SAT Prep: Test 2 Section 3 Part 3

WEBVTT
Kind: captions
Language: en

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Welcome back.

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Now let me apologize.

00:00:02.550 --> 00:00:04.170
That last problem, I think I
rushed it a little bit because

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I have to get under this
10-minute YouTube limit.

00:00:08.760 --> 00:00:09.360
But let me know.

00:00:09.360 --> 00:00:12.590
Send me a message on that last
video if you don't think the

00:00:12.590 --> 00:00:15.920
explanation was as satisfying as
you would like it to be or

00:00:15.920 --> 00:00:16.920
if you're still confused.

00:00:16.920 --> 00:00:19.400
With that said, let
me move on.

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Problem number 9 on page 461.

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The decimal number above, and
they wrote 5.101001000100001,

00:00:45.210 --> 00:00:46.820
and it keeps going.

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The decimal number above
consists of only 1's and 0's

00:00:49.780 --> 00:00:51.520
to the right of the
decimal point.

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The first 1 is followed
by one 0.

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The second 1 is followed
by two 0's.

00:00:55.650 --> 00:00:56.120
Right?

00:00:56.120 --> 00:00:58.005
The second 1 is followed by
two 0's, the third 1 is

00:00:58.005 --> 00:01:00.960
followed by three 0's,
and so forth.

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What is the total number of 0's
between the 98th and 101st

00:01:12.500 --> 00:01:15.830
1 in this decimal number?

00:01:15.830 --> 00:01:19.250
Well, the first 1 is followed
by one 0, the second 1 is

00:01:19.250 --> 00:01:20.650
followed by two 0's.

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The third 1 is followed
by three 0's.

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Right?

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So the 50th 1 is going to
be followed by 50 0's.

00:01:27.540 --> 00:01:28.380
Right?

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The nth 1 is followed
by n 0's.

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So the 98th is going to
be followed by 98 0's.

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Right?

00:01:40.725 --> 00:01:43.440
So you're going to have a 1,
then you're going to have 98

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0's, then you're going
to have the 99th 1.

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That's going to be followed
by 99 0's.

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That's just the pattern
that they set up.

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Then you're going to
have the 100th 1.

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That's going to be followed
by 100 0's.

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Then you're going to have
the 101st 1, right?

00:02:02.930 --> 00:02:05.850
And we don't care about the 0's
that are after the 101st

00:02:05.850 --> 00:02:11.250
1, because they said how many
0's are between the 98th 1 and

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the 101st 1.

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So I mean, another way to draw
it, as you know, we had a

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bunch of 0's on this, and
then this is the 98th 1.

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Then we're going to have
a bunch of 0's.

00:02:25.080 --> 00:02:28.880
We're going to have
98 0's here.

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Then you're going to have a 1.

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Then you're going to have
a bunch of 0's.

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This time you're going
to have 99 0's.

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This is going to
be the 99th 1.

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Then you have the 100th 1, which
is followed by 100 0's.

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Then you have the 101st 1.

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And we don't care about
this 0's after it.

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So this is how many 0's
are in between.

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We just have to add up this.

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See, 8 plus 9 is 17.

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So it's 98 plus 99 is 207.

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So it's 307 0's.

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Is that one of the choices?

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Sorry.

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I'm doing my math wrong.

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See, this is how I know it's
late at night and I

00:03:14.350 --> 00:03:15.400
should be in bed.

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But 98 plus 99 is 197
plus 100 is 297.

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So that's how many 0's are
between the 98th and 101st 1

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in this decimal number.

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Problem 10.

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Let me switch colors.

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This color is making
me sleepy.

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f of x is equal to 3 minus
2x squared over x for all

00:03:53.290 --> 00:03:56.760
non-zero x, then f of 2.

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Well, we just take a 2.

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f or 2, we're just taking a
2 and sticking it here.

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So everywhere we see an x,
we replace it with a 2.

00:04:03.840 --> 00:04:08.590
So it becomes 3 minus 2-- we
put a 2 in there, let me

00:04:08.590 --> 00:04:21.899
switch colors for emphasis-- 2
squared over-- that's not the

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same purple-- over 2.

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Just so you know where
that 2 came from,

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that's f of 2, right?

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Everywhere where there
was an x, I just

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replaced it with a 2.

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Now I just evaluate this.

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So 3 minus 2 times 2
squared-- so that 2

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squared is 4-- over 2.

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So 3 minus 8 over 2 equals--
well 3 minus 8 is minus 5-- so

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it's minus 5 over 2.

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That is choice d.

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OK, it looks like we
have a geometry

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problem for number 11.

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Let me draw this.

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There's two lines like
this and like this.

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These lines are perpendicular,
x degrees, y degrees.

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This line is l.

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This line m.

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In the figure above, l is
perpendicular to n.

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Right? l is perpendicular to n,
this is a 90 degree angle,

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and x is greater than
90 degrees.

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So this angle is larger
than 90 degrees.

00:05:55.830 --> 00:05:56.670
So they didn't draw
it that way.

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So this is actually probably
angling down a bit.

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Which of the following
must be true?

00:06:03.100 --> 00:06:05.100
y is less than 90 degrees?

00:06:05.100 --> 00:06:07.740
So let's see.

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So, it's interesting, because
who cares what's happening up

00:06:11.080 --> 00:06:12.150
here, right?

00:06:12.150 --> 00:06:15.500
If you have a line-- let me
just draw it like this.

00:06:15.500 --> 00:06:16.605
Let me draw it with the
lines actually.

00:06:16.605 --> 00:06:19.460
Let me draw it like if you have
one line like this, and

00:06:19.460 --> 00:06:21.580
then this line like this.

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If this right here is x,
x is greater than 90

00:06:29.450 --> 00:06:32.960
degrees, and this is y.

00:06:32.960 --> 00:06:35.820
We know that these two angles
are going to be supplementary

00:06:35.820 --> 00:06:37.970
to each other, that they
add up to 180, right?

00:06:37.970 --> 00:06:41.480
X plus y is equal
to 180 degrees.

00:06:41.480 --> 00:06:49.760
So if x is greater than 90, then
we know that y has to be

00:06:49.760 --> 00:06:53.150
less than 90, right?

00:06:53.150 --> 00:06:55.550
So that's what we
know is true.

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We could look at the
other choices.

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y is definitely not going
to be greater than 90.

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y is definitely not going
to be equal to 90.

00:07:00.920 --> 00:07:04.210
If y is equal to 90 and x is
greater than 90, then these

00:07:04.210 --> 00:07:06.210
two angles are going to be
greater than 180 degrees,

00:07:06.210 --> 00:07:09.740
which we know it can't be.

00:07:09.740 --> 00:07:11.780
n is perpendicular to m.

00:07:11.780 --> 00:07:14.540
We know that's not true because
if this is 90 degrees,

00:07:14.540 --> 00:07:16.670
then this is also 90 degrees.

00:07:16.670 --> 00:07:18.720
But this angle is a
corresponding angle, but it's

00:07:18.720 --> 00:07:19.530
not the same.

00:07:19.530 --> 00:07:21.700
So they're not parallel lines.

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This line is actually, I should
drawn it like this.

00:07:25.310 --> 00:07:27.200
Actually I drew a straight line
when I didn't want to.

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It's going to be skewed
down like this, right?

00:07:29.525 --> 00:07:32.820
Because x degrees
is more than 90.

00:07:32.820 --> 00:07:35.480
Then l is parallel to m, and
we also know that's not the

00:07:35.480 --> 00:07:37.320
case for the same reason.

00:07:37.320 --> 00:07:40.280
So it's choice a.

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All right.

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Let's see if I can fit in this
last problem within the

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10-minute window.

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Problem number 12.

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In the the xy plane, the line
with the equation y is equal

00:07:56.670 --> 00:08:04.680
to 5x minus 10 crosses the
x-axis at the point with the

00:08:04.680 --> 00:08:07.610
coordinates a, b.

00:08:07.610 --> 00:08:09.110
So this is where it crosses
the x-axis.

00:08:11.950 --> 00:08:15.820
So that's where it crosses
the x-axis.

00:08:15.820 --> 00:08:19.500
What is the value of a?

00:08:19.500 --> 00:08:23.330
OK, so when something crosses
the x-axis, what's

00:08:23.330 --> 00:08:24.340
the value of b?

00:08:24.340 --> 00:08:26.630
What's the y-value?

00:08:26.630 --> 00:08:30.470
Well, the y-value, we know,
is going to be 0, right?

00:08:30.470 --> 00:08:32.100
I mean, I could draw that.

00:08:32.100 --> 00:08:38.400
If I draw coordinates, and
if this is a line,

00:08:38.400 --> 00:08:39.049
let me draw the line.

00:08:39.049 --> 00:08:42.880
We'll say the line looks
something like this.

00:08:42.880 --> 00:08:46.330
Where something crosses the
x-axis, that's going to be

00:08:46.330 --> 00:08:48.910
right here, when y equals 0.

00:08:48.910 --> 00:08:50.600
So all we have to do is
say that y equals 0

00:08:50.600 --> 00:08:51.830
and solve for x.

00:08:51.830 --> 00:08:56.600
So y equals 0 is equal
to 5x minus 10.

00:08:56.600 --> 00:09:01.750
5x is equal to 10,
x is equal to 2.

00:09:01.750 --> 00:09:03.210
So this is the point
right here.

00:09:03.210 --> 00:09:07.290
This is the point 2 comma 0.

00:09:07.290 --> 00:09:15.190
So a is going to be equal to
choice d, which is 2, which is

00:09:15.190 --> 00:09:16.610
the answer.

00:09:16.610 --> 00:09:18.910
I think that's all the time
I have for in this video,

00:09:18.910 --> 00:09:19.930
because the next one
looks like I'll

00:09:19.930 --> 00:09:22.200
have to draw a table.

00:09:22.200 --> 00:09:24.510
So I will see you soon.


SAT Prep: Test 2 Section 3 part 1

WEBVTT
Kind: captions
Language: en

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It is late and I'm sleepy but I
feel strangely motivated to

00:00:04.810 --> 00:00:09.160
do SAT problems, so I will do
SAT problems. So we'll start a

00:00:09.160 --> 00:00:09.860
new test.

00:00:09.860 --> 00:00:12.760
This is going to be test 2, and
we're going to start on

00:00:12.760 --> 00:00:17.750
section 3, and that's
on page 459 of the--

00:00:17.750 --> 00:00:19.010
what's that book called?

00:00:19.010 --> 00:00:22.250
The Official SAT Study Guide
by The College Board.

00:00:22.250 --> 00:00:23.730
So let's get started.

00:00:23.730 --> 00:00:27.880
So the first problem,
problem number 1.

00:00:27.880 --> 00:00:35.740
It said, if 3b plus 1 is less
than 10, which of the

00:00:35.740 --> 00:00:38.640
following cannot to
be the value of b?

00:00:38.640 --> 00:00:40.410
So which of the following cannot
be the value of b?

00:00:40.410 --> 00:00:48.420
And they give a couple of
choices: a, minus 1; b, 0; c,

00:00:48.420 --> 00:00:53.190
1; d, 2; e, 3.

00:00:53.190 --> 00:00:57.310
And you could start with choice
a and just keep going

00:00:57.310 --> 00:01:00.120
through all of them and inspect
it really fast. The

00:01:00.120 --> 00:01:02.020
way I would do it is I would
automatically convert this

00:01:02.020 --> 00:01:04.180
equation, subtract one
from both sides.

00:01:04.180 --> 00:01:07.900
You say, well, this is the same
thing as 3 times b has to

00:01:07.900 --> 00:01:09.010
be less than 9.

00:01:09.010 --> 00:01:12.050
I just subtracted 1 from both
sides of this, right?

00:01:12.050 --> 00:01:14.950
And if you look at these, well
3 times negative 1, that's

00:01:14.950 --> 00:01:16.330
negative 3, that's
less than 9.

00:01:16.330 --> 00:01:19.110
3 times 0, that's 0,
that's less than 9.

00:01:19.110 --> 00:01:19.670
Let me write that.

00:01:19.670 --> 00:01:21.850
3 times negative 1
is negative 3.

00:01:21.850 --> 00:01:23.960
3 times 0 is 0.

00:01:23.960 --> 00:01:26.480
3 times 1 is 3, 3
times 2 is 6.

00:01:26.480 --> 00:01:28.140
These are all less
than 9, right?

00:01:28.140 --> 00:01:30.370
I just substituted each
of these for b.

00:01:30.370 --> 00:01:33.720
But 3 times 3, that is actually
equal to 9, right?

00:01:33.720 --> 00:01:38.760
3 times 3 is equal to 9, which
is not less than 9.

00:01:38.760 --> 00:01:41.530
So our choice is e.

00:01:41.530 --> 00:01:46.170
Number 2, I think I've space
to do it on this without

00:01:46.170 --> 00:01:48.510
having to clear the
chalkboard.

00:01:48.510 --> 00:01:54.220
It says, 2 to the 4x
is equal to 16.

00:01:54.220 --> 00:01:56.460
Whenever you see one of these
problems where the variable is

00:01:56.460 --> 00:02:00.700
in the exponent and you have
different bases here, the

00:02:00.700 --> 00:02:03.810
trick is really just to convert
them to the same base.

00:02:03.810 --> 00:02:08.479
So can I write 16
as a power of 2?

00:02:08.479 --> 00:02:09.509
Well sure.

00:02:09.509 --> 00:02:10.870
Two to the what power is 16?

00:02:10.870 --> 00:02:14.780
Well, 2 to the squared is 4, 2
to the 3rd power is 8, to the

00:02:14.780 --> 00:02:16.830
4th power is 16, right?

00:02:16.830 --> 00:02:21.840
So we could write this same
equation as 2 to the 4x is

00:02:21.840 --> 00:02:25.790
equal to, another way of writing
16 is 2 to the 4th

00:02:25.790 --> 00:02:27.380
power, right?

00:02:27.380 --> 00:02:30.110
And now, that this should
seem a little bit more

00:02:30.110 --> 00:02:31.370
straightforward.

00:02:31.370 --> 00:02:35.130
2 to the 4x is the same thing
as 2 to the 4th, so 4x must

00:02:35.130 --> 00:02:37.390
equal 4, right?

00:02:37.390 --> 00:02:44.470
4x must equal 4, so
x is equal to 1.

00:02:44.470 --> 00:02:48.410
And that choice exists, so I
feel confident in my answer.

00:02:48.410 --> 00:02:51.922
Let's move on.

00:02:51.922 --> 00:02:52.630
I think soon.

00:02:52.630 --> 00:02:55.340
I might be coming down
with the flu.

00:02:55.340 --> 00:02:58.290
Let's hope not, because that
would limit my ability to turn

00:02:58.290 --> 00:03:02.060
out SAT problems on YouTube.

00:03:02.060 --> 00:03:03.310
Number 3.

00:03:05.340 --> 00:03:08.920
How much greater is r minus
2 than r plus 5?

00:03:16.280 --> 00:03:19.610
Well, if I wanted to know how
much one number is greater

00:03:19.610 --> 00:03:25.310
than another, if I ask you how
much greater is 10 than 5,

00:03:25.310 --> 00:03:26.230
what do I do?

00:03:26.230 --> 00:03:28.890
Well, I take 10 minus 5
and that's my answer.

00:03:28.890 --> 00:03:32.450
So example: 10 minus
5 is 5, so 10 is 5

00:03:32.450 --> 00:03:34.320
greater than 5, right?

00:03:34.320 --> 00:03:39.160
So if I want to know how much
greater than r minus 2 is r

00:03:39.160 --> 00:03:41.550
plus 5, right-- and r plus 5
is going to be the greater

00:03:41.550 --> 00:03:43.370
one-- you just subtract them.

00:03:43.370 --> 00:03:50.270
So we say r plus 5
minus r minus 2.

00:03:50.270 --> 00:03:53.930
So r plus 5 minus r-- I'm just
distributing the negative

00:03:53.930 --> 00:03:55.505
number-- plus 2.

00:03:55.505 --> 00:03:59.040
The r's cancel out, and
you get the answer.

00:03:59.040 --> 00:04:00.020
It's 7.

00:04:00.020 --> 00:04:01.570
And you really didn't even
have to do it this way,

00:04:01.570 --> 00:04:04.260
because this r could be any
number, and the answer should

00:04:04.260 --> 00:04:05.550
be the same no matter
what r is.

00:04:05.550 --> 00:04:08.730
And obviously it worked out
because the r's canceled out.

00:04:08.730 --> 00:04:12.410
The easiest thing, let me just
assume that r is 0 because we

00:04:12.410 --> 00:04:15.060
can, because it shouldn't
matter what r is.

00:04:15.060 --> 00:04:27.890
If r was 0, then the question
is how much greater than

00:04:27.890 --> 00:04:30.970
negative 2 is plus 5, right?

00:04:30.970 --> 00:04:33.440
I just assumed that r was 0.

00:04:33.440 --> 00:04:34.540
And then you would do
the same thing.

00:04:34.540 --> 00:04:38.950
You would do 5 minus negative
2, which is equal to 7.

00:04:38.950 --> 00:04:42.060
And you could draw them on the
number line if you wanted to

00:04:42.060 --> 00:04:46.270
visually approach
this problem.

00:04:46.270 --> 00:04:48.480
All right, this next problem
looks visual.

00:04:51.910 --> 00:04:54.810
Problem number 4.

00:04:54.810 --> 00:04:58.300
OK, so it has a square.

00:04:58.300 --> 00:05:03.770
I don't need to draw it fancy,
let me just draw it freehand.

00:05:03.770 --> 00:05:05.900
So, it's a box, it
looks like this.

00:05:14.970 --> 00:05:17.200
They say that the box
is open at the top.

00:05:20.330 --> 00:05:21.745
So that's the bottom
right there.

00:05:25.380 --> 00:05:26.630
OK, and they give
measurements.

00:05:26.630 --> 00:05:31.690
This end is 4, this end is 3,
and then the height is 2.

00:05:31.690 --> 00:05:34.170
And they were to say if we were
to essentially cut the

00:05:34.170 --> 00:05:37.620
sides of this box and flatten
it all out, which of these

00:05:37.620 --> 00:05:40.410
choices are that this
box-- this is a pure

00:05:40.410 --> 00:05:41.370
visualization problem.

00:05:41.370 --> 00:05:43.530
But to flatten it out,
as I said, you would

00:05:43.530 --> 00:05:46.610
cut you this line.

00:05:46.610 --> 00:05:48.070
You'd cut this line.

00:05:48.070 --> 00:05:49.380
I'm just imagining
what I would do

00:05:49.380 --> 00:05:50.230
if I had some scissors.

00:05:50.230 --> 00:05:52.260
And there's no top to
the box, right?

00:05:52.260 --> 00:05:55.360
And then I would take this side
and I would flatten it

00:05:55.360 --> 00:05:55.980
out that way.

00:05:55.980 --> 00:05:57.735
This side, I would flatten
it out that way.

00:05:57.735 --> 00:06:00.880
That back side, I would flatten
it out there, and that

00:06:00.880 --> 00:06:03.740
side, I would flatten
it out there.

00:06:03.740 --> 00:06:06.060
So you'd have the base.

00:06:06.060 --> 00:06:14.435
The base would look like this
where you have 3 would be the

00:06:14.435 --> 00:06:16.120
width, right?

00:06:16.120 --> 00:06:22.820
This side corresponds
to this side, right?

00:06:22.820 --> 00:06:27.220
This side would correspond
to this side.

00:06:27.220 --> 00:06:29.970
And that would be of length 4.

00:06:29.970 --> 00:06:34.300
And then this height 2--
let me draw that.

00:06:34.300 --> 00:06:40.610
So this flap, which I
draw in brown, would

00:06:40.610 --> 00:06:41.860
come out like this.

00:06:46.420 --> 00:06:48.890
And this distance right
here would be 2.

00:06:48.890 --> 00:06:50.810
And then you would have
another-- let's say, the flap

00:06:50.810 --> 00:06:51.440
in the back.

00:06:51.440 --> 00:06:55.650
Let me draw it in another
color, in a dark purple.

00:06:55.650 --> 00:07:02.000
This flap in the back would
come out like this, flat.

00:07:02.000 --> 00:07:04.410
It would also have 2 length.

00:07:04.410 --> 00:07:10.220
And then the other 2 flaps,
like this one, would

00:07:10.220 --> 00:07:11.470
come out like this.

00:07:14.004 --> 00:07:16.130
And of course this has 2 here.

00:07:16.130 --> 00:07:18.630
And then finally I have
that flap in the back.

00:07:18.630 --> 00:07:21.750
I need to pick a color.

00:07:21.750 --> 00:07:26.330
This flap in the back would
come out like that, right?

00:07:26.330 --> 00:07:27.900
And I'll shade it in
like I shaded in

00:07:27.900 --> 00:07:29.150
the original picture.

00:07:29.150 --> 00:07:30.090
And that would also have 2.

00:07:30.090 --> 00:07:32.370
So this is a pure visualization
problem, there's

00:07:32.370 --> 00:07:34.340
not a lot more that I can
do to explain this.

00:07:34.340 --> 00:07:36.550
And there's no top
to this box.

00:07:36.550 --> 00:07:40.220
And we had a bottom.

00:07:40.220 --> 00:07:42.030
So this is the choice.

00:07:42.030 --> 00:07:46.770
See, if we rotated it around a
little bit, it's choice e.

00:07:46.770 --> 00:07:48.350
Actually rotated it around,
because choice

00:07:48.350 --> 00:07:49.580
e looks like this.

00:07:49.580 --> 00:07:59.780
Choice e looks like this.

00:07:59.780 --> 00:08:01.520
I'm not drawing it that
well, but it looks

00:08:01.520 --> 00:08:03.280
something like this.

00:08:03.280 --> 00:08:06.120
Where they say that this length
is 4, this length is 2,

00:08:06.120 --> 00:08:07.320
and this length is 3.

00:08:07.320 --> 00:08:11.500
So they took this and they
essentially rotated it so that

00:08:11.500 --> 00:08:16.390
this side becomes this side.

00:08:16.390 --> 00:08:19.630
And then this side becomes
that side.

00:08:19.630 --> 00:08:23.990
So they just rotated
it clockwise a

00:08:23.990 --> 00:08:25.550
little over 90 degrees.

00:08:25.550 --> 00:08:26.670
And that's how they
got that answer.

00:08:26.670 --> 00:08:29.670
So this is a pure visualization
problem.

00:08:29.670 --> 00:08:34.020
With that, I have about 90
seconds left in this video, so

00:08:34.020 --> 00:08:35.659
I'll wait to do it in
the next video.

00:08:35.659 --> 00:08:37.110
I'll see


SAT Prep: Test 2 Section 3 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:01.850
Welcome back.

00:00:01.850 --> 00:00:07.810
I'm now on page 460 and we'll
start with the number five.

00:00:07.810 --> 00:00:11.140
And they have a little
drawing for us.

00:00:11.140 --> 00:00:14.945
Let's see, if I had a little
box like that,

00:00:14.945 --> 00:00:17.170
I'll do it in yellow.

00:00:17.170 --> 00:00:21.780
So the box looks something
like this.

00:00:21.780 --> 00:00:24.685
And then it has a
bunch of lines.

00:00:24.685 --> 00:00:27.396
It has a line like that.

00:00:27.396 --> 00:00:28.730
It has a line like that.

00:00:32.166 --> 00:00:35.500
It has a line like that.

00:00:35.500 --> 00:00:40.400
It has a line like that.

00:00:40.400 --> 00:00:42.220
And then they have a couple
of points here.

00:00:42.220 --> 00:00:44.970
They have-- let me switch
to the pen.

00:00:44.970 --> 00:00:45.990
That's too dark.

00:00:45.990 --> 00:00:56.620
So you have point
A, B, C, and D.

00:00:56.620 --> 00:00:59.100
And the question is, in the
figure above, a path from

00:00:59.100 --> 00:01:03.670
point A to point D is determined
by moving upwards

00:01:03.670 --> 00:01:05.400
or to the right along the
gridlines-- so you have to

00:01:05.400 --> 00:01:08.470
move along the yellow lines--
how many different paths can

00:01:08.470 --> 00:01:12.200
be drawn from A to D that do
not include either B or C?

00:01:12.200 --> 00:01:16.150
So we can't have either B
or C in our path that

00:01:16.150 --> 00:01:17.760
goes from A to D.

00:01:17.760 --> 00:01:21.500
So the way I think about it is,
which of the-- if we call

00:01:21.500 --> 00:01:24.880
these yellow lines roads, I
guess-- which of these roads

00:01:24.880 --> 00:01:29.840
would be unusable if I can't
go to B or to C?

00:01:29.840 --> 00:01:34.540
Well, let me mark the
unusable roads.

00:01:34.540 --> 00:01:36.940
Well, the unusable roads would
be the roads that their only

00:01:36.940 --> 00:01:42.510
purpose is to get to B or go
away from B, or get to C or go

00:01:42.510 --> 00:01:43.300
away from C.

00:01:43.300 --> 00:01:47.230
So, for example, this road right
here, all it does is go

00:01:47.230 --> 00:01:50.310
towards B or go away from B, so
we can't use that because

00:01:50.310 --> 00:01:51.860
we're not going to
go through B.

00:01:51.860 --> 00:01:56.500
Similarly, we can't use that
road, can't use this road, and

00:01:56.500 --> 00:01:57.570
we can't use this road, right?

00:01:57.570 --> 00:01:59.660
Because those are all the roads
that come to and from B.

00:01:59.660 --> 00:02:00.690
So we're not going
to use those.

00:02:00.690 --> 00:02:04.050
The only reason why you would
use those roads is if you went

00:02:04.050 --> 00:02:05.460
through B, so you
can't use those.

00:02:05.460 --> 00:02:08.410
And similarly for C, same thing,
you can't use that one,

00:02:08.410 --> 00:02:11.440
can't use that one, can't
use that one, and

00:02:11.440 --> 00:02:13.340
can't use that one.

00:02:13.340 --> 00:02:14.400
So what are we left with?

00:02:14.400 --> 00:02:18.460
We might as well just erase
those roads, right?

00:02:18.460 --> 00:02:22.220
Because they're kind of just
messing up the picture, right?

00:02:22.220 --> 00:02:23.660
So let me just erase them.

00:02:30.190 --> 00:02:35.190
So let me erase all the roads
that I can't use.

00:02:35.190 --> 00:02:36.960
I can't use any of these
because I can't go

00:02:36.960 --> 00:02:39.470
through B or C.

00:02:39.470 --> 00:02:41.680
And then it's going to start
looking a lot clearer.

00:02:41.680 --> 00:02:44.150
Then the problem gets
really easy.

00:02:44.150 --> 00:02:47.660
So these are all the roads I
can use to go from A to D.

00:02:47.660 --> 00:02:50.110
So how many paths are there?

00:02:50.110 --> 00:02:57.230
Well, that's one path, one.

00:02:57.230 --> 00:02:59.310
I can do this.

00:02:59.310 --> 00:03:01.840
Two paths.

00:03:01.840 --> 00:03:04.120
Three paths.

00:03:04.120 --> 00:03:05.950
Four paths.

00:03:05.950 --> 00:03:07.500
And that's choice B.

00:03:07.500 --> 00:03:09.800
I hope I'm right.

00:03:09.800 --> 00:03:12.470
At least my choice exists.

00:03:12.470 --> 00:03:23.380
Problem number six, if 3/7 of
n is 42-- so if I write that

00:03:23.380 --> 00:03:27.230
algebraically, that's just
saying 3/7 times

00:03:27.230 --> 00:03:29.700
n is equal to 42.

00:03:29.700 --> 00:03:32.260
And they want to know what
5/7 of n is, so they

00:03:32.260 --> 00:03:33.510
want to know 5/7n.

00:03:35.730 --> 00:03:38.810
Well, to solve for n here, I
just multiply both sides times

00:03:38.810 --> 00:03:39.852
the reciprocal.

00:03:39.852 --> 00:03:41.110
So let me do that.

00:03:41.110 --> 00:03:45.190
So 7 over 3-- I'm just
multiplying both sides of the

00:03:45.190 --> 00:03:47.190
equation by 7 over 3.

00:03:47.190 --> 00:03:48.850
So this cancels out, right?

00:03:48.850 --> 00:03:54.500
So we could say n is equal
to 7 over 3 times 42.

00:03:54.500 --> 00:03:57.440
And now we want to know 5/7
times n, so we just take this

00:03:57.440 --> 00:03:59.830
n, place it here.

00:03:59.830 --> 00:04:03.980
So it'd be 5/7 times
this thing-- right?

00:04:03.980 --> 00:04:08.510
This is n-- times 7
over 3, times 42.

00:04:08.510 --> 00:04:11.140
And notice, I just kept it that
way because the 7's ended

00:04:11.140 --> 00:04:14.300
up cancelling out, so I have
to do less arithmetic.

00:04:14.300 --> 00:04:16.750
So scratch those out.

00:04:16.750 --> 00:04:20.529
And then let's see, five times
42 over 3-- let's see, what's

00:04:20.529 --> 00:04:23.020
42 divided by 3?

00:04:23.020 --> 00:04:23.550
It's what?

00:04:23.550 --> 00:04:24.880
14, right?

00:04:24.880 --> 00:04:27.000
Because 30 and 12.

00:04:27.000 --> 00:04:28.030
Right, it's 14.

00:04:28.030 --> 00:04:34.960
So this goes to 1, this goes to
14, and 5 times 14 is 70.

00:04:34.960 --> 00:04:36.520
And that is choice A.

00:04:36.520 --> 00:04:38.560
So all you do is you can
just solve for n and

00:04:38.560 --> 00:04:40.555
then multiply by 5/7.

00:04:40.555 --> 00:04:42.310
It's not too fancy.

00:04:42.310 --> 00:04:45.580
This is probably one just to see
how fast can you do this

00:04:45.580 --> 00:04:47.722
type of problem.

00:04:47.722 --> 00:04:52.310
OK, clear image, invert.

00:04:52.310 --> 00:04:54.910
All right, I'm on problem
number seven

00:04:54.910 --> 00:04:56.340
and I'm getting sleepy.

00:04:56.340 --> 00:04:58.460
I'm just going to go to
bed after this video.

00:04:58.460 --> 00:04:59.980
All right, the figure
above-- well, let me

00:04:59.980 --> 00:05:02.990
draw the figure above.

00:05:02.990 --> 00:05:13.625
We have that, and then we don't
know what this figure is

00:05:13.625 --> 00:05:14.875
for, just yet.

00:05:17.900 --> 00:05:28.440
And then they say A,
B, C, D, E, and F.

00:05:28.440 --> 00:05:30.440
And now let's read
the problem.

00:05:30.440 --> 00:05:33.980
The figure above shows the top
view of an open square box

00:05:33.980 --> 00:05:37.120
that is is divided into six
compartments with walls of

00:05:37.120 --> 00:05:38.050
equal height.

00:05:38.050 --> 00:05:43.300
Each of the rectangles D, E, and
F have twice the area of

00:05:43.300 --> 00:05:46.250
each of the equal squares
A, B, and C.

00:05:46.250 --> 00:05:49.820
So they're saying D is twice the
area of A, I guess, right?

00:05:49.820 --> 00:05:51.820
The way I drew it, it looks
like a little more, but we

00:05:51.820 --> 00:05:54.670
know this is twice the
area of this, OK.

00:05:54.670 --> 00:05:57.600
When a marble is dropped into
the box at random-- so it's

00:05:57.600 --> 00:06:01.850
kind of being dropped from above
into this box-- it falls

00:06:01.850 --> 00:06:03.910
into one of the compartments.

00:06:03.910 --> 00:06:07.060
What is the probability
that it will fall

00:06:07.060 --> 00:06:09.780
into compartment F?

00:06:09.780 --> 00:06:14.040
So the way you really just have
to think about it, the

00:06:14.040 --> 00:06:16.720
marble has an equal
chance of falling

00:06:16.720 --> 00:06:18.630
anywhere on this square.

00:06:18.630 --> 00:06:21.200
And so, if you want to know the
probability that it falls

00:06:21.200 --> 00:06:24.300
into F, you just have to figure
out the fraction of the

00:06:24.300 --> 00:06:30.470
area that F is of the
entire square.

00:06:30.470 --> 00:06:35.634
So if we said that A has an area
of x, then B also has an

00:06:35.634 --> 00:06:39.240
area of x, and C also has an
area of x, and they told us

00:06:39.240 --> 00:06:42.110
that these are twice these, so
this would be 2x, this would

00:06:42.110 --> 00:06:45.300
be 2x, and this would
be 2x, so the total

00:06:45.300 --> 00:06:46.780
area would be what?

00:06:46.780 --> 00:06:48.130
It would be x plus x, plus x.

00:06:48.130 --> 00:06:52.410
So 3x plus 2x, plus
2x, plus 2x.

00:06:52.410 --> 00:06:57.880
So these three are 6x, so the
total area is 9x, right?

00:06:57.880 --> 00:07:00.590
9x is the total area
of the box.

00:07:00.590 --> 00:07:03.370
I mean, we don't know how big x
is, but 9x is its total area

00:07:03.370 --> 00:07:03.670
of the box.

00:07:03.670 --> 00:07:05.740
And F is 2x, right?

00:07:05.740 --> 00:07:08.200
F is 2x.

00:07:08.200 --> 00:07:17.170
So the area of F over total
area-- I don't like this

00:07:17.170 --> 00:07:19.690
toothpaste blue color,
but we'll deal.

00:07:19.690 --> 00:07:21.850
So the area of F, I wrote
here before I

00:07:21.850 --> 00:07:23.900
scratched it out, is 2x.

00:07:23.900 --> 00:07:27.630
Over the total area is 9x,
and the x's cancel out.

00:07:27.630 --> 00:07:33.660
So F is 2/9 of this total box's
area, so there's a 2/9

00:07:33.660 --> 00:07:37.710
probability that a marble
dropped randomly on top of

00:07:37.710 --> 00:07:39.990
this box, with kind of a uniform
distribution anywhere

00:07:39.990 --> 00:07:42.480
in the box, there's a 2/9
chance that it will

00:07:42.480 --> 00:07:47.640
fall into cubby E.

00:07:47.640 --> 00:07:50.375
Sorry, cubby F, that's what they
asked us in the question.

00:07:50.375 --> 00:07:52.750
And actually, it would also be
2/9 that it would fall into

00:07:52.750 --> 00:07:55.390
cubby hole E, because
it's the same area.

00:07:55.390 --> 00:07:57.130
Problem number eight.

00:07:57.130 --> 00:08:04.380
So if a and b are odd integers,
which of the

00:08:04.380 --> 00:08:06.650
following must also
be an odd integer?

00:08:06.650 --> 00:08:08.590
Oh, these are fun.

00:08:08.590 --> 00:08:18.270
All right, so choice one they
give is a plus 1 times b.

00:08:18.270 --> 00:08:20.740
Well, can this be odd?

00:08:20.740 --> 00:08:22.250
Let's just try it with
some numbers.

00:08:22.250 --> 00:08:26.870
Let's say a and b are
3 and 5, right?

00:08:26.870 --> 00:08:28.380
These are two odd integers.

00:08:28.380 --> 00:08:35.120
If a 3, then a plus 1 is 4,
times 5, this equals 20.

00:08:35.120 --> 00:08:38.000
So we immediately, we ended
up getting an even answer.

00:08:38.000 --> 00:08:40.640
And that makes sense, because if
you take an odd number, add

00:08:40.640 --> 00:08:41.990
1, you get an even number.

00:08:41.990 --> 00:08:44.490
And anything times an even is
even, so it's not going to be

00:08:44.490 --> 00:08:45.680
choice one.

00:08:45.680 --> 00:08:48.110
Choice one is going to give us
an even number, actually, no

00:08:48.110 --> 00:08:49.300
matter what.

00:08:49.300 --> 00:08:50.550
Choice two.

00:08:53.860 --> 00:08:58.260
a plus 1 plus b.

00:08:58.260 --> 00:09:02.480
So we figured out that a plus
1, this is going to be even,

00:09:02.480 --> 00:09:05.750
and we know that b is odd.

00:09:05.750 --> 00:09:08.450
And you could try this out, but
if you take an even number

00:09:08.450 --> 00:09:11.350
and you were to add an odd
number to it, you're going to

00:09:11.350 --> 00:09:13.880
get an odd number.

00:09:13.880 --> 00:09:16.890
And that makes sense, I mean,
you could try it out with a

00:09:16.890 --> 00:09:18.300
bunch of different numbers,
but that's

00:09:18.300 --> 00:09:21.410
always going to happen.

00:09:21.410 --> 00:09:23.250
Because if you add an even
number to an even number you

00:09:23.250 --> 00:09:24.140
get another even number.

00:09:24.140 --> 00:09:27.950
And then if you add one more,
an odd number-- I think you

00:09:27.950 --> 00:09:29.730
get the point.

00:09:29.730 --> 00:09:33.160
And you could try it out with
our 3 and 5 examples.

00:09:33.160 --> 00:09:35.210
That wouldn't prove it for sure,
but it does work for

00:09:35.210 --> 00:09:36.760
that example.

00:09:36.760 --> 00:09:41.880
And then similarly,
a plus 1 minus b.

00:09:41.880 --> 00:09:42.850
It's the same notion.

00:09:42.850 --> 00:09:44.980
I mean, you could try it
out with some numbers.

00:09:44.980 --> 00:09:48.340
But if you have an even number
here-- because an odd plus 1

00:09:48.340 --> 00:09:54.420
is even-- and then if you were
to subtract another odd

00:09:54.420 --> 00:09:58.490
number, you'd also get
an odd number.

00:09:58.490 --> 00:10:00.770
So the choice is E.

00:10:00.770 --> 00:10:02.310
See you soon.


Functions (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.950
So one of the the YouTube
viewers asked me to

00:00:03.950 --> 00:00:04.610
do this problem.

00:00:04.610 --> 00:00:05.970
And it looked like a
pretty good problem.

00:00:05.970 --> 00:00:08.350
So I thought I would record
a quick video on it.

00:00:08.350 --> 00:00:15.136
So the problem's this: f of g
of x -- hope I get this right

00:00:15.136 --> 00:00:23.790
-- is equal to 2 times the
square root of x squared plus

00:00:23.790 --> 00:00:32.650
1 minus 1, all of that
over the square root of x

00:00:32.650 --> 00:00:36.760
squared plus 1 plus 1.

00:00:36.760 --> 00:00:47.970
And f of x is equal to 2x
minus 1 over x plus 1.

00:00:47.970 --> 00:00:50.890
And the question is, if we know
that f of g of x is equal to

00:00:50.890 --> 00:00:55.560
this fairly complex-looking
expression, and that f of x is

00:00:55.560 --> 00:00:59.980
equal to this, than
what is g of x?

00:00:59.980 --> 00:01:02.530
Well, we could actually
do this problem by kind

00:01:02.530 --> 00:01:04.850
of just looking at it.

00:01:04.850 --> 00:01:09.840
Because f of g of x, what we do
is, we took -- everywhere we

00:01:09.840 --> 00:01:13.220
see an x, we replace
it with g of x.

00:01:13.220 --> 00:01:15.750
So, f of g of x would
look like this.

00:01:15.750 --> 00:01:17.230
And I'll do it in
another color.

00:01:17.230 --> 00:01:20.400
So, let's see. f of g of x.

00:01:20.400 --> 00:01:22.690
And let's say we don't
know what g of x is.

00:01:22.690 --> 00:01:23.360
Would be like this.

00:01:23.360 --> 00:01:28.500
Everywhere where we saw an x,
we replace it with g of x.

00:01:28.500 --> 00:01:33.610
Because -- so it would
be 2 times g of x.

00:01:33.610 --> 00:01:41.250
Minus 1 over g of x plus 1.

00:01:41.250 --> 00:01:44.250
And we also know that f of g
of x is equal to this thing.

00:01:44.250 --> 00:01:44.870
Right?

00:01:44.870 --> 00:01:47.450
Here we just took the g of x
and put it in of f of x, and

00:01:47.450 --> 00:01:49.320
we wrote the g of x
in the expression.

00:01:49.320 --> 00:01:53.120
But we know that that is equal
to this expression up here.

00:01:53.120 --> 00:01:56.910
So that is also equal to 2
times the square root of x

00:01:56.910 --> 00:02:01.870
squared plus 1, minus
1, all of that over x

00:02:01.870 --> 00:02:06.900
squared plus 1 plus 1.

00:02:06.900 --> 00:02:09.340
And I think, now, at this
point, you can kind

00:02:09.340 --> 00:02:09.965
of pattern match.

00:02:09.965 --> 00:02:13.430
And you can see what g of x is.

00:02:13.430 --> 00:02:15.600
2 times something
minus 1, right?

00:02:15.600 --> 00:02:18.410
2 times something minus 1.

00:02:18.410 --> 00:02:21.670
2 times something minus 1 in
the numerator, and that's

00:02:21.670 --> 00:02:23.840
something plus 1 in
the denominator.

00:02:23.840 --> 00:02:26.460
And that's something plus
1 in the denominator.

00:02:26.460 --> 00:02:31.690
So in either case, we have g
of x is equal to the square

00:02:31.690 --> 00:02:34.130
root of x squared plus 1.

00:02:34.130 --> 00:02:37.240
And, actually, in the example
he sent me, there was

00:02:37.240 --> 00:02:38.110
a bunch of choices.

00:02:38.110 --> 00:02:40.610
And if you were given the
choices, then all you have to

00:02:40.610 --> 00:02:44.140
really do is take each of the
choices for g of x and replace

00:02:44.140 --> 00:02:49.000
them in for x in
this expression.

00:02:49.000 --> 00:02:52.200
And then see which of those
choices for g of x ends

00:02:52.200 --> 00:02:55.710
up with this expression.

00:02:55.710 --> 00:02:56.450
Actually, we can do it.

00:02:56.450 --> 00:02:59.160
I mean, he gave the choice
-- well, let me see.

00:02:59.160 --> 00:03:03.620
Let me delete some of this.

00:03:11.230 --> 00:03:17.390
I want to do it, let's
see if I can do it in

00:03:17.390 --> 00:03:18.270
-- oh, there we go.

00:03:18.270 --> 00:03:20.060
Better.

00:03:20.060 --> 00:03:21.660
All right.

00:03:21.660 --> 00:03:24.930
So, I wanted you to see that
you could do the problem

00:03:24.930 --> 00:03:26.590
even if you weren't
given any choices.

00:03:26.590 --> 00:03:29.110
But if you are given choices,
the problem actually even

00:03:29.110 --> 00:03:32.120
becomes a little bit
more straightforward.

00:03:32.120 --> 00:03:36.390
So the choices were --
actually, I erased some of it.

00:03:36.390 --> 00:03:41.730
It was given that f of x is
2x minus 1 over x plus 1.

00:03:41.730 --> 00:03:46.000
And then they said, which of
the following is g of x.

00:03:46.000 --> 00:03:49.640
So, g of x is equal to.

00:03:49.640 --> 00:03:51.130
And they give these choices.

00:03:51.130 --> 00:03:59.850
a), square root of x, b),
square root of x squared plus

00:03:59.850 --> 00:04:03.830
1, which we just figured out
was actually the answer.

00:04:03.830 --> 00:04:10.740
c), x, d), x squared.

00:04:10.740 --> 00:04:15.360
And then e), x squared plus 1.

00:04:15.360 --> 00:04:17.710
And the way you would do this
problem if, you, just by

00:04:17.710 --> 00:04:21.670
looking at it you didn't -- the
method we just saw, saw that if

00:04:21.670 --> 00:04:25.540
you just replaced x with square
root of x squared plus 1

00:04:25.540 --> 00:04:27.500
everywhere, then you'd
get f of g of x.

00:04:27.500 --> 00:04:30.440
The other option is, you just
take each of these and say well

00:04:30.440 --> 00:04:34.430
what is f of -- if you replaced
x with this term everywhere,

00:04:34.430 --> 00:04:34.990
what do you get?

00:04:34.990 --> 00:04:38.230
Well, then you'd get 2 times
square root of x minus 1 over

00:04:38.230 --> 00:04:40.050
the square root of x plus 1.

00:04:40.050 --> 00:04:43.090
And that's not what
we have up here.

00:04:43.090 --> 00:04:46.670
Then if you took this and you
replaced for x everywhere -- so

00:04:46.670 --> 00:04:49.690
if you took this expression,
you replaced it for x

00:04:49.690 --> 00:04:53.900
everywhere, then you would
get this expression.

00:04:53.900 --> 00:04:55.630
So you would know that
this would be the answer.

00:04:55.630 --> 00:04:59.680
If you just replaced x with x,
you would just get -- you would

00:04:59.680 --> 00:05:01.820
just get this over again,
which does not equal this.

00:05:01.820 --> 00:05:03.260
So that doesn't work.

00:05:03.260 --> 00:05:09.810
If g of x was this, then f of g
of x would be -- let's see,

00:05:09.810 --> 00:05:11.660
everywhere we see an x, you'd
put an x squared, so it

00:05:11.660 --> 00:05:16.320
would be 2x squared minus 1
over x squared plus 1.

00:05:16.320 --> 00:05:17.980
Which does not equal this.

00:05:17.980 --> 00:05:22.530
And similarly, if g of x was
this term right here, then f of

00:05:22.530 --> 00:05:32.600
g of x would be would be 2
times x squared plus 1 minus 1

00:05:32.600 --> 00:05:37.070
over x squared plus 1 plus 1.

00:05:37.070 --> 00:05:39.150
Which you can simplify
a bit, but that still

00:05:39.150 --> 00:05:40.600
doesn't equal this.

00:05:40.600 --> 00:05:42.890
And so this, once again, if
you're given the choices,

00:05:42.890 --> 00:05:43.780
you just try this out.

00:05:43.780 --> 00:05:46.200
And if you replace this
expression everywhere,

00:05:46.200 --> 00:05:47.920
where you see an x, right?

00:05:47.920 --> 00:05:51.840
Everywhere where you see an
x, and you replace it this

00:05:51.840 --> 00:05:55.200
expression, you get these and
that's what the question,

00:05:55.200 --> 00:05:57.000
essentially, was asking us.

00:05:57.000 --> 00:06:00.000
So, hopefully I didn't
confuse everyone too much.

00:06:00.000 --> 00:06:03.730
And hopefully this was
helpful for the viewer

00:06:03.730 --> 00:06:05.080
who asked me to do this.

00:06:05.080 --> 00:06:05.940
I'll see you all later.

00:06:05.940 --> 00:06:07.240
Bye.


SAT Prep: Test 1 Section 9  Part 5

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.960
OK, welcome back.

00:00:01.960 --> 00:00:05.280
We're now on problem number
15, and I tried my best to

00:00:05.280 --> 00:00:07.090
draw this graph ahead of time.

00:00:07.090 --> 00:00:09.890
This is probably one of the more
difficult for me with my

00:00:09.890 --> 00:00:11.630
shaky hand.

00:00:11.630 --> 00:00:12.510
Let's see what this says.

00:00:12.510 --> 00:00:16.180
So this is point O for the
origin right here.

00:00:16.180 --> 00:00:18.590
This is point P.

00:00:18.590 --> 00:00:21.561
This is point Q.

00:00:21.561 --> 00:00:27.160
Then of course this is the
y-axis, this is the x-axis.

00:00:27.160 --> 00:00:31.910
And they say that this graph,
right here, is y is equal to a

00:00:31.910 --> 00:00:34.380
minus x squared.

00:00:34.380 --> 00:00:42.150
And this curve right here is
y is equal to x squared.

00:00:42.150 --> 00:00:45.420
So the question says, in the
figure above, the graphs y

00:00:45.420 --> 00:00:48.670
equals x squared and y equals
a minus x squared for some

00:00:48.670 --> 00:00:50.450
constant a-- oh, those
are the graphs.

00:00:50.450 --> 00:00:51.030
Right.

00:00:51.030 --> 00:01:05.420
If the length of PQ is equal to
6, what is the value of a?

00:01:05.420 --> 00:01:06.710
Well, it's interesting.

00:01:06.710 --> 00:01:09.330
So the first thing that should
jump out at you is that these

00:01:09.330 --> 00:01:11.950
are completely symmetric
graphs, right?

00:01:11.950 --> 00:01:15.120
They both are kind
of mirror images

00:01:15.120 --> 00:01:17.300
around the y-axis, right?

00:01:17.300 --> 00:01:22.920
So if this length is 6, then the
length of each of these--

00:01:22.920 --> 00:01:26.340
then here is 3 and here is 3.

00:01:26.340 --> 00:01:27.280
Right?

00:01:27.280 --> 00:01:30.550
So what do we know about where
these graphs intersect?

00:01:30.550 --> 00:01:37.050
Well, we know that they
intersect at the point 3, and

00:01:37.050 --> 00:01:41.050
we know that they intersect as
well at the point minus 3.

00:01:41.050 --> 00:01:42.740
Those are where those two
graphs intersect.

00:01:42.740 --> 00:01:43.765
And that's a lot
of information.

00:01:43.765 --> 00:01:47.200
And the way, once again, we came
to this conclusion was

00:01:47.200 --> 00:01:50.020
that these are both symmetric
around the y-axis.

00:01:50.020 --> 00:01:54.960
So if this intersection
is 6 wide, soon. then

00:01:54.960 --> 00:01:56.110
it's 3 on each side.

00:01:56.110 --> 00:01:56.320
Right?

00:01:56.320 --> 00:01:57.920
3 at the positive side, 3
at the negative side.

00:01:57.920 --> 00:02:01.550
So it intersects at x equals
3 and x equals minus 3.

00:02:01.550 --> 00:02:03.670
And if we want to figure out the
intersection, we can also

00:02:03.670 --> 00:02:06.010
just set these two equal to
each other. y equals x

00:02:06.010 --> 00:02:08.150
squared, y equals a
minus x squared.

00:02:08.150 --> 00:02:17.210
So we know that x squared must
be equal to a minus-- what am

00:02:17.210 --> 00:02:20.330
I doing? a minus x squared.

00:02:20.330 --> 00:02:23.640
Or add x squared to both
sides of this equation.

00:02:23.640 --> 00:02:27.865
You get 2x squared
is equal to a.

00:02:27.865 --> 00:02:29.115
Right?

00:02:31.210 --> 00:02:36.540
And this'll be true at the two
points, at the two x points,

00:02:36.540 --> 00:02:38.530
where these two graphs
intersect.

00:02:38.530 --> 00:02:39.930
Well, what are the x points?

00:02:39.930 --> 00:02:40.760
We figured that out.

00:02:40.760 --> 00:02:43.990
It's 3 and minus 3.

00:02:43.990 --> 00:02:47.110
So if you put 3 in here,
you get 2 times 3

00:02:47.110 --> 00:02:48.960
squared is equal to a.

00:02:48.960 --> 00:02:52.450
It's 2 times 9 is equal to a.

00:02:52.450 --> 00:02:54.170
a is equal to 18.

00:02:54.170 --> 00:02:56.060
And if you put minus 3 in here,
it'd be the same thing,

00:02:56.060 --> 00:02:58.080
because when you square it
the negative goes away.

00:02:58.080 --> 00:02:59.820
So a is 18.

00:02:59.820 --> 00:03:00.880
Not too bad.

00:03:00.880 --> 00:03:07.100
The key conclusion you have to
make is that they intersect at

00:03:07.100 --> 00:03:09.480
the points positive
and negative 3.

00:03:09.480 --> 00:03:11.350
And then you just set the
equations equal to each other

00:03:11.350 --> 00:03:13.150
and solve for a.

00:03:13.150 --> 00:03:16.000
Let's do the next problem.

00:03:16.000 --> 00:03:19.240
Clear image.

00:03:19.240 --> 00:03:23.660
Set X has x members, and
set Y has y members.

00:03:23.660 --> 00:03:27.410
Set Z contains all the members
that are either in X or Y,

00:03:27.410 --> 00:03:30.130
with the exception of the
k common members.

00:03:30.130 --> 00:03:32.160
OK, so this sounds like
a Venn diagram.

00:03:32.160 --> 00:03:34.280
I don't know if you remember a
Venn diagram, but I'll show

00:03:34.280 --> 00:03:35.500
you what one is.

00:03:35.500 --> 00:03:37.290
So let's say that
this is set X.

00:03:40.230 --> 00:03:43.590
And let's say that this
is set Y right here.

00:03:46.900 --> 00:03:50.130
Looks like a MasterCard
logo. you

00:03:50.130 --> 00:03:56.230
So this is set Y,
this is set X.

00:03:56.230 --> 00:03:59.280
And it says, set Z consists of
all the members that are

00:03:59.280 --> 00:04:02.520
either in set X or set Y,
with the exception of

00:04:02.520 --> 00:04:04.180
the k common members.

00:04:04.180 --> 00:04:07.110
So this is the k common
members, right?

00:04:07.110 --> 00:04:12.160
These are the things that
overlap with set X and set Y.

00:04:12.160 --> 00:04:13.990
So what does Z consist of?

00:04:13.990 --> 00:04:19.990
Well, Z is going to be-- let
me pick a dark color.

00:04:19.990 --> 00:04:23.540
Z is going to be-- if I do the
fill tool-- it's going to be

00:04:23.540 --> 00:04:25.850
that plus that.

00:04:25.850 --> 00:04:26.610
Right?

00:04:26.610 --> 00:04:27.520
Without the k, right?

00:04:27.520 --> 00:04:30.910
Because it says Z does not
include the overlap.

00:04:30.910 --> 00:04:33.270
So what are those two things?

00:04:33.270 --> 00:04:37.260
Let me go back to
the paint tool.

00:04:37.260 --> 00:04:37.640
Orange.

00:04:37.640 --> 00:04:40.060
It's going to be tacky with
all the colors I'm using.

00:04:40.060 --> 00:04:44.060
So there's x members
in all of X.

00:04:44.060 --> 00:04:46.760
x members in all of capital X.

00:04:46.760 --> 00:04:48.670
So how many members are
in this blue region?

00:04:48.670 --> 00:04:53.170
So if there's x in the whole
region, there's going to be x

00:04:53.170 --> 00:04:56.190
minus k in this blue
region, right?

00:04:56.190 --> 00:04:59.100
I just subtracted
out this k part.

00:04:59.100 --> 00:05:03.080
Similarly, Y has y members,
as the problem tells us.

00:05:03.080 --> 00:05:06.750
Capital Y set has lowercase
y members.

00:05:06.750 --> 00:05:09.190
But if we take out these k that
overlap, this is going to

00:05:09.190 --> 00:05:11.920
be y minus k.

00:05:11.920 --> 00:05:13.800
I'm just subtracting it out.

00:05:13.800 --> 00:05:17.250
And so Z is this blue area
plus this blue area.

00:05:17.250 --> 00:05:23.440
So it'll be x minus
k plus y minus k.

00:05:23.440 --> 00:05:27.140
And that is equal to x plus y.

00:05:27.140 --> 00:05:31.420
We have two minus k's,
so it's minus 2k.

00:05:31.420 --> 00:05:34.720
And that is choice D.

00:05:34.720 --> 00:05:36.590
Not too bad, huh?

00:05:36.590 --> 00:05:37.150
All right.

00:05:37.150 --> 00:05:39.470
I think we are all done with
this practice test. Hopefully

00:05:39.470 --> 00:05:41.780
you found it vaguely useful.

00:05:41.780 --> 00:05:43.190
I'll see


SAT Prep: Test 1 Section 9 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.740
Good morning.

00:00:01.740 --> 00:00:03.280
We'll now start.

00:00:03.280 --> 00:00:05.860
We're still in the first test,
and we'll now start on Section

00:00:05.860 --> 00:00:08.410
9 on page 423.

00:00:08.410 --> 00:00:12.090
So let's start with
the first problem.

00:00:12.090 --> 00:00:15.600
Problem number 1.

00:00:15.600 --> 00:00:31.470
If 5t equals 45, and tk is
equal to 1, what is k?

00:00:31.470 --> 00:00:32.890
Well, 5t equals 45.

00:00:32.890 --> 00:00:33.990
This is an easy equation.

00:00:33.990 --> 00:00:38.080
You divide both sides by
5, you get t equals 9.

00:00:38.080 --> 00:00:39.820
And then you have tk equals 1.

00:00:39.820 --> 00:00:43.320
We know what t is equal to.
t is equal to 9, right?

00:00:43.320 --> 00:00:46.400
So we get 9k is equal to 1.

00:00:46.400 --> 00:00:51.532
Divide both sides by 9, you
get k is equal to 1/9.

00:00:51.532 --> 00:00:53.630
And that is choice B.

00:00:53.630 --> 00:00:55.370
That's pretty straightforward.

00:00:55.370 --> 00:00:57.520
All right.

00:00:57.520 --> 00:00:59.130
Number 2.

00:00:59.130 --> 00:01:00.380
I'll switch colors.

00:01:05.280 --> 00:01:10.610
It takes 4 complete rotations
of-- it's

00:01:10.610 --> 00:01:11.190
early in the morning.

00:01:11.190 --> 00:01:12.420
I can't even read properly.

00:01:12.420 --> 00:01:15.330
It takes 4 complete turns
of a crank to

00:01:15.330 --> 00:01:17.510
raise a window 2 inches.

00:01:17.510 --> 00:01:24.440
So 4 turns raise the
window 2 inches.

00:01:28.770 --> 00:01:33.030
At this rate, how many turns
does it take to raise the same

00:01:33.030 --> 00:01:36.990
window 3 and 1/2 inches?

00:01:36.990 --> 00:01:37.360
OK.

00:01:37.360 --> 00:01:39.970
So 4 turns, 2 inches.

00:01:39.970 --> 00:01:41.750
Just to simplify things, we
could say, well, that's the

00:01:41.750 --> 00:01:51.740
same thing as 2 turns
per inch.

00:01:51.740 --> 00:01:52.990
Right?

00:01:55.330 --> 00:01:58.520
2 turns per inch, or you could
even say 1 turn per 1/2 inch.

00:01:58.520 --> 00:01:59.870
But let's just say
2 turns per inch.

00:01:59.870 --> 00:02:02.400
I just divided the top by 2.

00:02:02.400 --> 00:02:04.600
If 4 turns to do 2 inches,
it's going to take 2

00:02:04.600 --> 00:02:05.740
turns to do 1 inch.

00:02:05.740 --> 00:02:11.830
And it wants to know, the same
window, 3 and 1/2 inches.

00:02:11.830 --> 00:02:13.170
Actually, when I first
read it, I read it

00:02:13.170 --> 00:02:14.070
as 3 and 1/2 feet.

00:02:14.070 --> 00:02:17.020
So actually, this isn't that.

00:02:17.020 --> 00:02:21.904
So we want to go 3
and 1/2 inches.

00:02:21.904 --> 00:02:22.950
It's early in the morning.

00:02:22.950 --> 00:02:24.200
My brain is still foggy.

00:02:27.440 --> 00:02:30.190
We could actually even
ignore this.

00:02:30.190 --> 00:02:32.850
So we can just set this
up as a ratio.

00:02:32.850 --> 00:02:38.510
4 turns goes 2 inches,
is equal to x

00:02:38.510 --> 00:02:40.440
turns for 3 and 1/2.

00:02:40.440 --> 00:02:44.950
And 3 and 1/2, we could write
that as 3.5, or we could write

00:02:44.950 --> 00:02:46.200
that as 7/2.

00:02:49.770 --> 00:02:51.670
And then we cross multiply.

00:02:51.670 --> 00:02:57.750
So 4 times 7/2 is equal to 2x.

00:02:57.750 --> 00:03:00.470
And then this cancels
with this.

00:03:00.470 --> 00:03:03.343
We have 14 is equal to 2x.

00:03:03.343 --> 00:03:07.440
x is equal to 7 turns.

00:03:07.440 --> 00:03:09.140
And that's choice B.

00:03:09.140 --> 00:03:10.580
Sorry for leading you in
the wrong direction.

00:03:10.580 --> 00:03:14.320
I thought they said 3 and
1/2 feet at first. OK.

00:03:14.320 --> 00:03:16.300
Turn the page.

00:03:16.300 --> 00:03:19.360
Number 3.

00:03:19.360 --> 00:03:21.730
Image here.

00:03:25.860 --> 00:03:27.190
I'll switch colors again.

00:03:29.970 --> 00:03:32.510
Keeps it interesting for
me, at least. OK.

00:03:32.510 --> 00:03:45.415
If x/y is equal to 2/3, what
is the value of 3x/2y?

00:03:51.470 --> 00:03:54.245
So we know this, and how do
we get the value of this?

00:03:54.245 --> 00:03:56.680
Well, we could just manipulate
this relationship

00:03:56.680 --> 00:03:57.750
algebraically.

00:03:57.750 --> 00:04:01.420
So we have 3x/2y on the
left-hand side, and then

00:04:01.420 --> 00:04:03.200
whatever we end up on the
right-hand side is our answer.

00:04:03.200 --> 00:04:04.350
So let's see how you do that.

00:04:04.350 --> 00:04:10.890
So we start off with x/y
is equal to 2/3.

00:04:10.890 --> 00:04:14.720
Well, to get here, let's
multiply both sides by 3.

00:04:14.720 --> 00:04:19.110
So if you do a 3 over here,
we have to also multiply

00:04:19.110 --> 00:04:20.709
this side by 3.

00:04:20.709 --> 00:04:24.910
And then if we multiply both
sides by 1/2-- because this 2

00:04:24.910 --> 00:04:28.090
in the denominator is
essentially 1/2-- then you end

00:04:28.090 --> 00:04:32.760
up with a 2 here,
and a 2 here.

00:04:32.760 --> 00:04:35.040
So now we have 3x and 2y.

00:04:35.040 --> 00:04:38.130
And now we have 2
times 3 is 6.

00:04:38.130 --> 00:04:40.330
3 times 2 is 6.

00:04:40.330 --> 00:04:42.190
That equals 1.

00:04:42.190 --> 00:04:44.150
Choice C.

00:04:44.150 --> 00:04:44.780
All right.

00:04:44.780 --> 00:04:46.250
The next one looks like I
have to do some drawing.

00:04:46.250 --> 00:04:47.960
So let me clear this
out a little bit.

00:04:50.790 --> 00:04:53.040
Number 4.

00:04:53.040 --> 00:04:56.060
In the figure above-- let me
draw the figure above.

00:04:56.060 --> 00:04:58.570
Looks complicated, huh?

00:04:58.570 --> 00:05:05.010
So I have one line that goes
straight like this.

00:05:05.010 --> 00:05:07.620
I have one line that goes
straight up like this.

00:05:11.770 --> 00:05:16.400
I have another line that crosses
that line like that.

00:05:20.740 --> 00:05:22.310
And then I got a bunch
of points.

00:05:22.310 --> 00:05:23.070
Let me draw the points.

00:05:23.070 --> 00:05:25.550
It takes more time to draw these
than to maybe do them,

00:05:25.550 --> 00:05:27.560
although I haven't done
this, so I don't know.

00:05:27.560 --> 00:05:28.470
Maybe it will take time.

00:05:28.470 --> 00:05:30.500
N.

00:05:30.500 --> 00:05:34.304
This is P.

00:05:34.304 --> 00:05:46.650
This is T, and R, S.

00:05:46.650 --> 00:05:50.070
Then I have O.

00:05:50.070 --> 00:05:51.940
Then I've got the angle.

00:05:51.940 --> 00:06:00.230
They're calling this angle y,
calling this y degrees.

00:06:00.230 --> 00:06:03.740
They say this angle here
is 20 degrees.

00:06:03.740 --> 00:06:06.700
And they say this is
a right angle.

00:06:06.700 --> 00:06:10.240
And they say that this angle
right here is x degrees.

00:06:10.240 --> 00:06:10.650
All right.

00:06:10.650 --> 00:06:11.690
So let's read what they're
saying now.

00:06:11.690 --> 00:06:17.350
In the figure above, PS--
so this is PS-- and TR

00:06:17.350 --> 00:06:18.470
intersect at O.

00:06:18.470 --> 00:06:18.910
Right.

00:06:18.910 --> 00:06:20.180
We drew that.

00:06:20.180 --> 00:06:23.295
ON is perpendicular to PS.

00:06:23.295 --> 00:06:25.400
This is my little perpendicular
signal,

00:06:25.400 --> 00:06:26.020
looks like an r.

00:06:26.020 --> 00:06:26.320
OK.

00:06:26.320 --> 00:06:28.070
That's perpendicular.

00:06:28.070 --> 00:06:33.690
What is the value
of y minus x?

00:06:33.690 --> 00:06:34.940
Interesting.

00:06:37.650 --> 00:06:40.660
So let me ask you a question.

00:06:40.660 --> 00:06:43.640
y is the opposite
angle to what?

00:06:43.640 --> 00:06:46.980
If you remember from the
geometry, opposite angles are

00:06:46.980 --> 00:06:47.590
equivalent, right?

00:06:47.590 --> 00:06:51.430
If I have two lines that
intersect like this, then this

00:06:51.430 --> 00:06:54.630
angle is going to be equal
to this angle.

00:06:54.630 --> 00:06:58.850
So we know this is y, so what
angle is opposite y?

00:06:58.850 --> 00:07:01.610
Well, this angle
is opposite y.

00:07:01.610 --> 00:07:04.460
This whole thing.

00:07:04.460 --> 00:07:05.240
So what is this angle?

00:07:05.240 --> 00:07:10.770
This angle is x degrees plus
this 90 degrees, right?

00:07:10.770 --> 00:07:12.260
It's x plus 90.

00:07:12.260 --> 00:07:13.570
That's this whole angle.

00:07:13.570 --> 00:07:15.570
I added this part of
it plus this part.

00:07:15.570 --> 00:07:18.240
That equals this whole angle,
and it's the same thing as y.

00:07:18.240 --> 00:07:21.160
And they draw the stuff to
really confuse you, wrote 20

00:07:21.160 --> 00:07:23.950
degrees here.

00:07:23.950 --> 00:07:25.170
None of that really matters.

00:07:25.170 --> 00:07:30.280
So y is going to be equal to
this whole angle, and that

00:07:30.280 --> 00:07:35.210
whole angle is going to be equal
to this 90 degrees-- is

00:07:35.210 --> 00:07:40.180
equal to this part plus
this part-- plus x.

00:07:40.180 --> 00:07:42.900
Subtract x from both sides,
you get y minus

00:07:42.900 --> 00:07:45.760
x is equal to 90.

00:07:45.760 --> 00:07:46.430
That's it.

00:07:46.430 --> 00:07:47.930
That was number 4.

00:07:47.930 --> 00:07:50.220
That's choice C.

00:07:50.220 --> 00:07:54.880
I do all that and the problem
took too little time.

00:07:54.880 --> 00:07:55.630
OK.

00:07:55.630 --> 00:07:57.020
Image.

00:07:57.020 --> 00:07:58.270
Invert.

00:07:59.920 --> 00:08:01.340
OK, so here they've
got a schedule.

00:08:01.340 --> 00:08:02.800
It's the schedule above,
and hopefully you

00:08:02.800 --> 00:08:03.460
have your book open.

00:08:03.460 --> 00:08:05.260
I'm on page 424.

00:08:05.260 --> 00:08:07.470
In the schedule above, each
session is to be 1 and 1/2

00:08:07.470 --> 00:08:08.380
hours long.

00:08:08.380 --> 00:08:10.790
Each break is to be 1/4 hour
long, and lunch is

00:08:10.790 --> 00:08:12.290
to be 1 hour long.

00:08:12.290 --> 00:08:12.770
All right.

00:08:12.770 --> 00:08:14.760
Session-- oh, boy.

00:08:14.760 --> 00:08:16.290
I should just draw this thing.

00:08:16.290 --> 00:08:24.540
Maybe it'll run into the-- no,
I didn't want to do that.

00:08:28.410 --> 00:08:32.360
This is my chart.

00:08:32.360 --> 00:08:33.610
Three columns.

00:08:36.940 --> 00:08:43.035
I've got this one, and
I've got this one.

00:08:46.180 --> 00:08:48.930
And then I'm going to have one,
two, three, four, five,

00:08:48.930 --> 00:08:50.180
six, seven.

00:09:05.950 --> 00:09:07.200
I'm going to run out of space.

00:09:10.210 --> 00:09:11.460
Trying to make this too neat.

00:09:14.785 --> 00:09:16.891
I don't know if I ran
out of space or not.

00:09:16.891 --> 00:09:20.190
I'll probably have to continue
this in the next video, but

00:09:20.190 --> 00:09:22.050
I'll just keep drawing until
the time runs out.

00:09:22.050 --> 00:09:23.900
In the next video, it'll
be all set up.

00:09:23.900 --> 00:09:33.760
So this is the event,
begin time, end.

00:09:33.760 --> 00:09:35.220
Actually, I'll just continue
this in the next video.

00:09:35.220 --> 00:09:36.490
I don't want to waste your
time drawing this.

00:09:36.490 --> 00:09:38.260
I'll draw it, and then
we'll start the next

00:09:38.260 --> 00:09:39.220
video on this problem.

00:09:39.220 --> 00:09:40.650
See you soon.


SAT Prep: Test 1 Section 9 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:01.550
Welcome back.

00:00:01.550 --> 00:00:03.320
So we're on problem number 9.

00:00:03.320 --> 00:00:07.220
In the figure above,
AD is equal to 1.

00:00:07.220 --> 00:00:12.300
So AD, this distance right
here, this is equal to 1.

00:00:12.300 --> 00:00:14.290
DC is equal to square
root of 3.

00:00:14.290 --> 00:00:20.680
So this distance right here
is square root of 3.

00:00:20.680 --> 00:00:22.640
What is the value of z?

00:00:22.640 --> 00:00:25.130
And they say, no, it is
not drawn to scale.

00:00:25.130 --> 00:00:25.460
Clearly.

00:00:25.460 --> 00:00:28.180
Because this is longer than
this, but it looks the same.

00:00:28.180 --> 00:00:30.660
OK, so they want to know
what the value of z is.

00:00:30.660 --> 00:00:32.820
So what can we figure out?

00:00:32.820 --> 00:00:35.520
Well, the one thing that pops
out at me is these are both

00:00:35.520 --> 00:00:36.320
right triangles.

00:00:36.320 --> 00:00:38.990
So this is going to deal
with right triangles.

00:00:38.990 --> 00:00:40.490
And the first thing we
see is that these two

00:00:40.490 --> 00:00:42.510
angles are the same.

00:00:42.510 --> 00:00:44.330
So these two angles
are the same, and

00:00:44.330 --> 00:00:45.580
this is a right angle.

00:00:48.340 --> 00:00:50.120
We don't even have to know this,
but we immediately know

00:00:50.120 --> 00:00:52.160
this is a 45-45-90
triangle, right?

00:00:52.160 --> 00:00:54.270
Because the angles have
to add up to 180.

00:00:54.270 --> 00:00:55.230
Two of them are the same.

00:00:55.230 --> 00:00:57.580
You could say 2x plus
90 is equal to 180,

00:00:57.580 --> 00:00:59.280
and solve for it.

00:00:59.280 --> 00:01:04.750
But what you also know is that
this is an isosceles triangle,

00:01:04.750 --> 00:01:06.040
where two sides are the same.

00:01:06.040 --> 00:01:08.360
I don't know if you remember
from geometry class, if you

00:01:08.360 --> 00:01:11.070
have a triangle like that, and
this angle is equal to this

00:01:11.070 --> 00:01:14.380
angle, then this side is going
to be equal to that side.

00:01:17.300 --> 00:01:18.820
I forgot what the property's
called.

00:01:18.820 --> 00:01:21.930
I think it's just equal angles,
equal sides, or base

00:01:21.930 --> 00:01:24.740
angles-- well, whatever.

00:01:24.740 --> 00:01:28.700
Hopefully you remember that
term in geometry.

00:01:28.700 --> 00:01:31.890
In this case, these two angles
are the base angles, so if

00:01:31.890 --> 00:01:35.440
this is 1, then this
is also 1.

00:01:35.440 --> 00:01:38.280
This BD is also going to be 1.

00:01:38.280 --> 00:01:42.370
And this is going to be square
root of 2 or something.

00:01:42.370 --> 00:01:45.000
So that now, if we look at
this triangle, something

00:01:45.000 --> 00:01:48.250
interesting should pop out at
you, if you're familiar with

00:01:48.250 --> 00:01:50.370
30-60-90 triangles.

00:01:50.370 --> 00:01:55.470
This side is square root
of 3 times this side.

00:01:55.470 --> 00:01:57.440
What do we know about
30-60-90 triangles?

00:01:57.440 --> 00:02:00.450
And you could even turn back
to page-- if you don't

00:02:00.450 --> 00:02:02.790
remember, they give
it to you, right?

00:02:02.790 --> 00:02:05.590
They even tell you, right
here, that on a 30-60-90

00:02:05.590 --> 00:02:09.570
triangle, on page 423 they draw
it out for you, actually.

00:02:09.570 --> 00:02:13.306
If you have a 30-60-90 triangle
like this, and this

00:02:13.306 --> 00:02:18.640
side is 30, and this side is x,
then this side is going to

00:02:18.640 --> 00:02:20.830
be x times the square
root of 3.

00:02:20.830 --> 00:02:23.580
They literally give that to you
on page 423 when they give

00:02:23.580 --> 00:02:25.990
you their formulas.

00:02:25.990 --> 00:02:28.360
So this side is 1.

00:02:28.360 --> 00:02:30.550
This side is 1 times the
square root of 3.

00:02:30.550 --> 00:02:34.070
So z must be 30 degrees.

00:02:34.070 --> 00:02:35.980
It's a 30-60-90 triangle.

00:02:35.980 --> 00:02:38.710
If they asked you why, this
would have been 60 degrees.

00:02:38.710 --> 00:02:42.570
And then, if you wanted to know
what this side would have

00:02:42.570 --> 00:02:45.780
been, it would have been 2, just
using 30-60-90 triangles.

00:02:45.780 --> 00:02:46.815
All right.

00:02:46.815 --> 00:02:53.310
Let's move to number 10.

00:02:53.310 --> 00:02:54.470
Hopefully that's not
too confusing.

00:02:54.470 --> 00:02:56.790
If it is, just review 30-60-90
triangles, and look at that

00:02:56.790 --> 00:03:01.540
formula on page 423.

00:03:01.540 --> 00:03:02.450
Number 10.

00:03:02.450 --> 00:03:16.750
If 30% of 40% of a positive
number is equal to-- so 0.3--

00:03:16.750 --> 00:03:19.990
30% of 40%.

00:03:19.990 --> 00:03:24.990
So 0.3 times-- but I don't want
to use times, because

00:03:24.990 --> 00:03:26.620
they want me to use
x for the number.

00:03:26.620 --> 00:03:35.800
So 0.3 times 0.4-- so that's
30% of 40% of a positive

00:03:35.800 --> 00:03:43.230
number-- so that's x,
is equal to 20% of

00:03:43.230 --> 00:03:45.910
w% of the same number.

00:03:45.910 --> 00:03:55.550
So that equals 20% of w%.

00:03:55.550 --> 00:04:02.640
So if it's w%, then we could
write that as w/100.

00:04:02.640 --> 00:04:09.650
So if w was 50, that'd be
50/100 or 0.5 times x.

00:04:09.650 --> 00:04:13.600
Well, we can divide both sides
of this equation by x.

00:04:13.600 --> 00:04:14.970
These cancel out.

00:04:14.970 --> 00:04:19.089
And then we just literally
solve for w.

00:04:19.089 --> 00:04:20.450
And actually, I probably didn't
even have to write it

00:04:20.450 --> 00:04:21.200
as a decimal.

00:04:21.200 --> 00:04:25.430
30% of 40% of something--
well, we can just

00:04:25.430 --> 00:04:26.780
multiply the decimals.

00:04:26.780 --> 00:04:28.340
What's 3 times 4?

00:04:28.340 --> 00:04:30.130
It's 12.

00:04:30.130 --> 00:04:31.585
There's two numbers behind
the decimal.

00:04:31.585 --> 00:04:38.630
So this is 12, which
is also 12%.

00:04:38.630 --> 00:04:45.980
Is equal to-- and then we
say 0.2 times what w?

00:04:45.980 --> 00:04:46.790
What number?

00:04:46.790 --> 00:04:48.520
Let's say times w.

00:04:48.520 --> 00:04:49.320
Well, I shouldn't write w.

00:04:49.320 --> 00:04:50.910
I'm going to write this as y.

00:04:50.910 --> 00:04:51.520
And I'll tell you why.

00:04:51.520 --> 00:04:57.330
Because this will give
w% as a decimal.

00:04:57.330 --> 00:05:00.790
So I'm just saying 0.3
times 0.4 is 0.12.

00:05:00.790 --> 00:05:03.740
And that's the same thing
as 0.2 times what?

00:05:03.740 --> 00:05:07.970
Well, you divide both sides by
0.2, you get y is equal to

00:05:07.970 --> 00:05:11.980
0.12 over 0.2.

00:05:11.980 --> 00:05:19.130
And that is equal to 0.6.

00:05:19.130 --> 00:05:22.320
And you could review decimals
if that didn't make a lot of

00:05:22.320 --> 00:05:23.020
sense to you.

00:05:23.020 --> 00:05:24.490
So that equals 0.6.

00:05:24.490 --> 00:05:30.510
So y is 0.6, or w% is 0.6.

00:05:30.510 --> 00:05:33.880
So the same thing
as 0.6 is 60%.

00:05:33.880 --> 00:05:35.130
So the choice is B.

00:05:39.070 --> 00:05:40.390
That might seem a little
complicated.

00:05:40.390 --> 00:05:42.410
And the tricky part is really
just making sure that you're

00:05:42.410 --> 00:05:46.040
not using decimals and then
assuming that's a percent, or

00:05:46.040 --> 00:05:46.660
vice versa.

00:05:46.660 --> 00:05:49.250
The important thing is
we took 30% and 40%.

00:05:49.250 --> 00:05:51.380
When you multiply those two
together, that's the same

00:05:51.380 --> 00:05:54.410
thing as 20% times w%.

00:05:54.410 --> 00:05:56.030
And I like to put it
all in decimals.

00:05:56.030 --> 00:05:57.765
You could have put it all in
fractions as well, actually.

00:05:57.765 --> 00:05:59.520
It actually probably would've
been the easiest. You would

00:05:59.520 --> 00:06:05.770
have said, 30% is 3/10,
times 4/10 is equal

00:06:05.770 --> 00:06:13.130
to 2/10 times w/100.

00:06:13.130 --> 00:06:14.360
I took a 0 out.

00:06:14.360 --> 00:06:18.990
You could have said 30/100,
40/100, like that.

00:06:18.990 --> 00:06:21.370
And then this actually would've
been easy, because

00:06:21.370 --> 00:06:23.670
you could have just taken the
100's off of both sides.

00:06:23.670 --> 00:06:26.050
You said 30 times 40
is 20 times what?

00:06:26.050 --> 00:06:29.010
And then you would have gotten
w is equal to 60.

00:06:29.010 --> 00:06:30.900
Actually, that's how I
should have done it.

00:06:30.900 --> 00:06:34.120
But anyway, hopefully I didn't
confuse you too much.

00:06:36.810 --> 00:06:39.690
Next problem.

00:06:39.690 --> 00:06:42.670
Notice I do these things
in real time.

00:06:42.670 --> 00:06:46.640
The negative is maybe I don't
give you the most optimal

00:06:46.640 --> 00:06:53.520
solution at first, but I think
the positive is that you're

00:06:53.520 --> 00:06:54.960
not seeing just a
clean solution.

00:06:54.960 --> 00:06:57.320
You're seeing how someone, the
thought process that they go

00:06:57.320 --> 00:06:58.510
through when they tackle
these problems.

00:06:58.510 --> 00:07:00.770
And I think that that's
actually a super

00:07:00.770 --> 00:07:01.680
useful thing to see.

00:07:01.680 --> 00:07:04.310
Because when you see these very
polished presentations,

00:07:04.310 --> 00:07:06.580
that person did the problem
ahead of time, and so they

00:07:06.580 --> 00:07:07.340
just give you the answer.

00:07:07.340 --> 00:07:09.780
But they're not showing you kind
of how they stumbled upon

00:07:09.780 --> 00:07:10.350
the answer.

00:07:10.350 --> 00:07:13.620
So I can only hope that
this is useful.

00:07:13.620 --> 00:07:13.880
All right.

00:07:13.880 --> 00:07:14.410
Number 11.

00:07:14.410 --> 00:07:17.750
This looks exciting.

00:07:17.750 --> 00:07:19.025
OK, I'll switch colors.

00:07:21.810 --> 00:07:24.030
Let me draw this thing,
and I have the perfect

00:07:24.030 --> 00:07:26.940
tool to draw it.

00:07:26.940 --> 00:07:28.340
Oh, no, I didn't want
it like that.

00:07:31.940 --> 00:07:33.580
So a big rectangle like that.

00:07:33.580 --> 00:07:37.590
And they took half
of it like that.

00:07:37.590 --> 00:07:38.500
No, no, no.

00:07:38.500 --> 00:07:41.830
Edit, undo.

00:07:41.830 --> 00:07:42.590
I've got to go half.

00:07:42.590 --> 00:07:44.992
My eyes aren't working.

00:07:44.992 --> 00:07:46.242
It's roughly half.

00:07:50.000 --> 00:07:54.360
And then they take half like
this, and then they

00:07:54.360 --> 00:07:57.130
take half like this.

00:07:57.130 --> 00:07:59.010
Well, they keep doing it.

00:07:59.010 --> 00:08:00.380
Like that.

00:08:00.380 --> 00:08:04.030
Then they take a half
like this, then they

00:08:04.030 --> 00:08:06.605
take a half like that.

00:08:06.605 --> 00:08:08.370
At least drawing this is fun.

00:08:08.370 --> 00:08:08.890
OK.

00:08:08.890 --> 00:08:11.580
I think this is what they have.
And then they actually

00:08:11.580 --> 00:08:12.750
shade in that last one.

00:08:12.750 --> 00:08:14.350
So let me shade in
that last one.

00:08:14.350 --> 00:08:15.550
I shaded it in in white.

00:08:15.550 --> 00:08:16.180
But anyway.

00:08:16.180 --> 00:08:19.410
In the figure above, rectangle
ABCD-- that's the whole

00:08:19.410 --> 00:08:24.270
thing-- is made up of seven
non-overlapping rectangles.

00:08:24.270 --> 00:08:26.380
The two smallest rectangles
have the same area.

00:08:26.380 --> 00:08:28.020
So these two have
the same area.

00:08:28.020 --> 00:08:30.620
Each of the other rectangles
have twice the area of the

00:08:30.620 --> 00:08:32.070
smaller rectangle.

00:08:32.070 --> 00:08:33.940
OK, and it looks that way.

00:08:33.940 --> 00:08:37.539
The area of the shaded rectangle
is what fraction of

00:08:37.539 --> 00:08:41.220
the area of the rectangle
ABCD?

00:08:41.220 --> 00:08:44.710
Well, let's just chug
through this.

00:08:44.710 --> 00:08:47.810
So let me find a
suitable color.

00:08:47.810 --> 00:08:50.040
So this is x.

00:08:50.040 --> 00:08:51.990
Oh, I'm still using the
rectangle tool.

00:08:51.990 --> 00:08:54.180
If this is x, and this
is going to be

00:08:54.180 --> 00:08:57.020
x, this is 2x, right?

00:08:57.020 --> 00:09:00.030
If this is 2x, then
this is 4x.

00:09:00.030 --> 00:09:01.750
Because they say that
each larger one

00:09:01.750 --> 00:09:03.540
is twice the previous.

00:09:03.540 --> 00:09:05.040
So this is 4x.

00:09:05.040 --> 00:09:06.740
So this is going to be 8x.

00:09:10.840 --> 00:09:12.760
Am I doing this right?

00:09:12.760 --> 00:09:13.900
8x.

00:09:13.900 --> 00:09:16.630
And then this is going
to be 16x.

00:09:16.630 --> 00:09:20.580
And then this is going
to be 32x.

00:09:20.580 --> 00:09:23.210
The area of the shaded
rectangle.

00:09:23.210 --> 00:09:25.080
And then, we could just
sum all of these.

00:09:25.080 --> 00:09:26.790
But we also know that
this is half of the

00:09:26.790 --> 00:09:28.700
entire rectangle, right?

00:09:28.700 --> 00:09:30.330
This rectangle right here.

00:09:30.330 --> 00:09:33.240
So the entire thing is 64x.

00:09:33.240 --> 00:09:34.520
This times 2.

00:09:34.520 --> 00:09:37.300
Because this makes up half
of the entire thing.

00:09:37.300 --> 00:09:43.680
So the entire thing is 64x, so
x is going to be 1/64 of the

00:09:43.680 --> 00:09:46.380
entire rectangle.

00:09:46.380 --> 00:09:46.680
There.

00:09:46.680 --> 00:09:47.580
That's it.

00:09:47.580 --> 00:09:48.370
All right.

00:09:48.370 --> 00:09:50.110
Number 12.

00:09:50.110 --> 00:09:51.690
Oh, no, I'm almost
out of time.

00:09:51.690 --> 00:09:54.160
So I'll do number 12
in the next video.


SAT Prep: Test 1 Section 9 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:01.700
Welcome back.

00:00:01.700 --> 00:00:03.500
So we're on problem number 5.

00:00:03.500 --> 00:00:05.880
In the schedule above,
each session is to be

00:00:05.880 --> 00:00:07.130
an hour and a half.

00:00:11.510 --> 00:00:13.285
Each break is to be 1/4 hour.

00:00:18.130 --> 00:00:19.830
And lunch is to be
1 hour long.

00:00:24.410 --> 00:00:31.170
If Session IV is to end at 4:30
PM, at what time should

00:00:31.170 --> 00:00:32.980
Session I begin?

00:00:32.980 --> 00:00:34.340
So there's actually
two ways to do it.

00:00:34.340 --> 00:00:36.730
When I just glanced at this, I
said, well, I'll just fill in

00:00:36.730 --> 00:00:37.210
this chart.

00:00:37.210 --> 00:00:38.740
And actually, I'm going to do
that as well, just to show you

00:00:38.740 --> 00:00:40.500
that they get the same answer.

00:00:40.500 --> 00:00:43.660
But there is a faster way
to do this problem.

00:00:43.660 --> 00:00:45.920
Essentially, what they're saying
is the day ends at

00:00:45.920 --> 00:00:46.660
4:30, right?

00:00:46.660 --> 00:00:50.620
Because 4:30 is the last thing
that happens in the day.

00:00:50.620 --> 00:00:53.130
And then they're saying, what
time should Session I begin?

00:00:53.130 --> 00:00:54.280
So they're essentially
saying, what time

00:00:54.280 --> 00:00:56.020
should the day begin?

00:00:56.020 --> 00:00:58.470
So if we can just figure out
how many hours are in this

00:00:58.470 --> 00:01:02.980
day, and then take that many
hours from 4:30 PM, we would

00:01:02.980 --> 00:01:05.580
get our answer for when Session
1 should begin.

00:01:05.580 --> 00:01:06.870
So how many hours
are in the day?

00:01:06.870 --> 00:01:12.200
Well, there's one, two,
three, four sessions.

00:01:12.200 --> 00:01:18.090
4 sessions times an hour
and a half each.

00:01:18.090 --> 00:01:21.800
Well, 4 times 1 and 1/2,
that's 6 hours.

00:01:21.800 --> 00:01:24.680
You can multiply the fractions
if you don't believe me.

00:01:24.680 --> 00:01:26.230
6 hours.

00:01:26.230 --> 00:01:28.250
There's 1 lunch.

00:01:28.250 --> 00:01:32.120
1 lunch is 1 hour.

00:01:32.120 --> 00:01:33.370
And then how many breaks
are there?

00:01:33.370 --> 00:01:35.280
There's 2 breaks.

00:01:35.280 --> 00:01:39.510
So there's 2 breaks times 1/4.

00:01:39.510 --> 00:01:42.260
So that equals 1/2.

00:01:42.260 --> 00:01:45.043
So if we were to add up the
hours for the sessions, the

00:01:45.043 --> 00:01:49.710
lunch and the breaks, we would
get 7 and 1/2 hours.

00:01:49.710 --> 00:01:51.880
And so then you just have to
figure out, well, what is 7

00:01:51.880 --> 00:01:56.430
and 1/2 hours before 4:30 PM?

00:01:56.430 --> 00:01:59.060
Sometimes people get
uncomfortable subtracting time

00:01:59.060 --> 00:02:01.070
past noon, because it
gets a little weird.

00:02:01.070 --> 00:02:02.780
So the way I think about
it, break it up.

00:02:02.780 --> 00:02:09.160
Well, if I go 4 and 1/2 hours
ago, if I go from 4:30 and I

00:02:09.160 --> 00:02:13.490
go minus 4 and 1/2 hours, then
I get to noon, right?

00:02:13.490 --> 00:02:15.515
And so we already took
out 4 and 1/2 hours.

00:02:15.515 --> 00:02:18.460
And we want to take out 7 and
1/2, so we have 3 hours left

00:02:18.460 --> 00:02:19.700
to take out.

00:02:19.700 --> 00:02:22.160
So 12-- you take out 3
hours, well, that's

00:02:22.160 --> 00:02:24.590
easy, that's 9:00 AM.

00:02:24.590 --> 00:02:27.670
And so that would
be our answer.

00:02:27.670 --> 00:02:29.320
That's the fast way to do it.

00:02:29.320 --> 00:02:30.770
Actually, it might not be the
fast way, because you'd have

00:02:30.770 --> 00:02:32.650
to think about all of these
things, and you're not sure if

00:02:32.650 --> 00:02:34.010
you subtracted right.

00:02:34.010 --> 00:02:36.800
The other option is
just fill it in.

00:02:36.800 --> 00:02:43.240
This session ends at 4:30, so
it has to begin at 4:00.

00:02:43.240 --> 00:02:45.000
Or no, sorry, it has to
begin at 3:00, right?

00:02:45.000 --> 00:02:45.900
Hour and a half.

00:02:45.900 --> 00:02:50.000
So that means this break would
have to end at 3 o'clock, and

00:02:50.000 --> 00:02:53.220
begin at 2:45.

00:02:53.220 --> 00:03:01.510
This would have had to end at
2:45, and begin at 1:15.

00:03:01.510 --> 00:03:06.590
This would have to end at
1:15, begin at 12:15.

00:03:06.590 --> 00:03:11.040
End at 12:15, if you go an hour
and a half before 12:15,

00:03:11.040 --> 00:03:16.760
2 hours would be 10:15,
so it would be 10:45.

00:03:16.760 --> 00:03:22.210
This ends at 10:45, and so it's
a 15-minute break, so it

00:03:22.210 --> 00:03:24.590
would have to start at 10:30.

00:03:24.590 --> 00:03:27.200
And then if this ends at 10:30,
and you subtract an

00:03:27.200 --> 00:03:29.660
hour and a half from that, you
get to 9:00 AM, which is our

00:03:29.660 --> 00:03:31.980
answer either way.

00:03:31.980 --> 00:03:34.560
So the second way, filling it
out, actually isn't that bad.

00:03:34.560 --> 00:03:36.820
If that immediately is what
comes to you, it still didn't

00:03:36.820 --> 00:03:37.580
take too much time.

00:03:37.580 --> 00:03:39.460
You could have done it
that way as well.

00:03:39.460 --> 00:03:40.710
Move on to the next problem.

00:03:44.730 --> 00:03:46.110
Number 6.

00:03:46.110 --> 00:03:47.360
I will change colors.

00:03:50.240 --> 00:04:05.070
If 2x minus 5, x plus 1, and 3x
minus 8 are all integers,

00:04:05.070 --> 00:04:07.600
and x plus 1 is the median
of these integers.

00:04:07.600 --> 00:04:09.035
So this is the median.

00:04:09.035 --> 00:04:10.510
Now, what does median mean?

00:04:10.510 --> 00:04:12.130
Median just means the middle
number, right?

00:04:12.130 --> 00:04:20.240
If I have a bunch of numbers--
if I have 1, 3, 7, 8, 9.

00:04:20.240 --> 00:04:22.050
The middle number here,
it's not the average.

00:04:22.050 --> 00:04:24.410
It's just what number-- half
the numbers are larger, and

00:04:24.410 --> 00:04:25.570
half the numbers are smaller.

00:04:25.570 --> 00:04:27.660
So in this example, the middle
number would be 7.

00:04:27.660 --> 00:04:29.280
So that's all median means.

00:04:29.280 --> 00:04:31.480
So it's telling us that x plus
1 is the median number.

00:04:31.480 --> 00:04:33.050
They say, which of the
following could

00:04:33.050 --> 00:04:33.895
be a value for x?

00:04:33.895 --> 00:04:36.070
And they give a bunch
of choices.

00:04:36.070 --> 00:04:37.930
In this type of situation, I
just try them out, because

00:04:37.930 --> 00:04:39.810
these are pretty straightforward
equations.

00:04:39.810 --> 00:04:41.150
They're giving us pretty
clean numbers.

00:04:41.150 --> 00:04:42.700
Let's just try it out.

00:04:42.700 --> 00:04:44.710
Choice A.

00:04:44.710 --> 00:04:52.770
So 2x minus 5 is 2
times 5 minus 5.

00:04:52.770 --> 00:04:53.950
That's 10 minus 5.

00:04:53.950 --> 00:04:56.542
That equals 5.

00:04:56.542 --> 00:05:01.410
x plus 1 is 5 plus 1,
which equals 6.

00:05:01.410 --> 00:05:04.320
And this is the x plus 1 term,
so it should be our median.

00:05:04.320 --> 00:05:07.430
And then 3 times 5 minus 8.

00:05:07.430 --> 00:05:11.350
3 times 5 minus 8, well,
that's 15 minus

00:05:11.350 --> 00:05:13.420
8 is equal to 7.

00:05:13.420 --> 00:05:14.650
Well, we lucked out.

00:05:14.650 --> 00:05:18.310
Looks like the x plus 1
term was the median.

00:05:18.310 --> 00:05:20.720
And that's probably a good
indication that they wanted us

00:05:20.720 --> 00:05:22.200
to try out the numbers, because
they actually gave the

00:05:22.200 --> 00:05:24.840
first choice as being the
one that worked out.

00:05:24.840 --> 00:05:28.310
So the choice is A.

00:05:28.310 --> 00:05:30.978
Move on.

00:05:30.978 --> 00:05:35.472
I'll try a different method
of deleting things.

00:05:35.472 --> 00:05:36.280
Look at that.

00:05:36.280 --> 00:05:39.550
That was pretty neat.

00:05:39.550 --> 00:05:41.695
So now we're on number 7.

00:05:45.610 --> 00:05:49.330
An exhibitor is selling
decorative wreaths at an arts

00:05:49.330 --> 00:05:50.270
and craft show.

00:05:50.270 --> 00:05:51.430
Useless sentence.

00:05:51.430 --> 00:05:54.620
The net profit p in dollars from
the sales of the wreath

00:05:54.620 --> 00:05:58.020
is given by-- now, this is a
useful piece of information.

00:05:58.020 --> 00:06:14.640
p of n is equal to 0.75n minus
50, where n is the number of

00:06:14.640 --> 00:06:17.070
wreaths sold.

00:06:17.070 --> 00:06:19.380
How many wreaths must the
exhibitor sell in order to

00:06:19.380 --> 00:06:24.330
earn a net profit of $100?

00:06:24.330 --> 00:06:27.570
Well, we want this thing
to equal $100, right?

00:06:27.570 --> 00:06:30.500
So we just set it
equal to 100.

00:06:30.500 --> 00:06:37.710
100 is equal to 0.75n
minus 50.

00:06:37.710 --> 00:06:39.270
Add 50 to both sides.

00:06:39.270 --> 00:06:44.350
150 is equal to 0.75n.

00:06:44.350 --> 00:06:49.910
And I like to write-- I like
to deal with fractions.

00:06:49.910 --> 00:06:53.620
I don't not like decimals,
but fractions are easier.

00:06:53.620 --> 00:06:57.720
This is the same
thing as 3/4 n.

00:06:57.720 --> 00:06:59.880
0.75 is the same thing as 3/4.

00:06:59.880 --> 00:07:03.200
So now we just multiply both
sides times-- something was

00:07:03.200 --> 00:07:06.210
ringing in my computer, but
anyway-- multiply both sides

00:07:06.210 --> 00:07:08.500
times the inverse of this
coefficient, so times 4/3.

00:07:13.100 --> 00:07:15.420
This of course cancels, so we
just have n left on the

00:07:15.420 --> 00:07:16.580
right-hand side.

00:07:16.580 --> 00:07:20.010
And on this side,
4/3 times 150/1.

00:07:20.010 --> 00:07:22.025
The 150 cancels out
with the 3.

00:07:25.300 --> 00:07:26.330
This becomes 1.

00:07:26.330 --> 00:07:29.040
So 4 times 50 is 200.

00:07:29.040 --> 00:07:30.440
You just have a 1 in
the denominator.

00:07:30.440 --> 00:07:31.560
Is equal to n.

00:07:31.560 --> 00:07:32.570
There you go.

00:07:32.570 --> 00:07:38.830
The exhibitor has to sell 200
wreaths, which is choice E.

00:07:38.830 --> 00:07:42.740
Next problem, number 8, which
I will do in yellow.

00:07:45.470 --> 00:07:55.675
If x squared plus y squared is
equal to 73, and xy is equal

00:07:55.675 --> 00:08:01.170
to 24, what is x
plus y squared?

00:08:05.420 --> 00:08:08.320
What happens when you multiply
x plus y squared?

00:08:08.320 --> 00:08:13.570
x plus y squared is the
same thing as x plus y

00:08:13.570 --> 00:08:16.490
times x plus y.

00:08:16.490 --> 00:08:18.470
And this might be second nature
to you after taking a

00:08:18.470 --> 00:08:20.090
lot of algebra, but I'll
just multiply it out

00:08:20.090 --> 00:08:22.380
just to show you.

00:08:22.380 --> 00:08:24.482
x times x is x squared.

00:08:24.482 --> 00:08:28.300
x times y is xy.

00:08:28.300 --> 00:08:31.690
y times x is xy.

00:08:31.690 --> 00:08:35.020
And then y times
y is y squared.

00:08:35.020 --> 00:08:41.539
So it equals x squared plus
2xy plus y squared.

00:08:41.539 --> 00:08:42.419
Well, that's interesting.

00:08:42.419 --> 00:08:50.130
We know that these two terms
combined, that's x squared

00:08:50.130 --> 00:08:51.540
plus y squared.

00:08:51.540 --> 00:08:54.650
So we know that that
equals 73.

00:08:54.650 --> 00:08:56.740
I know I'm running out of--
let me write it up here.

00:08:56.740 --> 00:09:02.390
So x squared plus 2xy
6 plus y squared.

00:09:02.390 --> 00:09:04.760
So these two terms are 73.

00:09:04.760 --> 00:09:09.760
So we have 73 plus-- and we
can cancel those two out.

00:09:09.760 --> 00:09:11.120
And then we have 2xy.

00:09:11.120 --> 00:09:16.000
Well, we know xy
is equal to 24.

00:09:16.000 --> 00:09:19.480
So it's 2 times 24.

00:09:19.480 --> 00:09:23.390
So it's 73 plus 48.

00:09:23.390 --> 00:09:24.410
And then, let's see.

00:09:24.410 --> 00:09:29.970
7-- I think this is going
to be what, 121?

00:09:29.970 --> 00:09:33.490
If my math is not-- let's see.

00:09:33.490 --> 00:09:36.200
7, so it's 110, plus
another 11.

00:09:36.200 --> 00:09:36.820
Right.

00:09:36.820 --> 00:09:38.400
121.

00:09:38.400 --> 00:09:39.150
That's all there was.

00:09:39.150 --> 00:09:42.350
You just really have to expand
out x plus y squared, and then

00:09:42.350 --> 00:09:44.430
you say, oh, well, they kind of
give you what the component

00:09:44.430 --> 00:09:47.080
terms, what the values are,
and then we're done.

00:09:47.080 --> 00:09:48.940
I'll see you in the
next video.


SAT Prep: Test 1 Section 9 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.440
Welcome back.

00:00:01.440 --> 00:00:03.665
We are now on problem
number 12, which

00:00:03.665 --> 00:00:04.915
I will do in magenta.

00:00:07.330 --> 00:00:08.180
All right.

00:00:08.180 --> 00:00:15.260
If 2x is less than y, which
is less than 0-- huh.

00:00:15.260 --> 00:00:19.460
So both of these are negative,
and 2x is more negative.

00:00:19.460 --> 00:00:20.420
All right.

00:00:20.420 --> 00:00:21.800
Which of the following
is greater?

00:00:21.800 --> 00:00:23.710
So they give us a bunch
of choices.

00:00:23.710 --> 00:00:24.140
So let's see.

00:00:24.140 --> 00:00:30.490
Choice A is minus 2x.

00:00:30.490 --> 00:00:37.390
Choice B is minus 2x plus y.

00:00:37.390 --> 00:00:41.890
Choice C is 2x.

00:00:41.890 --> 00:00:44.950
Choice D is 0.

00:00:44.950 --> 00:00:49.480
And choice E is minus y.

00:00:49.480 --> 00:00:50.540
OK.

00:00:50.540 --> 00:00:52.600
So the interesting thing that
should immediately jump out at

00:00:52.600 --> 00:00:55.170
you is that both 2x and
y are negative, right?

00:00:55.170 --> 00:00:57.400
Because they're all
less than 0.

00:00:57.400 --> 00:01:00.650
And 2x is even more
negative than y.

00:01:00.650 --> 00:01:00.850
Right?

00:01:00.850 --> 00:01:04.080
If I were to draw a number line,
if this is the number

00:01:04.080 --> 00:01:06.885
line-- we might be instructed
to even show it.

00:01:12.260 --> 00:01:14.830
This is 0.

00:01:14.830 --> 00:01:17.850
This is y.

00:01:17.850 --> 00:01:19.290
2x would be some place
over here.

00:01:19.290 --> 00:01:21.100
We don't know exactly where.

00:01:21.100 --> 00:01:24.500
This could be 0, this
could be-- I

00:01:24.500 --> 00:01:25.370
could pick random numbers.

00:01:25.370 --> 00:01:26.620
This could be negative 1.

00:01:29.240 --> 00:01:32.990
And then this could
be negative 2.

00:01:32.990 --> 00:01:33.400
Right?

00:01:33.400 --> 00:01:34.650
These are just random numbers.

00:01:34.650 --> 00:01:36.420
And actually, that might
be the best way

00:01:36.420 --> 00:01:38.270
of doing this problem.

00:01:38.270 --> 00:01:40.370
We could reason through it,
and actually, I will.

00:01:40.370 --> 00:01:42.990
But the easiest thing is, well,
let me just take these

00:01:42.990 --> 00:01:45.860
numbers that satisfy this
inequality and see what

00:01:45.860 --> 00:01:47.170
happens with them, right?

00:01:47.170 --> 00:01:49.110
And that probably is
the fastest way.

00:01:49.110 --> 00:01:51.720
So in our case, what's
minus 2x?

00:01:51.720 --> 00:01:57.140
So if 2x is minus 2, that would
be minus times minus 2.

00:01:57.140 --> 00:01:59.440
Because 2x is minus 2.

00:01:59.440 --> 00:02:03.170
So that equals plus 2.

00:02:03.170 --> 00:02:05.370
What's 2x plus y?

00:02:05.370 --> 00:02:09.410
So 2x is minus 2, plus
y is minus 1.

00:02:09.410 --> 00:02:11.000
So it becomes a minus 3.

00:02:11.000 --> 00:02:18.725
So this becomes minus minus 2
minus 1, so that's minus 3,

00:02:18.725 --> 00:02:21.980
but then you have a minus
outside so it's plus 3.

00:02:21.980 --> 00:02:24.070
So we can immediately
say that this is not

00:02:24.070 --> 00:02:24.840
the greatest, right?

00:02:24.840 --> 00:02:27.270
Because we already found
one that is greater.

00:02:27.270 --> 00:02:31.920
2x would be just minus 2.

00:02:31.920 --> 00:02:32.470
We said that. soon.

00:02:32.470 --> 00:02:34.820
2x is minus 2.

00:02:34.820 --> 00:02:36.610
0 is 0.

00:02:36.610 --> 00:02:39.150
Minus y in our example
would be 1.

00:02:39.150 --> 00:02:43.620
In all of these situations, this
is the largest. And you

00:02:43.620 --> 00:02:46.940
could try it with any set of
numbers that satisfy this.

00:02:46.940 --> 00:02:48.820
We could have called this--
instead of saying this is 2x,

00:02:48.820 --> 00:02:50.940
we could have called
this z and done it.

00:02:50.940 --> 00:02:54.870
But without doing the number
line, and without picking

00:02:54.870 --> 00:02:56.180
numbers, you could have
also reasoned.

00:02:56.180 --> 00:03:01.690
You could have said, 2x is
more negative than y.

00:03:01.690 --> 00:03:03.830
So if you put a negative in
front of 2x, it's going to be

00:03:03.830 --> 00:03:04.916
more positive.

00:03:04.916 --> 00:03:05.590
Right?

00:03:05.590 --> 00:03:07.180
Because a negative times a
negative is a positive.

00:03:07.180 --> 00:03:08.770
So it'll become a
more positive.

00:03:08.770 --> 00:03:12.620
So first of all, this number
will be a negative number, so

00:03:12.620 --> 00:03:13.580
that's not right.

00:03:13.580 --> 00:03:16.370
This number will be positive,
so 0 can't be the answer.

00:03:16.370 --> 00:03:17.600
And then you say, well,
what's bigger?

00:03:17.600 --> 00:03:19.960
Minus y or minus 2x?

00:03:19.960 --> 00:03:24.370
Well, minus 2x is more negative,
so minus-- or sorry.

00:03:24.370 --> 00:03:29.460
2x is more negative, so minus
2x will be more positive.

00:03:29.460 --> 00:03:30.530
So you know it's not
going to be this.

00:03:30.530 --> 00:03:32.710
So then you're picking
between these two.

00:03:32.710 --> 00:03:35.020
And then you have to say, well,
what's more negative?

00:03:35.020 --> 00:03:39.410
2x or 2x plus y?

00:03:39.410 --> 00:03:41.890
Well, 2x plus y is going to
be a more negative number,

00:03:41.890 --> 00:03:43.740
because both of these are
negative numbers.

00:03:43.740 --> 00:03:45.170
So when you add them, you're
going to get something even

00:03:45.170 --> 00:03:46.920
more negative than
either of them.

00:03:46.920 --> 00:03:49.290
Then we take the negative of
that, you get a larger

00:03:49.290 --> 00:03:50.610
positive number.

00:03:50.610 --> 00:03:52.400
That last explanation might
have been confusing.

00:03:52.400 --> 00:03:54.670
And really, the easiest way is
just pick some numbers that

00:03:54.670 --> 00:03:56.250
satisfy this inequality.

00:03:56.250 --> 00:04:00.440
And then substitute, and then
you have your answer.

00:04:00.440 --> 00:04:00.830
All right.

00:04:00.830 --> 00:04:10.090
Number 13, which I'll do
in this pale yellow.

00:04:10.090 --> 00:04:13.060
Carlos delivered n packages
on Monday.

00:04:13.060 --> 00:04:16.550
So Monday, n.

00:04:16.550 --> 00:04:20.170
4 times as many packages on
Tuesday as on Monday.

00:04:20.170 --> 00:04:25.160
So on Tuesday, he did 4 times
as many, so that's 4n.

00:04:25.160 --> 00:04:30.180
And 3 more-- not 3 times, 3
more-- packages on Wednesday

00:04:30.180 --> 00:04:31.720
than Monday.

00:04:31.720 --> 00:04:36.700
So Wednesday, 3 more packages
on Wednesday than Monday.

00:04:36.700 --> 00:04:41.810
So this is n plus 3, 3
more than on Monday.

00:04:41.810 --> 00:04:46.590
What is the average number of
packages he delivered per day

00:04:46.590 --> 00:04:47.940
over the three days?

00:04:47.940 --> 00:04:50.190
So what's the average of
these three numbers?

00:04:50.190 --> 00:04:50.830
So let's see.

00:04:50.830 --> 00:04:52.080
We sum them up.

00:04:52.080 --> 00:04:58.640
n plus 4n plus n plus 3, and
we divide by 3, because

00:04:58.640 --> 00:05:00.620
there's three days.

00:05:00.620 --> 00:05:04.120
n plus 4n is 5n, plus
3 is 6n. you

00:05:04.120 --> 00:05:07.990
So it's 6n plus 3 over 3.

00:05:07.990 --> 00:05:12.180
Divide all the terms by 3,
that equals 2n plus 1.

00:05:12.180 --> 00:05:13.500
And we are done.

00:05:13.500 --> 00:05:15.895
That was number 13,
which was C.

00:05:19.690 --> 00:05:27.490
Number 14.

00:05:27.490 --> 00:05:29.260
This looks exciting.

00:05:29.260 --> 00:05:45.020
a plus b to the 1/2 is equal to
a minus b to the minus 1/2.

00:05:45.020 --> 00:05:46.780
Which of the following
must be true?

00:05:46.780 --> 00:05:50.160
And they say, b is equal to 0.

00:05:50.160 --> 00:05:50.830
That's choice A.

00:05:50.830 --> 00:05:54.160
They say a plus b
is equal to 1.

00:05:54.160 --> 00:05:57.290
a minus b is equal to 1.

00:05:57.290 --> 00:06:00.880
a squared plus b squared
is equal to 1.

00:06:00.880 --> 00:06:04.910
a squared minus b squared
is equal to 1.

00:06:04.910 --> 00:06:07.500
Well, what I want to do is, I
just want to write this in a

00:06:07.500 --> 00:06:09.300
form that makes more
sense to me.

00:06:09.300 --> 00:06:14.870
So this is essentially saying
the square root of a plus b is

00:06:14.870 --> 00:06:22.780
equal to 1 over the square
root of a minus b.

00:06:22.780 --> 00:06:23.630
Right?

00:06:23.630 --> 00:06:25.640
Negative 1/2.

00:06:25.640 --> 00:06:28.840
So let's multiply both sides
of this equation times the

00:06:28.840 --> 00:06:30.970
square root of a minus b.

00:06:34.400 --> 00:06:36.770
This side will just
cancel out, right?

00:06:36.770 --> 00:06:38.440
And you're just left with 1.

00:06:38.440 --> 00:06:44.320
This is the same thing as the
square root of a minus b

00:06:44.320 --> 00:06:47.562
times a plus b.

00:06:47.562 --> 00:06:49.830
Right?

00:06:49.830 --> 00:06:51.360
And then on the right-hand
side, we're

00:06:51.360 --> 00:06:52.990
just left with a 1.

00:06:52.990 --> 00:06:57.300
And what's a minus
b times a plus b?

00:06:57.300 --> 00:07:02.350
So this is the square root of
a squared minus b squared is

00:07:02.350 --> 00:07:03.600
equal to 1.

00:07:05.870 --> 00:07:08.080
We could square both
sides, right?

00:07:08.080 --> 00:07:12.560
So if we square both sides, so
we take both to the second

00:07:12.560 --> 00:07:15.710
power, what are we left with?

00:07:15.710 --> 00:07:18.890
This radical sign goes away, so
we're left with a squared

00:07:18.890 --> 00:07:22.650
minus b squared is equal to 1.

00:07:22.650 --> 00:07:27.190
And that is choice E.

00:07:27.190 --> 00:07:27.700
Let's see.

00:07:27.700 --> 00:07:29.995
Do I have time to do
the next problem?

00:07:29.995 --> 00:07:31.570
It looks complicated.

00:07:31.570 --> 00:07:35.110
So let me wait to do the next
problem in the next video.

00:07:35.110 --> 00:07:36.360
See