GMAT: Math 30

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:07.670
We're on problem 154.

00:00:07.670 --> 00:00:10.550
An empty pool being filled with
water at a constant rate

00:00:10.550 --> 00:00:15.026
takes 8 hours to fill to
3/5 of its capacity.

00:00:15.026 --> 00:00:22.380
So you could say you can fill
3/5 of a pool every 8 hours,

00:00:22.380 --> 00:00:25.000
which is just another way of
saying that you can fill,

00:00:25.000 --> 00:00:30.610
let's see, 3/5 divided by 8 is
3 over-- what's 5 times 8?

00:00:30.610 --> 00:00:33.792
3/40 pools per hour.

00:00:33.792 --> 00:00:35.990
I have a feeling that they're
just going to ask us something

00:00:35.990 --> 00:00:38.140
along just the rate
of the fill.

00:00:38.140 --> 00:00:41.160
So then they say, how much
more time will it take to

00:00:41.160 --> 00:00:43.690
finish filling the pool?

00:00:43.690 --> 00:00:46.200
Oh, well, this actually is
a little bit easier.

00:00:46.200 --> 00:00:53.170
So if it takes 8 hours to fill
the pool to 3/5 of capacity,

00:00:53.170 --> 00:00:55.860
how many hours will it take to
finish filling the pool?

00:00:55.860 --> 00:01:00.360
So how many hours will it take
to fill it essentially what?

00:01:00.360 --> 00:01:01.170
What's left?

00:01:01.170 --> 00:01:02.475
2/5, right?

00:01:02.475 --> 00:01:03.900
To fill it 2/5 more.

00:01:07.690 --> 00:01:12.130
So this is going to be the same
thing as 8/3 is equal to

00:01:12.130 --> 00:01:15.870
x/2, And you could do that by
dividing both sides of this

00:01:15.870 --> 00:01:17.270
equation by 5, right?

00:01:17.270 --> 00:01:19.400
And you get a 5 in the
denominator in both places,

00:01:19.400 --> 00:01:21.100
which cancels out.

00:01:21.100 --> 00:01:28.440
And this is easy, 8/3 as x/2 so
you have 16 is equal to 3x.

00:01:28.440 --> 00:01:34.380
And then you get x is equal
to 5 1/3 hours.

00:01:34.380 --> 00:01:37.510
That's 5 hours and 20 minutes,
because a third of an hour is

00:01:37.510 --> 00:01:38.680
20 minutes.

00:01:38.680 --> 00:01:40.350
And that is choice B.

00:01:44.240 --> 00:01:48.220
Problem 155.

00:01:48.220 --> 00:01:54.220
A positive number x is
multiplied by 2 and this

00:01:54.220 --> 00:01:56.980
product is then divided by 3.

00:01:56.980 --> 00:01:58.470
And they told us
x was positive.

00:01:58.470 --> 00:02:01.460
If the positive square root
of the result of these two

00:02:01.460 --> 00:02:12.380
operations equals x, what
is the value of x?

00:02:12.380 --> 00:02:13.180
Let's solve this.

00:02:13.180 --> 00:02:17.520
Let's square both sides of this,
so we get 2x/3 is equal

00:02:17.520 --> 00:02:20.060
to x squared.

00:02:20.060 --> 00:02:25.170
And then multiply both sides by
3, you get 2x is equal to

00:02:25.170 --> 00:02:27.550
3x squared.

00:02:27.550 --> 00:02:29.950
And they tell us that
x is positive.

00:02:32.750 --> 00:02:34.920
Well, first of all, that tells
us that x does not equal 0.

00:02:34.920 --> 00:02:37.760
They tell us that x
is greater than 0.

00:02:37.760 --> 00:02:39.290
So let's see, let's divide
both sides of

00:02:39.290 --> 00:02:41.770
this equation by x.

00:02:41.770 --> 00:02:44.410
Well, there's a couple of ways
we could think about it.

00:02:44.410 --> 00:02:47.200
Maybe the easiest way-- yeah,
let's just divide both sides

00:02:47.200 --> 00:02:49.960
of this equation by x, and we
can do that because we know

00:02:49.960 --> 00:02:54.430
that x cannot equal 0 and you
get 2 is equal to 3x.

00:02:54.430 --> 00:02:59.480
And then divide both sides by 3,
you get x is equal to 2/3,

00:02:59.480 --> 00:03:00.820
and that's choice D.

00:03:00.820 --> 00:03:03.160
I was thinking about saying,
oh, well, let's subtract 2x

00:03:03.160 --> 00:03:03.840
from both sides.

00:03:03.840 --> 00:03:07.310
You get 3x squared minus
2x is equal to 0.

00:03:07.310 --> 00:03:12.540
You could factor out an x, and
then you get 3x minus 2 times

00:03:12.540 --> 00:03:14.140
x is equal to 0.

00:03:14.140 --> 00:03:17.050
And that would tell you that
either 3x minus 2 is equal to

00:03:17.050 --> 00:03:20.070
0, which would give you that
answer, or x is equal to 0.

00:03:20.070 --> 00:03:20.260
but?

00:03:20.260 --> 00:03:22.630
X can't equal 0, because they
told us that x is positive.

00:03:22.630 --> 00:03:26.930
But anyway, it gets you
the same answer, 2/3.

00:03:26.930 --> 00:03:29.960
Next question.

00:03:29.960 --> 00:03:34.030
Problem 156.

00:03:34.030 --> 00:03:42.760
A tank contains 10,000 gallons
of a solution that is 5%

00:03:42.760 --> 00:03:44.400
sodium chloride by volume.

00:03:50.200 --> 00:03:50.960
What is that?

00:03:50.960 --> 00:03:51.530
NaCl?

00:03:51.530 --> 00:03:52.770
This isn't chemistry.

00:03:52.770 --> 00:03:56.940
Let's see, 2,500 gallons of
water evaporate from the tank.

00:04:01.410 --> 00:04:03.770
The remaining solution will be
approximately what percent

00:04:03.770 --> 00:04:05.010
sodium chloride?

00:04:05.010 --> 00:04:08.345
So they're essentially saying,
well, if water evaporates, the

00:04:08.345 --> 00:04:11.650
salt in the solution doesn't
go away, right?

00:04:11.650 --> 00:04:13.800
The salt doesn't evaporate
with the water.

00:04:13.800 --> 00:04:22.110
So if we're 5% sodium chloride,
what's 5% of 10,000?

00:04:22.110 --> 00:04:33.580
5% of 100 is 5, 5% of 1,000 is
50, 5% of 10,000 is 500.

00:04:33.580 --> 00:04:35.560
I guess you could
say there's 500

00:04:35.560 --> 00:04:36.810
gallons of sodium chloride.

00:04:42.950 --> 00:04:47.290
If we lose 2,500 gallons of the
water, now the percentage

00:04:47.290 --> 00:04:50.435
of sodium chloride is still
500, but it's going

00:04:50.435 --> 00:04:52.180
to be 500 over what?

00:04:52.180 --> 00:04:53.970
We don't have 10,00 gallons
of water anymore.

00:04:53.970 --> 00:04:57.000
We have 10,000 gallons of water
minus 2,500 gallons of

00:04:57.000 --> 00:05:01.270
water, so we have 7,500
gallons of water.

00:05:01.270 --> 00:05:08.120
Let's see, that's equal to 5/75,
which is equal to 1/15,

00:05:08.120 --> 00:05:09.380
and what is that as
a percentage?

00:05:13.630 --> 00:05:16.550
15 goes into 100 6 times?

00:05:16.550 --> 00:05:17.060
It's not 7.

00:05:17.060 --> 00:05:18.440
Yeah, 6 times.

00:05:18.440 --> 00:05:20.380
6 times 5 is 30.

00:05:20.380 --> 00:05:22.620
6 times 1 is 6 plus 90.

00:05:22.620 --> 00:05:25.500
And then I have a
10, so it keeps

00:05:25.500 --> 00:05:28.410
going on, 0.0666 repeating.

00:05:28.410 --> 00:05:33.060
So it's 6 and 2/3 percentages,
or 6 point-- well, let's see,

00:05:33.060 --> 00:05:34.660
they have choice-- yeah,
choice D looks right.

00:05:34.660 --> 00:05:37.490
6.67% if you round this.

00:05:37.490 --> 00:05:38.930
It just keeps repeating
with 6.

00:05:38.930 --> 00:05:41.035
So that is choice D.

00:05:44.690 --> 00:05:48.360
Problem 157.

00:05:48.360 --> 00:05:59.160
If x plus 5y is equal to 16 and
x is equal to minus 3y,

00:05:59.160 --> 00:06:00.340
then what does y equal?

00:06:00.340 --> 00:06:02.830
So let's just substitute
this over here.

00:06:02.830 --> 00:06:08.950
So we get minus 3y plus
5y is equal to 16.

00:06:08.950 --> 00:06:11.980
Add those together, you get
2y is equal to 16, or

00:06:11.980 --> 00:06:14.470
y is equal to 8.

00:06:14.470 --> 00:06:18.610
And that is choice E.

00:06:18.610 --> 00:06:23.720
Problem 158.

00:06:23.720 --> 00:06:28.580
A committee is composed of w
women and m men, so w and m.

00:06:28.580 --> 00:06:32.510
If 3 women and 2 men are added
to the committee and if one

00:06:32.510 --> 00:06:35.450
person is selected at random
from the enlarged committee,

00:06:35.450 --> 00:06:37.790
then the probability that a
woman is selected can be

00:06:37.790 --> 00:06:39.960
represented as?

00:06:39.960 --> 00:06:43.910
OK, the probability of a woman
being selected is essentially

00:06:43.910 --> 00:06:48.000
the fraction of the population
that are women.

00:06:48.000 --> 00:06:49.840
So how many women are there?

00:06:49.840 --> 00:06:53.900
There's going to be
w plus 3 women.

00:06:53.900 --> 00:06:56.400
And what's the total
population?

00:06:56.400 --> 00:07:02.320
There's w plus 3 women, and
then there's m plus 2 men.

00:07:05.250 --> 00:07:07.720
So this is the probability we
select a woman, and then we

00:07:07.720 --> 00:07:08.930
can just simplify this a bit.

00:07:08.930 --> 00:07:15.750
That equals w plus 3 over
w plus m plus 5.

00:07:15.750 --> 00:07:19.710
And that is choice E.

00:07:19.710 --> 00:07:27.020
Problem 159.

00:07:27.020 --> 00:07:37.360
If the product of the integers
w, x, y, and z is 70.

00:07:37.360 --> 00:07:44.630
So let me just write that, w,
x, y, z is equal to 770.

00:07:44.630 --> 00:07:50.550
And if 1 is less than w, which
is less than x, which is less

00:07:50.550 --> 00:07:59.410
than y, which is less than z,
what is the value of w plus z?

00:07:59.410 --> 00:08:00.620
Ok, this is interesting.

00:08:00.620 --> 00:08:04.270
So they're telling us that
these are integers.

00:08:04.270 --> 00:08:09.230
We have four integers, and
they're all greater than 1.

00:08:09.230 --> 00:08:12.120
Let's see if we can do a prime
factorization of 770, because

00:08:12.120 --> 00:08:15.770
I have a feeling that that is
what it's going to come down

00:08:15.770 --> 00:08:18.380
to, if we just look at
the factors of 770.

00:08:18.380 --> 00:08:21.940
So let's see, that's
equal to 7.

00:08:21.940 --> 00:08:27.050
7 times 11 is 77.

00:08:27.050 --> 00:08:30.320
7 times 11 times 10, right?

00:08:30.320 --> 00:08:32.710
7 times 11 times 10 is 770.

00:08:32.710 --> 00:08:33.179
110.

00:08:33.179 --> 00:08:38.184
And then 10 is-- so that's equal
to 7 times 11 times--

00:08:38.184 --> 00:08:40.010
let's see, what's 10?

00:08:40.010 --> 00:08:45.450
10 we can rewrite
it as 2 times 5.

00:08:45.450 --> 00:08:47.540
So that's its prime
factorization.

00:08:47.540 --> 00:08:50.300
And lo and behold, we have
exactly four numbers, and

00:08:50.300 --> 00:08:50.920
let's put them in order.

00:08:50.920 --> 00:08:56.430
So that equals 2 times
5 times 7 times 11.

00:08:56.430 --> 00:09:00.980
And so this is what w, x, y,
and z have to be because

00:09:00.980 --> 00:09:01.810
they're different numbers.

00:09:01.810 --> 00:09:04.520
They're all greater than 1,
and they're all integers.

00:09:04.520 --> 00:09:09.330
So let's see, they want to know
w plus z, so w plus z is

00:09:09.330 --> 00:09:13.340
2 plus 11 is 13, and
that's choice B.

00:09:16.610 --> 00:09:19.680
And looks like we have time
even for another one.

00:09:19.680 --> 00:09:21.910
Let's do this one.

00:09:21.910 --> 00:09:25.690
The figure above shows
a circular flowerbed.

00:09:25.690 --> 00:09:29.160
Let me see if I can draw it.

00:09:29.160 --> 00:09:33.430
So that's the outside and then
they drew the inside.

00:09:33.430 --> 00:09:42.324
So if I do it like-- oh,
I didn't do a good job.

00:09:42.324 --> 00:09:44.230
Let me see if I can
draw the inside,

00:09:44.230 --> 00:09:45.770
maybe if I put it there.

00:09:50.960 --> 00:09:54.270
It's not showing up
the way I want to.

00:09:54.270 --> 00:09:57.700
Let me move it around
a little bit.

00:09:57.700 --> 00:10:00.400
Let me just erase it and try
to draw it one more time.

00:10:03.180 --> 00:10:05.350
Well, I think you
get the idea.

00:10:05.350 --> 00:10:10.620
And they tell us that the
distance from the center to

00:10:10.620 --> 00:10:13.315
here is 8 feet.

00:10:13.315 --> 00:10:16.070
The figure above shows a
circular flowerbed with it's

00:10:16.070 --> 00:10:18.230
center at 0 and surrounded
by a circular path

00:10:18.230 --> 00:10:19.590
that is 5 feet wide.

00:10:19.590 --> 00:10:20.662
So this is 5.

00:10:20.662 --> 00:10:22.030
That is 3 feet wide.

00:10:22.030 --> 00:10:22.840
I'm having trouble reading.

00:10:22.840 --> 00:10:23.920
3 feet wide.

00:10:23.920 --> 00:10:27.700
What is the area of the
path in square feet?

00:10:27.700 --> 00:10:31.570
So it's essentially the area of
this outside circle minus

00:10:31.570 --> 00:10:33.260
the area of this
inside circle.

00:10:33.260 --> 00:10:35.800
That would be the area
of this path.

00:10:35.800 --> 00:10:37.700
So what's the area of
the outside circle?

00:10:37.700 --> 00:10:41.310
Well, the area of the outside
circle, you go 8 to get to

00:10:41.310 --> 00:10:43.470
this edge, then you
have 3 more.

00:10:43.470 --> 00:10:49.130
So its radius is 8 plus 3 is 11,
so the area of the outside

00:10:49.130 --> 00:10:51.750
is equal to pi times
11 squared.

00:10:51.750 --> 00:10:53.145
And what's the area
of the inside?

00:10:53.145 --> 00:10:56.910
The area of the inside is equal
to pi times the inside

00:10:56.910 --> 00:11:00.080
radius times 64 squared.

00:11:00.080 --> 00:11:04.400
So the area of our path is
going to be area of the

00:11:04.400 --> 00:11:06.590
outside minus the area of
the inside, which is

00:11:06.590 --> 00:11:09.500
equal to 121 pi.

00:11:09.500 --> 00:11:13.710
That's 11 squared times pi
minus-- oh, sorry, it's 8

00:11:13.710 --> 00:11:15.620
squared, which is 64.

00:11:15.620 --> 00:11:16.810
I should have written
8 squared there, so

00:11:16.810 --> 00:11:19.420
that's minus 64 pi.

00:11:19.420 --> 00:11:22.350
And that is equal to--
what's 121 minus 64?

00:11:22.350 --> 00:11:31.990
121 minus 64 is-- let's see,
we have 21 plus 36, that's

00:11:31.990 --> 00:11:38.040
equal to 57 pi, and
that is choice D.

00:11:38.040 --> 00:11:39.880
See you in the next video.


GMAT: Data sufficiency 21 (correction)

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:03.730
Problem 94.

00:00:03.730 --> 00:00:08.090
In Jefferson School, 300
students study French or

00:00:08.090 --> 00:00:10.090
Spanish or both.

00:00:10.090 --> 00:00:10.430
OK.

00:00:10.430 --> 00:00:11.750
And they have to do
one of those two.

00:00:11.750 --> 00:00:13.890
If 100 of these students do
not study French-- so this

00:00:13.890 --> 00:00:17.180
sounds like a Venn Diagram.

00:00:17.180 --> 00:00:18.975
Let's see.

00:00:18.975 --> 00:00:20.966
So let's say that's French.

00:00:20.966 --> 00:00:23.270
And I'll do Spanish in
a different color.

00:00:23.270 --> 00:00:27.230
Let's say that is Spanish.

00:00:27.230 --> 00:00:28.870
And we have 300 students,
and they study

00:00:28.870 --> 00:00:31.360
either French or Spanish.

00:00:31.360 --> 00:00:33.870
If 100 of these students
do not study French--

00:00:33.870 --> 00:00:34.460
so what did I say?

00:00:34.460 --> 00:00:37.560
This was French and
this is Spanish.

00:00:37.560 --> 00:00:39.470
So 100 of these students
do not study French.

00:00:39.470 --> 00:00:45.730
So this area right
here is 100.

00:00:45.730 --> 00:00:46.120
Right?

00:00:46.120 --> 00:00:49.700
Those are people who study
Spanish but no French at all.

00:00:49.700 --> 00:00:52.290
How many of these students study
both French and Spanish?

00:00:52.290 --> 00:00:54.990
So what they want to know is
the intersection of who

00:00:54.990 --> 00:00:56.360
studies French and Spanish.

00:00:56.360 --> 00:00:59.120
So that's this blue
area right here.

00:00:59.120 --> 00:01:04.140
So statement number one tells
us, of the 300 students, 60 do

00:01:04.140 --> 00:01:05.487
not study Spanish.

00:01:13.840 --> 00:01:19.600
So people who study French but
no Spanish-- and I didn't mean

00:01:19.600 --> 00:01:20.570
they know Spanish.

00:01:20.570 --> 00:01:23.560
People who study French and do
not study Spanish, that's this

00:01:23.560 --> 00:01:24.000
right here.

00:01:24.000 --> 00:01:27.210
And that's 60.

00:01:27.210 --> 00:01:27.570
Right?

00:01:27.570 --> 00:01:29.950
And let's see if we can use this
information to figure out

00:01:29.950 --> 00:01:32.020
what the intersection is.

00:01:32.020 --> 00:01:34.780
So if you think about it,
what we want to do

00:01:34.780 --> 00:01:37.930
is the whole universe.

00:01:37.930 --> 00:01:41.290
So the whole universe is going
to be equal to the people who

00:01:41.290 --> 00:01:48.770
study-- so this 60 people plus--
we'll call that the

00:01:48.770 --> 00:01:52.496
intersection, or we could call
that French and Spanish-- plus

00:01:52.496 --> 00:01:54.930
this blue area, plus
this tan area.

00:01:54.930 --> 00:01:55.140
Right?

00:01:55.140 --> 00:01:56.750
That's the whole universe.

00:01:56.750 --> 00:01:59.740
And that is equal to 300.

00:01:59.740 --> 00:02:02.210
So people who study French
and Spanish plus

00:02:02.210 --> 00:02:06.210
160 is equal to 300.

00:02:06.210 --> 00:02:10.639
Subtract 160 from both sides
and you get the people who

00:02:10.639 --> 00:02:11.820
study just French and Spanish.

00:02:11.820 --> 00:02:12.800
That's what?

00:02:12.800 --> 00:02:14.390
That's 240.

00:02:14.390 --> 00:02:20.280
That's 140 people who study
both French and Spanish.

00:02:20.280 --> 00:02:22.730
So statement one
is sufficient.

00:02:22.730 --> 00:02:25.570
Let's see what statement
two gets us.

00:02:25.570 --> 00:02:29.390
Statement two tells us, a total
of 240 of the students

00:02:29.390 --> 00:02:32.250
study Spanish.

00:02:32.250 --> 00:02:34.140
So in statement two,
we don't know this.

00:02:34.140 --> 00:02:37.460
But we know that a total
of 240 students

00:02:37.460 --> 00:02:40.520
study Spanish, right?

00:02:40.520 --> 00:02:42.530
So we know that this whole
circle is 240.

00:02:42.530 --> 00:02:45.200
And if we're just trying to
figure out this blue part, we

00:02:45.200 --> 00:02:48.020
just have to subtract
out the tan part.

00:02:48.020 --> 00:02:50.400
So if we want to know French
and Spanish, French and

00:02:50.400 --> 00:02:54.420
Spanish is going to be equal to
the whole amount that study

00:02:54.420 --> 00:02:59.110
Spanish-- which they just told
us, 240-- minus just the

00:02:59.110 --> 00:03:01.060
people who study only Spanish.

00:03:01.060 --> 00:03:01.310
Right?

00:03:01.310 --> 00:03:03.390
Because you could study
Spanish and French.

00:03:03.390 --> 00:03:06.980
So 240 people who study Spanish
minus the people who

00:03:06.980 --> 00:03:08.100
just study Spanish.

00:03:08.100 --> 00:03:08.840
So that's 100.

00:03:08.840 --> 00:03:10.100
That's that tan area.

00:03:10.100 --> 00:03:12.150
That also equals 140.

00:03:12.150 --> 00:03:14.620
So both statements individually
are sufficient to

00:03:14.620 --> 00:03:15.480
answer this question.

00:03:15.480 --> 00:03:18.260
So the answer is D.


GMAT: Math 28

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:06.910
Problem 143.

00:00:06.910 --> 00:00:15.960
3.003 divided by 2.002 is equal
to-- well, this already

00:00:15.960 --> 00:00:17.920
looks suspicious.

00:00:17.920 --> 00:00:22.930
It looks like 3/2, but just to
confirm that, this is the same

00:00:22.930 --> 00:00:28.880
thing as 3 times 1.001.

00:00:28.880 --> 00:00:34.680
And this is the same thing
as 2 times 1.001.

00:00:34.680 --> 00:00:37.860
So you cancel those out, and you
get 3/2, which is the same

00:00:37.860 --> 00:00:39.970
thing is 1.5.

00:00:39.970 --> 00:00:42.430
And that's choice E.

00:00:42.430 --> 00:00:46.700
Problem 144.

00:00:46.700 --> 00:00:57.290
If 4 minus x over 2 plus x is
equal to x, what is the value

00:00:57.290 --> 00:01:04.200
of x squared plus 3x minus 4?

00:01:04.200 --> 00:01:05.710
Let's see if we could simplify
this little bit.

00:01:05.710 --> 00:01:09.080
Let's multiply both sides of
this equation times 2 plus x,

00:01:09.080 --> 00:01:13.430
so you get 4 minus x is equal
to 2 plus x times x

00:01:13.430 --> 00:01:17.450
is 2x plus x squared.

00:01:17.450 --> 00:01:24.030
Let's subtract x from both--
no, add x to both sides, so

00:01:24.030 --> 00:01:29.130
you get 4 is equal to
3x plus x squared.

00:01:29.130 --> 00:01:34.530
Subtract 4 from both sides,
and you get 0 is equal to

00:01:34.530 --> 00:01:38.990
minus 4 plus 3x plus x squared,
which is just

00:01:38.990 --> 00:01:41.110
rearranging that, which was the
same thing as x squared

00:01:41.110 --> 00:01:46.820
plus 3x minus 4, so that is
equal to 0, which is C.

00:01:50.260 --> 00:01:58.140
Next problem, 145.

00:01:58.140 --> 00:02:00.140
They've drawn us a trapezoid.

00:02:00.140 --> 00:02:03.935
Let me see if I can draw
the trapezoid as well.

00:02:03.935 --> 00:02:09.699
So they have the base, one
side, and then they have

00:02:09.699 --> 00:02:12.820
another side that looks
something like that, and they

00:02:12.820 --> 00:02:17.040
connect the two lines
like that.

00:02:17.040 --> 00:02:21.800
And then they labeled this side
as 2 feet, this side as 5

00:02:21.800 --> 00:02:26.320
feet, and they say this
is a, and this is b.

00:02:26.320 --> 00:02:27.420
All right.

00:02:27.420 --> 00:02:29.800
The trapezoid shown in the
figure above represents a

00:02:29.800 --> 00:02:32.720
cross-section of the
rudder of a ship.

00:02:32.720 --> 00:02:45.930
If the distance from a to b is
13 feet, what is the area of

00:02:45.930 --> 00:02:50.510
the cross-section of
the rudder in feet?

00:02:50.510 --> 00:02:52.920
So let's think about
this a little bit.

00:02:52.920 --> 00:02:55.805
Let's see if we can figure
out this area.

00:03:03.860 --> 00:03:06.490
So this area right here
is easy to figure out.

00:03:06.490 --> 00:03:10.430
If this is 13, this is
5, what is this side

00:03:10.430 --> 00:03:11.180
going to be equal to?

00:03:11.180 --> 00:03:12.270
We can use the Pythagorean
theorem.

00:03:12.270 --> 00:03:16.770
If we call this x, we could say
x squared plus this side

00:03:16.770 --> 00:03:20.020
squared, plus 25, is equal
to 13 squared.

00:03:20.020 --> 00:03:21.060
13 squared is what?

00:03:21.060 --> 00:03:22.420
169?

00:03:22.420 --> 00:03:23.780
I want to say that,
but let me check.

00:03:23.780 --> 00:03:26.160
I haven't memorized my
13 times tables.

00:03:26.160 --> 00:03:28.460
3 times 13 is 39.

00:03:28.460 --> 00:03:29.330
130.

00:03:29.330 --> 00:03:33.680
Yeah, 169.

00:03:33.680 --> 00:03:35.930
Subtract 25 from both
sides, you get x

00:03:35.930 --> 00:03:39.540
squared is equal to 144.

00:03:39.540 --> 00:03:40.800
That's tells us we're
on the right track.

00:03:40.800 --> 00:03:41.755
We got a perfect square.

00:03:41.755 --> 00:03:45.070
So x is equal to 12.

00:03:45.070 --> 00:03:48.020
So it's very easy to figure out
the area of this part of

00:03:48.020 --> 00:03:50.770
the trapezoid, right?

00:03:50.770 --> 00:03:53.160
It's just the base times
the height times 1/2.

00:03:56.190 --> 00:04:00.110
So it'd be 12 times 5, which
is 60, times 1/2, so this

00:04:00.110 --> 00:04:01.570
would be 30, this area here.

00:04:01.570 --> 00:04:03.900
But this is a little
bit trickier.

00:04:03.900 --> 00:04:05.150
Well, let's think about
it this way.

00:04:07.910 --> 00:04:13.580
Let's extend this trapezoid
into a rectangle.

00:04:13.580 --> 00:04:16.649
So if you did that, then you
can view the area of the

00:04:16.649 --> 00:04:18.620
rudder, so the original problem
that I have in magenta

00:04:18.620 --> 00:04:25.290
there as the area of this entire
rectangle minus this

00:04:25.290 --> 00:04:28.960
green area right here.

00:04:28.960 --> 00:04:30.580
So let's do that.

00:04:30.580 --> 00:04:33.510
First of all, what's the area
of this green area?

00:04:33.510 --> 00:04:35.810
So this side is what?

00:04:35.810 --> 00:04:37.960
If this is 2, this side is 3.

00:04:37.960 --> 00:04:40.650
And we already figured
out this width is 12.

00:04:40.650 --> 00:04:44.730
So the green area is 3 times 12
times 1/2, that's the area

00:04:44.730 --> 00:04:45.810
of a triangle.

00:04:45.810 --> 00:04:49.320
So 3 times 12 is 36
times 1/2 is 18.

00:04:49.320 --> 00:04:51.510
Do it in a different color.

00:04:51.510 --> 00:04:53.830
So the area of this
green area is 18.

00:04:53.830 --> 00:04:59.010
So the area of our rudder, this
whole magenta area that

00:04:59.010 --> 00:05:02.430
I'm shading n, is equal
to the area of the

00:05:02.430 --> 00:05:04.570
rectangle minus 18.

00:05:04.570 --> 00:05:09.190
So the area of the rectangle
is 12 times 5, which is 60.

00:05:09.190 --> 00:05:12.420
And then let's subtract out
the green area minus 18 is

00:05:12.420 --> 00:05:17.150
equal to 60 minus
10, that's 42.

00:05:17.150 --> 00:05:20.510
And that's choice C.

00:05:20.510 --> 00:05:26.830
Problem 146.

00:05:26.830 --> 00:05:33.340
If 0 is less than or equal to x,
which is less than or equal

00:05:33.340 --> 00:05:40.810
to 4, and y is less than 12,
which of the following cannot

00:05:40.810 --> 00:05:43.030
be the value of xy?

00:05:46.010 --> 00:05:48.080
Let me going to write down
all the choices.

00:05:48.080 --> 00:05:52.380
A, minus 2.

00:05:52.380 --> 00:05:58.510
and y is less than 12, I don't
see why not. x could be 2 and

00:05:58.510 --> 00:06:00.810
y could be minus 1.

00:06:00.810 --> 00:06:02.470
y could be arbitrarily small.

00:06:02.470 --> 00:06:05.760
So that's not true,
it's not A.

00:06:05.760 --> 00:06:08.770
B, 0.

00:06:08.770 --> 00:06:11.460
Well, x and y actually can
both be zero, so if you

00:06:11.460 --> 00:06:15.320
multiple them, no reason
why you can't get zero.

00:06:15.320 --> 00:06:18.810
C, 6.

00:06:18.810 --> 00:06:23.840
Well, if x is 2, and y is
3, I can easily get 6.

00:06:23.840 --> 00:06:25.370
And they don't even
tell us that x and

00:06:25.370 --> 00:06:28.320
y have to be different.

00:06:28.320 --> 00:06:29.940
They don't even tell us they
have to be integers.

00:06:29.940 --> 00:06:40.110
24, Well, the largest number
that x can be equal to is 4,

00:06:40.110 --> 00:06:46.110
and x can't be a negative, so
if x is 4, times y is 6, you

00:06:46.110 --> 00:06:48.950
get 24, so I'm already guessing
the answer is E

00:06:48.950 --> 00:06:50.500
without having looked at it.

00:06:50.500 --> 00:06:52.900
E, 48.

00:06:52.900 --> 00:06:56.650
Because the largest value of x
is 4, and so you'd have to

00:06:56.650 --> 00:07:01.380
multiply 4 times 12 to get to
48, but you can't get there.

00:07:01.380 --> 00:07:04.460
So the answer is E.

00:07:04.460 --> 00:07:07.160
Problem 147.

00:07:07.160 --> 00:07:08.410
Turn the page.

00:07:11.000 --> 00:07:13.180
I'll do it in blue since
it seems to be

00:07:13.180 --> 00:07:14.550
dealing with water.

00:07:14.550 --> 00:07:19.960
147: In the figure above-- let
me draw the figure above.

00:07:19.960 --> 00:07:26.270
So I have a V-axis, then I have
this other axis, and then

00:07:26.270 --> 00:07:29.950
they've drawn what looks like
water, at least before reading

00:07:29.950 --> 00:07:31.886
the problem.

00:07:31.886 --> 00:07:33.630
I've drawn that.

00:07:33.630 --> 00:07:36.490
They say this is V.

00:07:36.490 --> 00:07:41.360
They say that this distance
is 5 feet.

00:07:41.360 --> 00:07:46.660
They say that this distance is
10 feet, and they label us a

00:07:46.660 --> 00:07:47.430
couple of points.

00:07:47.430 --> 00:07:51.700
It's R and that's S.

00:07:51.700 --> 00:07:55.030
In the figure above, V
represents an observation

00:07:55.030 --> 00:07:58.830
point at one end of a pool.

00:07:58.830 --> 00:08:01.710
From V, an object that is
actually located on the bottom

00:08:01.710 --> 00:08:06.700
of the pool at point R, so an
object that is actually here,

00:08:06.700 --> 00:08:08.170
appears to be at point S.

00:08:12.070 --> 00:08:14.480
The light goes from here, but
then refracts, and goes like

00:08:14.480 --> 00:08:16.840
that, so if you just follow the
path, it would go straight

00:08:16.840 --> 00:08:17.990
to point S.

00:08:17.990 --> 00:08:24.100
If VR is equal to
10 feet-- what?

00:08:24.100 --> 00:08:26.190
Oh, VR is equal to 10 feet,
so let me draw VR.

00:08:31.050 --> 00:08:35.230
VR is equal to 10 feet.

00:08:35.230 --> 00:08:39.870
What is the distance RS, in
feet, between the actual

00:08:39.870 --> 00:08:45.610
position and the perceived
position of the object?

00:08:45.610 --> 00:08:49.350
So they want to know this
distance right here.

00:08:49.350 --> 00:08:52.020
Well let's think about
it a little bit.

00:08:52.020 --> 00:08:56.840
If we can figure out this
distance, then we subtract

00:08:56.840 --> 00:08:59.790
that distance from 10, we'll
have this distance.

00:08:59.790 --> 00:09:00.330
So what's this?

00:09:00.330 --> 00:09:01.840
Let's call this x.

00:09:01.840 --> 00:09:09.230
This is a right triangle, so
you have x squared plus 5

00:09:09.230 --> 00:09:13.230
squared, plus 25, is equal to
this hypotenuse squared, is

00:09:13.230 --> 00:09:15.880
equal to 100, which is the
Pythagorean theorem.

00:09:15.880 --> 00:09:22.780
x squared is equal to 75, so
then x is going to be equal

00:09:22.780 --> 00:09:26.180
to-- let's see, square
root of 75.

00:09:26.180 --> 00:09:28.820
That's someplace between
8 and 9, right?

00:09:28.820 --> 00:09:35.570
8 is 64, 9 is 81, so
x is greater than 8

00:09:35.570 --> 00:09:37.750
and less than 9.

00:09:37.750 --> 00:09:40.340
Oh, actually, they write it in
terms of square root of-- let

00:09:40.340 --> 00:09:42.180
me simplify this a
little bit more.

00:09:42.180 --> 00:09:45.590
So I can write that as-- the
square root of 75, I can write

00:09:45.590 --> 00:09:47.740
as 3 times 25.

00:09:47.740 --> 00:09:49.960
Square root of 25 is 5,
so that becomes 5

00:09:49.960 --> 00:09:51.860
square roots of 3.

00:09:51.860 --> 00:09:55.940
So this distance right here
is 5 square roots of 3.

00:09:55.940 --> 00:09:58.330
So if I want to figure out this
distance, I just subtract

00:09:58.330 --> 00:09:59.250
that from 10.

00:09:59.250 --> 00:10:06.090
So 10 minus 5 square roots of
3, and that is choice A.

00:10:06.090 --> 00:10:08.050
See you in the next video.


GMAT: Math 27

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.920
We're on problem 138.

00:00:02.920 --> 00:00:06.540
At a loading dock, each worker
on the night crew loaded 3/4

00:00:06.540 --> 00:00:09.970
as many boxes as each worker
on the day crew.

00:00:09.970 --> 00:00:14.450
So that means that the boxes per
night crew worker is equal

00:00:14.450 --> 00:00:19.850
to 3/4 the number of boxes
per day crew worker.

00:00:19.850 --> 00:00:32.150
If the night crew has 4/5 as
many workers as the day crew--

00:00:32.150 --> 00:00:35.140
the number of night crew workers
is going to be 4/5

00:00:35.140 --> 00:00:38.960
times the number of
day crew workers.

00:00:38.960 --> 00:00:42.370
What fraction of all the boxes
loaded by the two crews did

00:00:42.370 --> 00:00:44.090
the day crew load?

00:00:44.090 --> 00:00:46.540
So, how many boxes will
the day crew load?

00:00:46.540 --> 00:00:51.710
The day crew is going to load
the number of day crew workers

00:00:51.710 --> 00:00:55.480
times the boxes per
day crew worker.

00:00:55.480 --> 00:00:58.780
And they want to know what
fraction of all of the boxes

00:00:58.780 --> 00:00:59.650
loaded is this?

00:00:59.650 --> 00:01:01.900
So that'll be the numerator,
and the denominator will be

00:01:01.900 --> 00:01:02.870
all the boxes loaded.

00:01:02.870 --> 00:01:05.500
So all the boxes loaded are
going to be the boxes loaded

00:01:05.500 --> 00:01:06.860
by the day crew.

00:01:06.860 --> 00:01:09.330
So that's the number of workers
times the boxes per

00:01:09.330 --> 00:01:12.370
worker, plus the number
of boxes loaded

00:01:12.370 --> 00:01:13.550
by the night crew.

00:01:13.550 --> 00:01:17.310
That's going to be the number
of night crew workers times

00:01:17.310 --> 00:01:21.440
the number of boxes per
night crew worker.

00:01:21.440 --> 00:01:24.880
And to simplify this, if we
could just substitute these

00:01:24.880 --> 00:01:27.310
variables with things that look
a lot more like that.

00:01:27.310 --> 00:01:30.570
And lo and behold, we have easy
substitutions to tell us

00:01:30.570 --> 00:01:33.520
what b sub n is, the boxes
per night crew worker.

00:01:33.520 --> 00:01:36.970
And they tell us what the number
of night crew workers

00:01:36.970 --> 00:01:38.530
are in terms of the
day crew workers.

00:01:38.530 --> 00:01:39.380
So let's just substitute.

00:01:39.380 --> 00:01:44.360
So this is equal to day crew
workers times boxes per day

00:01:44.360 --> 00:01:49.440
crew worker, divided by day crew
workers times boxes per

00:01:49.440 --> 00:01:53.120
day crew workers, plus,
what is the number

00:01:53.120 --> 00:01:55.940
of night crew workers?

00:01:55.940 --> 00:02:00.330
It's w sub n, but that's the
same thing as 4/5 w sub d.

00:02:05.470 --> 00:02:07.360
And then what are the
number of boxes

00:02:07.360 --> 00:02:08.430
per night crew worker?

00:02:08.430 --> 00:02:11.200
That's 3/4 times the boxes
per day crew worker.

00:02:16.430 --> 00:02:19.510
The 4's cancel out.

00:02:19.510 --> 00:02:20.220
Actually, we could
simplify this.

00:02:20.220 --> 00:02:24.670
We can divide the top and the
bottom by this w sub d times b

00:02:24.670 --> 00:02:32.235
sub d, and you get 1 over 1
plus-- all of these cancel and

00:02:32.235 --> 00:02:33.485
you're just left with 3/5.

00:02:35.820 --> 00:02:44.100
That's equal to 1 over 8/5,
which is equal to 5/8.

00:02:44.100 --> 00:02:45.610
And that is choice E.

00:02:48.730 --> 00:02:55.840
Next problem, 139.

00:02:55.840 --> 00:03:04.320
A restaurant meal costs $35.50
and there is no tax.

00:03:04.320 --> 00:03:14.520
If the tip was more than 10% but
less than 15% of the cost

00:03:14.520 --> 00:03:17.656
of the meal, then the total
amount must have been-- So

00:03:17.656 --> 00:03:18.740
what's the range for the tip?

00:03:18.740 --> 00:03:21.440
10% of $35.50.

00:03:21.440 --> 00:03:25.770
That's just going to be $3.55.

00:03:25.770 --> 00:03:29.130
And 15%-- there's a couple ways
you can think about it.

00:03:29.130 --> 00:03:32.320
You could say it's half
more than this.

00:03:32.320 --> 00:03:33.250
What's half of this?

00:03:33.250 --> 00:03:42.970
Half of this is $1.77 1/2.

00:03:42.970 --> 00:03:45.180
I think that's right.

00:03:45.180 --> 00:03:49.300
soon.

00:03:49.300 --> 00:03:51.060
Really, you can't
get a half-cent.

00:03:51.060 --> 00:03:51.510
Is that right?

00:03:51.510 --> 00:03:53.300
2 times that, you get 2.

00:03:53.300 --> 00:03:57.830
And 2 times 70 is $1.40.

00:03:57.830 --> 00:03:58.960
So this would be 5%.

00:03:58.960 --> 00:04:01.030
I just took half of this That's
how people do it.

00:04:01.030 --> 00:04:02.700
Half of 10%.

00:04:02.700 --> 00:04:06.556
If where you live, the tax
is 10%, but I don't

00:04:06.556 --> 00:04:07.500
want to get into that.

00:04:07.500 --> 00:04:19.779
So if you add this to this, you
get 4, 5, 12, 13, and 5.

00:04:19.779 --> 00:04:27.500
So the tip is going to be
between $3.55 and $5.33, I

00:04:27.500 --> 00:04:29.380
guess we could say.

00:04:29.380 --> 00:04:32.450
And so the total bill, that's
what they want to know, we

00:04:32.450 --> 00:04:33.450
just add both of these.

00:04:33.450 --> 00:04:38.920
So at minimum, you're going to
pay what's this plus $35?

00:04:38.920 --> 00:04:44.050
So $35.50 and $3.55.

00:04:44.050 --> 00:04:47.610
5, 10, 9.

00:04:47.610 --> 00:04:50.760
$39.05.

00:04:50.760 --> 00:04:55.110
And at the high end,
you have $35.50.

00:04:55.110 --> 00:05:00.270
And then you have $5.33.

00:05:00.270 --> 00:05:07.850
And that is equal to
3, 8, 0, 1, 4.

00:05:07.850 --> 00:05:13.760
So your total bill is going
to be between $39 and $41.

00:05:13.760 --> 00:05:16.740
And that is exactly choice B.

00:05:16.740 --> 00:05:17.310
They just round up.

00:05:17.310 --> 00:05:19.900
They say $41 is here
and $39 is here.

00:05:19.900 --> 00:05:21.610
If it's between these 2 numbers,
it's definitely

00:05:21.610 --> 00:05:23.350
between $39 and $41.

00:05:23.350 --> 00:05:24.600
And that's choice B.

00:05:27.440 --> 00:05:32.510
Problem 140.

00:05:32.510 --> 00:05:34.880
In a weightlifting competition,
the total weight

00:05:34.880 --> 00:05:38.070
of Joe's two lifts was 750.

00:05:38.070 --> 00:05:46.380
So l1, his first lift plus the
second lift was 750 pounds.

00:05:46.380 --> 00:05:56.005
If twice the weight of the
first lift was 300-- If 2

00:05:56.005 --> 00:06:00.180
times the first lift was equal
to 300 more than the weight of

00:06:00.180 --> 00:06:09.340
his second lift, what was
the weight in pounds

00:06:09.340 --> 00:06:12.080
of his first lift?

00:06:12.080 --> 00:06:15.720
So we just want to
solve for l1.

00:06:15.720 --> 00:06:19.720
So this top equation we can
rewrite as l2 is equal

00:06:19.720 --> 00:06:23.080
to 750 minus l1.

00:06:23.080 --> 00:06:25.390
Then we can substitute
that here.

00:06:25.390 --> 00:06:31.600
So then you get 2 times l1 is
equal to 300 plus l2, which is

00:06:31.600 --> 00:06:36.190
this, plus 750 minus l1.

00:06:36.190 --> 00:06:37.680
Add l1 to both sides.

00:06:37.680 --> 00:06:42.220
You get 3 l1 is equal
to 300 plus 750.

00:06:42.220 --> 00:06:44.890
That's 1,050.

00:06:44.890 --> 00:06:52.070
And 3 goes into 1,050-- 3 times
3 is 9, 15, 3 goes into

00:06:52.070 --> 00:06:55.430
15 5 times, so it's 350.

00:06:55.430 --> 00:06:58.930
So his first lift
was 350 pounds.

00:06:58.930 --> 00:07:01.380
And that's choice D.

00:07:04.450 --> 00:07:09.820
Problem 141.

00:07:09.820 --> 00:07:14.740
A club collected exactly
$599 from its members.

00:07:14.740 --> 00:07:18.400
If each member contributed
at least $12, what is the

00:07:18.400 --> 00:07:21.630
greatest number of members
the club could have?

00:07:26.180 --> 00:07:30.180
So the more that each member
contributed, the smaller you

00:07:30.180 --> 00:07:32.110
the number of people
in the club.

00:07:32.110 --> 00:07:36.400
If each member contributed at
least $12-- so what is the

00:07:36.400 --> 00:07:39.190
greatest number of members
the club could have?

00:07:39.190 --> 00:07:47.690
So if they all contributed
exactly $12-- 12

00:07:47.690 --> 00:07:57.690
goes into 600 50 times.

00:07:57.690 --> 00:08:08.830
So if this number were $600,
you could say if $600 were

00:08:08.830 --> 00:08:13.240
collected in increments of at
least $12, then you could have

00:08:13.240 --> 00:08:18.860
at most 50 people.

00:08:18.860 --> 00:08:20.840
But we don't have $600.

00:08:20.840 --> 00:08:22.960
$600 would be 50 times 12.

00:08:22.960 --> 00:08:27.762
We have $599.

00:08:27.762 --> 00:08:32.679
$599 can be done by at
most not 50 people,

00:08:32.679 --> 00:08:36.860
but at most 49 people.

00:08:36.860 --> 00:08:37.570
Let's think about it.

00:08:37.570 --> 00:08:39.360
Let's make sure that
that's right.

00:08:39.360 --> 00:08:43.510
So my answer is 49,
which is C.

00:08:43.510 --> 00:08:46.060
And let's think about how much
money would be left over, so

00:08:46.060 --> 00:08:47.940
49 times 12.

00:08:47.940 --> 00:08:49.220
This is just a reality check.

00:08:49.220 --> 00:08:50.910
49 times 12.

00:08:50.910 --> 00:08:53.050
2 times 9 is 18.

00:08:53.050 --> 00:08:56.410
2 times 4 is 8 plus 1 is 9.

00:08:56.410 --> 00:08:58.830
0, 49.

00:08:58.830 --> 00:09:05.820
8, 18, 588.

00:09:05.820 --> 00:09:09.670
So if you have 49 people, and
they all contribute $12

00:09:09.670 --> 00:09:15.050
exactly, you'll raise $588.

00:09:15.050 --> 00:09:17.470
And then you could have had a
couple of other people, you

00:09:17.470 --> 00:09:20.480
could have had 11 other
people who instead of

00:09:20.480 --> 00:09:21.970
$12, they paid $13.

00:09:21.970 --> 00:09:24.260
And that's how you get
to $599, because they

00:09:24.260 --> 00:09:25.520
say at least $12.

00:09:25.520 --> 00:09:29.410
You can't have 50 people,
because 50 people, if they

00:09:29.410 --> 00:09:30.910
contribute at least $12,
you would have to

00:09:30.910 --> 00:09:33.010
raise at least $600.

00:09:33.010 --> 00:09:35.000
So it can't be 50, so
the answer is 49.

00:09:35.000 --> 00:09:38.490
And you might say why did you
go immediately to $600?

00:09:38.490 --> 00:09:40.320
And a lot of times on the GMAT,
where they have these

00:09:40.320 --> 00:09:43.480
numbers that are 12 and 599--
those are strange numbers.

00:09:43.480 --> 00:09:46.190
You say 599 is awfully
close to 600.

00:09:46.190 --> 00:09:48.650
Let's see if I can use that
information to deduce

00:09:48.650 --> 00:09:51.010
something interesting
about this problem.

00:09:51.010 --> 00:09:58.290
Anyway, next problem, 142.

00:09:58.290 --> 00:10:03.660
Of the 3,600 employees of
Company x, 1/3 are clerical.

00:10:03.660 --> 00:10:08.540
So clerical is equal
to 1/3 times 3,600,

00:10:08.540 --> 00:10:11.200
which is equal to 1,200.

00:10:11.200 --> 00:10:14.890
If the clerical staff were to
be reduced by 1/3, what

00:10:14.890 --> 00:10:18.690
percentage of the total number
of remaining employees would

00:10:18.690 --> 00:10:20.560
then be clerical?

00:10:20.560 --> 00:10:23.030
We're going to reduce the
clerical staff by 1/3, so how

00:10:23.030 --> 00:10:24.920
many people are we going
to get rid of?

00:10:24.920 --> 00:10:27.590
So 1/3 of 1,200 is what?

00:10:27.590 --> 00:10:29.280
It's 400.

00:10:29.280 --> 00:10:36.450
So the clerical staff is now
going to be 1,200 minus 1/3,

00:10:36.450 --> 00:10:39.340
minus 400, which is
equal to 800.

00:10:39.340 --> 00:10:42.960
So what proportion is this of
the remaining employees?

00:10:42.960 --> 00:10:45.770
So clerical is 800, and
how many remaining

00:10:45.770 --> 00:10:46.970
employees are there?

00:10:46.970 --> 00:10:48.960
Are there 3,600?

00:10:48.960 --> 00:10:52.440
No, we've gotten rid of 400
employees who were clerical.

00:10:52.440 --> 00:10:55.850
So now we have 3,200
employees.

00:10:55.850 --> 00:11:01.140
3,600 minus 400, so that
is equal to 1/4.

00:11:03.660 --> 00:11:07.700
And that's A, 25%.

00:11:07.700 --> 00:11:11.590
Problem 143.

00:11:11.590 --> 00:11:12.510
Actually, I'm out of time.

00:11:12.510 --> 00:11:13.740
I'll continue this in
the next video.

00:11:13.740 --> 00:11:14.990
See


GMAT: Math 29

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.590
We're on problem 148.

00:00:03.590 --> 00:00:06.260
What is the lowest integer
that is the sum of three

00:00:06.260 --> 00:00:09.570
different primes, each
greater than 20?

00:00:09.570 --> 00:00:11.480
So it's the lowest integer, so
we essentially just have to

00:00:11.480 --> 00:00:13.260
find the three primes
above 20.

00:00:13.260 --> 00:00:15.740
So let's see, 21 isn't
a prime, 22 isn't a

00:00:15.740 --> 00:00:18.230
prime, 23 is a prime.

00:00:18.230 --> 00:00:24.080
24, no, 25, no, 26,
no, 27, 28, nope.

00:00:24.080 --> 00:00:26.640
29, that's prime.

00:00:26.640 --> 00:00:29.980
30 isn't, 31 is.

00:00:29.980 --> 00:00:37.670
31 plus 29 is 60 plus 23 is
83, and that is choice E.

00:00:37.670 --> 00:00:39.650
I hope I haven't missed any
primes, no, I don't think I

00:00:39.650 --> 00:00:44.570
have. 24, 25, 26, 27, 28,
I think I got all

00:00:44.570 --> 00:00:45.650
the primes in there.

00:00:45.650 --> 00:00:47.210
What is the lowest integer of
the sum of three different

00:00:47.210 --> 00:00:48.670
primes, each greater than 20?

00:00:48.670 --> 00:00:50.780
Yep, that's the answer, E.

00:00:50.780 --> 00:00:53.480
Problem 149.

00:00:53.480 --> 00:00:58.720
The average of 6, 8, and 10
equals-- so the average of 6,

00:00:58.720 --> 00:01:01.010
8, and 10, so first of all,
what's the average

00:01:01.010 --> 00:01:02.250
of 6, 8, and 10?

00:01:02.250 --> 00:01:03.070
Well, let's just do it.

00:01:03.070 --> 00:01:04.890
6 plus 8 plus 10.

00:01:04.890 --> 00:01:07.300
You could immediately see the
average is going to be 8,

00:01:07.300 --> 00:01:08.980
because you have 8 in the
middle, and both of these

00:01:08.980 --> 00:01:10.590
numbers are 2 away from it.

00:01:10.590 --> 00:01:14.440
But that divided by 3 is
equal to 24 divided by

00:01:14.440 --> 00:01:15.720
3 is equal to 8.

00:01:19.170 --> 00:01:24.510
And that's equal to the average
of 7 plus 9 plus x

00:01:24.510 --> 00:01:29.420
over 3, so that tells us to
multiply both sides of this

00:01:29.420 --> 00:01:32.250
equation by 3, that says
that 24 is equal to--

00:01:32.250 --> 00:01:33.480
what's 7 plus 9?

00:01:33.480 --> 00:01:36.480
That's 16 plus x.

00:01:36.480 --> 00:01:39.590
Subtract 16 from both sides,
you get x is equal to 8.

00:01:39.590 --> 00:01:41.570
Choice C.

00:01:41.570 --> 00:01:46.510
And you could have said, OK, 7
and 9 or both half-- 8 is in

00:01:46.510 --> 00:01:48.280
the middle, and they have to
average 8, so you might have

00:01:48.280 --> 00:01:50.600
just been able to eyeball it and
say 8, but it didn't take

00:01:50.600 --> 00:01:53.670
that much math, it's not that
much of a shortcut.

00:01:53.670 --> 00:01:57.260
Problem 150.

00:01:57.260 --> 00:02:05.290
If x is equal to minus 1, then x
to the fourth minus x to the

00:02:05.290 --> 00:02:11.240
third plus x squared, all
of that over x minus

00:02:11.240 --> 00:02:13.290
1 is equal to what?

00:02:13.290 --> 00:02:15.670
x to the fourth is 1.

00:02:15.670 --> 00:02:19.930
x to the third is minus 1, but
then we're going to minus

00:02:19.930 --> 00:02:23.050
that, so this whole thing is
going to be plus 1, and x

00:02:23.050 --> 00:02:24.820
squared is plus 1.

00:02:24.820 --> 00:02:27.640
Negative 1 squared is 1.

00:02:27.640 --> 00:02:30.200
And you have minus 1 minus 1.

00:02:30.200 --> 00:02:34.300
So in the numerator you have
1, 2, 3, in the denominator

00:02:34.300 --> 00:02:35.800
you have minus 2.

00:02:35.800 --> 00:02:38.680
Let's put the minus out
front, minus 3/2.

00:02:38.680 --> 00:02:39.930
That's choice A.

00:02:42.310 --> 00:02:47.080
Problem 151.

00:02:47.080 --> 00:02:50.240
A toy store regularly
sells all stock at a

00:02:50.240 --> 00:02:56.940
discount of 20% to 40%.

00:02:56.940 --> 00:03:00.680
If an additional 25% were
deducted from the discount

00:03:00.680 --> 00:03:03.600
price during a special sale,
what would be the lowest

00:03:03.600 --> 00:03:06.490
possible price of a toy costing

00:03:06.490 --> 00:03:08.920
$16 before any discount?

00:03:08.920 --> 00:03:12.710
So we want to discount the
$16 as much as possible.

00:03:12.710 --> 00:03:16.430
So they say they're normally
discounting 20% to 40%, so if

00:03:16.430 --> 00:03:18.140
you want to discount this as
much as possible, you want to

00:03:18.140 --> 00:03:20.420
discount this by 40%.

00:03:20.420 --> 00:03:24.280
So if you discount this by 40%,
that's the same thing as

00:03:24.280 --> 00:03:28.410
saying it's going to be equal to
60% of its original price.

00:03:28.410 --> 00:03:35.440
So this would be the price after
being discounted by 40%,

00:03:35.440 --> 00:03:40.090
and then we would discount
by another 25%.

00:03:40.090 --> 00:03:42.160
Actually, I think it's easier
to go the other way.

00:03:42.160 --> 00:03:43.860
Well, let's just
do it this way.

00:03:43.860 --> 00:03:47.520
So we discount it by
another 25%, so we

00:03:47.520 --> 00:03:50.710
multiply it by 3/4 again.

00:03:50.710 --> 00:03:54.320
If you discount by 25%, the
price is going to be 3/4 of

00:03:54.320 --> 00:03:55.550
its original price.

00:03:55.550 --> 00:03:56.810
Actually, let's do
it as fractions.

00:03:56.810 --> 00:03:58.500
Fractions are always easier.

00:03:58.500 --> 00:04:02.220
60% percent of your
price is 3/5.

00:04:02.220 --> 00:04:06.070
If you discount by 2/5, or 40%,
that's the same thing as

00:04:06.070 --> 00:04:08.290
multiplying by 3/5, and
then we're going

00:04:08.290 --> 00:04:09.540
to multiply by 3/4.

00:04:12.410 --> 00:04:13.890
Set 25%.

00:04:13.890 --> 00:04:16.579
If you're discounting by 25%,
that's the same thing as being

00:04:16.579 --> 00:04:19.360
worth 75% of your
original price.

00:04:19.360 --> 00:04:22.530
So what does that get us?

00:04:22.530 --> 00:04:28.130
16 divided by 4, 4, you get a 1,
and then you have 4 times 3

00:04:28.130 --> 00:04:39.820
times 3, 4 times 9 is 36, over
5, which is equal to-- 5 goes

00:04:39.820 --> 00:04:43.410
into 36 7 and 1/5.

00:04:43.410 --> 00:04:45.860
So $7.20.

00:04:45.860 --> 00:04:49.120
And that's choice B.

00:04:49.120 --> 00:04:52.090
Problem 152.

00:04:52.090 --> 00:04:56.250
The shaded portion of the
rectangular lot shown above

00:04:56.250 --> 00:04:57.380
represents a flowerbed.

00:04:57.380 --> 00:04:59.450
OK, so let me draw this
rectangular lot.

00:05:05.290 --> 00:05:13.180
They drew this shaded portion
something like that, and I can

00:05:13.180 --> 00:05:14.430
even color it in.

00:05:19.630 --> 00:05:23.150
So they say the shaded portion
of the rectangular lot shown

00:05:23.150 --> 00:05:24.820
above represents a flowerbed.

00:05:24.820 --> 00:05:30.690
If the area of the bed is 24
square yards, and x is equal

00:05:30.690 --> 00:05:34.060
to y plus 2, so they tell us
this side right here is x,

00:05:34.060 --> 00:05:38.370
which now they're telling
us is equal to y plus 2.

00:05:38.370 --> 00:05:40.555
And this right here is going
to be equal to y.

00:05:44.710 --> 00:05:49.870
Then z equals what? z is
this side right there.

00:05:49.870 --> 00:05:53.670
So if this area is 24, that
tells us that x times y

00:05:53.670 --> 00:05:55.520
times 1/2 is 24.

00:05:55.520 --> 00:05:56.790
That's the area of that.

00:05:56.790 --> 00:06:01.360
So 1/2xy is equal to 24.

00:06:01.360 --> 00:06:05.820
Or you can even say that
xy is equal to 48.

00:06:05.820 --> 00:06:08.380
We know what xy is.

00:06:08.380 --> 00:06:11.380
We can just substitute for
x, so we get y plus

00:06:11.380 --> 00:06:13.490
2, that's x, right?

00:06:13.490 --> 00:06:19.150
X is equal to y plus 2 times
y is equal to 48.

00:06:19.150 --> 00:06:28.210
And so we get y squared plus
2y is equal to 48.

00:06:28.210 --> 00:06:36.230
Or y squared plus 2y minus
48 is equal to 0.

00:06:36.230 --> 00:06:37.480
We could factor this
quadratic.

00:06:40.210 --> 00:06:42.800
What two numbers when you
multiply by minus 48, and you

00:06:42.800 --> 00:06:44.180
add you get plus 6?

00:06:44.180 --> 00:06:54.110
So y plus 8 times y minus 6 is
equal to 0, 8 times minus 6 is

00:06:54.110 --> 00:06:58.410
minus 48, and then 8 minus 6 is
plus 2, so y could be equal

00:06:58.410 --> 00:07:01.790
to minus 8 or plus 6, right?

00:07:01.790 --> 00:07:04.510
Whatever numbers make either
of these equal to zero.

00:07:04.510 --> 00:07:07.130
Well, we can't have a negative
distance, so y is not equal to

00:07:07.130 --> 00:07:10.350
minus 8, so y is equal to 6.

00:07:10.350 --> 00:07:12.840
y is equal to 6, x
is equal to 8.

00:07:15.750 --> 00:07:18.530
Now, we can just confirm, 6
times 8 is 48 times 1/2 is

00:07:18.530 --> 00:07:19.990
equal to 24.

00:07:19.990 --> 00:07:21.380
And now we can use the
Pythagorean theorem

00:07:21.380 --> 00:07:22.400
to figure out z.

00:07:22.400 --> 00:07:26.670
So we'll get z squared
is equal to 36.

00:07:26.670 --> 00:07:30.920
That's y squared plus 64,
which is x squared, so z

00:07:30.920 --> 00:07:33.520
squared is equal to 100.

00:07:33.520 --> 00:07:36.700
z is equal to 10.

00:07:36.700 --> 00:07:37.950
Choice E.

00:07:40.100 --> 00:07:46.520
Problem 153.

00:07:46.520 --> 00:07:50.810
Jack is now 14 years older than
Bill, so Jack is equal to

00:07:50.810 --> 00:07:53.710
Bill plus 14.

00:07:53.710 --> 00:08:02.920
If in 10 years Jack will be
twice as old as Bill, so 2

00:08:02.920 --> 00:08:06.980
times Bill's age, how old
will Jack be in 5 years?

00:08:06.980 --> 00:08:10.410
So they want to know j plus 5.

00:08:10.410 --> 00:08:12.470
That's how old Jack will
be in 5 years.

00:08:12.470 --> 00:08:17.220
So let's just try to
solve for Jack.

00:08:17.220 --> 00:08:24.560
This top equation tells us that
Bill is equal to Jack

00:08:24.560 --> 00:08:27.900
minus 14, and then
we can substitute

00:08:27.900 --> 00:08:29.560
this in for Bill here.

00:08:29.560 --> 00:08:33.409
So then this equation will
become Jack plus 10 is equal

00:08:33.409 --> 00:08:36.840
to 2 times Jack minus 14.

00:08:36.840 --> 00:08:44.630
So Jack plus 10 is equal to
2 times Jack minus 28.

00:08:44.630 --> 00:08:48.120
If we subtract Jack from both
sides, you get 10 is equal to

00:08:48.120 --> 00:08:50.690
Jack minus 28.

00:08:50.690 --> 00:08:53.030
We can add 28 to both
sides, and you get

00:08:53.030 --> 00:08:55.890
38 is equal to Jack.

00:08:55.890 --> 00:08:57.190
So that's how old Jack is now.

00:08:57.190 --> 00:08:58.550
He's 38 years old.

00:08:58.550 --> 00:09:01.640
So how old is Jack going
to be in 5 years?

00:09:01.640 --> 00:09:07.280
38 plus 5 is equal
to 43 years old.

00:09:07.280 --> 00:09:10.530
And I've clearly made a mistake,
because that's not

00:09:10.530 --> 00:09:11.220
one of the options.

00:09:11.220 --> 00:09:13.080
So let's see where I'm
made an error.

00:09:13.080 --> 00:09:15.850
So Jack is now 14 years
older than Bill.

00:09:15.850 --> 00:09:17.990
Fair enough.

00:09:17.990 --> 00:09:21.570
So in 10 years, Jack will be
twice-- oh, I see my mistake.

00:09:21.570 --> 00:09:23.200
He's going to be twice
as old as Bill is

00:09:23.200 --> 00:09:25.100
going to be in 10 years.

00:09:25.100 --> 00:09:26.760
That was a bone-headed
mistake.

00:09:26.760 --> 00:09:29.780
So let me clear all of this.

00:09:29.780 --> 00:09:31.015
That was a bad mistake.

00:09:31.015 --> 00:09:34.580
I shouldn't have made it.

00:09:34.580 --> 00:09:39.070
I'm lucky that they didn't have
what I put down as one of

00:09:39.070 --> 00:09:41.230
the choices, and I would've
gotten it wrong.

00:09:41.230 --> 00:09:42.060
All right.

00:09:42.060 --> 00:09:46.350
So we know that in 10 years,
Jack is going to be 2 times

00:09:46.350 --> 00:09:47.900
what Bill is going to
be in 10 years.

00:09:47.900 --> 00:09:49.290
That was my mistake.

00:09:49.290 --> 00:09:51.900
So now let's see
what we can do.

00:09:51.900 --> 00:09:55.400
We'll use that same information
up there, so Bill

00:09:55.400 --> 00:09:58.330
is equal to Jack minus 14.

00:09:58.330 --> 00:10:03.220
Substitute that information
here, so we get Jack plus 10

00:10:03.220 --> 00:10:09.350
is equal to 2 times Jack minus
14, instead of Bill, plus 10.

00:10:09.350 --> 00:10:13.655
So you get Jack plus 10 is
equal to 2 times Jack.

00:10:13.655 --> 00:10:18.450
Let's see, minus 14 plus 10,
that's minus 4, so Jack plus

00:10:18.450 --> 00:10:23.490
10 is equal to 2 times
Jack minus 8.

00:10:23.490 --> 00:10:25.470
Let's subtract Jack
from both sides.

00:10:25.470 --> 00:10:28.830
10 is equal to Jack minus 8.

00:10:28.830 --> 00:10:30.300
Let's add 8 to both sides.

00:10:30.300 --> 00:10:32.816
You get 18 is equal to Jack.

00:10:32.816 --> 00:10:35.030
So that's how old Jack is
right now, he's 18.

00:10:35.030 --> 00:10:40.060
So in five years, Jack is going
to be 18 plus 5, which

00:10:40.060 --> 00:10:44.250
is 23 years old, choice D.

00:10:44.250 --> 00:10:45.190
And I'm out of time.

00:10:45.190 --> 00:10:47.010
See you in the next video.


GMAT: Math 26

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.232
Problem 133.

00:00:04.232 --> 00:00:09.330
A certain country had a total
annual expenditure of 1.2

00:00:09.330 --> 00:00:13.570
times 10 to the 1/12
power last year.

00:00:13.570 --> 00:00:16.810
If the population of the country
was 240 million last

00:00:16.810 --> 00:00:25.470
year, what was the per
capita expenditure?

00:00:25.470 --> 00:00:27.420
240 million, what is that?

00:00:27.420 --> 00:00:31.840
A million is 10 to
the sixth power.

00:00:31.840 --> 00:00:38.920
So this is 240 times
10 to the sixth.

00:00:38.920 --> 00:00:44.840
And that's the same thing as
24 times 10 to the seventh.

00:00:44.840 --> 00:00:47.910
I just took 10 from here
and multiply 10 there.

00:00:47.910 --> 00:00:53.690
And that is equal to 2.4 times
10 to the eighth power.

00:00:53.690 --> 00:00:55.590
Divide this by 10 and
multiply this by 10.

00:00:55.590 --> 00:00:56.410
You get the same number.

00:00:56.410 --> 00:00:58.030
So 2.4 times 10 to the eighth.

00:00:58.030 --> 00:01:03.120
So the per capita income is
going to be equal to the total

00:01:03.120 --> 00:01:08.250
annual expenditure, so 1.2 times
10 to the 12th, divided

00:01:08.250 --> 00:01:14.160
by the population, 2.4 times
10 to the eighth.

00:01:14.160 --> 00:01:18.720
1.2 divided by 2.4, that's equal
to 0.5, that's exactly

00:01:18.720 --> 00:01:21.710
half of that, times 10
to the 12th divided

00:01:21.710 --> 00:01:22.490
by 10 to the eighth.

00:01:22.490 --> 00:01:23.515
You just subtract the
exponents, so

00:01:23.515 --> 00:01:25.720
that's 10 to the fourth.

00:01:25.720 --> 00:01:30.810
And that's not one of the
answers, so we can multiply

00:01:30.810 --> 00:01:32.650
this 10 and divide this by 10.

00:01:32.650 --> 00:01:36.505
So this becomes 5 times
10 to the third.

00:01:36.505 --> 00:01:39.570
Ten to the third is 1,000, so
5 times 1,000 is equal to

00:01:39.570 --> 00:01:45.390
$5,000 in annual expenditure
per capita.

00:01:45.390 --> 00:01:47.870
And that's choice E.

00:01:47.870 --> 00:01:56.010
Problem 134.

00:01:56.010 --> 00:01:58.370
A certain rectangular
window is twice as

00:01:58.370 --> 00:01:59.620
long as it is wide.

00:02:05.340 --> 00:02:10.840
So if this is the width, its
length is 2 times the width.

00:02:10.840 --> 00:02:14.160
If its perimeter is 10 feet, and
its dimensions are, so its

00:02:14.160 --> 00:02:19.220
perimeter going to be w plus
w plus the 2 lengths.

00:02:19.220 --> 00:02:25.490
So its perimeter is going to be
2w plus 2w plus w plus w.

00:02:25.490 --> 00:02:29.110
And that's going to
be equal to 10.

00:02:29.110 --> 00:02:33.840
If we add up all the w's,
that's 2, 4, 5, 6.

00:02:33.840 --> 00:02:37.350
So we get 6w is equal to 10.

00:02:37.350 --> 00:02:45.440
w is equal to 10/6, which
is equal to 5/3 feet.

00:02:45.440 --> 00:02:49.630
And so its dimensions are
5/3 and then the 2w.

00:02:49.630 --> 00:02:54.240
So the length is 2 times
that, which is 10/3.

00:02:54.240 --> 00:02:59.180
So it's 10/3 by 5/3.

00:02:59.180 --> 00:03:00.110
And that's choice B.

00:03:00.110 --> 00:03:03.750
They say 5/3 by 10/3,
but same thing.

00:03:03.750 --> 00:03:06.540
So choice B.

00:03:06.540 --> 00:03:12.900
Problem 135.

00:03:12.900 --> 00:03:13.892
This is interesting.

00:03:13.892 --> 00:03:16.810
Let's see if I can draw
this diagram.

00:03:16.810 --> 00:03:21.370
So to start, this is x, then
it goes there and has a

00:03:21.370 --> 00:03:27.910
diamond like that, and
it goes like that.

00:03:27.910 --> 00:03:29.390
And we have another diamond.

00:03:32.380 --> 00:03:34.170
I don't know what this is all
for yet, but it looks

00:03:34.170 --> 00:03:35.956
interesting.

00:03:35.956 --> 00:03:41.290
It goes like that, then
goes like that.

00:03:41.290 --> 00:03:44.960
There's a diamond there, looks
like there, looks like there.

00:03:44.960 --> 00:03:48.980
There's a line that goes like
that, goes like that, like

00:03:48.980 --> 00:03:51.970
that, and like that.

00:03:51.970 --> 00:03:56.160
And then finally, we're at y.

00:03:56.160 --> 00:03:59.560
The diagram above shows of
various paths along which a

00:03:59.560 --> 00:04:06.340
mouse can travel from point x
where does it lead to point y,

00:04:06.340 --> 00:04:08.140
where there's a reward
with a food pellet.

00:04:08.140 --> 00:04:12.360
How many different paths from
x to y can the mouse take if

00:04:12.360 --> 00:04:15.980
it goes directly from x to y
without retracing any point

00:04:15.980 --> 00:04:18.610
along the path?

00:04:18.610 --> 00:04:19.829
There's only one way.

00:04:19.829 --> 00:04:21.709
It has to go here.

00:04:21.709 --> 00:04:24.840
So we're only at 1
path right now.

00:04:24.840 --> 00:04:27.630
And it can either go this
way to get here, or it

00:04:27.630 --> 00:04:29.560
could go this way.

00:04:29.560 --> 00:04:32.580
So to get here, there's how
many different paths?

00:04:32.580 --> 00:04:33.940
Like that, or like that.

00:04:33.940 --> 00:04:37.360
So at this point there's 2
ways to get right there.

00:04:37.360 --> 00:04:41.510
There's 1 way to get here,
2 ways to get here.

00:04:41.510 --> 00:04:44.110
Now, if there's 2 ways to get
here, that means that there's

00:04:44.110 --> 00:04:46.840
2 ways to get here.

00:04:46.840 --> 00:04:50.560
Now how many ways are
there to get here?

00:04:50.560 --> 00:04:54.360
There's each of the 2 ways to
get there, and now we each of

00:04:54.360 --> 00:04:57.610
those in combination with each
of these 2 ways, so there's 4

00:04:57.610 --> 00:04:58.840
ways to get here.

00:04:58.840 --> 00:05:00.220
You could think of
it like this.

00:05:00.220 --> 00:05:03.120
You could go to the top 1
and the top, that's 1.

00:05:03.120 --> 00:05:04.940
Top and the bottom, 2.

00:05:04.940 --> 00:05:06.820
Bottom and the bottom, 3.

00:05:06.820 --> 00:05:08.520
Bottom and the top, 4.

00:05:08.520 --> 00:05:11.620
So you could just multiply how
many ways it takes to get

00:05:11.620 --> 00:05:15.130
here, and then how many more
ways to get from here to here.

00:05:15.130 --> 00:05:17.610
From here to here, there's 2
possible ways, and there's

00:05:17.610 --> 00:05:20.490
already 2 ways to get here, so
you could multiply the 2.

00:05:20.490 --> 00:05:24.580
So there's 4 ways to get here,
4 ways to get here.

00:05:24.580 --> 00:05:30.650
To get from here to here, there
are 3 ways, 1, 2, and 3.

00:05:30.650 --> 00:05:32.880
But since there are already 4
ways to get here, with each of

00:05:32.880 --> 00:05:35.245
those 4 ways you can go
in 3 different ways.

00:05:35.245 --> 00:05:38.660
So we multiply by 4 times 3,
so there's 12 ways to get

00:05:38.660 --> 00:05:39.400
right here.

00:05:39.400 --> 00:05:41.460
If there's 12 ways to
get here, there's 12

00:05:41.460 --> 00:05:43.060
ways to get to y.

00:05:43.060 --> 00:05:46.900
So the answer is C, 12.

00:05:46.900 --> 00:05:50.770
Problem 136.

00:05:50.770 --> 00:05:56.610
If the operation circle with
a dot in it is defined by x

00:05:56.610 --> 00:06:03.620
circle with a dot y, is equal to
the square root of xy, for

00:06:03.620 --> 00:06:05.580
all positive numbers x and y.

00:06:05.580 --> 00:06:07.920
So they're saying x and
y are greater than 0.

00:06:07.920 --> 00:06:09.990
That's good, because that
ensures we have a positive

00:06:09.990 --> 00:06:11.890
under the square root sign.

00:06:11.890 --> 00:06:22.190
Then they want to know what 5
circle with a dot 45 circle

00:06:22.190 --> 00:06:25.770
with a dot 60 is equal to.

00:06:25.770 --> 00:06:29.290
5 circle 45 five, that's
equal to the square

00:06:29.290 --> 00:06:34.540
root of 5 times 45.

00:06:34.540 --> 00:06:39.880
That's that, and whatever that
is we're circling with that.

00:06:39.880 --> 00:06:43.330
So what you do when you have x
circle y, you multiply the 2

00:06:43.330 --> 00:06:44.430
and then you take
the square root.

00:06:44.430 --> 00:06:47.150
So we would multiply 60 times
this and take the square root

00:06:47.150 --> 00:06:48.400
of the whole thing.

00:06:51.610 --> 00:06:53.490
Let's just simplify this.

00:06:53.490 --> 00:06:58.490
This right here is equal to the
square root of 5 times 45,

00:06:58.490 --> 00:07:01.200
which doesn't look simplifiable
at face value.

00:07:01.200 --> 00:07:03.440
But if you think about,
45 is 9 times 5.

00:07:03.440 --> 00:07:04.910
You just want to take
out any perfect

00:07:04.910 --> 00:07:05.820
squares you have there.

00:07:05.820 --> 00:07:10.260
So that's equal to 5
times 9 times 5.

00:07:10.260 --> 00:07:16.590
And that's equal to square
root of 25 times 9.

00:07:16.590 --> 00:07:19.120
And that is equal to the square
root of 25 times the

00:07:19.120 --> 00:07:23.390
square root of 9, and that's
equal to 5 times 3, which is

00:07:23.390 --> 00:07:25.560
equal to 15.

00:07:25.560 --> 00:07:28.290
So this simplifies to 15.

00:07:28.290 --> 00:07:33.340
And 15 dot circle, whatever you
want to call it, 60, is

00:07:33.340 --> 00:07:39.310
equal to the square root
of 15 times 60.

00:07:39.310 --> 00:07:42.810
And let's see if we can somehow
turn this into some

00:07:42.810 --> 00:07:47.070
type of a perfect square.

00:07:47.070 --> 00:07:54.840
Let's rewrite this as-- 60 is
the same thing as 15 times 4.

00:07:54.840 --> 00:07:59.890
So this is equal to the square
root of 15 times 15 times the

00:07:59.890 --> 00:08:02.390
square root of 4.

00:08:02.390 --> 00:08:06.055
The square root of 15 times 15
is going to be 15 times 2,

00:08:06.055 --> 00:08:08.260
which is equal to 30.

00:08:08.260 --> 00:08:09.960
And that's choice A.

00:08:13.490 --> 00:08:15.970
Problem 137.

00:08:15.970 --> 00:08:17.220
I'll do it in yellow.

00:08:20.500 --> 00:08:23.900
A bar over a sequence of digits
in a decimal indicates

00:08:23.900 --> 00:08:26.050
that the sequence repeats
indefinitely.

00:08:26.050 --> 00:08:35.620
What is the value of 10 to the
fourth minus 10 to the 12

00:08:35.620 --> 00:08:43.159
times 0.0012, and then this 12
will just keep repeating over

00:08:43.159 --> 00:08:44.780
and over again.

00:08:44.780 --> 00:08:45.810
So let's see what we can do.

00:08:45.810 --> 00:08:48.330
What's 10 to the fourth?

00:08:48.330 --> 00:08:49.530
Let's think about it this way.

00:08:49.530 --> 00:08:52.140
Let's distribute this.

00:08:52.140 --> 00:08:53.830
So what's 10 to the fourth?

00:08:53.830 --> 00:08:55.242
That's 10,000.

00:08:55.242 --> 00:09:11.440
So what's 10,000 times
this number, 0.0012?

00:09:11.440 --> 00:09:13.270
This 12 keeps repeating,
so we'll think

00:09:13.270 --> 00:09:14.080
about that in a second.

00:09:14.080 --> 00:09:21.910
So what's 12 times 10,000?

00:09:21.910 --> 00:09:25.370
It's 120,000.

00:09:25.370 --> 00:09:30.360
It's 12 with four
0's, 1, 2, 3, 4.

00:09:30.360 --> 00:09:32.900
And we have 4 numbers behind
the decimal point,

00:09:32.900 --> 00:09:36.280
so it gets us there.

00:09:36.280 --> 00:09:37.230
You might say OK, we're done.

00:09:37.230 --> 00:09:37.770
It's 12.

00:09:37.770 --> 00:09:39.070
But it's actually
not 12, because

00:09:39.070 --> 00:09:41.320
it's actually 12121212.

00:09:41.320 --> 00:09:44.280
The 12's just keep going.

00:09:44.280 --> 00:09:51.430
It would actually be 121212, and
then 12's keep repeating

00:09:51.430 --> 00:09:54.580
on like that.

00:09:54.580 --> 00:09:57.420
I think you understand
why, right?

00:09:57.420 --> 00:10:00.760
This would be if we just had 1
12 times 10,000, but if the

00:10:00.760 --> 00:10:02.870
12's kept going, then these
wouldn't be 0's,

00:10:02.870 --> 00:10:06.170
these would be 12's.

00:10:06.170 --> 00:10:08.430
Hopefully that make sense to
you, maybe there's an easy way

00:10:08.430 --> 00:10:08.950
to think about it.

00:10:08.950 --> 00:10:10.580
So what's 100?

00:10:10.580 --> 00:10:11.400
We'll use the same logic.

00:10:11.400 --> 00:10:21.060
What's 100 times 0.0012?

00:10:21.060 --> 00:10:22.630
Actually, I probably could have
done it a simpler way,

00:10:22.630 --> 00:10:24.230
but let's just continue
this way.

00:10:24.230 --> 00:10:29.060
So 12 times 100 1,200, and we
have 4 numbers behind the

00:10:29.060 --> 00:10:32.310
decimal point, 1, 2, 3, 4.

00:10:35.450 --> 00:10:39.310
And of course, it's not just
0.12, it'll just keep going.

00:10:39.310 --> 00:10:42.420
0.1212 will just keep going.

00:10:42.420 --> 00:10:45.450
So we have this times
this, which is this

00:10:45.450 --> 00:10:46.840
minus this times this.

00:10:46.840 --> 00:10:47.700
I'm just distributing it.

00:10:47.700 --> 00:10:59.320
So if I just subtract this
from that, minus 0.12-- I

00:10:59.320 --> 00:10:59.980
wrote this right.

00:10:59.980 --> 00:11:05.120
This was 12, because we said
12 times 10,000 is 120,000.

00:11:05.120 --> 00:11:08.760
We had 1, 2, 3, 4 behind
the decimal.

00:11:08.760 --> 00:11:10.355
So it's 12.

00:11:10.355 --> 00:11:15.230
But since we had multiple 12's
behind it, it becomes 12.12.

00:11:15.230 --> 00:11:18.710
I don't know what I was doing
with this right here.

00:11:18.710 --> 00:11:20.190
So let me write that again.

00:11:20.190 --> 00:11:21.590
Sorry I'm doing this
a little bit messy.

00:11:21.590 --> 00:11:28.750
So 0.0012 repeating times 10 to
the fourth is going to be

00:11:28.750 --> 00:11:31.470
equal to 12.12 repeating.

00:11:31.470 --> 00:11:36.550
And then we're going to subtract
0.0012 repeating

00:11:36.550 --> 00:11:42.870
times 10 squared, and that is
equal to 0.12 repeating.

00:11:42.870 --> 00:11:45.970
So if I subtract out the 0.12
repeating from 12.12

00:11:45.970 --> 00:11:49.540
repeating, I'm just going
to be left with 12.

00:11:49.540 --> 00:11:51.310
And hopefully that's
one of the answers.

00:11:51.310 --> 00:11:52.140
It is.

00:11:52.140 --> 00:11:55.020
Choice E.

00:11:55.020 --> 00:11:57.020
See you in the next video.


GMAT: Math 25

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.019
We're on problem 129.

00:00:04.019 --> 00:00:10.480
On a recent trip, Cindy drove
her car 290 miles, rounded to

00:00:10.480 --> 00:00:15.430
the nearest 10 miles, and used
12 gallons of gasoline,

00:00:15.430 --> 00:00:16.830
rounded to the nearest
gallons.

00:00:20.720 --> 00:00:24.220
The actual number of miles per
gallon that Cindy's car got on

00:00:24.220 --> 00:00:27.500
this trip must have
been between?

00:00:27.500 --> 00:00:31.040
So they want us to kind of say
these are the rounded numbers,

00:00:31.040 --> 00:00:33.610
now what were the actual numbers
between, and then from

00:00:33.610 --> 00:00:35.720
that what's the range
of miles per gallon?

00:00:35.720 --> 00:00:41.530
So if this is rounded to the
nearest 10, then her miles had

00:00:41.530 --> 00:00:48.570
to have been greater than or
equal to 285, because anything

00:00:48.570 --> 00:00:52.080
285 or more would have
been rounded to 290.

00:00:52.080 --> 00:01:00.220
But then less than 295 because
anything less than 295 would

00:01:00.220 --> 00:01:01.820
be would be rounded down,
but 295 itself would be

00:01:01.820 --> 00:01:04.330
rounded up to 300.

00:01:04.330 --> 00:01:10.900
And her gallons had to have been
greater than or equal to

00:01:10.900 --> 00:01:13.800
11.5 gallons, by the
same logic, and

00:01:13.800 --> 00:01:18.900
less than 12.5 gallons.

00:01:18.900 --> 00:01:21.990
So what was the lowest possible
miles per gallon?

00:01:21.990 --> 00:01:23.930
So the lowest possible miles
per gallon would have the

00:01:23.930 --> 00:01:27.240
lowest miles and the
highest gallons.

00:01:27.240 --> 00:01:29.440
You want to minimize the
numerator and maximize the

00:01:29.440 --> 00:01:29.960
denominator.

00:01:29.960 --> 00:01:39.040
So lowest possible miles
would be 285.

00:01:39.040 --> 00:01:42.300
And the highest possible
gallons, at least it'll

00:01:42.300 --> 00:01:47.290
approach 12.5.

00:01:47.290 --> 00:01:53.320
So 12.499999 is the possible
number of gallons, so it'll

00:01:53.320 --> 00:01:54.750
approach this.

00:01:54.750 --> 00:01:58.610
But it can never be exactly 12.5
gallons, so her miles per

00:01:58.610 --> 00:02:01.930
gallon are going to be
greater than this.

00:02:01.930 --> 00:02:03.430
It would've been greater than
equal to this with like

00:02:03.430 --> 00:02:07.260
12.499999, but since 12.5 can
never get there, we're going

00:02:07.260 --> 00:02:10.820
to be greater than this in terms
of miles per gallon.

00:02:10.820 --> 00:02:13.700
And then the highest possible
miles per gallon is the

00:02:13.700 --> 00:02:16.910
highest possible miles,
which is 295.

00:02:16.910 --> 00:02:20.630
But we're never going to be able
to go twice quite there.

00:02:20.630 --> 00:02:22.720
We're always going to
be less than 295.

00:02:22.720 --> 00:02:26.530
And then the lowest number of
gallons-- going the highest

00:02:26.530 --> 00:02:28.720
mileage with the lowest possible
gallons, you want to

00:02:28.720 --> 00:02:32.370
minimize the demonimator--
divided by 11.5.

00:02:32.370 --> 00:02:33.620
Let's see if that is a choice.

00:02:37.810 --> 00:02:41.530
I don't see that exactly, but
choice D looks interesting

00:02:41.530 --> 00:02:43.910
because choice D says-- so they
say the actual number of

00:02:43.910 --> 00:02:46.110
miles per gallon that Cindy's
car got on this trip must have

00:02:46.110 --> 00:02:47.360
been between.

00:02:47.360 --> 00:02:55.590
Choice D looks close to what we
have because we have this.

00:02:55.590 --> 00:03:05.000
Choice D says between
284 divided by 12.5.

00:03:05.000 --> 00:03:07.820
That's going to be even less
than that, because you have

00:03:07.820 --> 00:03:09.390
the 12.5 but the numerator's
less.

00:03:09.390 --> 00:03:11.410
So that's going to be even lower
than the range that we

00:03:11.410 --> 00:03:12.440
have.

00:03:12.440 --> 00:03:19.920
And then on choice D,
295 divided 11.4.

00:03:19.920 --> 00:03:22.090
This number is larger than this
number, because you have

00:03:22.090 --> 00:03:24.570
the same numerator and you have
a smaller denominator.

00:03:24.570 --> 00:03:26.290
So you're dividing by
a smaller number.

00:03:26.290 --> 00:03:27.910
So this number's larger.

00:03:27.910 --> 00:03:31.360
So if her miles per gallon is
between these two numbers, and

00:03:31.360 --> 00:03:33.050
these two numbers are
between these two

00:03:33.050 --> 00:03:35.540
numbers, then D is right.

00:03:35.540 --> 00:03:39.560
Her miles per gallon need to be
between 284 divided by 12.5

00:03:39.560 --> 00:03:42.400
and 295 divided by 11.4.

00:03:42.400 --> 00:03:45.090
So it's choice D.

00:03:45.090 --> 00:03:49.990
Problem 130.

00:03:49.990 --> 00:03:52.230
Which of the following
inequalities is an algebraic

00:03:52.230 --> 00:03:56.060
expression for the shaded part
of the number line above?

00:03:56.060 --> 00:03:58.990
So this number line, let's
try to draw it.

00:03:58.990 --> 00:04:01.430
That's the line.

00:04:01.430 --> 00:04:04.510
And I'm not going to draw all
of the dots on it, but it

00:04:04.510 --> 00:04:09.280
essentially starts at positive
3 and it goes backwards.

00:04:09.280 --> 00:04:10.530
Let's switch colors.

00:04:13.050 --> 00:04:16.980
It starts essentially from
minus 5 to positive 3.

00:04:16.980 --> 00:04:18.835
That's all of what
this includes.

00:04:21.390 --> 00:04:28.950
This includes minus 5 to
positive 3, then you have 2,

00:04:28.950 --> 00:04:31.590
1, 0, and then minus 5.

00:04:31.590 --> 00:04:36.390
So you could say that x is
less than or equal to 3

00:04:36.390 --> 00:04:38.350
because it's filled in at
3, and greater than or

00:04:38.350 --> 00:04:40.200
equal to minus 5.

00:04:40.200 --> 00:04:41.640
And that's not one
of the choices.

00:04:41.640 --> 00:04:43.480
They want it in terms
of absolute value.

00:04:46.320 --> 00:04:49.560
So how we can view this, this is
a range around some number.

00:04:49.560 --> 00:04:51.050
Whenever you're dealing without
absolute value, you're

00:04:51.050 --> 00:04:53.390
essentially saying the distance
from some number.

00:04:53.390 --> 00:04:56.080
So this is a range
of minus 5 and 3.

00:04:56.080 --> 00:04:59.300
This has a length of 8,
so they're both 4

00:04:59.300 --> 00:05:02.220
away from minus 1.

00:05:02.220 --> 00:05:02.950
Think about it.

00:05:02.950 --> 00:05:07.180
You could go 4 backwards
or 4 forwards.

00:05:07.180 --> 00:05:11.350
So one way to think about it
is the difference between x

00:05:11.350 --> 00:05:19.480
and minus 1, and that might
be a negative or positive

00:05:19.480 --> 00:05:21.302
difference depending on whether
we're positive or

00:05:21.302 --> 00:05:25.910
negative, depending on whether
we're above negative 1 or

00:05:25.910 --> 00:05:26.850
below negative 1.

00:05:26.850 --> 00:05:28.850
But we just want the absolute
difference.

00:05:28.850 --> 00:05:33.000
We just want the distance
away from negative 1.

00:05:33.000 --> 00:05:36.510
And that's given by this, the
difference between any x and

00:05:36.510 --> 00:05:37.960
negative 1.

00:05:37.960 --> 00:05:39.690
And we took the absolute value
in case x is less than

00:05:39.690 --> 00:05:40.530
negative 1.

00:05:40.530 --> 00:05:44.410
The distance is going to be
less than or equal to 4,

00:05:44.410 --> 00:05:47.970
because if we go above negative
1 we're at most at 3,

00:05:47.970 --> 00:05:50.170
which is 4 more than
negative 1.

00:05:50.170 --> 00:05:52.700
If we go below negative 1, we're
at minus 5 which is 4

00:05:52.700 --> 00:05:53.540
less than negative 1.

00:05:53.540 --> 00:05:55.490
So it's less than
or equal to 4.

00:05:55.490 --> 00:05:58.210
So if you simplify this, you
get x plus 1, the absolute

00:05:58.210 --> 00:06:00.450
value is less than
or equal to 4.

00:06:00.450 --> 00:06:03.990
And that is choice E.

00:06:03.990 --> 00:06:05.900
And this is a useful skill.

00:06:05.900 --> 00:06:07.725
You have probably taken the SAT
already, but in general, a

00:06:07.725 --> 00:06:10.490
lot of standardized tests think
of absolute value as a

00:06:10.490 --> 00:06:12.870
distance in 1 dimension.

00:06:12.870 --> 00:06:14.350
So the distance along
the number line.

00:06:14.350 --> 00:06:18.620
So they're saying all of these
points, they're within some

00:06:18.620 --> 00:06:19.890
range from 1 point.

00:06:19.890 --> 00:06:21.620
And you can say if you take the
midpoint of this, you're

00:06:21.620 --> 00:06:22.610
at negative 1.

00:06:22.610 --> 00:06:25.500
And every point here's within
4 of negative 1.

00:06:25.500 --> 00:06:29.700
So the distance between every
point on this line, every x

00:06:29.700 --> 00:06:33.300
that satisfies this and negative
1, the distance is

00:06:33.300 --> 00:06:34.450
less than or equal to 4.

00:06:34.450 --> 00:06:37.720
Anyway, I kind of just did the
problem twice for you.

00:06:37.720 --> 00:06:44.630
Problem 131.

00:06:44.630 --> 00:06:51.390
A factory has 500 workers,
15% of whom are women.

00:06:51.390 --> 00:06:57.590
So 15% times 500 is
equal to women.

00:06:57.590 --> 00:07:00.190
If 50 additional workers are
to be hired and all of the

00:07:00.190 --> 00:07:03.970
present workers remain, how many
of the additional workers

00:07:03.970 --> 00:07:08.410
must be women in order to raise
the percent of women

00:07:08.410 --> 00:07:11.780
employees to 20%?

00:07:11.780 --> 00:07:13.480
So how many women workers
do we have right now?

00:07:13.480 --> 00:07:17.720
5 times 15, we have
75 workers.

00:07:17.720 --> 00:07:24.830
15 times 500.

00:07:24.830 --> 00:07:31.610
We have 75 women right now.

00:07:31.610 --> 00:07:34.280
If we wanted to find the
proportion of women, it's 75

00:07:34.280 --> 00:07:38.140
over 500, which is
once again 15.

00:07:38.140 --> 00:07:39.770
It's 15%.

00:07:39.770 --> 00:07:43.790
We're going to add a total of 50
workers to this population.

00:07:43.790 --> 00:07:47.480
And they want to know how many
of the additional workers must

00:07:47.480 --> 00:07:49.560
be women in order to raise
the percent of women

00:07:49.560 --> 00:07:50.770
population to 20%.

00:07:50.770 --> 00:07:53.180
So we're adding 50
total workers.

00:07:53.180 --> 00:07:56.740
So how many of those have to be
women in order for the new

00:07:56.740 --> 00:07:58.740
proportion-- this is going to
be the new number of women,

00:07:58.740 --> 00:08:00.740
the number of women that are
added to the 75 that were

00:08:00.740 --> 00:08:05.190
already there-- divided by the
500 workers who were there,

00:08:05.190 --> 00:08:07.940
plus the 50, some of whom
might be women.

00:08:07.940 --> 00:08:12.640
So this now has to equal 20%.

00:08:12.640 --> 00:08:21.930
We get 75 plus w over
550 is equal to 0.2.

00:08:21.930 --> 00:08:27.300
So we get 75 plus the number of
women that need to be added

00:08:27.300 --> 00:08:32.340
is equal to 0.02 times 550.

00:08:32.340 --> 00:08:34.970
We multiply both sides
times 550.

00:08:34.970 --> 00:08:44.900
That equals 110.

00:08:44.900 --> 00:08:47.320
And then subtract 75
from both sides.

00:08:47.320 --> 00:08:51.010
Women has to be equal
to 110 minus 75.

00:08:51.010 --> 00:08:57.920
And that's equal to-- 25 plus
10-- that's equal to 35.

00:08:57.920 --> 00:09:00.680
35 of the 50 employees
have to be women.

00:09:00.680 --> 00:09:01.470
And that's what they asked.

00:09:01.470 --> 00:09:03.930
How many of the additional
workers must be women?

00:09:03.930 --> 00:09:05.180
So that's choice E.

00:09:08.250 --> 00:09:14.810
Next question, 132.

00:09:14.810 --> 00:09:18.220
At a small snack shop, the
average arithmetic mean

00:09:18.220 --> 00:09:21.750
revenue was $400 per day
over a 10 day period.

00:09:21.750 --> 00:09:25.440
So the average over
10 days was $400.

00:09:25.440 --> 00:09:28.600
During this period, if the
average daily revenue was $360

00:09:28.600 --> 00:09:34.010
for the first 6 days-- so the
average of 6 days was $360--

00:09:34.010 --> 00:09:37.390
what was the average daily
revenue for the last 4 days?

00:09:37.390 --> 00:09:39.060
So we've done a lot of
problems like this.

00:09:39.060 --> 00:09:40.590
Hopefully this is a bit of
second nature if you've

00:09:40.590 --> 00:09:42.080
watched the solution
to the other ones.

00:09:42.080 --> 00:09:44.500
But what's the average
of all of them?

00:09:44.500 --> 00:09:49.020
It's the average of
the first 6 days.

00:09:49.020 --> 00:09:52.110
We can almost assume that the
first 6 days had exactly $360

00:09:52.110 --> 00:09:53.240
in revenue.

00:09:53.240 --> 00:09:55.680
So their average times 6.

00:09:55.680 --> 00:09:59.580
So this would be the total
revenue of the first 6 days

00:09:59.580 --> 00:10:02.950
plus what's the total revenue
of the next 4 days?

00:10:02.950 --> 00:10:07.140
It's going to be the average
of the next 4 days times 4.

00:10:07.140 --> 00:10:08.690
That would be the total
of those 4.

00:10:08.690 --> 00:10:10.500
We're not saying that they all
have to be exactly the

00:10:10.500 --> 00:10:12.550
average, but when you sum them
all up they're equal to the

00:10:12.550 --> 00:10:15.320
average times 4.

00:10:15.320 --> 00:10:18.120
Now if you divide all of those,
this is a total of the

00:10:18.120 --> 00:10:20.050
revenue all 10 days.

00:10:20.050 --> 00:10:22.580
If you take that sum and divide
it by 10, you get the

00:10:22.580 --> 00:10:25.010
average for the 10 days.

00:10:25.010 --> 00:10:27.555
And that is equal to $400.

00:10:27.555 --> 00:10:30.430
Let's simplify this
a little bit.

00:10:30.430 --> 00:10:31.725
Let's see what we could do.

00:10:34.840 --> 00:10:44.820
So we get 360 times 6 plus the
average for 4 days times 4 is

00:10:44.820 --> 00:10:50.570
equal to 10 times 400,
which is 4,000.

00:10:50.570 --> 00:10:53.390
Before I get too involved in
multiplication, let's divide

00:10:53.390 --> 00:10:55.720
both sides of this
equation by 4.

00:10:55.720 --> 00:10:59.990
So 360 divided by 4, that's 90,
times 6 plus this divided

00:10:59.990 --> 00:11:04.320
by 4 plus A4 is equal
to 1,000.

00:11:04.320 --> 00:11:09.890
90 times 6 is 540 plus
the average of the 4

00:11:09.890 --> 00:11:12.310
days is equal to 1,000.

00:11:12.310 --> 00:11:17.540
So the average of the 4 days is
equal to 1,000 minus 540.

00:11:17.540 --> 00:11:18.160
Which is what?

00:11:18.160 --> 00:11:22.460
That's $460 per day.

00:11:22.460 --> 00:11:26.180
And that's choice D.

00:11:26.180 --> 00:11:27.260
And I'm out of time.

00:11:27.260 --> 00:11:29.010
See you in the next video.


GMAT: Math 24

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.960
And problem 124.

00:00:03.960 --> 00:00:07.490
According to a car dealer sales
report, 1/3 of the cars

00:00:07.490 --> 00:00:11.640
sold during a certain period
were sedans, and 1/5 of the

00:00:11.640 --> 00:00:14.160
other cars sold were
station wagons.

00:00:14.160 --> 00:00:19.800
So 1/3 of the cars sold,
so the total,

00:00:19.800 --> 00:00:22.000
were equal to sedans.

00:00:22.000 --> 00:00:27.150
And 1/5 of the other cars sold
were station wagons, so 1/5 of

00:00:27.150 --> 00:00:28.260
the remainder of this.

00:00:28.260 --> 00:00:33.420
So that's 1/5 of the 2/3 cars
that weren't sedans.

00:00:33.420 --> 00:00:39.650
So 1/5 of the 2/3 of the total
were station wagons.

00:00:39.650 --> 00:00:42.490
Let's call it W for wagons.

00:00:42.490 --> 00:00:47.310
If N station wagons were sold
during that period-- so W is

00:00:47.310 --> 00:00:53.460
equal to N-- how many sedans
in terms of n were sold?

00:00:53.460 --> 00:01:03.000
So we know that 1/5 times 2/3
times the total is equal to N.

00:01:03.000 --> 00:01:05.489
Let's just substitute S for T.

00:01:05.489 --> 00:01:08.810
If we multiply both sides of
this top equation times 3, we

00:01:08.810 --> 00:01:11.030
know that the total is
equal to 3 times

00:01:11.030 --> 00:01:11.715
the number of sedans.

00:01:11.715 --> 00:01:14.270
So let's substitute
that here for T.

00:01:14.270 --> 00:01:22.370
So you get 1/5 times 2/3
times 3S is equal to N.

00:01:22.370 --> 00:01:24.250
3 cancels out.

00:01:24.250 --> 00:01:25.500
3 divided by 3.

00:01:25.500 --> 00:01:32.030
So you have 2/5 S is equal
to N, or-- let me

00:01:32.030 --> 00:01:33.200
just write it up here.

00:01:33.200 --> 00:01:34.970
So multiply both sides by 5/2.

00:01:34.970 --> 00:01:36.703
You have S is equal to 5/2 N.

00:01:39.950 --> 00:01:43.240
And that is choice D.

00:01:43.240 --> 00:01:48.570
Problem 125.

00:01:48.570 --> 00:01:56.810
If p/q is less than 1, and p and
q are positive integers,

00:01:56.810 --> 00:02:01.880
which of the following must
be greater than 1?

00:02:01.880 --> 00:02:02.670
So think about this.

00:02:02.670 --> 00:02:07.770
If p/q is less than 1,
that tells us that p

00:02:07.770 --> 00:02:09.430
is less than q.

00:02:09.430 --> 00:02:11.039
You can multiply both
sides of this times

00:02:11.039 --> 00:02:12.220
q, and you get that.

00:02:12.220 --> 00:02:14.660
p has to be less than q.

00:02:14.660 --> 00:02:15.315
And they're both positive
integers.

00:02:15.315 --> 00:02:17.860
So there's nothing tricky here
with negative numbers.

00:02:17.860 --> 00:02:19.800
So just the instincts, which
of the following must be

00:02:19.800 --> 00:02:20.595
greater than 1?

00:02:20.595 --> 00:02:22.830
I don't know if this is going to
be a choice, but if this is

00:02:22.830 --> 00:02:24.160
less than 1, if you invert
it it's going to be

00:02:24.160 --> 00:02:25.410
greater than 1.

00:02:27.800 --> 00:02:29.560
So divide both sides
of this by p.

00:02:29.560 --> 00:02:33.060
You'll get 1 is less than q/p.

00:02:33.060 --> 00:02:34.100
And that makes sense.

00:02:34.100 --> 00:02:39.360
If 1/2 is less than 1, then
2/1 is going to be

00:02:39.360 --> 00:02:40.010
greater than 1.

00:02:40.010 --> 00:02:43.800
If 2/3 is less than 1, then
3/2 is going to be

00:02:43.800 --> 00:02:45.400
greater than 1.

00:02:45.400 --> 00:02:48.310
So if p/q is less than 1, then
q over p is going to be

00:02:48.310 --> 00:02:48.920
greater than 1.

00:02:48.920 --> 00:02:51.990
And we did it algebraically by
just multiplying both sides by

00:02:51.990 --> 00:02:55.110
q and then multiplying
both sides by p.

00:02:55.110 --> 00:02:56.060
That is a choice.

00:02:56.060 --> 00:02:57.510
E, q/p.

00:02:57.510 --> 00:02:58.605
That made it easy.

00:02:58.605 --> 00:03:00.220
I thought I might have
to do more work if

00:03:00.220 --> 00:03:01.310
that didn't show up.

00:03:01.310 --> 00:03:04.630
So q/p is definitely
greater than 1.

00:03:04.630 --> 00:03:11.050
Next question, 126.

00:03:11.050 --> 00:03:14.280
It would take 1 machine 4 hours
to complete a large

00:03:14.280 --> 00:03:15.620
production order.

00:03:15.620 --> 00:03:18.240
So machine 1, it would
take 4 hours.

00:03:20.950 --> 00:03:27.780
So you could say 1 order
every 4 hours, or

00:03:27.780 --> 00:03:29.600
1/4 order per hour.

00:03:29.600 --> 00:03:31.705
That's machine 1.

00:03:31.705 --> 00:03:34.720
And another machine, 3 hours
to complete the same order.

00:03:34.720 --> 00:03:44.460
So machine 2 does 1 order in 3
hours, or 1/3 orders per hour.

00:03:44.460 --> 00:03:46.950
How many hours would it take
both machines working

00:03:46.950 --> 00:03:49.330
simultaneously at their
respective constant rates to

00:03:49.330 --> 00:03:50.850
complete the order?

00:03:50.850 --> 00:03:52.690
If they're working together,
you're going to have one

00:03:52.690 --> 00:03:57.130
machine producing at 1/4 order
per hour, plus another machine

00:03:57.130 --> 00:04:00.840
producing at 1/3
order per hour.

00:04:00.840 --> 00:04:04.410
So their combined rate is going
to be what over 12?

00:04:04.410 --> 00:04:07.510
And then 1/4 is the same thing
as 3/12, and 1/3 is the same

00:04:07.510 --> 00:04:08.540
thing as 4/12.

00:04:08.540 --> 00:04:10.120
So their combined rate
is going to be

00:04:10.120 --> 00:04:17.120
7/12 order per hour.

00:04:17.120 --> 00:04:19.190
And you want to know how long
it's going to take them to

00:04:19.190 --> 00:04:21.550
make 1 order.

00:04:21.550 --> 00:04:25.750
So you can think of production
is equal to rate times time.

00:04:25.750 --> 00:04:29.050
And time is equal to production
divided by rate.

00:04:29.050 --> 00:04:31.190
Or if you don't want to think
about them, just think if I

00:04:31.190 --> 00:04:33.980
can make 7/12 orders per hour,
if I have to make 1 order, I'm

00:04:33.980 --> 00:04:38.740
just going to be 1
order divided by

00:04:38.740 --> 00:04:44.600
7/12 order per hour.

00:04:44.600 --> 00:04:46.420
This might seem a little
confusing to you, but if these

00:04:46.420 --> 00:04:47.790
were constant numbers,
try to play with it.

00:04:47.790 --> 00:04:50.540
If I had to make 10 orders, and
I can make 2 orders per

00:04:50.540 --> 00:04:53.920
hour, then I just divide
10 by 2, which

00:04:53.920 --> 00:04:54.890
would tell me 5 hours.

00:04:54.890 --> 00:04:57.940
But it's just a little more
confusing when I have to make

00:04:57.940 --> 00:05:01.010
1 order and my rate is
7/12 orders for hour.

00:05:01.010 --> 00:05:03.760
But hopefully if you substitute
whole numbers in

00:05:03.760 --> 00:05:05.570
your head it makes little
bit of sense.

00:05:05.570 --> 00:05:11.460
So this is the same
thing as 12/7.

00:05:11.460 --> 00:05:14.410
And then of course the orders
cancel out, and you end up

00:05:14.410 --> 00:05:15.510
with hours in the numerator.

00:05:15.510 --> 00:05:19.280
So 12/7 hours or 1 5/7 hours.

00:05:19.280 --> 00:05:19.830
What do they have?

00:05:19.830 --> 00:05:21.160
Yes, that's choice C.

00:05:25.220 --> 00:05:30.390
Question 127.

00:05:30.390 --> 00:05:39.750
To mail a package, the rate is
x cents for the first pound,

00:05:39.750 --> 00:05:48.310
and y cents for each additional
pound, where they

00:05:48.310 --> 00:05:50.840
tell us that x is
greater than y.

00:05:50.840 --> 00:05:55.820
2 packages weighing 3 pounds and
5 pounds respectively can

00:05:55.820 --> 00:05:59.310
be mailed separately or combined
as one package, so 3

00:05:59.310 --> 00:06:02.490
pounds and 5 pounds.

00:06:02.490 --> 00:06:03.090
That's one package.

00:06:03.090 --> 00:06:08.080
Which method is cheaper, and
how much money is saved?

00:06:08.080 --> 00:06:09.060
Let's think about it.

00:06:09.060 --> 00:06:12.950
If you keep them separate, how
much are you going to pay?

00:06:12.950 --> 00:06:15.090
For the 3 pound packet,
you're going to pay x

00:06:15.090 --> 00:06:17.180
for the first pound.

00:06:17.180 --> 00:06:18.470
And then you're going to have
2 pounds left that you are

00:06:18.470 --> 00:06:20.310
going to pay y at.

00:06:20.310 --> 00:06:22.840
And then for this one, you
are going to pay x

00:06:22.840 --> 00:06:23.640
for the first pound.

00:06:23.640 --> 00:06:27.260
You're going to have 4 pounds
left, so plus 4y.

00:06:27.260 --> 00:06:29.645
So when you add them together,
your total expense is going to

00:06:29.645 --> 00:06:34.210
be 2x plus 6y.

00:06:34.210 --> 00:06:35.760
That's if you mail
them separately.

00:06:35.760 --> 00:06:37.910
Now if you combine them
into one 8 pound

00:06:37.910 --> 00:06:41.510
package, what do you have?

00:06:41.510 --> 00:06:44.190
You have to pay x for the first
pound, and then you have

00:06:44.190 --> 00:06:46.120
7 pounds left that you're
going to pay for y.

00:06:46.120 --> 00:06:47.490
So it's going to
cost x plus 7y.

00:06:52.120 --> 00:06:54.230
So first of all, we know that
this is going to be more.

00:06:54.230 --> 00:06:55.820
Why do we know this is more?

00:06:55.820 --> 00:06:58.960
Because essentially we're taking
one of the y's-- and

00:06:58.960 --> 00:07:03.130
when you pay them separately for
that one extra pound that

00:07:03.130 --> 00:07:07.310
here you're paying it as in x
cents, instead of y cents that

00:07:07.310 --> 00:07:08.750
you're paying in this case.

00:07:08.750 --> 00:07:10.360
And x is more than y.

00:07:10.360 --> 00:07:15.250
And that will hold out when we
actually subtract the 2 out.

00:07:15.250 --> 00:07:17.020
Actually, hopefully you
understand the reasoning why

00:07:17.020 --> 00:07:19.730
this is going to be larger
because x is greater than y.

00:07:19.730 --> 00:07:20.600
So let's figure it out.

00:07:20.600 --> 00:07:21.960
What's the difference?

00:07:21.960 --> 00:07:23.330
This is the higher number.

00:07:23.330 --> 00:07:24.880
And let's subtract out
the lower number,

00:07:24.880 --> 00:07:27.340
minus x minus 7y.

00:07:27.340 --> 00:07:30.600
I just subtracted x plus 7y.

00:07:30.600 --> 00:07:36.960
And that you end up
with x minus y.

00:07:36.960 --> 00:07:39.440
6 minus 7y is minus 1y.

00:07:39.440 --> 00:07:41.620
So x minus y.

00:07:41.620 --> 00:07:43.685
And the fact that x is greater
than y lets you

00:07:43.685 --> 00:07:44.490
know this is positive.

00:07:44.490 --> 00:07:48.610
So that also tells you that this
is the more expensive of

00:07:48.610 --> 00:07:49.400
the alternatives.

00:07:49.400 --> 00:07:51.150
Separate is the more
expensive.

00:07:51.150 --> 00:07:53.710
And the difference
is x minus y.

00:07:53.710 --> 00:07:56.780
So they ask which method
is cheaper?

00:07:56.780 --> 00:08:01.210
So the combined method
is cheaper and you

00:08:01.210 --> 00:08:03.760
save x minus y cents.

00:08:09.620 --> 00:08:13.000
And that's choice A.

00:08:13.000 --> 00:08:14.650
Choice B, they say combined
is cheaper with a

00:08:14.650 --> 00:08:16.270
savings of y minus x.

00:08:16.270 --> 00:08:17.860
But y minus x is going to
be a negative number.

00:08:17.860 --> 00:08:20.470
So you can't say you saved
a negative number.

00:08:20.470 --> 00:08:22.650
They tell us that x is greater
than y, so you

00:08:22.650 --> 00:08:24.590
could cancel that out.

00:08:24.590 --> 00:08:28.390
And the other ones don't
make any sense at all.

00:08:28.390 --> 00:08:34.720
Next problem, 128.

00:08:34.720 --> 00:08:39.669
If money is invested at r%
interest compounded annually,

00:08:39.669 --> 00:08:42.450
the amount of the investment
will double in approximately

00:08:42.450 --> 00:08:53.960
70/r years.

00:08:53.960 --> 00:09:01.420
If Pat's parents invested $5,000
in a long-term bond

00:09:01.420 --> 00:09:08.790
that pays 8% interest compounded
annually, what will

00:09:08.790 --> 00:09:11.530
be the approximate total amount
of the investment 18

00:09:11.530 --> 00:09:15.980
years later as Pat is
ready for college?

00:09:18.890 --> 00:09:25.470
It doubles every 70/r
years, where r is 8.

00:09:25.470 --> 00:09:27.000
They gave us that information,
so let's try to use it.

00:09:27.000 --> 00:09:29.540
So how long does it take to
double at an 8% rate?

00:09:29.540 --> 00:09:33.800
So it's 70 divided by
8 is equal to what?

00:09:33.800 --> 00:09:35.050
It's equal to 8.5?

00:09:44.410 --> 00:09:48.230
8 goes into 70 eight
times, 64.

00:09:48.230 --> 00:09:49.130
6, 0.

00:09:49.130 --> 00:09:52.360
8 goes into 60 seven times.

00:09:52.360 --> 00:09:54.460
7 times 8 is 56.

00:09:54.460 --> 00:09:56.030
And you have a 40.

00:09:56.030 --> 00:09:58.840
8.75 years.

00:09:58.840 --> 00:10:01.210
This isn't going to be exact,
but you're going to double

00:10:01.210 --> 00:10:05.650
your money roughly every 8.75
years if using this formula

00:10:05.650 --> 00:10:07.820
they give you at an
8% interest rate.

00:10:07.820 --> 00:10:12.210
And they say what will be the
total amount of the investment

00:10:12.210 --> 00:10:13.880
18 years later?

00:10:13.880 --> 00:10:17.110
So 18 years, that's roughly
two doublings.

00:10:17.110 --> 00:10:22.190
So you're going to start with
$5,000, and after 8.75 years

00:10:22.190 --> 00:10:24.520
you're going to be at $10,000.

00:10:24.520 --> 00:10:27.580
You're going to double
in 8.75 years.

00:10:27.580 --> 00:10:30.635
And then after another
8.75 years, you are

00:10:30.635 --> 00:10:33.600
going to be at $20,000.

00:10:33.600 --> 00:10:36.830
This isn't exactly 18 years.

00:10:36.830 --> 00:10:39.280
This is 17.5, but that's
about as close as

00:10:39.280 --> 00:10:40.110
we're going to get.

00:10:40.110 --> 00:10:42.290
So in 17.5 years, you
get to $20,000.

00:10:42.290 --> 00:10:44.210
They are asking 18 years,
and this is all about

00:10:44.210 --> 00:10:46.860
approximation because they
don't want you to do some

00:10:46.860 --> 00:10:48.620
hardcore exponent math here.

00:10:48.620 --> 00:10:51.940
And so the answer is A.

00:10:51.940 --> 00:10:54.510
And if you look at that 8.75,
that's roughly you're doubling

00:10:54.510 --> 00:10:55.260
every 9 years.

00:10:55.260 --> 00:10:56.390
They ask us how long.

00:10:56.390 --> 00:10:58.380
So you could say after 9 years
I'm at $10,000, after another

00:10:58.380 --> 00:11:00.040
9 years I'm at $20,000.

00:11:00.040 --> 00:11:03.760
So after 18 years, I'll
be at $20,000.

00:11:03.760 --> 00:11:04.610
And I'm all out of time.

00:11:04.610 --> 00:11:06.300
See you in the next video.