SAT Prep: Test 1 Section 6 Part 1

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Language: en

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Welcome back.

00:00:01.750 --> 00:00:03.650
We are now going to start
a new section.

00:00:03.650 --> 00:00:05.270
This is going to be section 6.

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It starts on page 407 of the
official SAT study guide.

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Let's get started with some
problems. So let's see this

00:00:14.120 --> 00:00:17.150
first problem.

00:00:17.150 --> 00:00:20.640
Just so you know, they tend to
be tricky, but really fast

00:00:20.640 --> 00:00:22.550
problems. And sometimes they're
just really easy and

00:00:22.550 --> 00:00:23.870
fast. But they tend to be
problems that you should be

00:00:23.870 --> 00:00:25.770
able to get, hopefully,
in a few seconds.

00:00:25.770 --> 00:00:30.020
So if your plan of attack
immediately tells you will

00:00:30.020 --> 00:00:31.970
take you three or four minutes,
your plan of attack

00:00:31.970 --> 00:00:34.130
might be non-optimal.

00:00:34.130 --> 00:00:36.190
It maybe correct,
but non-optimal.

00:00:36.190 --> 00:00:44.270
So the first question
is, x plus 2/x is

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equal to 5 plus 2/5.

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And I was actually just looking
at this problem right

00:00:51.520 --> 00:00:53.490
before we started this video.

00:00:53.490 --> 00:00:56.200
And well, anyway, this is a
pretty neat problem because

00:00:56.200 --> 00:00:59.020
this falls into the category
of, there's two plans of

00:00:59.020 --> 00:00:59.540
attacks here.

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One will take you some time,
and involves quadratics.

00:01:02.900 --> 00:01:07.660
The other plan of attack is
ultra fast, so this is, if x

00:01:07.660 --> 00:01:12.970
plus 2/x is equal to 5 plus 2/5,
then x can equal which of

00:01:12.970 --> 00:01:13.490
the following?

00:01:13.490 --> 00:01:14.560
And they give some choices.

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They give the choice, 1/5.

00:01:17.100 --> 00:01:19.300
They give the choice, 4/5.

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They give the choice, 1,
5/2, and 5, right?

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These are the choices
they give.

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So if you see this on a algebra
test, your natural

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reaction is, well, let
me solve for x.

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And so you would add 5 plus
2/5, you get some fraction

00:01:34.720 --> 00:01:36.990
there, 27/5.

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You'd multiply both
sides by x.

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You'd get x squared plus 2
is equal to 27 over 5x.

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You'd have a quadratic.

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You would want to factor that or
use quadratic equations, et

00:01:45.955 --> 00:01:47.620
cetera, et cetera, et cetera.

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That would take you some time,
and considering this is the

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first problem, that's
not going to

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be the plan of attack.

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So let me ask you, do you
see a pattern here?

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Do these two sides
look the same?

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Well, they kind of do, right?

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We have an x here, an x here.

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We have a 2 here, right?

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We have a 2 in both places.

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We have a plus in both places.

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But the only difference is,
wherever I see an x on the

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left hand side, I see a 5
on the right hand side.

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So does something
jump out at you?

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Well sure, if I just replace x
with 5, it's going to equal

00:02:22.580 --> 00:02:23.230
this, right?

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Because if I just put 5-- x
equals 5-- you get 5 plus 2/5

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is equal to 5 plus 2/5.

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So this is the answer.

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So it could have literally taken
you about two seconds.

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And I have to admit, when I
first looked at this, I

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thought, the first question
is a quadratic?

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I said, that can't be the way.

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And I looked at it a little
bit longer, about three

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seconds longer, and I
said, oh, this is

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just pattern matching.

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Let's move on to the next
question-- Image, clear image.

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Image, invert colors--
OK, so they have a

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right triangle here.

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Let be draw the right
triangle.

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Not a wrong triangle,
a right triangle.

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Let me get the paint tool.

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I'll draw this edge now
with my free hand,

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my shaky free hand.

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And they say this side
is 2, this side is y.

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this side is x.

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This , of course, is
the hypoteneuse.

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And they say, in the right
triangle above, If x is equal

00:03:26.570 --> 00:03:30.390
to 3, what is the value of y?

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Well, this is just straight up
Pythagorean theorem, right?

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This is the hypoteneuse,
y is the hypoteneuse.

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So y squared is going to equal
this side, 4-- 2 squared--

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plus this side, 3 squared.

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So y squared is going
to equal 4 plus 9.

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y squared is going
to equal 13.

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y is equal to the square
root of 13.

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And that is choice A.

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This was just straight up
Pythagorean theorem, nothing

00:04:03.690 --> 00:04:07.210
tricky there, you just kind of
kind of chug through it.

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I'm assuming I didn't
do something wrong.

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Well anyway, let's assume that
and then move forward.

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I will now switch colors for
the sake of variety.

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OK, number 3.

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It says all numbers that are
divisible by both 2 and 6, are

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also divisible by 4.

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Right.

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OK, that's cool.

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Which of the following numbers
can be used to show that the

00:04:41.370 --> 00:04:43.070
statement above is false?

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So they're saying all numbers
that are divisible by both 2

00:04:45.760 --> 00:04:53.470
and 6-- so this is what the
statement is saying.

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Then it means that you're
going to be

00:04:54.700 --> 00:04:57.320
divisible by 4, right?

00:04:57.320 --> 00:04:59.580
So we want to prove that this
statement is wrong.

00:04:59.580 --> 00:05:01.750
So what we have to find is, some
number that is divisible

00:05:01.750 --> 00:05:09.720
by 2 and 6, that's not
divisible by 4.

00:05:09.720 --> 00:05:11.210
And we could actually just
look at the choices.

00:05:11.210 --> 00:05:19.150
They give us some choices,
4, 8, 12, 18, 24.

00:05:19.150 --> 00:05:21.810
Well, 4 isn't even divisible
by 2 and 6, so

00:05:21.810 --> 00:05:23.750
that doesn't work.

00:05:23.750 --> 00:05:27.010
8 isn't even divisible by 2 and
6, so that doesn't work.

00:05:27.010 --> 00:05:30.020
12 is divisible by 2 and 6, but
it's also divisible by 4,

00:05:30.020 --> 00:05:32.400
so it doesn't disprove
this statement.

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So, you can't say that.

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18 is divisible by
2 and 6, right?

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Is it divisible by 4?

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No.

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So this is our answer.

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The example 18 is an example of
a number divisble by 2 and

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6, and is not divisible by 4.

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So this proves that this is
an incorrect statement.

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Number 4.

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In the figure above, let me draw
this figure above, and

00:06:02.640 --> 00:06:03.890
I'll switch colors.

00:06:07.030 --> 00:06:11.340
This is my rectangle, and then
there's going to be a circle.

00:06:17.126 --> 00:06:20.330
I have a shaky hand.

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The circle looks something
like that.

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OK, and then we have side
B, A, C, and D.

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It says, in the figure above,
the circle is tangent to sides

00:06:41.810 --> 00:06:43.790
BC and AD-- tangent means
it just touches

00:06:43.790 --> 00:06:45.960
at one point, right?

00:06:45.960 --> 00:06:49.820
of the 8 by 12 rectangle ABCD.

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So they're saying this is
an 8 by 12 rectangle.

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And is it's approxmately drawn--
well can assume that

00:06:55.160 --> 00:06:56.890
this is going to be the 8 side
and this is the 12 side

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because it's longer.

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What is the area
of the circle?

00:07:03.290 --> 00:07:05.700
Well, what's the formula
for area of a circle?

00:07:05.700 --> 00:07:11.476
Its area is equal to pi r
squared, where r is the radius

00:07:11.476 --> 00:07:13.010
of the circle.

00:07:13.010 --> 00:07:14.830
So let's draw that radius
of the circle.

00:07:14.830 --> 00:07:18.765
So the radius of the circle is
going to be this line right

00:07:18.765 --> 00:07:20.015
here, right?

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That's the radius, which
is half the diameter.

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Well, what's the diameter?

00:07:28.720 --> 00:07:31.030
Well, the diameter is
this whole length.

00:07:31.030 --> 00:07:33.030
So the diameter is 8, right?

00:07:33.030 --> 00:07:35.150
So the radius is going
to be half of 8.

00:07:35.150 --> 00:07:37.080
So we know the radius
is 4, and then we

00:07:37.080 --> 00:07:39.030
just use the formula.

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We know this is going to be 4
because this is also 4, right?

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This is also 4, because
its side is 8.

00:07:46.810 --> 00:07:50.580
This top side could have been
12 billion units wide and it

00:07:50.580 --> 00:07:52.230
wouldn't have mattered, right?

00:07:52.230 --> 00:07:57.640
So we know the radius is 4, so
we say, area is equal to pi

00:07:57.640 --> 00:07:59.450
times 4 squared.

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Area is equal to 16 pi,
and that is choice A.

00:08:07.680 --> 00:08:10.540
All right, the next problem
looks like it's more involved,

00:08:10.540 --> 00:08:13.500
and I only have two minutes left
in this video, so I will

00:08:13.500 --> 00:08:15.090
do it in the next video.

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I'll see you soon.


SAT Prep: Test 1 Section 6 Part 4

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All right, we left
off at number 11.

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This is a pretty straightforward
triangle.

00:00:04.740 --> 00:00:08.330
So I can draw it in my messy
style, without the line tool.

00:00:12.180 --> 00:00:16.530
We have A, B, C.

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This is 3, 4, and 6.

00:00:21.020 --> 00:00:27.990
Each angle of triangle ABC has
the same measure as an angle

00:00:27.990 --> 00:00:31.650
in triangle XYZ, which
has not been shown.

00:00:31.650 --> 00:00:33.060
Well, we should draw
that, though.

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It has the same angle.

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I have a feeling that this is
a similar triangle X, Y, Z.

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XYZ.

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If the length of one side of
XYZ is 24-- so we just know

00:00:50.420 --> 00:00:52.820
some random side here is 24.

00:00:52.820 --> 00:00:54.970
it could be this one.

00:00:54.970 --> 00:00:57.230
Maybe, I don't know.

00:00:57.230 --> 00:00:59.780
If the length of one side is
24, what is the possible

00:00:59.780 --> 00:01:01.420
perimeter of XYZ?

00:01:01.420 --> 00:01:03.950
What is one possible
perimeter?

00:01:03.950 --> 00:01:07.580
This probably has three correct
answers, which is fun,

00:01:07.580 --> 00:01:09.790
because in some ways, increases
your probability of

00:01:09.790 --> 00:01:11.080
getting this right.

00:01:11.080 --> 00:01:14.030
But when you know that two
triangles have all the same

00:01:14.030 --> 00:01:17.450
angles, you immediately know
that it is a similar triangle.

00:01:17.450 --> 00:01:19.280
Which means that they look the
same, they're just of a

00:01:19.280 --> 00:01:21.070
different scale.

00:01:21.070 --> 00:01:23.890
And you should watch the similar
triangle video that I

00:01:23.890 --> 00:01:25.840
made in the geometry
play list, if this

00:01:25.840 --> 00:01:27.150
makes no sense to you.

00:01:27.150 --> 00:01:29.370
Because this will make no sense
if you haven't heard the

00:01:29.370 --> 00:01:30.480
term before.

00:01:30.480 --> 00:01:34.630
But, since these are similar
triangles, the ratios of the

00:01:34.630 --> 00:01:37.820
corresponding sides are
going to be the same.

00:01:37.820 --> 00:01:41.380
They're letting us pick which
side has a length of 24.

00:01:41.380 --> 00:01:44.630
So let's say that the side that
corresponds to side 3 is

00:01:44.630 --> 00:01:47.480
of length of 24, right?

00:01:47.480 --> 00:01:49.340
We don't know that this is
definitely the case, but

00:01:49.340 --> 00:01:50.430
they're letting us pick.

00:01:50.430 --> 00:01:55.170
So I'm going to say that
X Y corresponds to A B.

00:01:55.170 --> 00:01:59.400
So if this side is 24, then this
side is going to be 4/3

00:01:59.400 --> 00:02:03.440
of this side, right?

00:02:03.440 --> 00:02:05.050
Hopefully, you remember similar
triangles, that the

00:02:05.050 --> 00:02:07.070
ratios of the sides
are the same.

00:02:07.070 --> 00:02:13.570
So if 3 is to 4, as 24 is to--
Well, to go from 3 to 24, we

00:02:13.570 --> 00:02:14.610
multiply by 8.

00:02:14.610 --> 00:02:17.920
So to go from 4 to this,
we multiply by 8.

00:02:17.920 --> 00:02:19.380
So 32.

00:02:19.380 --> 00:02:22.470
And to go from 6 to this,
we multiply by 8.

00:02:22.470 --> 00:02:23.440
48.

00:02:23.440 --> 00:02:28.070
So this is a possible
combination of sides for a

00:02:28.070 --> 00:02:29.850
similar triangle to
triangle ABC.

00:02:29.850 --> 00:02:31.720
And we can just add these up.

00:02:31.720 --> 00:02:38.970
So 24 plus 32 is 56.

00:02:38.970 --> 00:02:41.160
Plus 48.

00:02:41.160 --> 00:02:44.480
6 plus 8 is 14.

00:02:44.480 --> 00:02:45.230
104.

00:02:45.230 --> 00:02:49.950
Just bubble type that in, 104.

00:02:49.950 --> 00:02:54.180
This could have been 24.

00:02:54.180 --> 00:02:59.440
If this is 24, that means
we're multiplying

00:02:59.440 --> 00:03:01.030
by 6 to get to 24.

00:03:01.030 --> 00:03:02.850
So this would be 18.

00:03:02.850 --> 00:03:04.030
This would be 36.

00:03:04.030 --> 00:03:05.540
You could add all of
those up together.

00:03:05.540 --> 00:03:06.940
And you would have gotten
another answer.

00:03:06.940 --> 00:03:08.760
And that would have
been correct.

00:03:08.760 --> 00:03:13.350
And similarly, you could
have made this side 24.

00:03:13.350 --> 00:03:15.440
If this side was 24, then that
means we're multiplying

00:03:15.440 --> 00:03:16.980
every side by 4.

00:03:16.980 --> 00:03:19.130
So this would have been 12, and
this would have been 16.

00:03:19.130 --> 00:03:19.950
And you could have
added those up.

00:03:19.950 --> 00:03:22.000
And that would have been the
other, the third possible,

00:03:22.000 --> 00:03:24.050
correct answer for
this problem.

00:03:24.050 --> 00:03:25.750
Exciting.

00:03:25.750 --> 00:03:27.000
Let's move on.

00:03:30.630 --> 00:03:34.830
The sum of five consecutive
integers is a thousand.

00:03:34.830 --> 00:03:35.780
Number 12.

00:03:35.780 --> 00:03:38.680
If five consecutive integers
is a thousand, what is the

00:03:38.680 --> 00:03:40.350
value of the greatest
of these integers?

00:03:40.350 --> 00:03:42.220
So they're consecutive
integers.

00:03:42.220 --> 00:03:46.140
So there are five of them.

00:03:46.140 --> 00:03:48.830
So we want to know the greatest
of these integers.

00:03:48.830 --> 00:03:53.180
So let's say x is the greatest.
And so the second

00:03:53.180 --> 00:03:55.960
greatest is going
to be x minus 1.

00:03:55.960 --> 00:03:58.760
The third greatest
is x minus 2.

00:03:58.760 --> 00:04:01.540
The fourth greatest
is x minus 3.

00:04:01.540 --> 00:04:05.960
And then the fifth greatest
is x minus 4, right?

00:04:05.960 --> 00:04:07.340
These are five integers and
they're consecutive.

00:04:07.340 --> 00:04:10.080
And this is the largest,
right?

00:04:10.080 --> 00:04:11.920
And we're going to take
the sum of them.

00:04:14.970 --> 00:04:15.670
This is a tip.

00:04:15.670 --> 00:04:18.120
If they were asking for the
smallest number, I would have

00:04:18.120 --> 00:04:18.370
started with x.

00:04:18.370 --> 00:04:20.313
And I would have gone, x
plus one, x plus 2, x

00:04:20.313 --> 00:04:21.209
plus 3, like that.

00:04:21.209 --> 00:04:23.075
Because then x would have
been my smallest number.

00:04:23.075 --> 00:04:25.200
But they said x is the largest.
We want to solve for

00:04:25.200 --> 00:04:26.790
x, so the greatest of
these integers.

00:04:26.790 --> 00:04:28.770
So that's why I subtracted.

00:04:28.770 --> 00:04:31.390
If we add all these
up, we have 5 x's.

00:04:31.390 --> 00:04:36.650
So that's 5x minus 1
minus 2 is minus 3.

00:04:36.650 --> 00:04:38.990
Minus 3 is minus 6.

00:04:38.990 --> 00:04:41.800
Minus 4 is minus 10.

00:04:41.800 --> 00:04:43.130
5x minus 10.

00:04:43.130 --> 00:04:48.450
They say that the
sum is 1,000.

00:04:48.450 --> 00:04:51.390
And this is a fairly simple
algebra equation.

00:04:51.390 --> 00:04:55.040
5x is equal to-- add
10 to both sides.

00:04:55.040 --> 00:05:00.880
1,010, which is also 10 in
binary, if you are a geek.

00:05:00.880 --> 00:05:08.840
Divide both sides by 5. x
is equal to 205, right?

00:05:08.840 --> 00:05:10.770
No, I can't divide by 5.

00:05:10.770 --> 00:05:12.480
202, right?

00:05:12.480 --> 00:05:16.590
x is equal to 202.

00:05:16.590 --> 00:05:17.860
There you go.

00:05:17.860 --> 00:05:19.070
Not too bad.

00:05:19.070 --> 00:05:20.120
All right.

00:05:20.120 --> 00:05:22.610
Number 13.

00:05:22.610 --> 00:05:25.200
And I'm going to switch colors,
because pale yellow is

00:05:25.200 --> 00:05:26.990
getting monotonous.

00:05:26.990 --> 00:05:30.420
I'll switch to something
bold like blood red.

00:05:30.420 --> 00:05:32.810
No, that's tacky.

00:05:32.810 --> 00:05:36.430
Number 13.

00:05:36.430 --> 00:05:39.530
The figure above shows
y equals g of x.

00:05:42.880 --> 00:05:45.470
So let me draw that, because I
don't want to confuse you just

00:05:45.470 --> 00:05:49.100
saying a lot of random crap.

00:05:49.100 --> 00:05:51.350
You see what the graph looks
like, although you should have

00:05:51.350 --> 00:05:54.870
it turned to the same page.

00:05:54.870 --> 00:05:56.310
OK.

00:05:56.310 --> 00:05:57.560
I draw it like that.

00:06:01.130 --> 00:06:03.360
No, I don't have to draw
that far down.

00:06:03.360 --> 00:06:05.130
Like that.

00:06:05.130 --> 00:06:09.000
And it looks something
like this.

00:06:09.000 --> 00:06:10.412
Let me draw it in a
different color.

00:06:10.412 --> 00:06:14.300
The line itself-- something
goes from here,

00:06:14.300 --> 00:06:15.550
goes up like this.

00:06:18.240 --> 00:06:22.050
Then, it kind of comes
back down like here.

00:06:22.050 --> 00:06:23.640
Then it kind of bounces.

00:06:23.640 --> 00:06:26.400
It goes back up like this.

00:06:26.400 --> 00:06:31.440
And then they tell us that this
line, right here, is y is

00:06:31.440 --> 00:06:34.000
equal to g of x.

00:06:34.000 --> 00:06:35.130
It's going to be a functions
problem.

00:06:35.130 --> 00:06:35.810
I can tell you already.

00:06:35.810 --> 00:06:38.840
So you should watch the function
videos or do the

00:06:38.840 --> 00:06:40.410
function modules on
Khan Academy.

00:06:40.410 --> 00:06:43.560
So the figure above shows the
graph y equals g of x.

00:06:43.560 --> 00:06:53.880
If the function h is defined by,
h of x is equal to g of 2x

00:06:53.880 --> 00:06:59.840
plus 2, they want to know
what h of 1 is equal to.

00:06:59.840 --> 00:07:02.110
Well, with functions,
they're putting a 1

00:07:02.110 --> 00:07:04.000
into this black box.

00:07:04.000 --> 00:07:05.920
So wherever we see the x,
we put a 1 in, right?

00:07:05.920 --> 00:07:15.350
So h of 1 is equal to-- let's
just use this definition-- g

00:07:15.350 --> 00:07:17.830
of 2 times x.

00:07:17.830 --> 00:07:22.830
Well, in this case, x is 1.
g of 2 times 1 plus 2.

00:07:22.830 --> 00:07:27.900
So that equals g of 3 plus 2.

00:07:27.900 --> 00:07:30.510
Now, we have to figure
out what g of 3 is.

00:07:30.510 --> 00:07:35.240
And if you look at this graph--
and in the book it's

00:07:35.240 --> 00:07:35.700
more obvious.

00:07:35.700 --> 00:07:42.100
There's 1, 2, and then it hits
0 right at 3, right?

00:07:42.100 --> 00:07:46.050
So g of 3, when x is equal to
3, g of 3 is right here.

00:07:46.050 --> 00:07:47.660
That's 0.

00:07:47.660 --> 00:07:48.910
g of 3 is 0.

00:07:52.740 --> 00:07:53.700
We're just left with a 2.

00:07:53.700 --> 00:07:55.040
h of 1 is 2.

00:07:55.040 --> 00:07:56.810
That's all.

00:07:56.810 --> 00:07:57.960
Amazing.

00:07:57.960 --> 00:07:59.210
All right, next problem.

00:08:06.380 --> 00:08:09.420
Number 14.

00:08:09.420 --> 00:08:19.700
Exactly 4 actors try out
for 4 parts in a play.

00:08:19.700 --> 00:08:25.070
If each actor can perform any
1 part, and no one will

00:08:25.070 --> 00:08:28.380
perform more than 1 part, how
many different assignments of

00:08:28.380 --> 00:08:30.640
actors are possible?

00:08:30.640 --> 00:08:33.059
Well, let's just say you're just
randomly doing it, right?

00:08:33.059 --> 00:08:35.590
So let's put up the actors.

00:08:38.440 --> 00:08:39.710
You take the first guy, right?

00:08:39.710 --> 00:08:41.720
And you say, well, I
can put him in any

00:08:41.720 --> 00:08:43.150
1 of 4 roles, right?

00:08:43.150 --> 00:08:44.940
For the first guy, I could
put him in any

00:08:44.940 --> 00:08:47.700
1 of 4 roles, right?

00:08:47.700 --> 00:08:49.936
And then the second guy-- I
shouldn't assume they're all

00:08:49.936 --> 00:08:54.180
men-- and the second actor,
or woman, or actress.

00:08:54.180 --> 00:08:57.080
1 role has been taken
already, right?

00:08:57.080 --> 00:08:58.030
There were 4 roles.

00:08:58.030 --> 00:08:59.370
This first person took it.

00:08:59.370 --> 00:09:02.000
So now, there's only 3
roles left, right?

00:09:02.000 --> 00:09:04.640
There's 3 possibilities.

00:09:04.640 --> 00:09:06.380
There's 1 of 3 possibilities
for that

00:09:06.380 --> 00:09:08.530
second person, correct?

00:09:08.530 --> 00:09:11.700
OK, so the third person-- how
many possibilities does he or

00:09:11.700 --> 00:09:12.770
she have left?

00:09:12.770 --> 00:09:14.470
Well, 2 of the roles have
been taken already.

00:09:14.470 --> 00:09:17.870
So there's only 2 possibilities
for that person.

00:09:17.870 --> 00:09:20.400
And then the last person
to be picked has only

00:09:20.400 --> 00:09:22.210
1 role left, right?

00:09:22.210 --> 00:09:25.240
So the combinations are
literally just 4 times 3

00:09:25.240 --> 00:09:27.040
times 2 times 1.

00:09:27.040 --> 00:09:28.600
So that's 12 times 2.

00:09:28.600 --> 00:09:34.810
That's 24 combinations or
assignments of roles.

00:09:34.810 --> 00:09:37.415
And I actually haven't made a
probability, and combinations,

00:09:37.415 --> 00:09:39.060
and permutations video yet.

00:09:39.060 --> 00:09:40.420
I will.

00:09:40.420 --> 00:09:44.460
And once I do, watch that
if this problem seemed

00:09:44.460 --> 00:09:45.530
confusing to you.

00:09:45.530 --> 00:09:47.130
But hopefully, it makes a little
bit of intuitive sense.

00:09:47.130 --> 00:09:49.690
I'll see you in the
next video.


SAT Prep: Test 1 Section 6 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:01.580
And welcome back.

00:00:01.580 --> 00:00:08.000
We are now on number
seven on page 409.

00:00:08.000 --> 00:00:11.790
Dwayne has a newspaper route
for which he collects k

00:00:11.790 --> 00:00:13.040
dollars per day.

00:00:21.370 --> 00:00:26.400
From this amount, he pays out--
so this is how much he

00:00:26.400 --> 00:00:35.180
makes-- k/3 dollars per day
for the cost of the paper.

00:00:35.180 --> 00:00:38.210
So this is cost of papers.

00:00:38.210 --> 00:00:41.200
And he saves the rest
of the money.

00:00:41.200 --> 00:00:44.690
In terms of k, how many
days will it take

00:00:44.690 --> 00:00:48.230
Dwayne to save $1,000?

00:00:48.230 --> 00:00:50.290
That's his goal.

00:00:50.290 --> 00:00:52.710
Well, how much does
he make per day?

00:00:52.710 --> 00:00:57.650
Well, he makes k, and he has
to give out k/3 per day.

00:00:57.650 --> 00:01:06.370
So each day, per day, he
makes k minus k/3.

00:01:06.370 --> 00:01:11.060
We could write this
as 3k minus k/3.

00:01:14.200 --> 00:01:16.790
k is the same thing is 3k/3.

00:01:16.790 --> 00:01:18.225
So this is equal to 2k/3.

00:01:20.820 --> 00:01:22.760
That's how much he
makes per day.

00:01:22.760 --> 00:01:27.760
So how much does he
make in d days?

00:01:27.760 --> 00:01:29.520
If d is the number of days,
how much does he make?

00:01:29.520 --> 00:01:34.920
Well, he makes 2k/3 times d.

00:01:34.920 --> 00:01:38.550
That's how much he makes,
right? d days.

00:01:38.550 --> 00:01:46.390
Well, he wants to make $1,000,
so how many days

00:01:46.390 --> 00:01:47.640
is it going to take?

00:01:49.850 --> 00:01:54.692
Hopefully, you realize this is
a level one linear equation,

00:01:54.692 --> 00:01:57.210
so you multiply both sides
times a reciprocal.

00:01:57.210 --> 00:02:01.815
So 3/2k times this,
times 3/2k.

00:02:04.470 --> 00:02:05.260
This cancels this.

00:02:05.260 --> 00:02:05.840
This cancels.

00:02:05.840 --> 00:02:06.410
That cancels.

00:02:06.410 --> 00:02:08.300
You're just left with a d
on the left-hand side.

00:02:08.300 --> 00:02:10.509
And on the right-hand side, we
can do little canceling, too.

00:02:10.509 --> 00:02:15.290
This 2 will cancel and
make this into 500.

00:02:15.290 --> 00:02:17.620
You have 500 times 3/k.

00:02:17.620 --> 00:02:20.040
So that equals-- I don't know
why I started tilting-- that

00:02:20.040 --> 00:02:33.880
equals 1500/k days.

00:02:33.880 --> 00:02:38.830
And that is the choice
number D.

00:02:38.830 --> 00:02:39.910
All right.

00:02:39.910 --> 00:02:40.350
I get excited.

00:02:40.350 --> 00:02:41.360
Number eight has
a number line.

00:02:41.360 --> 00:02:45.030
I get excited whenever there's
a number line problem.

00:02:45.030 --> 00:02:45.800
I don't know why.

00:02:45.800 --> 00:02:49.820
It's a psychological
disorder, perhaps.

00:02:49.820 --> 00:02:51.740
But a good one.

00:02:51.740 --> 00:02:54.260
Some abnormalities are positive
abnormalities.

00:02:54.260 --> 00:02:54.610
Let's see.

00:02:54.610 --> 00:02:56.780
So let me draw this
number line.

00:03:04.680 --> 00:03:05.320
All right.

00:03:05.320 --> 00:03:07.615
I drew that number line.

00:03:07.615 --> 00:03:10.300
That's a number line.

00:03:10.300 --> 00:03:14.370
Most of the time is probably
just going to take drawing it.

00:03:14.370 --> 00:03:16.460
Let me do this in a
different color.

00:03:16.460 --> 00:03:18.160
So I have point A.

00:03:18.160 --> 00:03:20.440
Let me do it in a more
vivid color.

00:03:20.440 --> 00:03:25.250
Point A is here, and then it's
right next to negative 2.

00:03:25.250 --> 00:03:27.240
It's a little bit less
than negative 2,

00:03:27.240 --> 00:03:29.100
which is right there.

00:03:29.100 --> 00:03:33.680
Then I have negative
1 right here.

00:03:33.680 --> 00:03:34.680
Then I got a bunch of points.

00:03:34.680 --> 00:03:36.740
I have B.

00:03:36.740 --> 00:03:38.800
I have P.

00:03:38.800 --> 00:03:41.550
I have C.

00:03:41.550 --> 00:03:44.680
Then I have 0.

00:03:44.680 --> 00:03:48.020
Then I have 1.

00:03:48.020 --> 00:03:51.920
And then I have a couple
points more.

00:03:51.920 --> 00:03:52.530
D.

00:03:52.530 --> 00:03:57.970
And then I have Q.

00:03:57.970 --> 00:04:03.760
Then I have the number
2, and I have E.

00:04:03.760 --> 00:04:04.760
All right.

00:04:04.760 --> 00:04:05.660
So let's read the problem.

00:04:05.660 --> 00:04:08.440
Which of the letter points, on
the number line above, could

00:04:08.440 --> 00:04:10.790
represent the result when the
coordinate of point P is

00:04:10.790 --> 00:04:13.880
multiplied by the coordinate
of point Q.

00:04:13.880 --> 00:04:14.910
All right.

00:04:14.910 --> 00:04:18.850
So this is pretty much just an
approximation problem, right?

00:04:18.850 --> 00:04:22.950
So we're trying to figure out
P times Q, essentially.

00:04:22.950 --> 00:04:26.940
So point P, well, let's eyeball
it, it's a little bit

00:04:26.940 --> 00:04:28.670
to the left of 0, to the
right of negative 1.

00:04:28.670 --> 00:04:30.320
It looks like maybe, I
don't know, I'm just

00:04:30.320 --> 00:04:32.130
guessing, minus 0.5?

00:04:32.130 --> 00:04:33.980
minus 1/2, right?

00:04:33.980 --> 00:04:35.570
Minus 1/2.

00:04:35.570 --> 00:04:37.170
And what's Q?

00:04:37.170 --> 00:04:44.260
Well, Q looks a little bit
closer to 2 than it is to 1,

00:04:44.260 --> 00:04:47.880
so I would say maybe,
1 and 3/4?

00:04:47.880 --> 00:04:48.820
I'm just guessing.

00:04:48.820 --> 00:04:50.470
You could just approximate
it as you want.

00:04:50.470 --> 00:04:51.720
So 1 and 3/4.

00:04:54.690 --> 00:04:56.550
Or if you wanted to write that
as an improper fraction, which

00:04:56.550 --> 00:04:59.410
I prefer to deal with, 7/4.

00:04:59.410 --> 00:05:00.610
I'm just eyeballing it.

00:05:00.610 --> 00:05:02.760
Nothing fancy here.

00:05:02.760 --> 00:05:09.540
So if P is minus 1/2, and this
number is 7/4, what happens

00:05:09.540 --> 00:05:10.810
when you multiply the two?

00:05:10.810 --> 00:05:21.020
So minus 1/2 times 7/4 is equal
to minus 7/8, right?

00:05:21.020 --> 00:05:27.490
So minus 7/8, that's
almost minus 8/8.

00:05:27.490 --> 00:05:31.800
Minus 8/8 would be here.

00:05:31.800 --> 00:05:35.780
Minus 8/8 would be at negative
1, and so this is a little

00:05:35.780 --> 00:05:38.305
less negative, so it's a
slightly larger number.

00:05:38.305 --> 00:05:42.050
So minus 7/8 is going to be
someplace around this area.

00:05:42.050 --> 00:05:44.090
Well, it's interesting that
I've bumped into a number

00:05:44.090 --> 00:05:46.030
there while I was looking
in that area.

00:05:46.030 --> 00:05:47.780
I bumped into point B.

00:05:47.780 --> 00:05:50.050
And maybe my approximation
is wrong.

00:05:50.050 --> 00:05:52.380
Maybe it could be point P.

00:05:52.380 --> 00:05:54.020
But when I look at the
choices, P isn't a

00:05:54.020 --> 00:05:55.530
choice, and only B is.

00:05:55.530 --> 00:06:01.570
So B is what I would bubble in,
as what happens when you

00:06:01.570 --> 00:06:03.120
multiply P times Q.

00:06:03.120 --> 00:06:05.190
And that was kind of fun,
because it's just pure

00:06:05.190 --> 00:06:06.410
approximation.

00:06:06.410 --> 00:06:08.990
And you kind of get something
that approaches the answer.

00:06:08.990 --> 00:06:10.080
Well, it is the answer I guess.

00:06:10.080 --> 00:06:11.550
Hopefully.

00:06:11.550 --> 00:06:12.800
All right.

00:06:15.560 --> 00:06:19.060
I'm having too much fun,
absolutely too much fun.

00:06:19.060 --> 00:06:19.830
OK.

00:06:19.830 --> 00:06:22.720
Number nine.

00:06:22.720 --> 00:06:37.220
If 5y plus 2x is equal to 23,
and x is equal to y plus 1,

00:06:37.220 --> 00:06:39.120
what is the value of y?

00:06:39.120 --> 00:06:41.140
What does y equal?

00:06:41.140 --> 00:06:42.490
Well, this we could
do straight up

00:06:42.490 --> 00:06:43.560
substitution to solve.

00:06:43.560 --> 00:06:45.610
We have x is equal
to y plus 1.

00:06:45.610 --> 00:06:47.130
Substitute.

00:06:47.130 --> 00:06:49.030
Substitute for x.

00:06:49.030 --> 00:06:55.670
So then we're left with
5y plus 2 times x.

00:06:55.670 --> 00:06:57.910
For x, we substitute
this term.

00:06:57.910 --> 00:07:02.290
2 times y plus 1
is equal to 23.

00:07:02.290 --> 00:07:05.730
We just do algebra, or
even arithmetic.

00:07:05.730 --> 00:07:11.750
5y plus-- distribute the 2--
2y plus 2 is equal to 23.

00:07:11.750 --> 00:07:17.720
5y plus 2y is 7y, plus
2 is equal to 23.

00:07:17.720 --> 00:07:19.260
Subtract 2 from both sides.

00:07:19.260 --> 00:07:22.500
7y is equal to 21.

00:07:22.500 --> 00:07:25.740
y is equal to 3.

00:07:25.740 --> 00:07:27.850
This was just a question to see
if you even took Algebra

00:07:27.850 --> 00:07:30.540
I, not to make you feel bad
if you don't remember.

00:07:30.540 --> 00:07:31.500
People forget things.

00:07:31.500 --> 00:07:33.600
But anyway, let's move on.

00:07:33.600 --> 00:07:37.960
Number 10.

00:07:37.960 --> 00:07:42.230
A company produced 300
appliances in the first week

00:07:42.230 --> 00:07:43.430
of the month.

00:07:43.430 --> 00:07:45.810
So this is number
10 we're doing.

00:07:45.810 --> 00:07:48.160
Let me switch colors because
that mauve is too dark.

00:07:48.160 --> 00:07:49.410
So week one.

00:07:54.240 --> 00:07:56.540
Appliances.

00:07:56.540 --> 00:08:01.580
Produced 500 in week one.

00:08:01.580 --> 00:08:05.030
Because it received additional
machinery, its production

00:08:05.030 --> 00:08:09.650
increased 50% from the first
week to the second week.

00:08:09.650 --> 00:08:12.390
How many appliances did
the company produce

00:08:12.390 --> 00:08:13.030
in the second week?

00:08:13.030 --> 00:08:15.210
Well, I didn't even have to
make a table here, right?

00:08:15.210 --> 00:08:16.830
So it improved 50%.

00:08:16.830 --> 00:08:19.910
So plus 50%.

00:08:19.910 --> 00:08:20.690
So it's important.

00:08:20.690 --> 00:08:22.030
It improved 50% percent.

00:08:22.030 --> 00:08:27.650
Week 2 isn't 50% percent of 500,
it's 50% more than 500.

00:08:27.650 --> 00:08:34.630
So it's 500 plus 50% of 500.

00:08:34.630 --> 00:08:36.760
What's 50% percent of 500?

00:08:36.760 --> 00:08:38.370
Well, that's 250, right?

00:08:38.370 --> 00:08:44.250
So it's 250 plus 500 is 750.

00:08:44.250 --> 00:08:47.650
I was looking forward to
something more challenging.

00:08:47.650 --> 00:08:49.190
Anyway, number 11.

00:08:49.190 --> 00:08:56.480
Let's see if I have time to do
number 11 in this video.

00:08:56.480 --> 00:08:57.090
You know what?

00:08:57.090 --> 00:08:58.780
I only have one minute left.

00:08:58.780 --> 00:09:00.610
I will do this in
the next video.

00:09:00.610 --> 00:09:01.860
See you soon.


SAT Prep: Test 1 Section 6 Part 5

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.050
Welcome back.

00:00:02.050 --> 00:00:03.240
I scared you, I'm sure.

00:00:03.240 --> 00:00:07.040
I have more energy.

00:00:07.040 --> 00:00:08.700
I'm actually fasting
today, but I won't

00:00:08.700 --> 00:00:12.090
go into it too much.

00:00:12.090 --> 00:00:16.790
I don't want to go too much into
my personal life, but I

00:00:16.790 --> 00:00:18.900
eat one meal a day now and
I have so much energy.

00:00:18.900 --> 00:00:21.220
It's unbelievable.

00:00:21.220 --> 00:00:23.870
Hope you don't think I'm crazy,
but it's back to the

00:00:23.870 --> 00:00:26.810
SAT videos.

00:00:26.810 --> 00:00:33.950
So let's see, we're on
problem number 15.

00:00:33.950 --> 00:00:35.750
So I drew it out ahead of time,
because I didn't want to

00:00:35.750 --> 00:00:37.810
waste your time because
it's more valuable

00:00:37.810 --> 00:00:39.350
than mine, I think.

00:00:39.350 --> 00:00:42.170
Considering I have so much time
now that I only eat one

00:00:42.170 --> 00:00:43.480
meal a day.

00:00:43.480 --> 00:00:52.720
They say, in the figure above,
triangle PQR is equilateral

00:00:52.720 --> 00:01:03.340
and SR and TV intersect
at point P.

00:01:03.340 --> 00:01:05.140
What is the value of y?

00:01:05.140 --> 00:01:06.860
So let's just use all the
information they give us.

00:01:06.860 --> 00:01:08.190
This is an equilateral
triangle.

00:01:08.190 --> 00:01:09.930
So what do you know about
equilateral triangles?

00:01:09.930 --> 00:01:11.270
All the angles in
an equilateral

00:01:11.270 --> 00:01:11.970
triangle are the same.

00:01:11.970 --> 00:01:15.690
This is going to be 60 degrees,
this going to be 60

00:01:15.690 --> 00:01:18.090
degrees, and this whole
thing is going to

00:01:18.090 --> 00:01:20.410
be 60 degrees, right?

00:01:20.410 --> 00:01:22.950
Well, if this whole thing is
60 degrees, then what's x?

00:01:22.950 --> 00:01:24.860
Well, there's three x's,
and they're equally

00:01:24.860 --> 00:01:25.940
splitting it, right?

00:01:25.940 --> 00:01:30.970
So this is 20 degrees, this is
20 degrees and this is 20

00:01:30.970 --> 00:01:32.220
degrees, right?

00:01:34.560 --> 00:01:37.620
Well, what's equal to angle y?

00:01:37.620 --> 00:01:41.940
From our geometry module, we
know that angle y is equal to

00:01:41.940 --> 00:01:43.275
the opposite angle, right?

00:01:43.275 --> 00:01:47.860
Angle y is going to be equal
to-- so you got to pay close

00:01:47.860 --> 00:01:50.570
attention to which
lines this is.

00:01:50.570 --> 00:01:54.060
So that line and that
line, right?

00:01:54.060 --> 00:01:56.010
So the opposite angle
to angle y is this

00:01:56.010 --> 00:01:58.610
angle right here, right?

00:01:58.610 --> 00:02:01.320
So that angle is just two of the
three x's, so it's 20 and

00:02:01.320 --> 00:02:07.225
20, so angle y is equal
to 40 degrees.

00:02:07.225 --> 00:02:09.940
It just equals two of
these x degrees.

00:02:09.940 --> 00:02:11.060
That's it.

00:02:11.060 --> 00:02:13.470
All right, number 16.

00:02:13.470 --> 00:02:16.480
Maybe I could do it here with
this kind of toothpaste

00:02:16.480 --> 00:02:19.220
colored ink.

00:02:19.220 --> 00:02:22.430
Let the operations triangle and
square be defined for all

00:02:22.430 --> 00:02:26.220
real numbers a and b as follows,
a-- I like these

00:02:26.220 --> 00:02:37.420
problems-- a triangle b
is equal to a plus 3b.

00:02:37.420 --> 00:02:49.985
And then a square b is
equal to a plus 4b.

00:02:52.870 --> 00:02:54.330
And then they have this thing.

00:02:54.330 --> 00:03:16.200
If 4 triangle of 5y is equal
to 5y square 4, what is the

00:03:16.200 --> 00:03:17.270
value of y?

00:03:17.270 --> 00:03:19.760
Well this might seem bizarre
to you, but we just have to

00:03:19.760 --> 00:03:23.570
map these functions to-- And
this is just functions in

00:03:23.570 --> 00:03:24.610
disguise, right?

00:03:24.610 --> 00:03:26.980
A function of two variables.

00:03:26.980 --> 00:03:29.510
Let's see, so what
is 4 triangle 5y?

00:03:29.510 --> 00:03:31.630
So let's go to the triangle
function.

00:03:31.630 --> 00:03:32.940
That's this one, right?

00:03:32.940 --> 00:03:36.360
So in this case,
a is 4, right?

00:03:36.360 --> 00:03:38.900
Because it's the number before
the triangle, so that's equal

00:03:38.900 --> 00:03:43.020
to 4 plus 3 times b.

00:03:43.020 --> 00:03:44.170
Well, what's b?

00:03:44.170 --> 00:03:48.000
b is whatever's after
the triangle, right?

00:03:48.000 --> 00:03:49.640
In this case it's 5y.

00:03:49.640 --> 00:03:52.030
So 3 times b would
be 3 times 5y.

00:03:55.890 --> 00:03:58.190
Let me do some coloring here.

00:03:58.190 --> 00:04:02.570
This is just this, right?

00:04:02.570 --> 00:04:08.000
And then if I do the other side
of that-- 5y square 4.

00:04:08.000 --> 00:04:10.775
So in this case, 5-- so we're
using the square.

00:04:10.775 --> 00:04:15.400
So in this case 5y
is a, right?

00:04:15.400 --> 00:04:21.980
So that equals 5y plus 4b.

00:04:21.980 --> 00:04:22.500
And what's b?

00:04:22.500 --> 00:04:28.790
Well, b is whatever's after
the rectangle, right?

00:04:28.790 --> 00:04:32.450
So 4b is 4 times 4.

00:04:32.450 --> 00:04:33.370
And now we just solve.

00:04:33.370 --> 00:04:33.920
So let's see.

00:04:33.920 --> 00:04:39.100
4 times-- 4 plus-- let me just
rewrite that-- 4 plus 3 times

00:04:39.100 --> 00:04:46.980
5 is 15y is equal
to 5y plus 16.

00:04:46.980 --> 00:04:53.210
Let's take 4 from both
sides, so you get 15y

00:04:53.210 --> 00:04:58.200
equals 5y plus 12.

00:04:58.200 --> 00:05:04.320
Take 5y from both sides, you
get 10y is equal to 12.

00:05:04.320 --> 00:05:08.680
y is equal to 12/10, and then
we can reduce that.

00:05:08.680 --> 00:05:15.950
Divide the top to bottom by 2,
so y is equal to 6/5, and that

00:05:15.950 --> 00:05:17.820
is our answer.

00:05:17.820 --> 00:05:19.640
These are these kind of fill in
the blank ones, so you can

00:05:19.640 --> 00:05:20.980
actually write that--
bubble it in.

00:05:20.980 --> 00:05:23.000
The SAT wasn't like that
I when I took it.

00:05:23.000 --> 00:05:25.970
Actually, these problems
are really good.

00:05:25.970 --> 00:05:27.220
Let's move on.

00:05:33.990 --> 00:05:36.680
Number 17.

00:05:36.680 --> 00:05:40.840
Let me switch to magenta,
bright magenta.

00:05:40.840 --> 00:05:46.990
In the xy-coordinate plane, the
graph of x is equal to y

00:05:46.990 --> 00:06:01.460
squared minus 4 intersects the
line l, at 0 comma p, and the

00:06:01.460 --> 00:06:05.830
point 5 comma t, right?

00:06:05.830 --> 00:06:08.560
These are two points
on the line l.

00:06:08.560 --> 00:06:14.200
What is the greatest possible
value of the slope of l?

00:06:14.200 --> 00:06:15.130
So this is interesting.

00:06:15.130 --> 00:06:18.790
So x is equal to y
squared minus 4.

00:06:18.790 --> 00:06:20.640
So, first of all, you need
realize this isn't how you

00:06:20.640 --> 00:06:22.930
normally write a function,
right?

00:06:22.930 --> 00:06:25.545
You normally write it y is
equal to x something.

00:06:25.545 --> 00:06:30.495
But, what this is going to look
like-- x plus 4 is equal

00:06:30.495 --> 00:06:37.840
to y squared-- And you don't
have to draw this actually on

00:06:37.840 --> 00:06:39.480
the SAT test. It's going to
take you too much time.

00:06:39.480 --> 00:06:41.550
But I think it's useful here
to have a visualization of

00:06:41.550 --> 00:06:42.800
this, right?

00:06:45.710 --> 00:06:47.830
So the graph is going to look
something like this.

00:06:52.030 --> 00:06:54.780
It's going to actually
start at minus 4.

00:06:54.780 --> 00:06:58.020
It's going to look something
like this,

00:06:58.020 --> 00:06:59.380
the graph-- oh, woops.

00:06:59.380 --> 00:07:01.691
I'm still using the line tool.

00:07:01.691 --> 00:07:03.950
The graph is going to look
something like this.

00:07:03.950 --> 00:07:07.570
Start back here at minus 4.

00:07:07.570 --> 00:07:08.640
I think.

00:07:08.640 --> 00:07:12.410
And then the line l-- we can
only assume since it

00:07:12.410 --> 00:07:15.100
intersects it at two points,
it's going to look something

00:07:15.100 --> 00:07:18.650
like this, right?

00:07:18.650 --> 00:07:21.670
Where this could be 0 comma
p, and this is 5 comma t.

00:07:21.670 --> 00:07:24.530
What's important to realize is
that both of these points

00:07:24.530 --> 00:07:28.710
satisfy this relationship,
right?

00:07:28.710 --> 00:07:31.720
Because they're on this curve.

00:07:31.720 --> 00:07:34.655
So we can do that with
our math, right?

00:07:34.655 --> 00:07:40.905
We could write this as
0 comma-- Well, if x

00:07:40.905 --> 00:07:43.360
is 0, what is y?

00:07:43.360 --> 00:07:48.900
Well, 0 is equal to
y squared minus 4.

00:07:48.900 --> 00:07:50.580
We add 4 to both sides.

00:07:50.580 --> 00:07:52.810
Y squared is equal to 4.

00:07:52.810 --> 00:07:56.160
So y is equal to plus
or minus 2.

00:07:56.160 --> 00:07:57.790
And that's the tricky part,
right? y is equal to

00:07:57.790 --> 00:07:59.040
plus or minus 2.

00:08:02.620 --> 00:08:04.360
And then what does this equal?

00:08:04.360 --> 00:08:07.800
If x is equal to 5, what is t?

00:08:07.800 --> 00:08:10.330
What could this term
right here be?

00:08:10.330 --> 00:08:18.490
So let's see, 5 is equal
to y squared minus 4.

00:08:18.490 --> 00:08:20.370
y squared is equal to 9.

00:08:20.370 --> 00:08:22.815
y is equal to plus or minus 3.

00:08:27.830 --> 00:08:29.080
Fascinating.

00:08:33.620 --> 00:08:36.770
Actually I drew this
original graph.

00:08:36.770 --> 00:08:38.659
Since it's written this way,
since it's not written as a

00:08:38.659 --> 00:08:40.789
square root, it actually
would kind of do

00:08:40.789 --> 00:08:42.039
something like this.

00:08:45.750 --> 00:08:47.180
You don't have to draw this.

00:08:47.180 --> 00:08:47.870
So anyway.

00:08:47.870 --> 00:08:50.600
Plus or minus 3.

00:08:50.600 --> 00:08:53.240
So these are the possible
two points we could have

00:08:53.240 --> 00:08:54.120
intersected.

00:08:54.120 --> 00:08:57.130
And what the question is
asking is, what is the

00:08:57.130 --> 00:09:00.130
greatest possible value
of the slope of l?

00:09:00.130 --> 00:09:04.460
What we want to figure out--
If we want the greatest

00:09:04.460 --> 00:09:09.170
possible value, we want to go
from a very low number to a

00:09:09.170 --> 00:09:12.350
very high number, right?

00:09:12.350 --> 00:09:14.880
The x values are fixed.

00:09:14.880 --> 00:09:18.200
We want as great a possible
change in y, given that change

00:09:18.200 --> 00:09:19.840
in x, right?

00:09:19.840 --> 00:09:22.240
So if we want the lowest
possible value here, we'll go

00:09:22.240 --> 00:09:23.540
with the negative 2.

00:09:23.540 --> 00:09:25.100
If we want the highest
possible value of x,

00:09:25.100 --> 00:09:27.082
we'll go with 3.

00:09:27.082 --> 00:09:28.320
So let's see.

00:09:28.320 --> 00:09:31.940
So if we assume that, the
change in y is 3 minus

00:09:31.940 --> 00:09:37.330
negative 2 over change
in x, 5 minus 0.

00:09:37.330 --> 00:09:39.050
Adds up, right?

00:09:39.050 --> 00:09:41.830
So this is 3 plus
2 is 5 over 5.

00:09:41.830 --> 00:09:43.420
It equals 1.

00:09:43.420 --> 00:09:46.170
That is the greatest
possible slope.

00:09:46.170 --> 00:09:48.210
I'll see you in the
next video.


SAT Prep: Test 1 Section 6 Part 6

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.350
Actually, before moving on to
the next problem, sometimes at

00:00:02.350 --> 00:00:04.980
the end of a video I really rush
it, just because I have

00:00:04.980 --> 00:00:08.330
that silly YouTube 10
minute deadline.

00:00:08.330 --> 00:00:11.270
And I want you to really
understand this point here, of

00:00:11.270 --> 00:00:14.200
why I picked the negative
2 and the positive 3.

00:00:14.200 --> 00:00:19.670
Let me erase some of this and
kind of show you what my

00:00:19.670 --> 00:00:21.805
thought process was.

00:00:21.805 --> 00:00:23.740
So I'm using the paintbrush
tool.

00:00:23.740 --> 00:00:28.460
Let me just blank it all out.

00:00:28.460 --> 00:00:35.830
So if I just blank that
out, and I could

00:00:35.830 --> 00:00:37.730
actually blank this out.

00:00:37.730 --> 00:00:39.240
I'm just blanking
everything out.

00:00:41.930 --> 00:00:43.790
I could have just erased
everything, but anyway.

00:00:49.970 --> 00:00:56.860
So the possible terms again,
once again, were 0 plus or

00:00:56.860 --> 00:01:00.550
minus 2, and 5 plus
or minus 3.

00:01:00.550 --> 00:01:03.790
And I want to show you why I
picked the lowest value here,

00:01:03.790 --> 00:01:05.510
and the highest value here.

00:01:05.510 --> 00:01:07.160
So let me graph those points.

00:01:10.120 --> 00:01:12.710
That's my y-axis.

00:01:12.710 --> 00:01:14.020
This is my x-axis.

00:01:16.660 --> 00:01:23.140
And so, first I have
the point, 0 comma

00:01:23.140 --> 00:01:24.410
plus 2 or minus 2.

00:01:24.410 --> 00:01:30.580
So this is 0, 2.

00:01:30.580 --> 00:01:31.830
This is 0, minus 2.

00:01:35.880 --> 00:01:36.670
And then we have 5.

00:01:36.670 --> 00:01:40.340
Let's say this is 5 out here.

00:01:40.340 --> 00:01:42.960
I'm going to say this
is 3 up here.

00:01:42.960 --> 00:01:50.310
This is 5, positive 3, and then
5, negative 3 will be

00:01:50.310 --> 00:01:51.960
someplace around here,
just not scaled up.

00:01:51.960 --> 00:01:53.760
I have to squeeze at the bottom,
but you get the point.

00:01:56.970 --> 00:02:00.050
So there's a bunch of lines that
could connect this whole

00:02:00.050 --> 00:02:01.880
set of points, right?

00:02:01.880 --> 00:02:03.660
I'm going to do them in
different colors.

00:02:03.660 --> 00:02:06.380
You have this line.

00:02:06.380 --> 00:02:10.150
You have this line.

00:02:10.150 --> 00:02:13.010
You have this line.

00:02:13.010 --> 00:02:16.650
And then you have this line.

00:02:16.650 --> 00:02:19.070
Well, we want the highest
possible slope.

00:02:19.070 --> 00:02:21.290
And you could probably just
inspect it and say, well, the

00:02:21.290 --> 00:02:23.540
highest possible slope is the
one that moves up the

00:02:23.540 --> 00:02:26.310
steepest. So it'll
be this line.

00:02:26.310 --> 00:02:28.850
And this line and this line are
negative slopes, so that's

00:02:28.850 --> 00:02:29.820
not a high slope.

00:02:29.820 --> 00:02:32.230
And the way I did it,
intuitively, I said, well, in

00:02:32.230 --> 00:02:34.280
order to have a high slope,
I want as high a

00:02:34.280 --> 00:02:35.690
change in y as possible.

00:02:35.690 --> 00:02:49.480
I want to maximize my change
in y, given my change in x.

00:02:49.480 --> 00:02:53.280
And that's why I picked the
lowest y and the highest y.

00:02:53.280 --> 00:02:56.330
If I wanted the lowest possible
slope, I would have

00:02:56.330 --> 00:03:00.150
picked positive 2
and negative 3.

00:03:00.150 --> 00:03:03.350
Hopefully that makes a little
bit more sense.

00:03:03.350 --> 00:03:11.720
Let's see, let's do
problem number 18.

00:03:11.720 --> 00:03:14.930
Esther drove to work in the
morning at an average speed of

00:03:14.930 --> 00:03:16.610
45 miles per hour.

00:03:16.610 --> 00:03:22.690
45 miles per hour is how
fast Esther drove.

00:03:22.690 --> 00:03:23.940
So this is to work.

00:03:27.420 --> 00:03:30.540
She returned home in the
evening, along the same route

00:03:30.540 --> 00:03:33.370
at an average of 30
miles per hour.

00:03:33.370 --> 00:03:34.900
30 miles per hour-- maybe
there was more

00:03:34.900 --> 00:03:36.980
traffic-- from work.

00:03:40.460 --> 00:03:44.460
If Esther spent a total of one
hour commuting to and from

00:03:44.460 --> 00:03:54.710
work, so one hour total, how
many miles did Esther drive to

00:03:54.710 --> 00:03:58.750
work in the morning?

00:03:58.750 --> 00:04:01.620
So this is a simple distance
is equal to rate

00:04:01.620 --> 00:04:05.890
times to time problem.

00:04:05.890 --> 00:04:06.635
But it's a little fancy.

00:04:06.635 --> 00:04:08.380
It gets a little confusing.

00:04:08.380 --> 00:04:15.020
So let's just say that d is
the distance to work.

00:04:15.020 --> 00:04:17.290
And then we'll just solve
for d, right?

00:04:17.290 --> 00:04:24.190
So if d is the distance to work,
how far did she go in

00:04:24.190 --> 00:04:25.180
the morning?

00:04:25.180 --> 00:04:29.840
So in the morning we have the
distance she went in the

00:04:29.840 --> 00:04:32.790
morning is going to be equal
to 45 miles per hour.

00:04:32.790 --> 00:04:35.750
That's the rate.

00:04:35.750 --> 00:04:39.290
45 miles per hour
times the time.

00:04:39.290 --> 00:04:42.880
So let's just call
that times t.

00:04:42.880 --> 00:04:45.510
That's what she did
in the morning.

00:04:45.510 --> 00:04:47.500
She went 45 miles an
hour times however

00:04:47.500 --> 00:04:49.160
long it took her.

00:04:49.160 --> 00:04:51.300
Then in the afternoon, she's
going to go to the same

00:04:51.300 --> 00:04:54.340
distance, right?

00:04:54.340 --> 00:04:56.020
This time she went 30
miles per hour.

00:04:58.890 --> 00:05:00.540
Now, how long did it take her?

00:05:00.540 --> 00:05:02.410
Well, this is the interesting
thing.

00:05:02.410 --> 00:05:07.090
It took her a total of one
hour to go to work

00:05:07.090 --> 00:05:09.670
and back from work.

00:05:09.670 --> 00:05:18.110
If it took her t hours in the
morning, then it took her 1

00:05:18.110 --> 00:05:22.370
minus t hours.

00:05:22.370 --> 00:05:23.320
Actually, let's do
it in minutes.

00:05:23.320 --> 00:05:27.230
Let's assume t is in minutes.

00:05:27.230 --> 00:05:32.740
So then one hour is 60,
60 minus t minutes.

00:05:32.740 --> 00:05:33.990
Actually, I have to stay within
hours, because these

00:05:33.990 --> 00:05:35.280
are all miles per hour.

00:05:35.280 --> 00:05:38.010
So forgive me.

00:05:38.010 --> 00:05:40.620
1 minus t.

00:05:40.620 --> 00:05:42.220
Does that make sense?

00:05:42.220 --> 00:05:43.570
This one, I think, will
make sense to you.

00:05:43.570 --> 00:05:45.540
45 miles per hour times however

00:05:45.540 --> 00:05:46.480
many hours she traveled.

00:05:46.480 --> 00:05:48.390
It tells you how many
miles she went.

00:05:48.390 --> 00:05:52.260
This time you have her speed,
30 miles per hour, times the

00:05:52.260 --> 00:05:53.530
time she traveled.

00:05:53.530 --> 00:05:56.350
Well, the time she traveled in
the afternoon is going to be

00:05:56.350 --> 00:06:01.150
one hour minus whatever she
traveled in the morning.

00:06:01.150 --> 00:06:04.500
Because when you combine in the
morning and afternoon, she

00:06:04.500 --> 00:06:07.490
traveled for a total
of one hour.

00:06:07.490 --> 00:06:11.485
For example, if she took 15
minutes to go in the morning,

00:06:11.485 --> 00:06:14.060
then that means it took
her 45 minutes to

00:06:14.060 --> 00:06:15.980
come back in the afternoon.

00:06:15.980 --> 00:06:17.320
Hopefully that makes sense.

00:06:17.320 --> 00:06:19.510
But these are the
same d's, right?

00:06:19.510 --> 00:06:21.550
So we could set them equal
to each other.

00:06:21.550 --> 00:06:31.500
So we could say 45 times t is
equal to 30 times 1 minus t.

00:06:31.500 --> 00:06:31.870
Let me see.

00:06:31.870 --> 00:06:38.290
45t is equal to 30 minus 30t.

00:06:38.290 --> 00:06:43.960
And then we can add 30t to both
sides and you get 75t is

00:06:43.960 --> 00:06:45.550
equal to 30.

00:06:45.550 --> 00:06:51.000
Then we get t is equal
to 30/75 hours.

00:06:51.000 --> 00:06:56.905
And if we divide the top and
the bottom by 5, we get 6

00:06:56.905 --> 00:07:01.070
over-- 5 goes into this
how many times?

00:07:01.070 --> 00:07:02.490
Actually, yeah, let
me divide it by 5.

00:07:02.490 --> 00:07:06.430
5 goes into 10, 15, right?

00:07:06.430 --> 00:07:09.460
I could divide by a better
number than 5.

00:07:09.460 --> 00:07:11.815
Actually, I could divide the
top and the bottom by 15.

00:07:11.815 --> 00:07:17.110
And you get 2/5.

00:07:17.110 --> 00:07:18.600
Or just take this and divide
the top and bottom

00:07:18.600 --> 00:07:20.420
by 3, you get 2/5.

00:07:20.420 --> 00:07:22.680
So time is equal to 2/5 hours.

00:07:25.280 --> 00:07:28.590
That's 24 minutes,
roughly, right?

00:07:28.590 --> 00:07:31.600
So we go back to the distance.

00:07:31.600 --> 00:07:35.240
And remember, t is how long it
took her in the morning.

00:07:35.240 --> 00:07:37.460
In the afternoon, it's going to
be 1 minus this, so in the

00:07:37.460 --> 00:07:38.920
afternoon it took her
1 minus this.

00:07:38.920 --> 00:07:41.420
This would be 3/5 hours.

00:07:41.420 --> 00:07:43.940
The question is, how
many miles did she

00:07:43.940 --> 00:07:45.110
go to and from work?

00:07:45.110 --> 00:07:48.390
So that's d.

00:07:48.390 --> 00:07:50.510
Well, d is 45 times t.

00:07:50.510 --> 00:07:52.950
t is 2/5.

00:07:52.950 --> 00:07:59.333
45 times 2/5 is d.

00:07:59.333 --> 00:08:03.720
It comes to 9, cancel that,
so it comes to 8.

00:08:03.720 --> 00:08:04.670
18, I mean.

00:08:04.670 --> 00:08:07.050
9 times 2 is 18.

00:08:07.050 --> 00:08:10.440
So d is equal to 18 miles.

00:08:10.440 --> 00:08:13.910
And we are done, our
second SAT section.

00:08:13.910 --> 00:08:16.530
I hope I didn't confuse
you too much.

00:08:16.530 --> 00:08:18.000
See you soon.


SAT Prep: Test 1 Section 3 Part 7

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:01.290
All right.

00:00:01.290 --> 00:00:04.530
We're in the home stretch of
this first, I guess we could

00:00:04.530 --> 00:00:06.310
have called almost experimental
section, but

00:00:06.310 --> 00:00:08.820
hopefully it's been working
out decently.

00:00:08.820 --> 00:00:12.980
So we're on number 17.

00:00:12.980 --> 00:00:26.830
And it says, if a plus 2, times
x plus 1 is equal to s,

00:00:26.830 --> 00:00:31.800
what is x plus 1 in
terms of s and a?

00:00:31.800 --> 00:00:34.770
So x plus 1 is essentially
equal to what?

00:00:34.770 --> 00:00:39.250
So let's just take this
equation-- so what happens if

00:00:39.250 --> 00:00:41.690
we subtract a from both sides?

00:00:41.690 --> 00:00:49.550
Well, we get 2 times x plus
1 is equal to s minus a.

00:00:49.550 --> 00:00:51.550
I just subtracted a
from both sides.

00:00:51.550 --> 00:00:53.880
Let me switch colors.

00:00:53.880 --> 00:00:57.820
And then if we divide both sides
by 2, we get x plus 1 is

00:00:57.820 --> 00:01:02.150
equal to s minus a over 2.

00:01:02.150 --> 00:01:03.690
And I think we're done.

00:01:03.690 --> 00:01:04.390
That's it.

00:01:04.390 --> 00:01:06.450
That was choice B.

00:01:06.450 --> 00:01:08.520
It was actually surprising to
see a problem that took so

00:01:08.520 --> 00:01:10.430
little time this late in the
test. Because they normally

00:01:10.430 --> 00:01:12.360
get kind of harder as
you go through.

00:01:12.360 --> 00:01:14.240
And I guess the reason why this
is a little tricky is a

00:01:14.240 --> 00:01:18.420
lot of people's initial
motivation would be to try to

00:01:18.420 --> 00:01:20.510
solve for x, and then
try to figure out

00:01:20.510 --> 00:01:21.870
what x plus 1 equals.

00:01:21.870 --> 00:01:25.440
But the trick is to just not
break apart this x plus 1.

00:01:25.440 --> 00:01:27.390
I mean, you could kind of view
this as a variable, once

00:01:27.390 --> 00:01:30.820
again, like a couple of problems
ago, where you have

00:01:30.820 --> 00:01:33.070
to avoid distributing
this 2 out.

00:01:33.070 --> 00:01:36.900
If you just keep it as x plus 1
and you solve for x plus 1,

00:01:36.900 --> 00:01:38.860
you get very quickly to
s minus a over 2,

00:01:38.860 --> 00:01:41.240
which is choice B.

00:01:41.240 --> 00:01:43.250
The next problem, number 18.

00:01:43.250 --> 00:01:44.350
We are in the home stretch.

00:01:44.350 --> 00:01:46.560
And this is normally when
the problems get on

00:01:46.560 --> 00:01:47.950
the difficult side.

00:01:47.950 --> 00:01:50.050
So let me clear this, because
it looks like it might take

00:01:50.050 --> 00:01:51.300
some space.

00:01:53.420 --> 00:01:55.350
The shaded region above,
so let me draw the

00:01:55.350 --> 00:01:56.600
shaded region above.

00:01:59.190 --> 00:02:03.570
That's the y-axis-- oh man, I
want to draw it straight, but

00:02:03.570 --> 00:02:06.130
anyway, I'll deal.

00:02:06.130 --> 00:02:06.745
I'll deal.

00:02:06.745 --> 00:02:08.030
You can't always be perfect.

00:02:10.660 --> 00:02:18.570
So this is the y-axis,
this is the x-axis.

00:02:18.570 --> 00:02:23.130
And then I have this neat
looking curve that looks

00:02:23.130 --> 00:02:24.380
something like this.

00:02:30.770 --> 00:02:33.840
The curve keeps going.

00:02:33.840 --> 00:02:35.620
Let's see, it draws
off some points.

00:02:35.620 --> 00:02:40.640
So it's like 1, 2, 3, 4.

00:02:40.640 --> 00:02:42.370
And it draws this boundary
line here.

00:02:42.370 --> 00:02:45.550
It draws this boundary
line at 4.

00:02:45.550 --> 00:02:46.800
So it's like this.

00:02:50.090 --> 00:02:53.510
And then they shade the region,
and I have a feeling

00:02:53.510 --> 00:02:56.990
you're going to ask about
the shaded region.

00:02:56.990 --> 00:02:59.010
Actually, let me-- because
I'm going to use

00:02:59.010 --> 00:03:01.290
the shaded in tools.

00:03:01.290 --> 00:03:05.250
Let me make sure that I don't
end up filling up the whole

00:03:05.250 --> 00:03:08.800
screen, so I want to make sure
that I close all the gaps.

00:03:08.800 --> 00:03:10.590
So now let me use the
shaded in tool.

00:03:13.160 --> 00:03:14.930
Ah, there, it worked out.

00:03:14.930 --> 00:03:17.120
All right, so what
do they say?

00:03:17.120 --> 00:03:22.380
The shaded region in the figure
above is bounded by the

00:03:22.380 --> 00:03:24.300
x-axis, that's this.

00:03:24.300 --> 00:03:28.280
The line x equals 4,
that's this line.

00:03:28.280 --> 00:03:30.580
And the graph y equals f of x.

00:03:30.580 --> 00:03:32.480
That's this, that's
the brown line.

00:03:32.480 --> 00:03:36.140
y is equal to-- I never use
grey, maybe I should use grey.

00:03:36.140 --> 00:03:37.270
I don't like grey.

00:03:37.270 --> 00:03:39.010
Anyway, green.

00:03:39.010 --> 00:03:42.330
If the point a, b lies in the
shaded region-- so we don't

00:03:42.330 --> 00:03:42.660
know where.

00:03:42.660 --> 00:03:45.660
It's some place in this-- let
me use a darker color.

00:03:45.660 --> 00:03:47.330
Let me use navy blue.

00:03:47.330 --> 00:03:52.290
This point a, b is some place
in the shaded region.

00:03:52.290 --> 00:03:54.300
Which of the following
must be true?

00:03:54.300 --> 00:03:56.960
And they have choice number
one-- now I have to use a

00:03:56.960 --> 00:04:03.630
lighter color-- choice number
one is that a is less than or

00:04:03.630 --> 00:04:06.740
equal to 4.

00:04:06.740 --> 00:04:14.130
Choice number two is that b is
less than or equal to a.

00:04:14.130 --> 00:04:22.990
And choice number three is
that b is less than or

00:04:22.990 --> 00:04:26.700
equal to f of a.

00:04:26.700 --> 00:04:28.100
Interesting.

00:04:28.100 --> 00:04:30.855
So we know that a, b is in this
grey region, so it's some

00:04:30.855 --> 00:04:32.040
place around there.

00:04:32.040 --> 00:04:37.410
So if that's a, b, so that means
that this point-- and

00:04:37.410 --> 00:04:39.510
it's not exact. a, b could have
been anywhere in this

00:04:39.510 --> 00:04:42.090
grey area, but let's say
that this point is a.

00:04:42.090 --> 00:04:46.470
And we know that this
point is b.

00:04:46.470 --> 00:04:47.830
So we know a couple of things.

00:04:47.830 --> 00:04:53.765
We know that a is some place
between 0 and 4.

00:04:58.070 --> 00:05:01.540
So that means that a is less
than or equal to 4, because it

00:05:01.540 --> 00:05:04.420
could be 4, because we're
including the boundary.

00:05:04.420 --> 00:05:06.000
So this is correct.

00:05:06.000 --> 00:05:08.800
We know this is true.

00:05:08.800 --> 00:05:09.800
Let's see.

00:05:09.800 --> 00:05:16.180
We also know-- well, we don't
know whether-- let's see, b is

00:05:16.180 --> 00:05:17.740
less than a.

00:05:17.740 --> 00:05:19.480
How can we come to
that conclusion?

00:05:19.480 --> 00:05:26.320
Because if we had the point that
was, like, right-- well,

00:05:26.320 --> 00:05:27.920
even where it's here, but
let's say that this

00:05:27.920 --> 00:05:31.800
point was our a, b.

00:05:31.800 --> 00:05:34.580
And I mean, just eyeballing it,
just looking at the graph,

00:05:34.580 --> 00:05:37.570
it just looks-- I mean,
we don't know.

00:05:37.570 --> 00:05:39.350
It looks like b in this
situation would

00:05:39.350 --> 00:05:41.150
be larger than a.

00:05:41.150 --> 00:05:44.170
So I definitely don't think we
can make that conclusion.

00:05:44.170 --> 00:05:48.420
At least, there's no obvious way
to make that conclusion.

00:05:48.420 --> 00:05:50.790
Because we don't know what f
of x is. f of x could be

00:05:50.790 --> 00:05:54.210
really steep and have a really
high slope, in which case, b

00:05:54.210 --> 00:05:57.440
would grow faster than a at
this point earlier on.

00:05:57.440 --> 00:05:59.460
So I don't think that
this is right.

00:05:59.460 --> 00:06:01.690
So this is not right.

00:06:01.690 --> 00:06:03.980
And then let's see, is
b less than f of a?

00:06:03.980 --> 00:06:04.840
So what is f of a?

00:06:04.840 --> 00:06:07.440
If this is point a,
what is f of a?

00:06:07.440 --> 00:06:11.760
Well, f of a is going
to be right here.

00:06:11.760 --> 00:06:13.010
This is f of a.

00:06:17.440 --> 00:06:19.840
And no matter where you draw it,
b is definitely going to

00:06:19.840 --> 00:06:20.430
be less than f of a.

00:06:20.430 --> 00:06:24.370
If you picked this as our point,
and if you said that

00:06:24.370 --> 00:06:34.050
this is b, and that this is a,
then this would be f of a in

00:06:34.050 --> 00:06:36.210
bloody red.

00:06:36.210 --> 00:06:40.210
And b would still be less than
f of a, because b is in the

00:06:40.210 --> 00:06:43.500
shaded region that is less
than f of a, essentially.

00:06:43.500 --> 00:06:48.590
So we also know that this
last term is correct.

00:06:48.590 --> 00:06:49.410
Hopefully that makes sense.

00:06:49.410 --> 00:06:50.600
If this one doesn't make
sense, I want you to

00:06:50.600 --> 00:06:52.150
experiment with it.

00:06:52.150 --> 00:06:53.290
And you'll see.

00:06:53.290 --> 00:06:59.420
But we're essentially saying
that the point is all the y

00:06:59.420 --> 00:07:03.260
values less than f of x.

00:07:03.260 --> 00:07:05.650
This is the shaded region
essentially.

00:07:05.650 --> 00:07:06.820
So let's see, what
choice is that?

00:07:06.820 --> 00:07:08.230
One and three.

00:07:08.230 --> 00:07:11.320
One and three only,
that is choice D.

00:07:11.320 --> 00:07:14.460
D as in dog.

00:07:14.460 --> 00:07:16.020
Number 19.

00:07:16.020 --> 00:07:17.050
I don't know if we're going
to have time for this.

00:07:17.050 --> 00:07:21.050
I might have to do it
in the next video.

00:07:21.050 --> 00:07:22.850
Actually, let's just do it in
the next video, because I have

00:07:22.850 --> 00:07:24.680
two more problems, and I'm going
to have to have another

00:07:24.680 --> 00:07:25.930
video no matter what.

00:07:25.930 --> 00:07:28.380
So I'll see you in
the next video.


SAT Prep: Test 1 Section 3 Part 6

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:01.500
Welcome back.

00:00:01.500 --> 00:00:04.510
So we're on problem number 15.

00:00:04.510 --> 00:00:07.210
Boy, this SAT can be tiring.

00:00:07.210 --> 00:00:08.790
I know you probably rolled your
eyes when I said that

00:00:08.790 --> 00:00:11.060
because you're the one that
has to take it, not me.

00:00:11.060 --> 00:00:12.330
Anyway, number 15.

00:00:12.330 --> 00:00:15.730
The graph above shows-- and this
is on page 399-- shows

00:00:15.730 --> 00:00:17.280
Alena's height in inches.

00:00:17.280 --> 00:00:18.820
So the graph looks like this.

00:00:29.960 --> 00:00:31.570
And then I'll draw the
rest. I don't have to

00:00:31.570 --> 00:00:32.580
draw these fancy lines.

00:00:32.580 --> 00:00:33.470
You get the point.

00:00:33.470 --> 00:00:36.690
So this is her age.

00:00:36.690 --> 00:00:45.490
Let me do it at 6,
8, 10, and 12.

00:00:45.490 --> 00:00:50.070
And then when she's 6,
she is 45 inches.

00:00:50.070 --> 00:00:51.150
This is in inches.

00:00:51.150 --> 00:00:52.990
So at 6 she's at 45.

00:00:52.990 --> 00:01:00.060
At 8 she's at 50.

00:01:00.060 --> 00:01:04.830
At 10 she's at 55-- no, a little
less than-- well, this

00:01:04.830 --> 00:01:09.320
is like-- on the graph
it looks like 55.

00:01:09.320 --> 00:01:11.740
And the way they draw it, it's
like a little bit lower than

00:01:11.740 --> 00:01:16.840
55, so it's like a little bit
lower than 55 on the graph.

00:01:16.840 --> 00:01:25.800
And then at 12, she's
right at 60.

00:01:25.800 --> 00:01:29.500
And then they draw a line
connecting all of them so it

00:01:29.500 --> 00:01:31.650
looks something like this.

00:01:31.650 --> 00:01:35.220
So the line looks like
this, like this.

00:01:35.220 --> 00:01:37.360
It's nothing fancy.

00:01:37.360 --> 00:01:42.540
And it says Alena's height at
age 12 was what percent

00:01:42.540 --> 00:01:45.390
greater than a height
at age 6?

00:01:45.390 --> 00:01:49.030
So when she is 12,
age, height.

00:01:52.540 --> 00:01:54.600
So at age 12, she's what?

00:01:54.600 --> 00:01:56.910
At age 12, she's
60 centimeters.

00:01:56.910 --> 00:01:57.930
That's this point, right?

00:01:57.930 --> 00:02:00.300
60.

00:02:00.300 --> 00:02:03.960
And at age 6 she's what?

00:02:03.960 --> 00:02:06.100
45 centimeters, right?

00:02:06.100 --> 00:02:09.669
Age 6, 45.

00:02:09.669 --> 00:02:12.750
So this question is really-- so
we've got to make sure that

00:02:12.750 --> 00:02:13.400
we're reading right.

00:02:13.400 --> 00:02:18.220
Alena's height at the age of
12 was what percent greater

00:02:18.220 --> 00:02:19.850
than her height at
the age of 6?

00:02:19.850 --> 00:02:26.610
So they're really saying 60 is
what percent greater than 45?

00:02:26.610 --> 00:02:30.690
And so I actually do this
all the time in my job.

00:02:30.690 --> 00:02:32.060
I work in a hedge fund.

00:02:32.060 --> 00:02:33.450
But this is what I
do all the time.

00:02:33.450 --> 00:02:36.220
And a really quick and dirty way
of doing this, you would

00:02:36.220 --> 00:02:39.660
literally say, well, this is the
same thing as 60 divided

00:02:39.660 --> 00:02:45.480
by 45 minus 1 to get you the
percentage, because-- well, I

00:02:45.480 --> 00:02:46.930
don't want to confuse
you too much.

00:02:46.930 --> 00:02:49.820
But 60 divided by 45
will tell you what

00:02:49.820 --> 00:02:53.750
percentage 60 is of 45.

00:02:53.750 --> 00:02:56.310
So 60 divided by 45 is what?

00:02:56.310 --> 00:03:03.290
It's 1 and 15/45, or
1.333333, right?

00:03:03.290 --> 00:03:04.720
I think you can confirm that.

00:03:04.720 --> 00:03:08.200
So that equals 1.333333.

00:03:08.200 --> 00:03:22.440
Or 60 is 133% of 45.

00:03:22.440 --> 00:03:23.640
We can make all of
these statements.

00:03:23.640 --> 00:03:26.860
60 over 45, that's a percent
that 60 is of 45.

00:03:26.860 --> 00:03:28.180
That's 1.333333.

00:03:28.180 --> 00:03:29.760
It keeps repeating, right?

00:03:29.760 --> 00:03:31.340
Repeating.

00:03:31.340 --> 00:03:34.070
But they want to know what
percent greater than her

00:03:34.070 --> 00:03:36.120
height at age 6.

00:03:36.120 --> 00:03:40.390
So 60 is 133% of 45.

00:03:40.390 --> 00:03:47.210
This is the same thing as
saying that 60 is 33%

00:03:47.210 --> 00:03:53.420
greater than 45.

00:03:53.420 --> 00:03:55.380
Hopefully, you understand
what I'm saying.

00:03:55.380 --> 00:03:58.930
Saying one thing is 133% of
another thing is the same

00:03:58.930 --> 00:04:02.650
thing as saying that this
first thing is 33%

00:04:02.650 --> 00:04:04.890
greater than 45.

00:04:04.890 --> 00:04:13.690
Or we could say that saying
something 133% is equal to 33%

00:04:13.690 --> 00:04:16.040
more than 100%, right?

00:04:16.040 --> 00:04:17.400
More than 100.

00:04:17.400 --> 00:04:19.820
I hope I'm not confusing you.

00:04:19.820 --> 00:04:23.100
I try to do this in a couple of
different ways to help you

00:04:23.100 --> 00:04:27.370
understand what 33%
greater means.

00:04:27.370 --> 00:04:35.120
If x is 33% greater than y,
that means x is 133% of y.

00:04:35.120 --> 00:04:37.520
So they want to know how much
greater is 60 than 45?

00:04:37.520 --> 00:04:41.750
Well, it's 33% greater.

00:04:41.750 --> 00:04:43.983
And so the answer is C,
33 and 1/3, because

00:04:43.983 --> 00:04:45.950
it's a repeating 33.

00:04:45.950 --> 00:04:47.200
Next problem.

00:04:51.730 --> 00:04:54.100
This looks fun.

00:04:54.100 --> 00:04:58.486
So we have this cylinder
looking thing.

00:04:58.486 --> 00:05:02.720
Let's see how well I can
draw a cylinder.

00:05:02.720 --> 00:05:06.490
Well, my cylinder's going to
be a little tilted, but you

00:05:06.490 --> 00:05:07.630
get the point.

00:05:07.630 --> 00:05:12.520
I should have drawn these
vertical lines first. This is

00:05:12.520 --> 00:05:15.258
the cylinder.

00:05:15.258 --> 00:05:17.030
So that's the cylinder.

00:05:17.030 --> 00:05:21.500
And they're saying that on the
base of the cylinder, we have

00:05:21.500 --> 00:05:26.720
this line x, or this
distance x.

00:05:26.720 --> 00:05:28.040
It goes to the center.

00:05:28.040 --> 00:05:31.390
It's essentially the center
of the cylinder.

00:05:31.390 --> 00:05:33.390
Let me draw it in a more muted
color to show that's it's in

00:05:33.390 --> 00:05:37.640
the inside, and the height of
the cylinder goes to the top.

00:05:40.530 --> 00:05:43.680
They're essentially asking which
of the following-- and

00:05:43.680 --> 00:05:48.020
they give you a bunch of
choices-- has the same volume

00:05:48.020 --> 00:05:51.340
as the cylinder shown with
radius x and height to x?

00:05:51.340 --> 00:05:58.690
So they're saying this is x,
that's x and this is 2x.

00:05:58.690 --> 00:06:01.000
So before figuring out which of
these choices has the same

00:06:01.000 --> 00:06:03.110
volume as this thing, let's
figure out what the volume of

00:06:03.110 --> 00:06:04.680
this thing is.

00:06:04.680 --> 00:06:09.050
Well, if you know the area of a
circle, you essentially know

00:06:09.050 --> 00:06:12.140
the volume of a cylinder,
because it's really the area

00:06:12.140 --> 00:06:14.510
of one-- you can kind of figure
out the area of this

00:06:14.510 --> 00:06:17.450
base and then just multiply it
times the height, right?

00:06:17.450 --> 00:06:20.000
Multiply it times this third
dimension of the height.

00:06:20.000 --> 00:06:22.360
So what's the area
of this base?

00:06:22.360 --> 00:06:26.920
So volume is equal to area of
the base, which is pi times

00:06:26.920 --> 00:06:29.150
the radius squared, so
that's pi x squared.

00:06:32.630 --> 00:06:34.460
And then we're going to multiply
that times the

00:06:34.460 --> 00:06:36.830
height, times this, times 2x.

00:06:40.870 --> 00:06:44.540
So the volume of this is
equal to 2x times this.

00:06:44.540 --> 00:06:48.710
You get 2 pi x to the
third, right?

00:06:48.710 --> 00:06:51.300
Because this is x to the first.
Add the exponents.

00:06:51.300 --> 00:06:53.990
So volume is 2 pi
x to the third.

00:06:53.990 --> 00:06:55.840
We know the volume
of our cylinder.

00:06:55.840 --> 00:06:58.370
So let's see what these
other choices are.

00:06:58.370 --> 00:07:06.120
So choice A says a cylinder with
radius 2x and height x.

00:07:06.120 --> 00:07:11.580
So radius is equal to 2x and
the height is equal to x.

00:07:11.580 --> 00:07:13.300
So what would this be?

00:07:13.300 --> 00:07:21.250
The volume would be this squared
times pi times this

00:07:21.250 --> 00:07:25.010
because we take the area of the
base, so the area of the

00:07:25.010 --> 00:07:26.880
base would be what?

00:07:26.880 --> 00:07:33.020
It would be pi times 2x squared,
and we multiply that

00:07:33.020 --> 00:07:35.130
times the height.

00:07:35.130 --> 00:07:36.740
And this equals-- I can already
tell you this isn't

00:07:36.740 --> 00:07:39.140
going to be right because
we've got a 4 in here.

00:07:39.140 --> 00:07:40.120
Oh, this is x, right?

00:07:40.120 --> 00:07:41.410
The height is x.

00:07:41.410 --> 00:07:48.180
So this will get 4 pi x to the
third if you multiply it out,

00:07:48.180 --> 00:07:52.050
so that's not it, B.

00:07:52.050 --> 00:07:58.710
A cylinder with radius 2 pi,
radius is equal to 2 pi x, and

00:07:58.710 --> 00:08:00.650
height is equal to x.

00:08:00.650 --> 00:08:02.110
And I can already tell you this
is going to be it because

00:08:02.110 --> 00:08:04.300
we're going to get something
with a pi squared in it

00:08:04.300 --> 00:08:10.500
because if you take the area
of the base, you get-- the

00:08:10.500 --> 00:08:18.340
volume will just be pi times
2 pi x squared times x.

00:08:18.340 --> 00:08:21.835
And we're going to get 4 pi
squared, and then you're going

00:08:21.835 --> 00:08:23.410
to have another pi, and
it's 4 pi cubed.

00:08:23.410 --> 00:08:25.800
It's not going to be the
same thing as this.

00:08:25.800 --> 00:08:28.080
Choice C.

00:08:28.080 --> 00:08:30.280
A cube with edge 2x.

00:08:30.280 --> 00:08:31.670
This is easy.

00:08:31.670 --> 00:08:35.429
If a cube has side 2x, it's
volume is just that side

00:08:35.429 --> 00:08:36.789
cubed, right?

00:08:36.789 --> 00:08:41.120
So volume would be 2x
to the third, and

00:08:41.120 --> 00:08:43.130
that equals 8x cubed.

00:08:43.130 --> 00:08:46.350
That's definitely not right.

00:08:46.350 --> 00:08:47.600
D.

00:08:49.200 --> 00:08:52.730
A cube with edge 2 pi x.

00:08:52.730 --> 00:08:53.420
So the same thing.

00:08:53.420 --> 00:08:57.220
The volume for this cube would
be 2 pi x to the third.

00:08:57.220 --> 00:08:58.100
And this is going to be what?

00:08:58.100 --> 00:09:00.130
8 pi to the third,
x to the third.

00:09:00.130 --> 00:09:01.810
Once again, not right.

00:09:01.810 --> 00:09:04.860
So assuming I haven't done
anything wrong yet, it's

00:09:04.860 --> 00:09:07.830
probably going to be choice
E, but let's confirm.

00:09:07.830 --> 00:09:09.460
Choice E.

00:09:09.460 --> 00:09:12.710
A rectangular solid with the
dimensions-- ah, see this

00:09:12.710 --> 00:09:13.770
looks very suspicious.

00:09:13.770 --> 00:09:17.088
So let me draw the rectangular
solid.

00:09:17.088 --> 00:09:27.210
A rectangular solid with the
dimensions x, 2x and pi x.

00:09:27.210 --> 00:09:29.920
So we just multiply width times
height times depth.

00:09:29.920 --> 00:09:35.990
So x times 2x times pi x.

00:09:35.990 --> 00:09:38.275
You would get 2 pi x cubed.

00:09:41.960 --> 00:09:46.230
And the answer is E.

00:09:46.230 --> 00:09:47.420
Very good.

00:09:47.420 --> 00:09:49.490
And you could have done that
faster just kind of inspecting

00:09:49.490 --> 00:09:50.880
some of these earlier ones.

00:09:50.880 --> 00:09:53.290
Anyway, I'll see you
in the next video.


SAT Prep: Test 1 Section 3 Part 8

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:01.630
Welcome back.

00:00:01.630 --> 00:00:02.710
Almost done.

00:00:02.710 --> 00:00:06.040
We're on problem number 19.

00:00:06.040 --> 00:00:07.330
Let me read it out loud.

00:00:07.330 --> 00:00:12.870
At a bottling company, machine
A fills a bottle with spring

00:00:12.870 --> 00:00:17.580
water, and machine B accepts the
bottle only if the fluid

00:00:17.580 --> 00:00:21.930
ounces is between 11
7/8 and 12 and 1/8.

00:00:21.930 --> 00:00:24.770
So let me say O for ounces.

00:00:24.770 --> 00:00:32.970
So the ounces have to be
between 11 and 7/8.

00:00:32.970 --> 00:00:34.450
That's O, not 0.

00:00:34.450 --> 00:00:38.010
Maybe I'll use x.
x for ounces.

00:00:38.010 --> 00:00:40.570
That's an x, that weird
looking symbol.

00:00:40.570 --> 00:00:43.790
And 12 and 1/8.

00:00:43.790 --> 00:00:47.130
That's what B accepts if the
ounces are between that.

00:00:47.130 --> 00:00:53.490
If B accepts a bottle containing
n fluid ounces,

00:00:53.490 --> 00:00:55.400
which of the following
describes all

00:00:55.400 --> 00:00:56.910
possible values of n?

00:00:56.910 --> 00:00:58.830
So they're actually saying
n for ounces.

00:00:58.830 --> 00:01:00.020
We could just say this.

00:01:00.020 --> 00:01:02.420
So n is between these two terms.
And they give a bunch

00:01:02.420 --> 00:01:05.300
of choices that involve
absolute values.

00:01:05.300 --> 00:01:07.250
And what I'm going to do is
actually-- I'm going to draw

00:01:07.250 --> 00:01:09.730
this on a number line, and
then I think the absolute

00:01:09.730 --> 00:01:12.010
values are going
to make sense.

00:01:12.010 --> 00:01:16.330
Let me do another color just to
change things a little bit.

00:01:16.330 --> 00:01:38.760
So if this is the number line
and this is 11 and 7/8.

00:01:38.760 --> 00:01:43.670
And this is 12 and 1/8.

00:01:43.670 --> 00:01:47.680
And n has to be between
these two numbers.

00:01:47.680 --> 00:01:50.800
n can be anything between
these two numbers.

00:01:50.800 --> 00:01:53.660
So when they're doing absolute
value, I guess you can see

00:01:53.660 --> 00:01:56.670
that they're saying that-- they
have all these 12's and

00:01:56.670 --> 00:01:58.750
all that-- and I think you could
immediately see that 12

00:01:58.750 --> 00:02:04.660
is right in the middle
of these two numbers.

00:02:04.660 --> 00:02:06.070
12 is right there
in the middle.

00:02:06.070 --> 00:02:07.110
Let me do a different color.

00:02:07.110 --> 00:02:08.020
This is 12.

00:02:08.020 --> 00:02:09.550
It's right smack dab
in the middle.

00:02:09.550 --> 00:02:15.010
And as long as n is a little bit
less than 12 or a little

00:02:15.010 --> 00:02:18.500
bit more than 12, we're OK.

00:02:18.500 --> 00:02:21.540
The machine B is going
to accept it, right?

00:02:21.540 --> 00:02:27.890
So how much more than
12 can n be?

00:02:27.890 --> 00:02:30.120
This is n. n can be anywhere
in this range.

00:02:30.120 --> 00:02:32.980
Obviously if n is 12,
it's like perfect.

00:02:32.980 --> 00:02:34.250
It's just like right
in the middle.

00:02:34.250 --> 00:02:37.960
It's definitely nowhere close
to one of the bad ounces.

00:02:37.960 --> 00:02:39.225
It's kind of the
perfect ounces.

00:02:39.225 --> 00:02:41.600
The perfect number of
ounces in the bottle

00:02:41.600 --> 00:02:42.980
of the spring water.

00:02:42.980 --> 00:02:43.880
Let's see.

00:02:43.880 --> 00:02:48.910
So how much more can--
it can be 1/8 more.

00:02:48.910 --> 00:02:50.140
This distance is 1/8.

00:02:50.140 --> 00:02:52.780
How much more can
n be than 12?

00:02:52.780 --> 00:02:53.680
It can be 1/8 more.

00:02:53.680 --> 00:02:54.490
And how much less?

00:02:54.490 --> 00:02:55.800
Well this distance
is also 1/8.

00:02:58.390 --> 00:03:04.240
So we know that n minus 12-- the
difference between n and

00:03:04.240 --> 00:03:09.750
12-- and we don't know if n
is greater than 12, or n

00:03:09.750 --> 00:03:11.080
is less than 12.

00:03:11.080 --> 00:03:12.560
So we'll say just
the difference.

00:03:12.560 --> 00:03:14.830
And we're going to take
the absolute value.

00:03:14.830 --> 00:03:18.850
Because if n is greater, the
absolute value doesn't matter.

00:03:18.850 --> 00:03:23.380
And if n is less, then n minus
12 will be a negative number.

00:03:23.380 --> 00:03:26.560
But we just care about the
absolute difference.

00:03:26.560 --> 00:03:29.500
We don't care whether one is
negative or one is positive.

00:03:29.500 --> 00:03:33.200
So we say that n minus 12 has
to be less-- the difference

00:03:33.200 --> 00:03:35.650
between n and 12-- and that's
just the absolute value of n

00:03:35.650 --> 00:03:38.210
minus 12-- has to be less
than or equal to

00:03:38.210 --> 00:03:40.065
this kind of tolerance.

00:03:40.065 --> 00:03:42.810
You probably haven't done much
manufacturing yet, but if you

00:03:42.810 --> 00:03:45.270
do this is the kind of thing
you'll deal with.

00:03:45.270 --> 00:03:49.080
So it has to be less than
or equal to 1/8.

00:03:49.080 --> 00:03:50.250
And I want you to play
around with this.

00:03:50.250 --> 00:03:53.760
I mean you could say, well
n can't be more than

00:03:53.760 --> 00:03:55.470
1/8 more than 12.

00:03:55.470 --> 00:04:00.670
So you could say that n minus
12 has to be less than or

00:04:00.670 --> 00:04:02.590
equal to 1/8.

00:04:02.590 --> 00:04:08.360
And then you could also say
that on this side, n minus

00:04:08.360 --> 00:04:12.360
12-- right, if n is around
here-- n minus 12.

00:04:12.360 --> 00:04:17.399
So this if n is greater
than 12.

00:04:17.399 --> 00:04:22.560
And then if n is less than 12,
then n minus 12 has to be

00:04:22.560 --> 00:04:25.150
greater than or equal
to minus 1/8.

00:04:25.150 --> 00:04:28.390
This might be more confusing
to you, and I'm debating

00:04:28.390 --> 00:04:29.510
whether I should
even write it.

00:04:29.510 --> 00:04:31.970
But these two things would
also translate into this.

00:04:31.970 --> 00:04:34.140
But the big picture, absolute
value just means the

00:04:34.140 --> 00:04:35.440
difference between
two numbers.

00:04:35.440 --> 00:04:37.590
I don't care if one is bigger
than the other.

00:04:37.590 --> 00:04:39.340
I'm just saying that the
difference between these two

00:04:39.340 --> 00:04:42.660
numbers cannot be
more than 1/8.

00:04:42.660 --> 00:04:45.160
So n minus 12, the absolute
value is less

00:04:45.160 --> 00:04:46.810
than or equal to 1/8.

00:04:46.810 --> 00:04:50.760
And this choice-- well actually,
it's between these

00:04:50.760 --> 00:04:51.120
two values.

00:04:51.120 --> 00:04:52.380
It can't equal these values.

00:04:52.380 --> 00:04:54.020
So it can't equal these
values, so it's not

00:04:54.020 --> 00:04:55.990
less than or equal.

00:04:55.990 --> 00:04:57.230
It's just less than.

00:04:57.230 --> 00:05:00.170
So n minus 12 is
less than 1/8.

00:05:00.170 --> 00:05:02.740
That's choice C.

00:05:02.740 --> 00:05:05.045
And we are on the
last problem.

00:05:12.360 --> 00:05:14.020
Number 20.

00:05:14.020 --> 00:05:16.150
Almost there.

00:05:16.150 --> 00:05:19.160
The least integer of a
set of consecutive

00:05:19.160 --> 00:05:22.340
integers is minus 25.

00:05:22.340 --> 00:05:27.970
So the least, the smallest
is equal to minus 25.

00:05:27.970 --> 00:05:30.340
The least is minus 25.

00:05:30.340 --> 00:05:41.650
If the sum of these integers is
26, how many integers are

00:05:41.650 --> 00:05:43.080
in this set?

00:05:43.080 --> 00:05:44.150
This is interesting.

00:05:44.150 --> 00:05:51.940
So I have minus 25 plus
something plus

00:05:51.940 --> 00:05:54.290
something plus something.

00:05:54.290 --> 00:06:04.340
And when I add up all of
the integers, I get 26.

00:06:04.340 --> 00:06:07.530
So it's the least integer
of a set of consecutive

00:06:07.530 --> 00:06:10.020
integers is minus 25.

00:06:10.020 --> 00:06:14.400
If the sum of these integers is
26, how many integers are

00:06:14.400 --> 00:06:16.230
in this set?

00:06:16.230 --> 00:06:19.560
Well we start with minus 25.

00:06:19.560 --> 00:06:26.120
So the rest of the integers are
going to have to equal--

00:06:26.120 --> 00:06:28.610
this is an equation--
minus 25 plus all of

00:06:28.610 --> 00:06:29.330
these random things.

00:06:29.330 --> 00:06:33.040
So let's add 25 to both
sides of these things.

00:06:33.040 --> 00:06:35.880
So the rest of the integers are
going to have to equal--

00:06:35.880 --> 00:06:41.400
so you know, let's call it this
is I2 because this is I1,

00:06:41.400 --> 00:06:48.320
integer 3, integer 4 plus
integer n, because we don't

00:06:48.320 --> 00:06:49.610
know how many there are.

00:06:49.610 --> 00:06:56.090
So that means that I2 plus I3
plus bup, bup, bup, bup, bup.

00:06:56.090 --> 00:06:57.000
We don't know, a bunch.

00:06:57.000 --> 00:07:01.400
In is going to equal-- I'm just
adding 25 to both sides

00:07:01.400 --> 00:07:02.740
of this equation.

00:07:02.740 --> 00:07:10.980
So that is equal to 51.

00:07:10.980 --> 00:07:13.860
Let's see, the least integer
of a set of consecutive

00:07:13.860 --> 00:07:15.260
integers is minus 25.

00:07:15.260 --> 00:07:15.980
Oh, I forgot.

00:07:15.980 --> 00:07:17.510
That's consecutive integers.

00:07:17.510 --> 00:07:20.310
So we would actually know what
these values are, consecutive.

00:07:20.310 --> 00:07:25.910
So I2 is going to be minus 24--
you could almost ignore

00:07:25.910 --> 00:07:27.730
what I said just now.

00:07:27.730 --> 00:07:29.800
Let me actually clear this.

00:07:29.800 --> 00:07:31.040
I didn't read that
consecutive.

00:07:31.040 --> 00:07:33.000
It's important to read your
problems properly.

00:07:33.000 --> 00:07:42.280
So it's minus 25 plus minus 24
plus minus 23 is all-- and

00:07:42.280 --> 00:07:49.360
then we don't know how high it
goes, but that is equal to 26.

00:07:49.360 --> 00:07:51.150
So what do we know?

00:07:51.150 --> 00:08:01.420
We know that we go from minus
25 plus minus 24 plus minus

00:08:01.420 --> 00:08:05.090
23, and we're going to
go all the way to 0.

00:08:05.090 --> 00:08:09.660
And then we're going to keep
adding that to 1 plus 2 plus 3

00:08:09.660 --> 00:08:13.190
plus 4, and we're going
to add-- so you know.

00:08:13.190 --> 00:08:14.260
Well let me write it this way.

00:08:14.260 --> 00:08:22.110
Minus 25 plus minus 24 plus
minus 23 plus bam, bam, bam,

00:08:22.110 --> 00:08:38.120
bam, bam plus minus 1 plus 0
plus 1 plus 2 plus 24 plus 25.

00:08:38.120 --> 00:08:42.110
So I claim that the sum of all
of the integers from minus 25

00:08:42.110 --> 00:08:44.260
to 25 is going to be 0.

00:08:44.260 --> 00:08:45.610
And how do I know that?

00:08:45.610 --> 00:08:48.930
Well because minus 1-- let me
switch colors-- minus 1 is

00:08:48.930 --> 00:08:50.770
going to cancel out with 1.

00:08:50.770 --> 00:08:53.490
Minus 2 is going to
cancel out with 2.

00:08:53.490 --> 00:08:56.120
Minus 23 is going to cancel
out with positive 23.

00:08:56.120 --> 00:08:58.560
Minus 24 is going to
cancel out with 24.

00:08:58.560 --> 00:09:05.840
And negative 25 is going to
cancel out with positive 25.

00:09:05.840 --> 00:09:10.100
And so the last term, 26, has
nothing to cancel out with it.

00:09:10.100 --> 00:09:12.860
And that's the sum because
all of the other

00:09:12.860 --> 00:09:13.780
terms canceled out.

00:09:13.780 --> 00:09:15.440
So how many terms are there?

00:09:15.440 --> 00:09:23.430
So to go from minus 25 to minus
1, that is 25 terms.

00:09:23.430 --> 00:09:30.650
Then to go from 1 to 25, that's
25 terms. And then we

00:09:30.650 --> 00:09:35.380
have 0 is one term, and then
26 is another term.

00:09:35.380 --> 00:09:40.130
So 25 plus 25 is 50
plus 1 plus 1.

00:09:40.130 --> 00:09:43.580
52 terms. And that
is choice E.

00:09:43.580 --> 00:09:45.920
And we are done that
first section.

00:09:45.920 --> 00:09:47.040
I hope you had fun.

00:09:47.040 --> 00:09:48.630
I'll see you soon.