00:26.570
So the ratio of apples to
peaches to grapes is equal to

00:00:26.570 --> 00:00:31.470
6:5:2 by weight.

00:00:31.470 --> 00:00:34.510
If 39 pounds of the mixture is
prepared, the mixture include

00:00:34.510 --> 00:00:37.980
how many more pounds of
apples than grapes?

00:00:37.980 --> 00:00:40.760
So they essentially want us to
figure out what the apples

00:00:40.760 --> 00:00:44.050
minus the grapes are equal to.

00:00:44.050 --> 00:00:48.250
So we have 39 pounds, so the
pounds of apples plus the

00:00:48.250 --> 00:00:53.540
peaches plus the grapes,
that equals 39 pounds.

00:00:53.540 --> 00:00:55.690
And to do a ratio problem like
this, the easiest way is to

00:00:55.690 --> 00:01:01.850
say let's just define the number
of pounds of apples

00:01:01.850 --> 00:01:04.910
being 6 times some number x.

00:01:04.910 --> 00:01:07.400
If the apples are 6 times
some number x,

00:01:07.400 --> 00:01:09.010
what are the peaches?

00:01:09.010 --> 00:01:12.120
We know that the ratio of the
apples to the peaches are 6:5,

00:01:12.120 --> 00:01:13.660
so the peaches are going
to have to be 5

00:01:13.660 --> 00:01:15.150
times some number x.

00:01:15.150 --> 00:01:18.350
By the same logic, the grapes
are going to have to be 2

00:01:18.350 --> 00:01:19.850
times some number x.

00:01:19.850 --> 00:01:23.240
And if you try out the ratio the
apples to the grapes is 6x

00:01:23.240 --> 00:01:25.860
over 2x, which is 6:2.

00:01:25.860 --> 00:01:27.170
So it all works out.

00:01:27.170 --> 00:01:29.390
So let's use this information
to substitute back there and

00:01:29.390 --> 00:01:31.230
see what x is equal to.

00:01:31.230 --> 00:01:39.670
So we get 6x plus 5x plus
2x is equal to 39.

00:01:39.670 --> 00:01:43.580
6 plus 5 is 11 plus 2 is 13.

00:01:43.580 --> 00:01:47.860
13x is equal to 39.

00:01:47.860 --> 00:01:50.840
x is equal to 3.

00:01:50.840 --> 00:01:52.630
And so how many apples
are there?

00:01:52.630 --> 00:01:55.860
6x, there are 18 apples.

00:01:55.860 --> 00:01:58.000
We don't have to figure out the
peaches, but we can do it

00:01:58.000 --> 00:02:03.030
fast. There are 15 peaches, and
then there are 6 grapes.

00:02:03.030 --> 00:02:05.000
And if you want to know how
many more apples there are

00:02:05.000 --> 00:02:10.620
than grapes, 18 minus 6 is equal
to 12-- actually these

00:02:10.620 --> 00:02:11.720
are all in pounds.

00:02:11.720 --> 00:02:15.250
There are 12 pounds more
of apples than grapes.

00:02:15.250 --> 00:02:17.040
And that's choice B.

00:02:20.050 --> 00:02:24.370
Problem 164.

00:02:24.370 --> 00:02:27.430
Louise has x more dollars
than Jim has.

00:02:27.430 --> 00:02:33.240
So L is equal to Jim plus x, and
together they have a total

00:02:33.240 --> 00:02:34.060
of y dollars.

00:02:34.060 --> 00:02:38.650
So L plus J is equal to y.

00:02:38.650 --> 00:02:40.610
Which of the following
represents the number of

00:02:40.610 --> 00:02:42.410
dollars that Jim has?

00:02:42.410 --> 00:02:46.350
So they want Jim equals,
and they want it in

00:02:46.350 --> 00:02:47.240
terms of x and y.

00:02:47.240 --> 00:02:51.700
So let's take this L and
substitute it here so that we

00:02:51.700 --> 00:02:55.340
have an equation that this has
J's, x's, and y's in it.

00:02:55.340 --> 00:02:57.690
So if L is equal to J plus
x, we can substitute.

00:02:57.690 --> 00:03:04.830
So we get J plus x, instead of
an L, plus J is equal to y.

00:03:04.830 --> 00:03:06.630
That is that, just that.

00:03:06.630 --> 00:03:11.810
And you get 2J plus
x is equal to y.

00:03:11.810 --> 00:03:14.180
Subtract x from both sides.

00:03:14.180 --> 00:03:17.820
2J is equal to y minus x.

00:03:17.820 --> 00:03:23.720
And we get J is equal
to y minus x over 2.

00:03:23.720 --> 00:03:25.660
And that is choice A.

00:03:28.170 --> 00:03:29.430
Problem 165.

00:03:29.430 --> 00:03:30.680
Let's switch colors.

00:03:36.060 --> 00:03:39.885
During a certain season, a team
won 80% of its first 100

00:03:39.885 --> 00:03:46.486
games, and 50% of
the remaining.

00:03:50.050 --> 00:03:54.910
If the team won 70% of its games
for the entire season,

00:03:54.910 --> 00:04:03.230
so they won 70% of the total,
what was the total number of

00:04:03.230 --> 00:04:07.000
games that the team played?

00:04:07.000 --> 00:04:10.560
So how do we figure out the
total number of games that

00:04:10.560 --> 00:04:11.000
they played?

00:04:11.000 --> 00:04:14.900
It'll be 100 plus remainder.

00:04:14.900 --> 00:04:16.410
So let's think of it this way.

00:04:16.410 --> 00:04:19.260
If we wanted to figure out the
total percentage that they

00:04:19.260 --> 00:04:22.500
won, we would want the
number that they won.

00:04:22.500 --> 00:04:24.390
So how many games
did they win?

00:04:24.390 --> 00:04:32.000
They won 80% of their first 100
games, so that's 80 games,

00:04:32.000 --> 00:04:34.930
plus 50% of the remainder
games.

00:04:34.930 --> 00:04:38.100
So plus 0.5 times
the remainder.

00:04:38.100 --> 00:04:39.450
That's how many games
they've won.

00:04:39.450 --> 00:04:42.220
Now, how many games
did they play?

00:04:42.220 --> 00:04:46.680
The total number of games is
going to be 100 plus whatever

00:04:46.680 --> 00:04:48.330
this remainder number
of games are.

00:04:48.330 --> 00:04:49.990
That's their total
number of games.

00:04:49.990 --> 00:04:53.050
And the problem tells us that
this ratio, the total

00:04:53.050 --> 00:04:55.830
percentage-- this is of all the
games they won divided by

00:04:55.830 --> 00:04:57.920
all the games they played--
that that is

00:04:57.920 --> 00:05:01.720
equal to 0.7 or 70%.

00:05:01.720 --> 00:05:05.530
If we multiply both sides of
this equation by 100 plus r,

00:05:05.530 --> 00:05:13.480
we get 80 plus 0.5r is equal
to-- 0.7 times 100 is equal to

00:05:13.480 --> 00:05:17.735
70 plus 0.7r.

00:05:17.735 --> 00:05:19.970
Now let's subtract 70
from both sides.

00:05:19.970 --> 00:05:26.570
You get 10 plus 0.5r
is equal to 0.7r.

00:05:26.570 --> 00:05:30.490
And then if we subtract 0.5r
from both sides, we get 10 is

00:05:30.490 --> 00:05:38.490
equal to 0.2r, or r is
equal to 10 over 0.2.

00:05:38.490 --> 00:05:41.640
And 10 over 0.2, 0.2 is
the same thing as 1/5.

00:05:41.640 --> 00:05:45.000
So this is 10 times 5, so
that's equal to 50.

00:05:45.000 --> 00:05:49.710
So the remainder games were 50,
so they want to know the

00:05:49.710 --> 00:05:52.100
total number of games
the team plays.

00:05:52.100 --> 00:05:54.000
So it's going to be the
first 100 games plus

00:05:54.000 --> 00:05:55.320
the remainder games.

00:05:55.320 --> 00:05:56.870
So 100 plus 50.

00:05:56.870 --> 00:06:00.040
They played 150 games,
which is choice D.

00:06:05.140 --> 00:06:10.730
We're on problem 166.

00:06:10.730 --> 00:06:15.260
Of 30 applicants for a job,
14 had at least 4 years of

00:06:15.260 --> 00:06:15.960
experience.

00:06:15.960 --> 00:06:22.780
So experience greater
than 4, we have 14.

00:06:22.780 --> 00:06:26.620
I just make up notation
as we go.

00:06:26.620 --> 00:06:29.430
Of 30 applicants for a job,
14 had at least 4 years of

00:06:29.430 --> 00:06:34.170
experience, 18 had degrees--
so we say degrees, we have

00:06:34.170 --> 00:06:39.120
18-- and 3 had less than 4 years
of experience and did

00:06:39.120 --> 00:06:40.370
not have a degree.

00:06:43.450 --> 00:06:51.490
So experience was less than
4 years' experience and no

00:06:51.490 --> 00:06:59.390
degree, and that is 3 people.

00:06:59.390 --> 00:07:03.250
How many of the applicants had
at least 4 years of experience

00:07:03.250 --> 00:07:04.500
and a degree?

00:07:08.470 --> 00:07:10.550
So let's see how we can
think about it.

00:07:10.550 --> 00:07:13.380
There's 30 applicants
for a job.

00:07:13.380 --> 00:07:17.250
So let me draw our universe.

00:07:17.250 --> 00:07:19.770
If I did all of the
applicants, let me

00:07:19.770 --> 00:07:21.540
draw it like this.

00:07:21.540 --> 00:07:27.910
So that's the whole applicant
universe, and that's 30.

00:07:27.910 --> 00:07:33.940
And we know that 14 had more
than 4 years of experience.

00:07:33.940 --> 00:07:36.590
Let me do that in this color.

00:07:36.590 --> 00:07:47.810
So this attribute, I'll make it
look like that, so that's

00:07:47.810 --> 00:07:52.535
the 14 that have experience
greater than 4 years.

00:07:52.535 --> 00:07:55.470
And that's 14 of them.

00:07:55.470 --> 00:08:01.810
And then 18 have a degree.

00:08:01.810 --> 00:08:04.940
And I'm assuming that there's
some overlap between the

00:08:04.940 --> 00:08:05.900
degrees in the 14.

00:08:05.900 --> 00:08:09.310
We don't know that for sure, but
let's say that this point

00:08:09.310 --> 00:08:15.270
right here I'll draw
another circle.

00:08:15.270 --> 00:08:18.730
So let's say that circle looks
something like that.

00:08:18.730 --> 00:08:21.746
So let's call this the
degreed people.

00:08:21.746 --> 00:08:25.710
And there's 18 of them
in this whole circle.

00:08:25.710 --> 00:08:30.190
And then they say that there
were 3 people with experience

00:08:30.190 --> 00:08:31.610
less than 4 and no degree.

00:08:31.610 --> 00:08:33.700
So they're outside of both
of these circles.

00:08:33.700 --> 00:08:38.750
So this right here is 3.

00:08:38.750 --> 00:08:39.820
Now, what are they asking?

00:08:39.820 --> 00:08:41.590
I've already forgotten that.

00:08:41.590 --> 00:08:44.190
How many of the applicants had
at least 4 years experience

00:08:44.190 --> 00:08:45.370
and a degree?

00:08:45.370 --> 00:08:52.710
So they're asking for this
intersection right here.

00:08:52.710 --> 00:08:55.190
So there's one question
we can answer.

00:08:55.190 --> 00:08:58.870
What's the sum of this
population, the population

00:08:58.870 --> 00:09:03.535
that has more than 4 years of
experience or a degree?

00:09:03.535 --> 00:09:04.340
So let me circle.

00:09:04.340 --> 00:09:06.010
What's this sum right here?

00:09:06.010 --> 00:09:09.580
This and this.

00:09:09.580 --> 00:09:12.880
It's going to be whatever's left
after you take out the 3

00:09:12.880 --> 00:09:14.680
people who have nothing.

00:09:14.680 --> 00:09:17.530
So that's going to
be 27 people.

00:09:17.530 --> 00:09:22.780
So 27 people have at
least 4 years of

00:09:22.780 --> 00:09:24.880
experience or a degree.

00:09:24.880 --> 00:09:27.920
Now, how can relate that
to the 14 and 18?

00:09:27.920 --> 00:09:33.780
So what happens if we just
add the 14 and the 18?

00:09:33.780 --> 00:09:35.390
When we have the 14, we're
going to count

00:09:35.390 --> 00:09:37.720
these people once.

00:09:37.720 --> 00:09:39.940
And if we were add that
to 18, we would count

00:09:39.940 --> 00:09:42.180
these people twice.

00:09:42.180 --> 00:09:44.260
So if we wanted this number--
if we wanted the number of

00:09:44.260 --> 00:09:47.210
people who are in this circle or
this circle-- what we would

00:09:47.210 --> 00:09:54.050
want to do is add the 14 to the
18, but then subtract out

00:09:54.050 --> 00:09:57.390
once the people who are
this intersection.

00:09:57.390 --> 00:09:59.380
Let's call them B.

00:09:59.380 --> 00:10:00.180
That's a B.

00:10:00.180 --> 00:10:01.680
I know you can't read it.

00:10:01.680 --> 00:10:03.280
Let's say there B people
in that intersection.

00:10:03.280 --> 00:10:05.270
You'll subtract that out once.

00:10:05.270 --> 00:10:08.120
I want that to really
make sense for you.

00:10:08.120 --> 00:10:10.530
If we want to know the total
number of people in this

00:10:10.530 --> 00:10:16.110
circle and this circle combined,
when we say 18

00:10:16.110 --> 00:10:18.180
people, that's including
some people here.

00:10:18.180 --> 00:10:21.270
And if we were add that to 14
people, where once again some

00:10:21.270 --> 00:10:23.660
of those 14 are here again,
are these people.

00:10:23.660 --> 00:10:27.030
So when we just add 18 to 14,
we're counting this twice.

00:10:27.030 --> 00:10:30.090
So if we want to know the actual
number in the circle or

00:10:30.090 --> 00:10:33.220
the circle, since we add 18 to
14 we count it twice, we wants

00:10:33.220 --> 00:10:35.060
to subtract these
people out once.

00:10:35.060 --> 00:10:36.010
Hopefully that makes sense.

00:10:36.010 --> 00:10:37.650
But anyway, now we're
ready to solve.

00:10:37.650 --> 00:10:40.710
We get 27 is equal to--
14 plus 18 is

00:10:40.710 --> 00:10:47.050
equal to 32 minus B.

00:10:47.050 --> 00:10:50.640
27 from both sides, you get
0 is equal 5 minus B.

00:10:50.640 --> 00:10:51.510
Add B the both sides.

00:10:51.510 --> 00:10:53.610
You get B is equal to 5.

00:10:53.610 --> 00:10:58.030
And that is choice E.

00:10:58.030 --> 00:10:59.910
See you in the next video.


GMAT: Math 31

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.030
We're on problem 161.

00:00:04.030 --> 00:00:07.190
The positive integer n
is divisible by 25.

00:00:07.190 --> 00:00:15.040
If the square root of n is
greater than 25, which of the

00:00:15.040 --> 00:00:19.150
following could be the value
of n divided by 25?

00:00:19.150 --> 00:00:22.650
So what could be a valid value
for n divided by 25?

00:00:22.650 --> 00:00:24.130
So let's see if we can
manipulate this a little bit.

00:00:24.130 --> 00:00:28.990
If we square both sides, we get
n, right-- the square of

00:00:28.990 --> 00:00:34.270
the square root of n is n--
is greater than 625.

00:00:34.270 --> 00:00:40.090
And then if we divide both sides
by 25, we get n/25 is

00:00:40.090 --> 00:00:42.440
greater than 25.

00:00:42.440 --> 00:00:47.970
So a valid value for n/25 has
to be greater than 25.

00:00:47.970 --> 00:00:50.100
If we look at all the choices,
there's only one value that's

00:00:50.100 --> 00:00:56.690
greater than 25, and
that's E, 26.

00:00:56.690 --> 00:01:00.280
Problem 162.

00:01:00.280 --> 00:01:01.600
It looks they want to make
sure we know how

00:01:01.600 --> 00:01:03.300
to deal with fractions.

00:01:03.300 --> 00:01:11.510
1 over 1 plus 1 over
2 plus 1/3.

00:01:11.510 --> 00:01:13.770
Let's just drive through
this math.

00:01:13.770 --> 00:01:19.080
That equals 1 over 1 plus 1
over-- let's see, 2 is the

00:01:19.080 --> 00:01:26.630
same thing as 6/3 right--
6/3 plus 1/3 is 7/3.

00:01:26.630 --> 00:01:30.806
So this is equal to 1 over 1
plus-- 1 over 7/3 is the same

00:01:30.806 --> 00:01:37.810
thing as 3/7-- so that's equal
to 1 over 7/7 plus

00:01:37.810 --> 00:01:41.170
3/7, so it's 10/7.

00:01:41.170 --> 00:01:46.000
Which is equal to 7/10,
which is choice B.

00:01:48.500 --> 00:01:51.030
Let's switch colors.

00:01:51.030 --> 00:01:54.670
163.

00:01:54.670 --> 00:01:58.880
A fruit salad mixture consists
of apples, peaches, and grapes

00:01:58.880 --> 00:02:04.880
in the ratio-- so the so the
apples to peaches to grapes--

00:02:04.880 --> 00:02:10.729
is equal to 6 to 5 to 2,
respectively by weight.

00:02:10.729 --> 00:02:13.740
If 39 pounds of the mixture is
prepared, the mixture includes

00:02:13.740 --> 00:02:15.760
how many more pounds of
apples than grapes?

00:02:15.760 --> 00:02:20.850
So they want to figure out the
apples minus the grapes.

00:02:20.850 --> 00:02:24.410
OK, so they tell us that the
apples plus the peaches plus

00:02:24.410 --> 00:02:27.560
the grapes is equal
to 39 pounds.

00:02:27.560 --> 00:02:30.360
When you're dealing with a
ratio, it's useful to say well

00:02:30.360 --> 00:02:33.495
each of these is going to be
equal to, I guess you'd call

00:02:33.495 --> 00:02:36.950
it their ratio number times
some number, right?

00:02:36.950 --> 00:02:38.850
Let me explain what
I'm saying.

00:02:38.850 --> 00:02:42.420
Let's just say that apples
are equal to 6 times

00:02:42.420 --> 00:02:44.250
some number x, right?

00:02:44.250 --> 00:02:47.930
Well if apples are equal to 6
times some number x, then

00:02:47.930 --> 00:02:50.370
peaches are going to
be equal to 5 times

00:02:50.370 --> 00:02:52.140
that number x, right?

00:02:52.140 --> 00:02:54.380
Because the ratio of the apples
to the peaches is a 6

00:02:54.380 --> 00:02:55.430
to 5, right?

00:02:55.430 --> 00:02:58.070
So if you did 6x over 5x,
you'd get 6 to 5.

00:02:58.070 --> 00:03:03.940
And by the same logic, then the
grapes would be 2 times

00:03:03.940 --> 00:03:05.110
that same number x.

00:03:05.110 --> 00:03:06.780
And all the ratios
would work out.

00:03:06.780 --> 00:03:09.570
So we can substitute all of
these back into that equation.

00:03:09.570 --> 00:03:17.780
And you get 6x plus 5x plus
2x is equal to 39.

00:03:17.780 --> 00:03:21.850
6 plus 5 is 11, plus 2--

00:03:21.850 --> 00:03:22.230
Well

00:03:22.230 --> 00:03:23.860
actually I have a phone call
that I was expecting.

00:03:23.860 --> 00:03:25.110
So let me continue this
in the next video.