0:00:00.670,0:00:03.700
We're on problem number 7.

0:00:03.700,0:00:08.189
If the average of x and 3x is[br]12, what is the value of x?

0:00:08.189,0:00:12.370
So the average, so x plus 3x--[br]and I'm averaging two numbers

0:00:12.370,0:00:15.270
so divide by 2-- that's going[br]to be equal to 12.

0:00:15.270,0:00:16.180
So we just solve for this.

0:00:16.180,0:00:19.800
Multiply both sides of the[br]equation by 2 and you get 2

0:00:19.800,0:00:22.090
times, times 2, this[br]cancels with this.

0:00:22.090,0:00:26.020
You get x plus 3x[br]is equal to 24.

0:00:26.020,0:00:26.890
And what's x plus 3x.

0:00:26.890,0:00:28.670
That's 4x, right?

0:00:28.670,0:00:30.645
4x is equal to 24.

0:00:30.645,0:00:35.720
x is equal to 6, and[br]that's choice C.

0:00:35.720,0:00:36.970
Next problem.

0:00:40.080,0:00:42.970
Problem 8.

0:00:42.970,0:00:46.700
At Maple Creek High School, some[br]members of the chess club

0:00:46.700,0:00:50.320
are also on the swim team, and[br]no members of the swim team

0:00:50.320,0:00:52.130
are tenth graders.

0:00:52.130,0:00:54.830
Which of the following[br]must be true.

0:00:54.830,0:00:57.220
This seems like it'll call[br]for a Venn diagram.

0:00:57.220,0:01:02.680
So let's say that that[br]represents the chess club.

0:01:02.680,0:01:04.790
And they say some members[br]of the chess club

0:01:04.790,0:01:06.210
are on the swim team.

0:01:06.210,0:01:08.930
So some members are[br]on the swim team.

0:01:08.930,0:01:12.620
Maybe I should put the swim[br]team in like blue.

0:01:12.620,0:01:16.100
So let's say the swim team.

0:01:16.100,0:01:17.350
That's the swim team.

0:01:19.740,0:01:21.510
And these are the members,[br]right, that are

0:01:21.510,0:01:23.980
in both right here.

0:01:23.980,0:01:27.200
But then it tells us no members[br]of the swim team are

0:01:27.200,0:01:30.170
tenth graders.

0:01:30.170,0:01:32.970
So if I draw another circle[br]for the tenth graders, it

0:01:32.970,0:01:35.560
can't intersect with the swim[br]team, but it could intersect

0:01:35.560,0:01:36.080
with the chess team.

0:01:36.080,0:01:36.540
I don't know.

0:01:36.540,0:01:38.510
I mean it could be like that.

0:01:38.510,0:01:41.240
That could be tenth graders.

0:01:41.240,0:01:42.830
It could be like that.

0:01:42.830,0:01:45.930
Or it could be out[br]here some place.

0:01:45.930,0:01:47.150
But we don't know.

0:01:47.150,0:01:49.880
There could be chess and tenth[br]graders, just not the same

0:01:49.880,0:01:52.440
people who are on[br]the swim team.

0:01:52.440,0:01:53.690
So let's see.

0:01:55.820,0:01:57.180
So which of the following[br]must be true?

0:01:57.180,0:01:58.960
No members of the chess club[br]are tenth graders.

0:01:58.960,0:01:59.370
No.

0:01:59.370,0:02:01.800
This is a situation where you[br]could have some members of the

0:02:01.800,0:02:03.500
chess club who aren't[br]on the swim team who

0:02:03.500,0:02:05.420
could be tenth graders.

0:02:05.420,0:02:09.039
B, some members of the chess[br]club are tenth graders.

0:02:09.039,0:02:11.840
Well some members could be, but[br]we don't know for sure.

0:02:11.840,0:02:12.880
This could be tenth grade.

0:02:12.880,0:02:13.690
We don't know.

0:02:13.690,0:02:16.530
This could be the tenth grade[br]kind of set or this could be

0:02:16.530,0:02:16.970
the tenth grade.

0:02:16.970,0:02:19.590
There might be no tenth graders[br]in either the chess

0:02:19.590,0:02:20.310
team or the swim team.

0:02:20.310,0:02:21.890
We don't know for sure.

0:02:21.890,0:02:25.940
And then choice C, some members[br]of the chess club are

0:02:25.940,0:02:27.590
not tenth graders.

0:02:27.590,0:02:28.890
This we know for sure.

0:02:28.890,0:02:30.240
How do we know it for sure?

0:02:30.240,0:02:37.030
Because these kids who are on[br]both, they're in the chess

0:02:37.030,0:02:39.210
club, but they're also[br]on the swim team.

0:02:39.210,0:02:41.960
The fact that they're in swim[br]team, we know that they can't

0:02:41.960,0:02:43.550
be tenth graders.

0:02:43.550,0:02:46.670
So this is some members of the[br]chess club-- this little

0:02:46.670,0:02:50.330
intersection here-- that[br]are not tenth graders.

0:02:50.330,0:02:52.440
So choice C is the[br]correct choice.

0:02:55.280,0:02:56.530
Next problem.

0:03:01.830,0:03:08.380
If 3x plus n is equal[br]to x plus 1, what is

0:03:08.380,0:03:09.540
n in terms of x?

0:03:09.540,0:03:11.555
So we essentially just[br]solve for n.

0:03:11.555,0:03:13.790
Let's subtract 3x[br]from both sides.

0:03:13.790,0:03:17.100
You get n is equal to--[br]what's x minus 3x?

0:03:17.100,0:03:18.580
It's minus 2x.

0:03:18.580,0:03:21.000
And n plus 1.

0:03:21.000,0:03:21.710
And we're done.

0:03:21.710,0:03:23.870
And that choice isn't there, but[br]if you just switch these

0:03:23.870,0:03:27.000
two terms you just get that[br]equals 1 minus 2x and

0:03:27.000,0:03:28.900
that's choice D.

0:03:28.900,0:03:31.850
Pretty quick problem, especially[br]for one that's the

0:03:31.850,0:03:32.450
ninth problem.

0:03:32.450,0:03:34.110
They normally get a little[br]harder by this point.

0:03:34.110,0:03:36.630
Problem 10.

0:03:36.630,0:03:41.290
If k is a positive integer, let[br]k be defined as a set of

0:03:41.290,0:03:42.470
all multiples of k.

0:03:42.470,0:03:46.480
So k with a square around it[br]is equal to the set of

0:03:46.480,0:03:52.490
multiples of k.

0:03:52.490,0:03:56.900
All of the numbers in which of[br]the following sets are also in

0:03:56.900,0:03:59.770
all three of the set-- OK.

0:03:59.770,0:04:04.690
All of the numbers in which of[br]the following sets are also in

0:04:04.690,0:04:10.900
all three of the sets[br]of 2, 3 and 5?

0:04:10.900,0:04:22.330
So the what they're saying is 2,[br]3, 5, this donates all the

0:04:22.330,0:04:23.580
multiples of 2.

0:04:26.730,0:04:30.450
This is all multiples of 3.

0:04:30.450,0:04:37.340
This is all multiples of 5.

0:04:37.340,0:04:41.800
So what they're essentially[br]saying is let's find a number

0:04:41.800,0:04:45.970
where all of its multiples, all[br]of this number's multiples

0:04:45.970,0:04:49.740
are also going to be multiples[br]of each of these.

0:04:49.740,0:04:53.290
So it has to be a multiple--[br]so every number that--

0:04:53.290,0:04:55.920
whatever this mystery number[br]is, let's call it x-- every

0:04:55.920,0:05:00.790
multiple of x has to be a[br]multiple of 2, 3 and 5.

0:05:00.790,0:05:05.950
Well the simple way is if x is a[br]multiple of 2, 3 and 5, then

0:05:05.950,0:05:07.340
every multiple of x[br]is going to be a

0:05:07.340,0:05:08.970
multiple of 2, 3 and 5.

0:05:08.970,0:05:11.160
So what's 2 times 3 times 5?

0:05:11.160,0:05:13.680
It's 2 times 3 times 5.

0:05:13.680,0:05:16.090
That's 6 times 5, that's 30.

0:05:16.090,0:05:20.160
So 30 is a multiple of all of[br]them, so any multiple of 30

0:05:20.160,0:05:21.990
will be a multiple[br]of all of these.

0:05:21.990,0:05:25.050
When we look at the choices[br]we don't see 30.

0:05:25.050,0:05:27.360
But do we see any other[br]number that is a

0:05:27.360,0:05:30.010
multiple of 2, 3 and 5?

0:05:30.010,0:05:32.160
Well sure, 60 is, right?

0:05:32.160,0:05:33.620
We just multiply by 2 again.

0:05:33.620,0:05:36.100
But 60 is still a multiple[br]of 2, 3 and 5.

0:05:36.100,0:05:38.530
If you were to do 2, 4, 6, 8 all[br]the way you'd get 60, if

0:05:38.530,0:05:41.260
you go 3, 9, 12, 15 all the[br]way, you'd get to 60.

0:05:41.260,0:05:44.580
You go 5, 10, 15, 20,[br]25, you'd get to 60.

0:05:44.580,0:05:46.630
So 60 is a multiple[br]of all of them.

0:05:46.630,0:05:50.600
So what we're saying is-- so[br]what's the set of all the

0:05:50.600,0:05:51.410
multiples of 60?

0:05:51.410,0:05:58.260
It's 60, 120, 180, 240,[br]et cetera, right?

0:05:58.260,0:06:02.510
And all of these numbers are[br]in each of these sets.

0:06:02.510,0:06:06.020
Because all of these numbers are[br]multiples of 2, 3 and 5.

0:06:06.020,0:06:07.190
So our answer is 60.

0:06:07.190,0:06:09.320
If you look at the other[br]choices, some of them are

0:06:09.320,0:06:11.740
divisible by 5, some are[br]divisible by 2 or 3,

0:06:11.740,0:06:13.100
some are 3 and 5.

0:06:13.100,0:06:17.870
But none of them are divisible[br]by 2, 3 and 5, only 60 is.

0:06:17.870,0:06:19.120
Next problem.

0:06:21.760,0:06:23.890
That problem was a little hard[br]to read initially though.

0:06:23.890,0:06:25.140
That's how they confuse you.

0:06:31.210,0:06:33.383
So we're going to go from A to[br]D-- I should have drawn all

0:06:33.383,0:06:38.570
the lines first. Let me draw the[br]lines first. It's like a

0:06:38.570,0:06:40.741
hexagon kind of.

0:06:40.741,0:06:43.900
The top, the outside of[br]the hexagon there.

0:06:47.100,0:06:53.330
A, B, C, D, E, F.

0:06:53.330,0:06:54.740
And then this is the origin.

0:06:54.740,0:06:59.640
And the figure above,[br]AD is equal to BE.

0:06:59.640,0:07:00.020
Oh, no, no.

0:07:00.020,0:07:00.810
They don't tell us that.

0:07:00.810,0:07:01.820
I'm hallucinating.

0:07:01.820,0:07:05.720
In the figure above AD, BE,[br]and CF intersect at 0.0.

0:07:05.720,0:07:07.880
The intersect's here[br]at the origin.

0:07:07.880,0:07:12.710
If the measure of AOB, the[br]measure of that, is 80

0:07:12.710,0:07:22.520
degrees, and CF bisects[br]BOD, so it

0:07:22.520,0:07:26.110
bisects this larger angle.

0:07:26.110,0:07:28.880
CF bisect BOD, that angle.

0:07:28.880,0:07:32.230
So that tells us that[br]this angle has to be

0:07:32.230,0:07:34.100
equal to this angle.

0:07:34.100,0:07:35.520
That's the definition of[br]bisecting an angle.

0:07:35.520,0:07:37.310
You're splitting this larger[br]angle in half.

0:07:37.310,0:07:41.110
So these angles have to be[br]equal to each other.

0:07:41.110,0:07:43.270
So what is the measure of EOF?

0:07:47.600,0:07:51.770
So we want to figure[br]out this angle.

0:07:51.770,0:07:54.285
Well this angle is opposite to[br]this angle, so they're going

0:07:54.285,0:07:54.840
to be equal.

0:07:54.840,0:07:56.970
So if we can figure out[br]this angle we're done.

0:07:56.970,0:07:59.370
So let's call this angle x.

0:07:59.370,0:08:03.770
If that angle's x this[br]angle is also x.

0:08:03.770,0:08:05.900
This x, this x, and this[br]80 degrees, they're all

0:08:05.900,0:08:10.440
supplementary because they all[br]go halfway around the circle.

0:08:10.440,0:08:16.140
So x plus x plus 80 is going[br]to be equal to 180 degrees.

0:08:16.140,0:08:20.090
2x plus 80 is equal to 180.

0:08:20.090,0:08:24.590
2x is equal to 100,[br]x is equal to 50.

0:08:24.590,0:08:29.390
And as we said before, x is[br]equal to 50, the angle EOF,

0:08:29.390,0:08:31.750
which you're trying to figure[br]out, is opposite to it so it's

0:08:31.750,0:08:32.850
going to be equal.

0:08:32.850,0:08:34.909
So this is also going[br]to be 50 degrees.

0:08:34.909,0:08:38.610
And that's choice B.

0:08:38.610,0:08:40.559
Next problem.

0:08:40.559,0:08:43.590
I don't know if I have time[br]for this, but I'll try.

0:08:43.590,0:08:45.780
Problem 12.

0:08:45.780,0:08:47.310
k is a positive integer.

0:08:47.310,0:08:50.890
What is the least value[br]of k for which the

0:08:50.890,0:08:53.310
square root of-- OK.

0:08:53.310,0:08:59.390
So what is the least value[br]of k for which 5k

0:08:59.390,0:09:02.390
over 3 is an integer.

0:09:02.390,0:09:04.600
So this has to be a whole[br]number, right?

0:09:04.600,0:09:07.720
So essentially if we want to[br]find the least value of k, we

0:09:07.720,0:09:09.700
essentially want to say, well[br]what's the least integer that

0:09:09.700,0:09:11.910
this could be?

0:09:11.910,0:09:15.240
And they're telling us that[br]k is a positive integer.

0:09:15.240,0:09:19.330
So first of all, in order for[br]the square root to be an

0:09:19.330,0:09:24.230
integer, this whole thing has[br]to be an integer, right?

0:09:24.230,0:09:27.710
So let's see, k has to[br]be a multiple of 3.

0:09:27.710,0:09:31.525
In order for this expression to[br]be an integer, k has to be

0:09:31.525,0:09:32.670
a multiple of 3.

0:09:32.670,0:09:37.500
If k is 3, we get square root[br]of 15 over 3-- well that

0:09:37.500,0:09:40.140
doesn't work.

0:09:40.140,0:09:44.080
If k is 3 we just[br]get 5 in there.

0:09:44.080,0:09:45.850
Actually, let me continue this[br]into the next problem because

0:09:45.850,0:09:46.730
I don't want to rush this.

0:09:46.730,0:09:48.470
I'll see you in the[br]next video.