SAT Prep: Test 8 Section 5 Part 4

WEBVTT
Kind: captions
Language: en

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We're on problem 16.

00:00:04.540 --> 00:00:07.225
They draw the axes.

00:00:07.225 --> 00:00:10.880
x-axis, that's the y-axis.

00:00:10.880 --> 00:00:12.250
I wish there was the parabola
drawing tool

00:00:12.250 --> 00:00:14.430
here, but there isn't.

00:00:14.430 --> 00:00:16.240
There's the parabola, it looks
something like this.

00:00:19.750 --> 00:00:23.840
They tell us that this
is the point 1.

00:00:23.840 --> 00:00:24.450
Fair enough.

00:00:24.450 --> 00:00:27.460
This is, of course, the y-axis,
this is the x-axis.

00:00:27.460 --> 00:00:29.870
The figure above show that the
graph of a quadratic function

00:00:29.870 --> 00:00:33.000
h who's maximum value
is h of 2.

00:00:33.000 --> 00:00:36.560
So this is its maximum
value right here.

00:00:36.560 --> 00:00:39.760
When x is equal to 2, its
maximum is h of 2.

00:00:43.350 --> 00:00:51.510
If h of a is equal to 0--
so where does this

00:00:51.510 --> 00:00:52.400
function equal 0?

00:00:52.400 --> 00:00:54.630
It equals 0 two places.

00:00:54.630 --> 00:00:55.540
Here and here, right?

00:00:55.540 --> 00:00:57.950
Its y value is 0 in
those two places.

00:00:57.950 --> 00:01:00.610
So this could be a or that could
be a, we don't know.

00:01:00.610 --> 00:01:02.150
Which of the following
could be a?

00:01:02.150 --> 00:01:04.050
So we're saying this could
be a or this could

00:01:04.050 --> 00:01:06.420
be a, we don't know.

00:01:06.420 --> 00:01:08.780
And we know that this is 2,
so a is either going to be

00:01:08.780 --> 00:01:13.260
greater than 2 or less than 0.

00:01:13.260 --> 00:01:17.340
Or actually, we can actually
figure out what a is, right?

00:01:17.340 --> 00:01:21.620
Because if this is the maximum
point, it's going to be

00:01:21.620 --> 00:01:25.070
symmetric around the point 2.

00:01:25.070 --> 00:01:28.990
So the distance from 2 to a on
this side has to be the same

00:01:28.990 --> 00:01:30.810
as the distance from 2 to a on
this one-- Actually that

00:01:30.810 --> 00:01:31.740
doesn't help.

00:01:31.740 --> 00:01:32.960
So let's look at the choices.

00:01:32.960 --> 00:01:36.040
It could be negative 1--
that looks fair because

00:01:36.040 --> 00:01:38.060
it's less than 0.

00:01:38.060 --> 00:01:40.290
a can't be 0-- 0
is this point.

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Let me pick a color.

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That's 0.

00:01:43.840 --> 00:01:46.330
That's definitely not where you
inter-- they draw it so

00:01:46.330 --> 00:01:47.710
that you can't be there.

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It can't be 2 because that's
actually where the maximum

00:01:51.120 --> 00:01:54.800
point is, so a definitely
isn't 2.

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Can a be 3?

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And this is where the symmetry
comes into play.

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We know that it does not
intersect the x-axis for at

00:02:02.220 --> 00:02:06.850
least-- on the left-hand side
you go 1, 2, and then some

00:02:06.850 --> 00:02:09.250
distance which we don't really
know, but you're going more

00:02:09.250 --> 00:02:10.930
than 2 on this side.

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So you're also going
to have to go more

00:02:12.060 --> 00:02:12.760
than two on this side.

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You're going to have to go 1,
2, and then some distance.

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So this is 2, 3, 4.

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So you know that a has to be
less than 0, or a has to be

00:02:24.940 --> 00:02:26.430
greater than 4.

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And if you look at all of the
choices, the only-- negative 1

00:02:31.550 --> 00:02:32.500
is less than 0.

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None of the other ones
are greater than 4.

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And how did I know
greater than 4?

00:02:36.020 --> 00:02:37.440
Because it's symmetric.

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If this is the maximum point
at 2, and one of the places

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where I intersect the x-axis
is more than 2 to the left,

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the other place where I
intersect the x-axis has to be

00:02:48.050 --> 00:02:49.380
more than 2 to the right.

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So it has to be greater
than 4.

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This has to be less than 0, this
has to be greater than 4.

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And that's the answer.

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Negative 1.

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Next problem.

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And you really didn't have to--
once you felt comfortable

00:02:59.750 --> 00:03:01.330
with negative 1 you could have
just said well that's the

00:03:01.330 --> 00:03:03.970
answer because it can very
easily have intersected the

00:03:03.970 --> 00:03:06.400
x-axis at x equals negative 1.

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Problem 17-- and that was the
first choice, which tends to

00:03:08.490 --> 00:03:08.900
be convenient.

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Let me get a brighter color.

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Problem 17.

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If k and h are constants and
x squared plus kx plus 7 is

00:03:19.870 --> 00:03:28.100
equivalent to, equals, x plus 1
times x plus h, what is the

00:03:28.100 --> 00:03:30.500
value of k?

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Well at first sight it doesn't
seem like we could solve this,

00:03:33.330 --> 00:03:35.950
but maybe we can, let's try.

00:03:35.950 --> 00:03:45.470
x squared plus kx plus 7 is
equal to x squared plus x

00:03:45.470 --> 00:03:52.800
plus hx plus h.

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And so this equals x squared.

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What's 1x-- right, x is
the same thing as 1x--

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what's 1x plus xh?

00:04:00.420 --> 00:04:08.810
Well that's 1 plus hx plus h,
right, I just combined these

00:04:08.810 --> 00:04:10.610
two terms, added their
coefficients.

00:04:10.610 --> 00:04:15.780
And we say x squared plus kx
plus-- oh, well actually,

00:04:15.780 --> 00:04:19.209
maybe we can solve this because
we can just match up

00:04:19.209 --> 00:04:26.840
terms. We say this term matches
up to this term.

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This term matches
up to that term.

00:04:29.130 --> 00:04:32.590
So we know, h is equal to 7.

00:04:32.590 --> 00:04:35.790
And if h is equal to 7, what
is this term equal to?

00:04:35.790 --> 00:04:40.860
That term is going to be
equal to 7 plus 1 is 8.

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And this term, what does
this term match up to?

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It matches up to this.

00:04:44.270 --> 00:04:45.970
We're just kind of matching
coefficients.

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If this is 8 than
k is equal to 8.

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And that's choice D.

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At first I was like how am
I going to solve this?

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That's why you should always
just move forward and see if

00:04:53.710 --> 00:04:55.340
you see any patterns.

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Next problem.

00:05:00.490 --> 00:05:22.530
Problem 18.

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y, x, and then they
draw a line here,

00:05:29.590 --> 00:05:30.840
something like that.

00:05:33.610 --> 00:05:45.440
You go from A to C-- oh, the
line keeps going, and then

00:05:45.440 --> 00:05:50.010
there's the point up here,
which is the point

00:05:50.010 --> 00:05:52.510
here, this is 4,10.

00:05:52.510 --> 00:05:56.740
So this is 10 and this is 4.

00:05:56.740 --> 00:06:00.080
This is A, B, C.

00:06:00.080 --> 00:06:02.700
In the figure above if the
legs of triangle ABC are

00:06:02.700 --> 00:06:05.540
parallel to the axes-- this is
parallel to that, this is

00:06:05.540 --> 00:06:08.620
parallel to that, so this is
going to be a right angle.

00:06:08.620 --> 00:06:10.950
Which of the following could
be the lengths of

00:06:10.950 --> 00:06:14.230
the sides of ABC.

00:06:14.230 --> 00:06:15.110
Interesting.

00:06:15.110 --> 00:06:18.200
So this is really just a slope
problem because what they want

00:06:18.200 --> 00:06:20.990
you do is figure out the ratio
of this side to this side.

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So what is the slope
of this line?

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Well it rose 10-- change in y
over change in x is equal to

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10 minus 0, which is 10 over
4 minus 0, which is 4.

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That equals 5 over 2.

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So for every one it moved
to the right, it's going

00:06:44.640 --> 00:06:46.030
to go 5 over 2 up.

00:06:46.030 --> 00:06:49.220
So if this is x, this is
going to be 5 over 2x.

00:06:52.120 --> 00:06:54.510
So first we should look for--
the two shorter sides are

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going to have this ratio.

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1 to 5 over 2.

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So if you look at the first
choice, choice A, 2 and 5, if

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this is 2, if x is equal to 2,
then what's 2 times 5 or 2?

00:07:08.760 --> 00:07:11.210
Well this will be 5, right?

00:07:11.210 --> 00:07:12.370
That'll be 5.

00:07:12.370 --> 00:07:14.240
So that satisfies that ratio.

00:07:14.240 --> 00:07:18.300
And you could even figure out
rise is 5 when run is 2, which

00:07:18.300 --> 00:07:19.870
is the slope of 5 over 2.

00:07:19.870 --> 00:07:22.400
Now you just have to confirm
that they gave the right

00:07:22.400 --> 00:07:23.960
length for this longer side.

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So we just us Pythagorean
Theorem.

00:07:25.830 --> 00:07:28.050
The length of that side's going
to be the square root of

00:07:28.050 --> 00:07:29.490
this squared plus
this squared.

00:07:29.490 --> 00:07:32.980
The square root of 4 plus
25, which equals the

00:07:32.980 --> 00:07:35.310
square root of 29.

00:07:35.310 --> 00:07:38.370
So the sides could be
2, 5 squared to 29.

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And that is choice A.

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Choice B has 2 and 5, but then
it gives 7, assuming that you

00:07:44.405 --> 00:07:46.360
didn't know how to do the
Pythagorean Theorem.

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Next problem.

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Problem 19.

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Let f be defined by f of x
is equal to 2x minus 1.

00:08:00.140 --> 00:08:08.020
If 1/2 times f of the square
root of t is equal to 4, what

00:08:08.020 --> 00:08:10.200
is the value of t?

00:08:10.200 --> 00:08:12.490
So let's just evaluate
this expression.

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So it's 1/2 times what?

00:08:14.620 --> 00:08:17.265
f of the square root of t-- so
everywhere we see x we stick

00:08:17.265 --> 00:08:18.870
in a square root of t.

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So 1/2-- that 1/2 is
just this 1/2.

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f of the square root of t
is 2 times the square

00:08:24.960 --> 00:08:27.570
root of t minus 1.

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I just replaced x with
the square root of t.

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And we know that
this equals 4.

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Let's multiply both sides by 1/2
and you get-- or multiply

00:08:35.280 --> 00:08:37.510
both sides by 2-- the 1/2
gets canceled out here.

00:08:37.510 --> 00:08:41.429
So you get 2 square root of
t minus 1 is equal to 8.

00:08:41.429 --> 00:08:45.480
And you get 2 square root of 5,
adding 1 to both sides is

00:08:45.480 --> 00:08:47.420
equal to 9.

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Let me continue it here.

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Divide both sides by 2 you get
the square root of t is equal

00:08:52.960 --> 00:08:55.140
to 9 over 2.

00:08:55.140 --> 00:08:57.130
And now we can square
both sides of this.

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So you get t is equal to--
what's 9 over 2 squared?

00:09:00.220 --> 00:09:03.910
It's 81 over 4, and
that is choice E.

00:09:07.130 --> 00:09:08.610
And then let's see
if I have time.

00:09:08.610 --> 00:09:09.520
I have 50 seconds.

00:09:09.520 --> 00:09:14.040
Let me see if I can squeeze
in video 20 in here.

00:09:14.040 --> 00:09:16.450
If k is a positive integer,
which of the following must

00:09:16.450 --> 00:09:19.330
represent an even integer
that is twice the

00:09:19.330 --> 00:09:21.260
value of an odd integer.

00:09:24.580 --> 00:09:26.840
So k is any positive integer.

00:09:26.840 --> 00:09:29.060
So I want an even number
that is twice the

00:09:29.060 --> 00:09:30.930
value of an odd integer.

00:09:30.930 --> 00:09:34.920
So an odd integer-- if this is
any integer, an odd integer

00:09:34.920 --> 00:09:38.080
can be represented this--
2 times k plus 1.

00:09:38.080 --> 00:09:39.970
This is definitely going
to be odd integer.

00:09:39.970 --> 00:09:42.580
You can try it out with
k is 1, 2 or 3.

00:09:42.580 --> 00:09:44.840
So if I want double this,
I just multiply it by 2,

00:09:44.840 --> 00:09:50.110
I get 4k plus 2.

00:09:50.110 --> 00:09:51.590
And that is choice E.

00:09:51.590 --> 00:09:53.330
And you could try it out
with some numbers.

00:09:53.330 --> 00:09:55.380
You just have to take my word
that that has to be an odd

00:09:55.380 --> 00:09:57.550
integer and then
you double it.

00:09:57.550 --> 00:09:59.350
I'll see you in the
next section.


SAT Prep: Test 8 Section 2 Part 1

WEBVTT
Kind: captions
Language: en

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Welcome back.

00:00:01.450 --> 00:00:04.590
We're now in the final
test, test eight.

00:00:04.590 --> 00:00:08.660
And we'll start in Section
2, problem one.

00:00:08.660 --> 00:00:11.790
The total cost of three equally
priced mechanical

00:00:11.790 --> 00:00:13.770
pencils is $4.50.

00:00:13.770 --> 00:00:17.210
If the cost per pencil is
increased by $0.50, how much

00:00:17.210 --> 00:00:19.260
will five of these
pencils cost?

00:00:19.260 --> 00:00:24.600
So three pencils, three times
the cost of the pencil is

00:00:24.600 --> 00:00:28.290
equal to $4.50.

00:00:28.290 --> 00:00:32.150
So pencil is equal to
$4.50 divided by 3.

00:00:32.150 --> 00:00:34.810
Just divide both sides
of that equation by 3

00:00:34.810 --> 00:00:37.710
and that equals $1.50.

00:00:37.710 --> 00:00:41.340
Then they say the cost per
pencil is increased by $0.50.

00:00:41.340 --> 00:00:46.050
So we're going to go from $1.50
plus $0.50, so that'll

00:00:46.050 --> 00:00:50.880
get us to $2,00 per pencil.

00:00:50.880 --> 00:00:53.990
How much will five of these
pencils cost at the new rate?

00:00:53.990 --> 00:00:58.636
So it's $2.00 per pencil, times
5, and that's $10.00.

00:00:58.636 --> 00:01:01.980
And that's choice E.

00:01:01.980 --> 00:01:03.230
Problem two.

00:01:07.140 --> 00:01:09.140
OK, the table above represents
a relationship

00:01:09.140 --> 00:01:10.510
between x and y.

00:01:10.510 --> 00:01:17.555
So x can be-- x and y,
so it's 1, 2, 3, 4.

00:01:20.280 --> 00:01:28.870
And then y, when x is
1, is 3, 7, 11, 15.

00:01:28.870 --> 00:01:31.510
Which of the following linear
equations describes the

00:01:31.510 --> 00:01:32.725
relationship?

00:01:32.725 --> 00:01:36.530
So let's see if we can think of
something here on our own.

00:01:39.710 --> 00:01:44.510
Let's see, this looks like--
well, I don't know how you

00:01:44.510 --> 00:01:47.410
would get to 4 times-- let's see
the choices, actually, see

00:01:47.410 --> 00:01:49.230
which ones work.

00:01:49.230 --> 00:01:54.660
So A is y is equal
to x plus 1.

00:01:54.660 --> 00:01:56.160
Try this out, x plus 1.

00:01:56.160 --> 00:01:58.640
1 plus 1 is 2, so that's
not right.

00:01:58.640 --> 00:02:00.180
They've got 3 here.

00:02:00.180 --> 00:02:01.670
So that fails.

00:02:01.670 --> 00:02:05.900
B, y is equal to x plus 4.

00:02:05.900 --> 00:02:10.020
When x is 1-- if it was x plus
4, then this would be 5.

00:02:10.020 --> 00:02:12.750
But this is 3, so that's
not right.

00:02:12.750 --> 00:02:16.120
Choice C, y is equal to 3x.

00:02:16.120 --> 00:02:20.410
That works here, right? x
is 1, y is 3 times 1.

00:02:20.410 --> 00:02:23.250
But it fails here, because
when x is 2, then

00:02:23.250 --> 00:02:24.190
this should be 6.

00:02:24.190 --> 00:02:26.340
But it's 7, so this
isn't right.

00:02:26.340 --> 00:02:28.040
These are all wrong.

00:02:28.040 --> 00:02:31.440
Choice D, y is equal to 4x.

00:02:31.440 --> 00:02:32.710
Well, that fails on
this first one.

00:02:32.710 --> 00:02:36.320
When x is 1, if it was
4x, this would be 4.

00:02:36.320 --> 00:02:38.110
But it's not, so that fails.

00:02:38.110 --> 00:02:39.630
Choice E is probably going
to be our answer.

00:02:39.630 --> 00:02:40.720
Let's try.

00:02:40.720 --> 00:02:43.620
y is equal to 4x minus 1.

00:02:43.620 --> 00:02:48.360
So when x is 1, 4 times
1 is 4 minus 1 is 3.

00:02:48.360 --> 00:02:51.950
4 times 2 is 8 minus 1 is 7.

00:02:51.950 --> 00:02:55.920
4 times 3 is 12,
minus 1 is 11.

00:02:55.920 --> 00:02:58.395
4 times 4 is 16,
minus 1 is 15.

00:02:58.395 --> 00:03:00.370
So it works.

00:03:00.370 --> 00:03:06.830
Next problem, soon. problem 3.

00:03:06.830 --> 00:03:08.250
OK, they drew two circles.

00:03:08.250 --> 00:03:10.710
They look tangent
to each other.

00:03:10.710 --> 00:03:12.420
One circle looks like that.

00:03:12.420 --> 00:03:16.790
Let me see if my skills of
drawing tangent circles, how

00:03:16.790 --> 00:03:18.350
good they are.

00:03:18.350 --> 00:03:21.606
You got one circle, and
not bad, not bad if I

00:03:21.606 --> 00:03:23.180
have to say so myself.

00:03:23.180 --> 00:03:26.140
And then they draw a line that
looks like from the center of

00:03:26.140 --> 00:03:27.140
one circle to the other.

00:03:27.140 --> 00:03:30.520
I'll do it in a different
color.

00:03:30.520 --> 00:03:38.020
So we are going from here
roughly to there.

00:03:38.020 --> 00:03:40.020
I know it doesn't look like
center to center, but I think

00:03:40.020 --> 00:03:41.820
you get the idea.

00:03:41.820 --> 00:03:46.080
They're saying this
point here is A.

00:03:46.080 --> 00:03:48.300
This point right here is B.

00:03:48.300 --> 00:03:51.040
And then this point
right here is C.

00:03:51.040 --> 00:03:53.380
In the figure above, the two
circles are tangent at the

00:03:53.380 --> 00:03:57.125
point B, and AC is equal to 6,
so this whole distance is

00:03:57.125 --> 00:03:58.920
equal to 6, and they're tangent
right here, so these

00:03:58.920 --> 00:04:01.130
circles just touch point B.

00:04:01.130 --> 00:04:04.210
If the circumference of circle
with center A is twice the

00:04:04.210 --> 00:04:07.250
circumference of the circle
with center C, what is the

00:04:07.250 --> 00:04:09.700
length of BC?

00:04:09.700 --> 00:04:12.260
So we want to know what BC is.

00:04:12.260 --> 00:04:16.200
So this circumference is twice
the length of this

00:04:16.200 --> 00:04:17.529
circumference.

00:04:17.529 --> 00:04:19.260
So what's the formula
for circumference?

00:04:19.260 --> 00:04:24.650
Circumference is equal
to 2 pi r, right?

00:04:24.650 --> 00:04:27.660
So let's say that this is r.

00:04:27.660 --> 00:04:29.030
Let's call this distance r.

00:04:31.690 --> 00:04:35.580
What do we know about
this distance?

00:04:35.580 --> 00:04:39.320
Let's call that x,
I guess, right?

00:04:39.320 --> 00:04:42.180
So if this distance is r, the
radius of the large circle is

00:04:42.180 --> 00:04:47.220
r, its circumference is going
to be 2 pi r, right?

00:04:47.220 --> 00:04:48.640
So what's the circumference
of the smaller circle?

00:04:48.640 --> 00:04:49.940
Well, it's 1/2 of that.

00:04:49.940 --> 00:04:57.450
So c/2 is equal to 1/2 of the
big circle circumference is

00:04:57.450 --> 00:05:00.765
equal to the circumference of
the smaller circle, which is 2

00:05:00.765 --> 00:05:02.320
pi x, right?

00:05:02.320 --> 00:05:03.375
Because the radius is x.

00:05:03.375 --> 00:05:05.650
But what's c/2?

00:05:05.650 --> 00:05:07.760
That's this divided by 2.

00:05:07.760 --> 00:05:11.050
So c/2 is the same thing
as this divided

00:05:11.050 --> 00:05:13.020
by 2, which is what?

00:05:13.020 --> 00:05:16.480
The 2 would just disappear,
so it equals pi r.

00:05:16.480 --> 00:05:23.080
You could take both pi's away,
and you get 2x is equal to r,

00:05:23.080 --> 00:05:25.350
or x is equal to r/2.

00:05:25.350 --> 00:05:27.100
And you really didn't
have to do that.

00:05:27.100 --> 00:05:29.920
You could just know that
circumference is directly

00:05:29.920 --> 00:05:32.540
proportional to radius, unlike
area, which is proportional to

00:05:32.540 --> 00:05:33.490
the square.

00:05:33.490 --> 00:05:35.750
So circumference is directly
proportional to radius.

00:05:35.750 --> 00:05:37.900
So if this circle has twice
the circumference as this

00:05:37.900 --> 00:05:40.860
circle, its radius is also going
to be twice the radius

00:05:40.860 --> 00:05:42.230
of this circle.

00:05:42.230 --> 00:05:43.350
But that's interesting now.

00:05:43.350 --> 00:05:46.660
So now we know that if this is
x, this distance here is x, BC

00:05:46.660 --> 00:05:50.460
is x, this distance here is
going to be 2x, right?

00:05:50.460 --> 00:05:53.440
It's going to be 2 times
the small radius.

00:05:53.440 --> 00:05:57.580
And we know that when you add
all of them together, 2x plus

00:05:57.580 --> 00:06:02.200
x, you get the whole length from
AC, A to C, and you so

00:06:02.200 --> 00:06:03.590
that equals y.

00:06:03.590 --> 00:06:06.990
So you get 3x is equal
to 6 and then you get

00:06:06.990 --> 00:06:08.460
x is equal to 2.

00:06:08.460 --> 00:06:09.830
And that's our answer,
because we wanted to

00:06:09.830 --> 00:06:11.190
know just what x is.

00:06:11.190 --> 00:06:13.610
And so the really fast way of
doing this is, if you said,

00:06:13.610 --> 00:06:17.040
well, this is going to be
twice-- let's define x as this

00:06:17.040 --> 00:06:18.480
small radius.

00:06:18.480 --> 00:06:21.140
And if this circle has twice the
circumference, it's going

00:06:21.140 --> 00:06:22.000
to have twice the radius.

00:06:22.000 --> 00:06:23.490
So this would be 2x.

00:06:23.490 --> 00:06:25.530
And then you could just go
straight to this step without

00:06:25.530 --> 00:06:27.620
having to do all of this.

00:06:27.620 --> 00:06:34.200
Next problem, problem 4.

00:06:34.200 --> 00:06:35.835
OK, they drew a bunch
of points.

00:06:35.835 --> 00:06:38.730
Let's see, a lot of
drawing for me.

00:06:38.730 --> 00:06:40.600
I'm a little tired.

00:06:40.600 --> 00:06:44.570
My goal is to finish these
problems this weekend, and

00:06:44.570 --> 00:06:46.740
it's midnight and
past my bedtime.

00:06:46.740 --> 00:06:51.010
But I'm hanging in there and
trying to finish all of these.

00:06:51.010 --> 00:06:55.175
OK, so let's see, they have
point C, which is up here.

00:06:55.175 --> 00:06:57.120
And I don't know if I have
to be accurate with the

00:06:57.120 --> 00:06:59.110
coordinates yet.

00:06:59.110 --> 00:07:02.610
Point B is roughly here.

00:07:02.610 --> 00:07:08.090
Point A is here.

00:07:08.090 --> 00:07:13.542
So then you have point E that's
right around here.

00:07:13.542 --> 00:07:16.950
And then you have point D,
which is right there.

00:07:16.950 --> 00:07:18.310
What are they going to ask us?

00:07:18.310 --> 00:07:20.550
Which of the letter points
in the figure above has

00:07:20.550 --> 00:07:26.480
coordinates x, y, such that the
absolute value of x minus

00:07:26.480 --> 00:07:30.600
the absolute value of
y is equal to 3?

00:07:30.600 --> 00:07:32.570
So we could just try out the
points, but they're really

00:07:32.570 --> 00:07:37.250
just saying that no matter what
the x and y-- they say it

00:07:37.250 --> 00:07:38.410
doesn't matter what
quadrant it is.

00:07:38.410 --> 00:07:42.850
We want the difference between
the x and the y to be 3.

00:07:42.850 --> 00:07:47.210
So the intuition here is that
the x is going to be larger

00:07:47.210 --> 00:07:49.460
than the y, right?

00:07:49.460 --> 00:07:53.800
Or the distance that x is away
from the origin is going to be

00:07:53.800 --> 00:07:57.030
larger by 3 than the distance
that y is away from the

00:07:57.030 --> 00:07:57.720
origin, right?

00:07:57.720 --> 00:07:59.840
Because the absolute value of x,
you kind of view it as the

00:07:59.840 --> 00:08:03.800
distance x is from the origin,
and absolute value of y is the

00:08:03.800 --> 00:08:05.790
distance y is from the origin.

00:08:05.790 --> 00:08:08.980
So which ones have a larger
distance x from the origin

00:08:08.980 --> 00:08:10.910
than a y from the origin?

00:08:10.910 --> 00:08:12.200
Well, it looks like D, right?

00:08:12.200 --> 00:08:13.920
D looks further in
this direction.

00:08:13.920 --> 00:08:15.360
Let me switch colors.

00:08:15.360 --> 00:08:20.110
D is further in this direction
than it is in this direction.

00:08:20.110 --> 00:08:24.370
And if we look at the other
choices, B also is like that.

00:08:24.370 --> 00:08:25.660
And so let's look at
the coordinates.

00:08:25.660 --> 00:08:26.970
Those are my prime
coordinates.

00:08:26.970 --> 00:08:28.920
If I actually look at the
graph, B looks like

00:08:28.920 --> 00:08:31.700
it's 1, 2, 3, 4.

00:08:31.700 --> 00:08:35.419
So this looks like minus
4 and minus 1.

00:08:35.419 --> 00:08:38.282
So B looks like minus
4, minus 1.

00:08:38.282 --> 00:08:40.049
And actually, that looks
like our answer, right?

00:08:40.049 --> 00:08:42.049
Because what's the absolute
value of 4?

00:08:42.049 --> 00:08:45.720
That's the absolute value of
minus 4 minus the absolute

00:08:45.720 --> 00:08:47.440
value of minus 1.

00:08:47.440 --> 00:08:51.040
That equals 4 minus
1, which equals 3.

00:08:51.040 --> 00:08:54.170
So that is choice B.

00:08:54.170 --> 00:08:56.020
We didn't have to try D.

00:08:56.020 --> 00:08:57.870
Next problem.

00:08:57.870 --> 00:09:00.370
Actually, I'm coming
up on nine minutes.

00:09:00.370 --> 00:09:02.450
So I'll do the next problem
in the next video.

00:09:02.450 --> 00:09:03.870
I'll see


SAT Prep: Test 8 Section 2 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.100
We're on problem number five.

00:00:02.100 --> 00:00:04.985
Let's see, we have a pie graph
here of survey results.

00:00:10.206 --> 00:00:15.030
And let's see, that looks like
almost a straight line.

00:00:15.030 --> 00:00:17.030
Dividing it in half there.

00:00:17.030 --> 00:00:21.690
And then this gets divided
in half again.

00:00:21.690 --> 00:00:25.635
It looks like that, roughly.

00:00:25.635 --> 00:00:28.010
It looks like that.

00:00:28.010 --> 00:00:32.479
And then this is slightly
bigger like that.

00:00:32.479 --> 00:00:37.960
All right, and this tells
us x is less than 20

00:00:37.960 --> 00:00:40.090
for 30% of the time.

00:00:40.090 --> 00:00:41.740
I don't even know what
these represent.

00:00:41.740 --> 00:00:48.430
x is greater than or equal
to 60 25% of the time.

00:00:48.430 --> 00:00:55.670
x is between 40 and less than
60, this is 25% of the time.

00:00:55.670 --> 00:00:58.500
So it can equal 40, but
it's less than 60.

00:00:58.500 --> 00:01:05.019
And then it's greater than or
equal to 20 and less than 40

00:01:05.019 --> 00:01:06.475
20% of the time.

00:01:06.475 --> 00:01:07.835
That's the survey results.

00:01:07.835 --> 00:01:11.120
The chart above shows the
results when 1,000 people were

00:01:11.120 --> 00:01:13.610
asked how old are you?

00:01:13.610 --> 00:01:16.210
The age they gave is
represented by x.

00:01:16.210 --> 00:01:19.960
How many people said their
age was less than 40?

00:01:19.960 --> 00:01:23.240
So x is less than 40.

00:01:23.240 --> 00:01:27.780
Well, this category right here
is x is between 20 and 40, so

00:01:27.780 --> 00:01:37.070
it's this one plus all the
people who said that they were

00:01:37.070 --> 00:01:38.940
younger than 20 because if
you're less than 20, you're

00:01:38.940 --> 00:01:41.130
definitely less than
40 as well, so it's

00:01:41.130 --> 00:01:42.970
both of these combined.

00:01:42.970 --> 00:01:45.490
And this is 30%, this is 20%.

00:01:45.490 --> 00:01:51.620
So if you combine it, it's 30%
plus 20% equals 50% of the

00:01:51.620 --> 00:01:53.105
entire population asked.

00:01:53.105 --> 00:01:58.760
And 1,000 people were asked,
so 50% times 1,000, well,

00:01:58.760 --> 00:01:59.690
that's straightforward.

00:01:59.690 --> 00:02:01.960
That equals 500 people.

00:02:01.960 --> 00:02:04.490
And that's choice D.

00:02:04.490 --> 00:02:09.840
Next problem, problem 6.

00:02:09.840 --> 00:02:13.630
Which of the following could
be the remainders when four

00:02:13.630 --> 00:02:20.320
consecutive positive integers
are each divided by 3?

00:02:20.320 --> 00:02:21.550
Interesting.

00:02:21.550 --> 00:02:28.650
So let's say we have x plus 1,
x plus 2, x plus 3, right?

00:02:28.650 --> 00:02:31.216
These are four consecutive
positive integers.

00:02:35.980 --> 00:02:37.940
Well, let's assume that
this first one is

00:02:37.940 --> 00:02:39.830
divisible by 3, right?

00:02:39.830 --> 00:02:42.850
Let's say that x divided by 3,
there's no remainder, so the

00:02:42.850 --> 00:02:45.260
remainder is 0.

00:02:45.260 --> 00:02:50.780
So when this number's divisible
by this number, x is

00:02:50.780 --> 00:02:53.450
divisible by 3, so now when
you divide x plus 1 by 3,

00:02:53.450 --> 00:02:55.200
you're going to have
1 left over.

00:02:55.200 --> 00:02:57.660
Similarly, you're going
to have 2 left over.

00:02:57.660 --> 00:03:00.810
Here, you're not going to
3 left over, right?

00:03:00.810 --> 00:03:06.440
Because if x is divisible by
3, then x plus 3 is also

00:03:06.440 --> 00:03:08.670
divisible by 3, right?

00:03:08.670 --> 00:03:10.150
You're going to have a cycle.

00:03:10.150 --> 00:03:12.700
It's going to go back to
a remainder of zero.

00:03:12.700 --> 00:03:15.250
So this is a possible
situation.

00:03:15.250 --> 00:03:19.360
And similarly, if x had a
remainder of 1, then x plus 1

00:03:19.360 --> 00:03:20.880
would have a remainder of 2.

00:03:20.880 --> 00:03:24.790
Then x plus 2 would have a
remainder of 0, and then this

00:03:24.790 --> 00:03:26.360
would have a remainder
of 1 again.

00:03:26.360 --> 00:03:27.800
So I don't know, let's
look at the choices.

00:03:27.800 --> 00:03:29.810
Are any of these out there?

00:03:29.810 --> 00:03:30.310
Well, sure.

00:03:30.310 --> 00:03:31.510
This is choice D, actually.

00:03:31.510 --> 00:03:37.440
The first thing we did, choice
D was 0, 1, 2, 0, which is D.

00:03:37.440 --> 00:03:42.770
And the key here is realizing
that you can't have a

00:03:42.770 --> 00:03:48.210
remainder of 3 or 4 when you're
dividing by 3, right?

00:03:48.210 --> 00:03:51.170
Your remainder can only
be 0, 1 or 2.

00:03:51.170 --> 00:03:54.780
So with that realization alone,
you can cancel out all

00:03:54.780 --> 00:03:56.460
the choices that have
a 3 or a 4 in it.

00:03:56.460 --> 00:03:58.150
You can't have a remainder
of 3 or 4 if

00:03:58.150 --> 00:03:59.510
you're dividing by 3.

00:03:59.510 --> 00:04:01.250
And you can try that
out, right?

00:04:01.250 --> 00:04:02.810
Because if you have a remainder
by 3, that means you

00:04:02.810 --> 00:04:05.310
could divide one more 3 into
the number and have a

00:04:05.310 --> 00:04:06.500
remainder of 0.

00:04:06.500 --> 00:04:11.670
So you could cancel
out A, B, C and E.

00:04:11.670 --> 00:04:12.900
You can actually cancel
out everything.

00:04:12.900 --> 00:04:15.430
So you would have just had to
have that one realization, and

00:04:15.430 --> 00:04:18.079
you would have said the choice
is D because you can't have a

00:04:18.079 --> 00:04:21.970
remainder of 3 or 4 if
you're dividing by 3.

00:04:21.970 --> 00:04:28.430
Next problem, problem 7.

00:04:31.090 --> 00:04:36.710
If y is inversely proportional
to x, so that means that y is

00:04:36.710 --> 00:04:39.620
proportional to the inverse of
x, so it's equal to some

00:04:39.620 --> 00:04:40.910
constant times 1/x.

00:04:40.910 --> 00:04:43.390
Because that's what inversely
proportional means.

00:04:43.390 --> 00:04:47.030
If we said proportional, it'd
be y equals k times x, but

00:04:47.030 --> 00:04:49.050
it's inversely proportional.

00:04:49.050 --> 00:04:56.120
And they tell us that
y is 15 when x is 5.

00:04:56.120 --> 00:05:00.320
So y is 15 is equal to
k times 1/5, right?

00:05:00.320 --> 00:05:03.300
y is 15 when x is equal to 5.

00:05:03.300 --> 00:05:06.810
Let's see, let's multiply
both sides of this by 5.

00:05:06.810 --> 00:05:14.290
So you get 5 times 15 is 75
is equal to k, right?

00:05:14.290 --> 00:05:17.890
So y is equal to 75/x.

00:05:17.890 --> 00:05:19.900
I just rewrote this.

00:05:19.900 --> 00:05:22.910
What is the value of
y when x is 25?

00:05:22.910 --> 00:05:25.210
So y is equal to 75.

00:05:25.210 --> 00:05:27.430
x is now 25.

00:05:27.430 --> 00:05:28.780
What's 75 divided by 25?

00:05:28.780 --> 00:05:30.150
Well, it's 3.

00:05:30.150 --> 00:05:33.680
And that's choice C.

00:05:33.680 --> 00:05:34.930
Problem 8.

00:05:40.620 --> 00:05:50.950
If 2x plus z is equal to 2y,
and 2x-- and then they also

00:05:50.950 --> 00:05:57.880
tell us that 2x plus 2y
plus z is equal to 20.

00:05:57.880 --> 00:06:00.005
What is the value of y?

00:06:02.620 --> 00:06:05.170
y is equal to what?

00:06:05.170 --> 00:06:06.820
So there's something
interesting here.

00:06:06.820 --> 00:06:14.330
we can rewrite this second
equation as subtract 2y from

00:06:14.330 --> 00:06:18.430
both sides of this equation
right here, and you get 2x

00:06:18.430 --> 00:06:24.560
plus z is equal to 20
minus y, right?

00:06:24.560 --> 00:06:27.880
So essentially, you have a 2x
plus z here, and you have a 2x

00:06:27.880 --> 00:06:29.540
plus z here.

00:06:29.540 --> 00:06:31.920
Let's make a new variable.

00:06:31.920 --> 00:06:34.120
You don't have to do this step,
but I think this'll

00:06:34.120 --> 00:06:35.550
simplify things.

00:06:35.550 --> 00:06:39.880
Let's call the variable Q.

00:06:39.880 --> 00:06:43.280
Let's say Q is equal
to 2x plus z.

00:06:43.280 --> 00:06:44.360
That's where they're trying
to confuse you.

00:06:44.360 --> 00:06:46.230
They're giving you two equations
with three unknowns.

00:06:46.230 --> 00:06:48.520
And you're like, how can I
solve for one of them?

00:06:48.520 --> 00:06:52.610
Well, what's interesting is they
have-- you're solving for

00:06:52.610 --> 00:06:55.240
one of them, and then the
relationship has a 2x plus z

00:06:55.240 --> 00:06:56.480
in both equations.

00:06:56.480 --> 00:06:59.830
So if you say Q is equal to 2x
plus z, everything starts to

00:06:59.830 --> 00:07:03.990
make sense because then this top
equation will become Q is

00:07:03.990 --> 00:07:05.360
equal to 2y.

00:07:05.360 --> 00:07:07.780
And what will this bottom
equation be?

00:07:07.780 --> 00:07:12.560
That would be Q is equal
to 20 minus y, right?

00:07:12.560 --> 00:07:15.040
And now you could set these
equal to each other.

00:07:15.040 --> 00:07:19.670
2y is equal to 20 minus y.

00:07:19.670 --> 00:07:25.960
Add y to both sides, you
get 3y is equal to 20.

00:07:25.960 --> 00:07:27.210
Am I doing that right?

00:07:27.210 --> 00:07:30.740
3y is equal to 20?

00:07:30.740 --> 00:07:34.005
You add y to both sides.
y is equal to 20/3.

00:07:34.005 --> 00:07:35.690
And they don't have
that choice, so I

00:07:35.690 --> 00:07:38.810
must have made a mistake.

00:07:38.810 --> 00:07:39.915
Let me redo the problem.

00:07:39.915 --> 00:07:41.260
I must've made a mistake.

00:07:41.260 --> 00:07:42.680
Problem 8.

00:07:42.680 --> 00:07:47.600
So they're telling us 2x plus
z is equal to 2y, and then

00:07:47.600 --> 00:07:56.140
they tell us 2x plus 2y
plus z is equal to 20.

00:07:56.140 --> 00:07:57.290
What is the value of y?

00:07:57.290 --> 00:08:02.510
OK, the second equation can be
written as 2x plus z is equal

00:08:02.510 --> 00:08:06.010
to 20 minus 2y.

00:08:06.010 --> 00:08:10.470
And this top equation is still
2x plus z is equal to 2y.

00:08:10.470 --> 00:08:11.510
Oh, I see.

00:08:11.510 --> 00:08:13.870
I had dropped a y someplace.

00:08:13.870 --> 00:08:16.450
So this must equal this,
because they both

00:08:16.450 --> 00:08:17.530
equal 2x plus z.

00:08:17.530 --> 00:08:19.630
I don't even have to do all
that substitution Q.

00:08:19.630 --> 00:08:22.830
I think that you can see that 2x
plus z equals this, 2x plus

00:08:22.830 --> 00:08:26.130
z equals this, so this
must equal this.

00:08:26.130 --> 00:08:30.030
So 2y is equal to 20 minus 2y.

00:08:30.030 --> 00:08:35.000
Add 2y to both sides, you
get 4y is equal to 20,

00:08:35.000 --> 00:08:36.820
y is equal to 5.

00:08:36.820 --> 00:08:40.100
And that is choice A.

00:08:40.100 --> 00:08:41.500
I'll see you in the
next video.

00:08:55.690 --> 00:08:56.090
Oh, whoops!

00:08:56.090 --> 00:08:57.350
I didn't delete.

00:08:57.350 --> 00:08:58.600
Sorry.


SAT Prep: Test 8 Section 2 Part 3

WEBVTT
Kind: captions
Language: en

00:00:01.250 --> 00:00:05.460
OK, we are in problem
number 9.

00:00:09.070 --> 00:00:14.490
If 2 times x minus 3 is equal to
7, what is the value of x?

00:00:14.490 --> 00:00:15.550
Well, this is straightforward.

00:00:15.550 --> 00:00:20.660
Just solve for x problem,
divide both sides by 2.

00:00:20.660 --> 00:00:23.100
Or actually, yeah, we could
do it either way.

00:00:23.100 --> 00:00:26.550
We could say this is 2x
minus 6 is equal to 7.

00:00:26.550 --> 00:00:30.420
Add 6 to both sides, you
get 2x is equal to 13.

00:00:30.420 --> 00:00:34.740
Divide both sides by 2. x is
equal to 13/2, which is the

00:00:34.740 --> 00:00:36.710
same thing as 6 and 1/2,
which is the same

00:00:36.710 --> 00:00:39.510
thing as 6.5, right?

00:00:39.510 --> 00:00:41.210
That's 12, yep.

00:00:41.210 --> 00:00:48.820
Problem 10.

00:00:48.820 --> 00:00:55.470
Point P lies on the line with
the equation y minus 4 is

00:00:55.470 --> 00:01:00.060
equal to 3 times x minus 2.

00:01:00.060 --> 00:01:08.910
If the x-coordinate of P is 4--
OK, so P is the 0.4, what

00:01:08.910 --> 00:01:10.510
is the y-coordinate?

00:01:10.510 --> 00:01:13.820
So we don't know what the
y-coordinate is, so we

00:01:13.820 --> 00:01:17.525
literally just have to solve--
we have to just input 4 for x

00:01:17.525 --> 00:01:19.290
and then solve for y.

00:01:19.290 --> 00:01:25.690
So you get y minus 4 is equal
to 3 times 4 minus 2.

00:01:25.690 --> 00:01:30.980
y minus 4 is equal to 3
times 4 minus 2 is 2.

00:01:30.980 --> 00:01:33.740
y minus 4 is equal to 6.

00:01:33.740 --> 00:01:34.805
y is equal to 10.

00:01:34.805 --> 00:01:36.930
And that's our answer.

00:01:36.930 --> 00:01:38.840
The point is 4, 10, but they
just wanted to know the

00:01:38.840 --> 00:01:41.490
y-coordinate, and that's 10.

00:01:41.490 --> 00:01:46.970
Problem 11.

00:01:46.970 --> 00:01:52.620
Car A traveled 60 miles and
averaged 20 miles per gallon

00:01:52.620 --> 00:01:53.380
of gasoline.

00:01:53.380 --> 00:02:03.910
So car A traveled 60 miles and
got 20 miles per gallon, mpg,

00:02:03.910 --> 00:02:05.660
miles per gallon.

00:02:05.660 --> 00:02:13.450
Car B traveled 15 miles
for each gallon.

00:02:13.450 --> 00:02:16.810
OK, so it got 15 miles
per gallon, right?

00:02:16.810 --> 00:02:21.310
It traveled 15 miles for each
gallon of gasoline it used.

00:02:21.310 --> 00:02:25.040
How many miles had car B
traveled when it had used the

00:02:25.040 --> 00:02:30.490
same amount of gasoline that car
A used to travel 60 miles?

00:02:30.490 --> 00:02:34.340
So they're saying we need to
figure out how many gallons A

00:02:34.340 --> 00:02:36.890
used, and then how far
can B get with that

00:02:36.890 --> 00:02:37.990
same number of gallons?

00:02:37.990 --> 00:02:39.790
That's essentially what
they're asking.

00:02:39.790 --> 00:02:42.240
So how many gallons did
A use to go 60 miles?

00:02:42.240 --> 00:02:45.860
Well, we've got 20 miles per
gallon and went 60 miles.

00:02:45.860 --> 00:02:53.750
So car A could do this
on 3 gallons, right?

00:02:53.750 --> 00:03:03.120
The way you can view this,
gallons times miles per gallon

00:03:03.120 --> 00:03:05.490
is equal to miles, right?

00:03:05.490 --> 00:03:09.920
And so gallons, we'll say g
times 20 miles per gallon is

00:03:09.920 --> 00:03:13.170
equal to 60 miles.

00:03:13.170 --> 00:03:16.540
So gallons, divide both sides
by 20, is equal to 30.

00:03:16.540 --> 00:03:18.330
That's times right there.

00:03:18.330 --> 00:03:19.790
I'm sorry, it's equal to 3.

00:03:19.790 --> 00:03:21.460
60 divided by 20 is 3.

00:03:21.460 --> 00:03:25.000
So car A used 3 gallons
to go 60 miles.

00:03:25.000 --> 00:03:39.800
And so car B can go 15 miles
per gallon times 3 gallons

00:03:39.800 --> 00:03:46.090
equals 15 times 3
is 45 gallons.

00:03:46.090 --> 00:03:47.340
Next problem.

00:03:54.020 --> 00:03:56.500
OK, see if I can draw that.

00:03:56.500 --> 00:03:59.680
It's a straight line here.

00:03:59.680 --> 00:04:05.025
Straight line, point A comes
up to this point to B.

00:04:05.025 --> 00:04:07.850
It comes down a little bit.

00:04:07.850 --> 00:04:09.660
And then point C comes
down here.

00:04:12.390 --> 00:04:16.190
OK, and they're telling us
that this is 65 degrees.

00:04:16.190 --> 00:04:19.180
This is 100 degrees.

00:04:19.180 --> 00:04:21.940
This is 120.

00:04:21.940 --> 00:04:25.430
And this, right here,
is x degrees.

00:04:25.430 --> 00:04:28.700
In the figure above, points A,
D, and E lie on the same line.

00:04:28.700 --> 00:04:30.190
This is A.

00:04:30.190 --> 00:04:32.050
This is D.

00:04:32.050 --> 00:04:33.220
This is E.

00:04:33.220 --> 00:04:34.330
They lie on the same line.

00:04:34.330 --> 00:04:37.690
What is the value of x?

00:04:37.690 --> 00:04:39.700
So we really just need to figure
out the value of this,

00:04:39.700 --> 00:04:42.160
and then we can say x is
supplementary to that.

00:04:42.160 --> 00:04:45.410
So let's think about how we can
think about what the value

00:04:45.410 --> 00:04:48.240
of this angle is.

00:04:48.240 --> 00:04:54.150
So the way I think about it is
draw a line here, and you

00:04:54.150 --> 00:04:57.500
could automatically know what
all the angles in a

00:04:57.500 --> 00:04:58.770
quadrilateral are.

00:04:58.770 --> 00:05:00.710
But I'll prove it to you that
all the angles in a

00:05:00.710 --> 00:05:03.110
quadrilateral are going
to add up to 360.

00:05:03.110 --> 00:05:04.590
How do I know that?

00:05:04.590 --> 00:05:11.010
Because think of it this way:
this angle plus this angle

00:05:11.010 --> 00:05:14.630
plus this angle is
going to be 180.

00:05:14.630 --> 00:05:18.280
And then this angle plus this
angle plus this angle

00:05:18.280 --> 00:05:20.200
is going to be 180.

00:05:20.200 --> 00:05:22.730
And if you add all of those
together, you're essentially

00:05:22.730 --> 00:05:26.570
adding all of the angles of
the quadrilateral-- it's

00:05:26.570 --> 00:05:30.190
late-- because this and this is
equal to the angle of the

00:05:30.190 --> 00:05:30.900
whole quadrilateral.

00:05:30.900 --> 00:05:34.390
So the angles in the entire
quadrilateral are 360 degrees.

00:05:34.390 --> 00:05:36.570
And you can memorize
that if you want.

00:05:36.570 --> 00:05:39.190
So if we know that all of these
angles add up to 360

00:05:39.190 --> 00:05:42.060
degrees, what is this big
angle right here?

00:05:42.060 --> 00:05:50.150
Well, we know that 65 plus 100
plus 120 plus-- let's call

00:05:50.150 --> 00:05:54.010
this angle y, and y is this
whole thing-- plus y is equal

00:05:54.010 --> 00:05:55.980
to 360 degrees.

00:05:55.980 --> 00:05:56.660
And this is what?

00:05:56.660 --> 00:06:05.220
This is 220 plus 65, 285
plus y is equal to 360.

00:06:05.220 --> 00:06:13.300
So y is equal to 360 minus
285, and that is what?

00:06:13.300 --> 00:06:15.620
That is 75 degrees, right?

00:06:15.620 --> 00:06:17.890
Because 70 would
be 350, right?

00:06:17.890 --> 00:06:21.100
So y is 75 degrees.

00:06:21.100 --> 00:06:25.220
So this whole angle
right there is 75.

00:06:25.220 --> 00:06:29.860
And we know that x plus this
big angle here, 75-- and

00:06:29.860 --> 00:06:30.840
ignore this green line.

00:06:30.840 --> 00:06:32.660
I'm looking about
this big angle.

00:06:32.660 --> 00:06:35.470
x plus 75 is equal to 180.

00:06:35.470 --> 00:06:39.280
So x has to be equal
to 105 degrees.

00:06:39.280 --> 00:06:41.820
Subtract 75 from both sides.

00:06:41.820 --> 00:06:50.700
Next question, problem 13.

00:06:50.700 --> 00:06:52.770
The first term of a sequence
is 20 and the

00:06:52.770 --> 00:06:53.830
second term is 8.

00:06:53.830 --> 00:06:56.450
So it goes from 20,
then it goes to 8.

00:06:56.450 --> 00:07:00.090
The third term and each term
thereafter is the average of

00:07:00.090 --> 00:07:02.380
the two terms immediately
preceding it.

00:07:02.380 --> 00:07:04.990
What is the value of the first
term in the sequence that is

00:07:04.990 --> 00:07:06.490
not an integer?

00:07:06.490 --> 00:07:06.790
OK.

00:07:06.790 --> 00:07:08.700
So we just have to average
these two.

00:07:08.700 --> 00:07:10.540
So this is going to be what?

00:07:10.540 --> 00:07:12.500
It's going to be 28
divided by 2.

00:07:12.500 --> 00:07:16.940
28/2, which equals 14, right?

00:07:16.940 --> 00:07:20.950
I just took 20 plus 8
divided by 2 is 14.

00:07:20.950 --> 00:07:23.910
And now, what's the average
of 8 and 14?

00:07:23.910 --> 00:07:28.202
Well, 8 plus 14 is 22.

00:07:28.202 --> 00:07:31.840
So 22 divided by 2, which
is equal to 11.

00:07:31.840 --> 00:07:33.260
Delete my work.

00:07:33.260 --> 00:07:36.750
Now, what's the average
of 14 and 11?

00:07:36.750 --> 00:07:39.900
You add them, you get
25 divided by

00:07:39.900 --> 00:07:43.480
2, that equals 12.5.

00:07:43.480 --> 00:07:44.180
So here we go.

00:07:44.180 --> 00:07:47.150
This is the first number in
the series that is not an

00:07:47.150 --> 00:07:48.500
integer, 12.5.

00:07:48.500 --> 00:07:50.310
And you could fill that in.

00:07:50.310 --> 00:07:52.970
Problem 14.

00:07:52.970 --> 00:07:56.050
If x is 1/5 of y-- so
x is equal to 1/5y.

00:07:58.600 --> 00:08:00.970
y is 3/10 of z.

00:08:00.970 --> 00:08:07.870
y is equal to 3/10z and z is
greater than 0, then x is what

00:08:07.870 --> 00:08:09.200
fraction of z?

00:08:09.200 --> 00:08:12.160
So you could literally just
substitute here, right?

00:08:12.160 --> 00:08:14.720
You can just substitute
this y.

00:08:14.720 --> 00:08:17.330
You could substitute
3/10z for y, right?

00:08:17.330 --> 00:08:20.510
Because we say y is equal to
3 times z, so x is equal

00:08:20.510 --> 00:08:23.600
to 1/5 times y.

00:08:23.600 --> 00:08:29.220
We're going to substitute
for y, 3/10z.

00:08:29.220 --> 00:08:31.670
So x is equal to--
what is this?

00:08:31.670 --> 00:08:33.475
3/50z.

00:08:33.475 --> 00:08:35.830
So x is 3/50 of z.

00:08:35.830 --> 00:08:40.370
So you would fill in 3/50
in your answer box.

00:08:40.370 --> 00:08:42.580
OK, Next problem.

00:08:42.580 --> 00:08:44.895
Well, I don't know if I have
time to do it in this video.

00:08:44.895 --> 00:08:46.145
Well, let me see.

00:08:50.410 --> 00:08:51.380
No, I don't want to rush it.

00:08:51.380 --> 00:08:52.930
I'll do it in the next video.

00:08:52.930 --> 00:08:54.180
See you soon.


SAT Prep: Test 8 Section 2 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.646
Problem 15.

00:00:03.646 --> 00:00:06.501
Let's see, they have a square.

00:00:06.501 --> 00:00:07.560
It looks like a square.

00:00:07.560 --> 00:00:08.940
I don't know if it's
a square just yet.

00:00:08.940 --> 00:00:11.540
And they have a triangle
that comes off of

00:00:11.540 --> 00:00:12.790
the square like this.

00:00:15.220 --> 00:00:17.300
Ah, close enough.

00:00:17.300 --> 00:00:18.420
And what do they tell us?

00:00:18.420 --> 00:00:20.790
They tell us that this
is 90 degrees.

00:00:20.790 --> 00:00:22.585
This is 60 degrees.

00:00:22.585 --> 00:00:30.445
And this is A, B, C, D, and E.

00:00:30.445 --> 00:00:31.870
And what other thing?

00:00:31.870 --> 00:00:38.640
In the figure above, E, B, C, D
is a square, OK, and A, E is

00:00:38.640 --> 00:00:40.210
equal to 8.

00:00:40.210 --> 00:00:45.030
What is the area
of E, B, C, D?

00:00:45.030 --> 00:00:46.760
So we just have to figure out
one side of the square.

00:00:46.760 --> 00:00:48.040
And it's a square, so all
the sides are equal

00:00:48.040 --> 00:00:49.130
and then we're done.

00:00:49.130 --> 00:00:53.080
And this is actually just a
30-60-90 triangle problem.

00:00:53.080 --> 00:00:53.600
How do I know it?

00:00:53.600 --> 00:00:55.700
Because this angle is 60,
this is 90, so this

00:00:55.700 --> 00:00:56.590
one has to be 30.

00:00:56.590 --> 00:00:58.450
Or I could add it up to 180.

00:00:58.450 --> 00:01:00.950
And if you don't even remember
what a 30-60-90 triangle is,

00:01:00.950 --> 00:01:03.000
where they add up, you could
go back to page 838,

00:01:03.000 --> 00:01:03.670
and they tell you.

00:01:03.670 --> 00:01:06.020
If you look in the middle of
that page, they have special

00:01:06.020 --> 00:01:11.920
right triangles, and they tell
you that if the hypotenuse is

00:01:11.920 --> 00:01:20.040
2x that the side opposite the 30
degree side is x, and then

00:01:20.040 --> 00:01:24.060
the side opposite the 60 degree
side is x square root

00:01:24.060 --> 00:01:26.430
of 3, right?

00:01:26.430 --> 00:01:29.670
So if x is equal to 8, what's
x squared of 3?

00:01:29.670 --> 00:01:32.500
x squared of 3 is going to be
equal to 8 square roots of 3.

00:01:32.500 --> 00:01:35.680
And, of course, this side will
be 2 times 8, which is 16.

00:01:35.680 --> 00:01:37.760
I always think of the 30 degree
side as half of the

00:01:37.760 --> 00:01:40.680
hypotenuse, and then the
60-degree side as square root

00:01:40.680 --> 00:01:42.430
of 3 times the 30-degree side.

00:01:42.430 --> 00:01:43.420
That's how I think about it.

00:01:43.420 --> 00:01:46.410
But you can just look at
what they do on page

00:01:46.410 --> 00:01:48.140
838 and you'll know.

00:01:48.140 --> 00:01:51.280
If this is 2x, the side opposite
the 30-degree side is

00:01:51.280 --> 00:01:55.186
x, And the side opposite the
60-degree side is x times the

00:01:55.186 --> 00:01:57.153
square root of 3, so this
has to be 8 times the

00:01:57.153 --> 00:01:58.660
square root of 3.

00:01:58.660 --> 00:02:00.790
If this side is 8 square root
times the square root of 3,

00:02:00.790 --> 00:02:01.990
then so is this side.

00:02:01.990 --> 00:02:05.150
And so to figure out the area of
the square, you just say 8

00:02:05.150 --> 00:02:08.170
times the square root of
3 times 8 times the

00:02:08.170 --> 00:02:08.860
square root of 3.

00:02:08.860 --> 00:02:09.930
And what does that equal?

00:02:09.930 --> 00:02:13.450
That's 8 times 8, that's
64, times the

00:02:13.450 --> 00:02:15.560
square root of 3 squared.

00:02:15.560 --> 00:02:17.540
And what's the square root
of three squared?

00:02:17.540 --> 00:02:19.170
That's just 3.

00:02:19.170 --> 00:02:22.520
So it would become 64 times
3, and that's what?

00:02:22.520 --> 00:02:27.870
180 plus 12, that's
equal to 192.

00:02:27.870 --> 00:02:29.120
Next problem.

00:02:32.450 --> 00:02:35.660
It's really good to become
comfortable with

00:02:35.660 --> 00:02:38.140
the 30-60-90 triangles.

00:02:38.140 --> 00:02:41.800
You'll probably get one or two
extra problems on the SAT if

00:02:41.800 --> 00:02:43.550
you get really proficient
at that.

00:02:43.550 --> 00:02:47.750
In a mixture of peanuts and
cashews, the ratio by weight

00:02:47.750 --> 00:02:49.640
of peanuts to cashews is 5:2.

00:02:49.640 --> 00:02:54.620
Peanuts to cashews, and that's
really the weight is 5:2.

00:02:54.620 --> 00:02:58.380
How many pounds of cashews will
there be in four pounds

00:02:58.380 --> 00:02:59.670
of this mixture?

00:02:59.670 --> 00:03:04.600
How many pounds of cashews
there'll be in 4 pounds of

00:03:04.600 --> 00:03:05.700
this mixture?

00:03:05.700 --> 00:03:09.300
So we know that cashews
plus peanuts is

00:03:09.300 --> 00:03:11.590
going to be 4 pounds.

00:03:11.590 --> 00:03:15.890
And we also know-- I could
multiply both sides of this

00:03:15.890 --> 00:03:17.660
equation by C, this top one.

00:03:17.660 --> 00:03:22.400
We also know that peanuts are
going to be 5/2 the number of

00:03:22.400 --> 00:03:24.160
cashews, right?

00:03:24.160 --> 00:03:27.290
Because the weight of peanuts
to cashews is 5:2.

00:03:27.290 --> 00:03:29.440
So peanuts is going to
be 5/2 of cashews.

00:03:29.440 --> 00:03:31.890
And the cashews plus the
peanuts is equal to 4.

00:03:31.890 --> 00:03:36.850
So why don't we just substitute
this in for this?

00:03:36.850 --> 00:03:44.550
So we get cashews plus 5/2
of cashews is equal to 4.

00:03:44.550 --> 00:03:45.830
And what is this equal to?

00:03:45.830 --> 00:03:49.850
This is equal to dot common
denominator 2.

00:03:49.850 --> 00:03:51.400
This is like one
cashew, right?

00:03:51.400 --> 00:03:57.080
So that's 2/2 plus 5/2 cashews
is equal to 4.

00:03:57.080 --> 00:04:02.120
So going up here, that's
7/2 times cashews is

00:04:02.120 --> 00:04:03.850
equal to the 4.

00:04:03.850 --> 00:04:06.800
Let's multiply both sides
of this times 2.

00:04:06.800 --> 00:04:11.650
So then you get 7 cashews
is equal to 8.

00:04:11.650 --> 00:04:18.329
Divide both sides by 7, you get
cashews is equal to 8/7

00:04:18.329 --> 00:04:22.770
pounds, right?

00:04:22.770 --> 00:04:27.380
8/7 pound, so 1 and 1/7 pound
is going to be cashews.

00:04:27.380 --> 00:04:29.930
And this stuff here, I could
just multiply both sides by

00:04:29.930 --> 00:04:34.590
2/7, times 2/7.

00:04:34.590 --> 00:04:36.040
This would have canceled out,
and I would have gotten the

00:04:36.040 --> 00:04:37.000
same thing, 8/7.

00:04:37.000 --> 00:04:41.110
So there are 8/7 pounds
of cashews here.

00:04:41.110 --> 00:04:42.360
Next problem.

00:04:45.040 --> 00:04:48.860
Kind of an odd number, but I
think on these free answer,

00:04:48.860 --> 00:04:52.480
they sometimes give you problems
that might have not

00:04:52.480 --> 00:04:58.810
the most clean numbers
just to trip you up.

00:04:58.810 --> 00:05:01.240
OK, so that's the y-axis.

00:05:01.240 --> 00:05:03.650
That's my x-axis.

00:05:03.650 --> 00:05:06.680
After this problem and the
next, I will go to bed.

00:05:06.680 --> 00:05:10.420
So, let's see, that's
the y-axis.

00:05:10.420 --> 00:05:13.326
And then they drew
a line here.

00:05:13.326 --> 00:05:15.510
It looks something like that.

00:05:15.510 --> 00:05:16.760
They call that line l.

00:05:19.580 --> 00:05:22.185
They draw another
line like this.

00:05:27.070 --> 00:05:31.770
Let's see, this is A, this is
B, and they say that this

00:05:31.770 --> 00:05:34.610
point right here is
the point 8, 3.

00:05:34.610 --> 00:05:35.580
OK.

00:05:35.580 --> 00:05:39.980
line m, not shown, passes
through the origin, that's o,

00:05:39.980 --> 00:05:43.370
and intersects A, B
between A and B.

00:05:43.370 --> 00:05:46.320
What is one possible value
of the slope of line m?

00:05:46.320 --> 00:05:49.440
So line m intersects at the
origin and intersects this

00:05:49.440 --> 00:05:50.110
line someplace.

00:05:50.110 --> 00:05:53.980
It's going to look something
like this.

00:05:53.980 --> 00:05:57.630
So the whole issue here is that
this line is going to

00:05:57.630 --> 00:06:01.450
have a lower slope than this
line, than line l.

00:06:01.450 --> 00:06:02.860
So what is the slope
of line l?

00:06:02.860 --> 00:06:05.210
Well, it goes through the point
8, 3, and it also goes

00:06:05.210 --> 00:06:09.330
through the point 0, 0.

00:06:09.330 --> 00:06:11.040
So what's the slope of line l?

00:06:11.040 --> 00:06:12.780
Change in y over change in x.

00:06:15.340 --> 00:06:25.570
3 minus 0 over 8 minus 0
is equal to 3/8, right?

00:06:25.570 --> 00:06:28.780
So this slope is 3/8.

00:06:28.780 --> 00:06:31.950
So in order for this slope-- it
has to be less than this.

00:06:31.950 --> 00:06:33.750
And it's also going to be
greater than 1, right?

00:06:33.750 --> 00:06:35.300
Because it's still in the first
quadrant, so it's still

00:06:35.300 --> 00:06:36.580
going to be in this range.

00:06:36.580 --> 00:06:39.250
So what's a slope that's less
than 3/8 and greater than 1?

00:06:39.250 --> 00:06:40.960
Well, I don't know.

00:06:40.960 --> 00:06:43.240
2/8, which equals 1/4?

00:06:43.240 --> 00:06:44.190
That would work.

00:06:44.190 --> 00:06:46.690
That's less than 3/8
and greater than 1.

00:06:46.690 --> 00:06:49.460
I mean, you could say 1/8.

00:06:49.460 --> 00:06:50.010
That would work.

00:06:50.010 --> 00:06:51.100
1/16 would work.

00:06:51.100 --> 00:06:52.410
1/1,000 would work.

00:06:52.410 --> 00:06:56.120
Any of those, that's all
possible values for the slope

00:06:56.120 --> 00:06:58.770
of m because they just say
it intersects A, B

00:06:58.770 --> 00:06:59.610
between A and B.

00:06:59.610 --> 00:07:01.000
It doesn't say where.

00:07:01.000 --> 00:07:04.685
So any line, you know, it
could look like this.

00:07:04.685 --> 00:07:07.090
The line could look really
small, so that would be like a

00:07:07.090 --> 00:07:11.100
slope of 1/1,000, but
it would still work.

00:07:11.100 --> 00:07:19.430
Next problem, problem 18.

00:07:19.430 --> 00:07:20.990
OK, so they gave us a table.

00:07:25.030 --> 00:07:34.808
So they say year, number
of students.

00:07:34.808 --> 00:07:43.410
And then they go 92,
93, 94, 95, 96.

00:07:43.410 --> 00:07:58.680
In 92, there was x, then 1,552,
1,238 1,459, and 1,351

00:07:58.680 --> 00:08:01.320
The table above shows student
enrollment in Weston High

00:08:01.320 --> 00:08:03.780
School from 1992 to 1996.

00:08:03.780 --> 00:08:07.920
If the median enrollment for the
five years was 1,351-- so

00:08:07.920 --> 00:08:12.490
this is the median; this is the
middle number-- and no two

00:08:12.490 --> 00:08:14.140
years have the same enrollment,
what is the

00:08:14.140 --> 00:08:16.480
greatest possible value for x?

00:08:16.480 --> 00:08:20.270
So 1,351 has to be the
middle number.

00:08:20.270 --> 00:08:22.440
Let me make another
list. 1,351 has to

00:08:22.440 --> 00:08:24.150
be the middle number.

00:08:24.150 --> 00:08:25.890
And let's see, what are the
numbers that we know are

00:08:25.890 --> 00:08:26.910
definitely greater?

00:08:26.910 --> 00:08:35.169
We have 1,451, and then
we have 1,552.

00:08:35.169 --> 00:08:37.220
And the numbers that
are less, we know--

00:08:37.220 --> 00:08:38.630
so this is the median.

00:08:38.630 --> 00:08:43.440
We know that 1,238 is less.

00:08:43.440 --> 00:08:49.610
And we know that if 1,351 is
the median, then x is also

00:08:49.610 --> 00:08:51.850
going to have to be less than--
there has to be two

00:08:51.850 --> 00:08:57.470
numbers less than 1,351, so x
has to be less than 1,351,

00:08:57.470 --> 00:08:59.290
although x could be
here as well.

00:08:59.290 --> 00:09:00.880
It could be larger than 1,238.

00:09:00.880 --> 00:09:10.110
So we know that x is
less than 1,351.

00:09:10.110 --> 00:09:12.750
And it could have been equal
to 1,351, but they tell us

00:09:12.750 --> 00:09:15.010
that no two years have
the same enrollment.

00:09:15.010 --> 00:09:17.770
So what is the largest
possible value for x?

00:09:17.770 --> 00:09:21.100
Well, it's 1,350, right?

00:09:21.100 --> 00:09:23.040
Because you know that's the
largest number that's less

00:09:23.040 --> 00:09:24.180
than 1,351.

00:09:24.180 --> 00:09:26.410
And then this would
be 1,350 here.

00:09:26.410 --> 00:09:28.505
And then 1,351 would still
be the median.

00:09:28.505 --> 00:09:29.900
And we are done!

00:09:29.900 --> 00:09:31.890
I'll see you in the
next section.


SAT Prep: Test 7 Section 8 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.590
We are in Section 8 on the
seventh test. We're almost

00:00:03.590 --> 00:00:04.960
done with this book,
so let's start.

00:00:04.960 --> 00:00:06.210
Problem number one.

00:00:08.760 --> 00:00:19.520
If 6,700 is equal to 100 times
6k plus 7, then k equals what?

00:00:19.520 --> 00:00:22.060
So let's divide both sides
of this by 100.

00:00:22.060 --> 00:00:25.060
So you get 100 and you get these
two zeroes, so you get

00:00:25.060 --> 00:00:28.310
67 is equal to 6k plus 7.

00:00:28.310 --> 00:00:30.520
I just divided both
sides by 100.

00:00:30.520 --> 00:00:31.980
Subtract 7 from both sides.

00:00:31.980 --> 00:00:34.490
You get 60 is equal to 6k.

00:00:34.490 --> 00:00:38.310
Divide both sides by 6, you
get 10 is equal to k.

00:00:38.310 --> 00:00:40.740
That's choice C.

00:00:40.740 --> 00:00:43.040
Problem 2.

00:00:43.040 --> 00:00:49.850
If 3 more than n is a negative
number-- so n plus 3 is a

00:00:49.850 --> 00:00:52.270
negative number-- that's
what that says, right?

00:00:52.270 --> 00:00:57.210
And 5 more than n is a positive
number, so n plus 5

00:00:57.210 --> 00:01:01.320
is greater than 0, which of
the following could be the

00:01:01.320 --> 00:01:02.770
value of n?

00:01:02.770 --> 00:01:07.810
So n plus 3 is negative, n plus
5 is greater, so n plus 3

00:01:07.810 --> 00:01:12.710
has to be less than
negative 3, right?

00:01:12.710 --> 00:01:15.500
This means that n is less
than negative 3.

00:01:15.500 --> 00:01:16.180
How did I get that?

00:01:16.180 --> 00:01:19.970
I subtracted 3 from both sides
of this equation, right?

00:01:19.970 --> 00:01:23.520
If I subtract 5 from both sides
of this equation, I

00:01:23.520 --> 00:01:28.570
would get that n is greater
than negative 5.

00:01:28.570 --> 00:01:30.460
So you have to be greater than
negative 5 and less than

00:01:30.460 --> 00:01:31.780
negative 3.

00:01:31.780 --> 00:01:34.240
What's the only number that I
can think of that's that?

00:01:34.240 --> 00:01:37.140
Well, there's a lot of numbers,
but a simple one is

00:01:37.140 --> 00:01:39.230
minus 4, right?

00:01:39.230 --> 00:01:42.140
Minus 4 is less than
negative 3, right?

00:01:42.140 --> 00:01:46.540
If I draw a number line, this is
minus 3, minus 4, minus 5.

00:01:46.540 --> 00:01:49.280
So minus 4 is less than negative
3 and it's greater

00:01:49.280 --> 00:01:50.100
than minus 5.

00:01:50.100 --> 00:01:51.120
So that's the answer.

00:01:51.120 --> 00:01:53.310
Choice B.

00:01:53.310 --> 00:01:55.100
And you can try it
out with minus 4.

00:01:55.100 --> 00:01:57.110
Minus 4 plus 3 is minus 1.

00:01:57.110 --> 00:01:58.520
So that works.

00:01:58.520 --> 00:02:00.830
Minus 4 plus 5 is plus 1.

00:02:00.830 --> 00:02:03.270
So that works.

00:02:03.270 --> 00:02:04.520
Problem three.

00:02:08.960 --> 00:02:14.090
OK in the figure above-- OK let
me see if I can draw this.

00:02:14.090 --> 00:02:16.640
I'll draw it big because
it looks complicated.

00:02:16.640 --> 00:02:26.800
A line like that, a line like
that, and one like that.

00:02:26.800 --> 00:02:41.180
They tell us this is x, this is
y, and then they tell us in

00:02:41.180 --> 00:02:44.020
the figure above, if x is equal
to 70, so this is equal

00:02:44.020 --> 00:02:49.710
to 70, and y is equal to 40,
so this is equal to 40, and

00:02:49.710 --> 00:02:53.030
the dotted lines bisect the
angles with measure x and y

00:02:53.030 --> 00:02:55.180
degrees, what is
the value of z?

00:02:55.180 --> 00:02:57.370
OK, so the dotted lines, instead
of drawing dotted

00:02:57.370 --> 00:03:00.950
lines, I'm going to draw green
lines, so this green line will

00:03:00.950 --> 00:03:05.450
bisect this angle, so it
goes right in between.

00:03:05.450 --> 00:03:08.600
So if it bisects the angle, what
is the measure of this

00:03:08.600 --> 00:03:11.350
angle right here?

00:03:11.350 --> 00:03:13.630
Well, it's going to be half of
this angle that it bisected.

00:03:13.630 --> 00:03:16.240
So this whole thing
is 70, so this is

00:03:16.240 --> 00:03:18.860
going to be 35 degrees.

00:03:18.860 --> 00:03:21.990
Similarly, this other green line
is going to bisect this

00:03:21.990 --> 00:03:25.960
y-- oh, I thought I was
using the line tool.

00:03:25.960 --> 00:03:30.490
This other green line is going
to bisect y right here, right?

00:03:30.490 --> 00:03:31.380
So it bisects it.

00:03:31.380 --> 00:03:35.100
So if it bisects it, what's
this angle going to be?

00:03:35.100 --> 00:03:37.280
Well, it's going to be half of
angle y, and they tell us

00:03:37.280 --> 00:03:38.240
angle y is 40.

00:03:38.240 --> 00:03:38.810
I wrote that.

00:03:38.810 --> 00:03:39.800
I just wrote over it.

00:03:39.800 --> 00:03:41.910
So this has to be 20
degrees, right?

00:03:41.910 --> 00:03:44.600
Because it's half
of the full y.

00:03:44.600 --> 00:03:45.770
And they say, what is
the values of z?

00:03:45.770 --> 00:03:48.840
Well, z is just this
whole thing.

00:03:48.840 --> 00:03:52.890
So it's this angle plus this
angle, so it's 55 degrees.

00:03:52.890 --> 00:03:54.140
And that's choice E.

00:03:56.840 --> 00:03:58.090
Problem four.

00:04:02.800 --> 00:04:04.960
A piece of fruit is to
be chosen at random

00:04:04.960 --> 00:04:06.700
from a basket a fruit.

00:04:06.700 --> 00:04:09.560
The probability that the piece
of fruit chosen will be an

00:04:09.560 --> 00:04:11.610
apple is 2/5.

00:04:11.610 --> 00:04:19.589
So the probability of an
apple is equal to 2/5.

00:04:19.589 --> 00:04:23.620
Which of the following could not
be the number of pieces of

00:04:23.620 --> 00:04:25.760
fruit in the basket?

00:04:25.760 --> 00:04:27.350
Could not be.

00:04:27.350 --> 00:04:35.280
So the secret here is you take
2/5 and whatever 2/5 is times

00:04:35.280 --> 00:04:37.610
the number of fruit, right?

00:04:37.610 --> 00:04:43.410
2/5 times the number of
fruit has to equal

00:04:43.410 --> 00:04:44.660
the number of apples.

00:04:47.970 --> 00:04:53.430
And implicitly, there's not
going to be a fractional

00:04:53.430 --> 00:04:54.980
number of apples
in the basket.

00:04:54.980 --> 00:04:56.840
You can't pick up a
piece of an apple.

00:04:56.840 --> 00:04:59.450
So this is going to have to be
an integer, the number of

00:04:59.450 --> 00:05:01.140
apples, right?

00:05:01.140 --> 00:05:07.020
So if 2/5 times some number is
going to be an integer, then

00:05:07.020 --> 00:05:10.020
this number has to be
divisible by 5.

00:05:10.020 --> 00:05:13.140
It makes sense, you know?

00:05:13.140 --> 00:05:14.810
If there are 10 fruit,
then this is going

00:05:14.810 --> 00:05:15.640
to be equal to 4.

00:05:15.640 --> 00:05:18.040
If there are 20 fruit, then
this is going to be what?

00:05:18.040 --> 00:05:19.740
What's 2/5 of 20?

00:05:19.740 --> 00:05:21.350
It's going to be 8.

00:05:21.350 --> 00:05:25.605
But if there were 6 fruit, or
if we thought the number of

00:05:25.605 --> 00:05:28.860
fruit was 6, 2/5 times
6 is 12/5.

00:05:28.860 --> 00:05:32.650
That means that there are
2 and 2/5 apples.

00:05:32.650 --> 00:05:33.650
That's not right.

00:05:33.650 --> 00:05:35.740
So what we have to find is the
number of fruit, and it has to

00:05:35.740 --> 00:05:37.822
be some number divisible by 5.

00:05:37.822 --> 00:05:40.140
And if you look at the choices,
they give you 20,

00:05:40.140 --> 00:05:41.200
that's divisible by 5.

00:05:41.200 --> 00:05:43.720
35 is divisible by 5.

00:05:43.720 --> 00:05:45.840
Choice C, 52.

00:05:45.840 --> 00:05:48.020
That's not divisible by 5.

00:05:48.020 --> 00:05:50.100
And then the other two choices,
70 and 80 are

00:05:50.100 --> 00:05:51.010
divisible by 5.

00:05:51.010 --> 00:05:53.650
So only C is not
divisible by 5.

00:05:53.650 --> 00:05:54.920
So that's our choice.

00:05:54.920 --> 00:05:57.180
That could not be the number
of fruit in the basket.

00:05:57.180 --> 00:05:58.260
And you could figure it out.

00:05:58.260 --> 00:06:01.770
If there were 52 fruit, and 2/5
were apples, you'd get a

00:06:01.770 --> 00:06:03.020
fractional number of apples.

00:06:03.020 --> 00:06:04.740
And that's not cool.

00:06:04.740 --> 00:06:05.990
Next problem.

00:06:10.080 --> 00:06:12.590
Problem five.

00:06:12.590 --> 00:06:14.960
A square and an equilateral
triangle have equal

00:06:14.960 --> 00:06:16.130
perimeters.

00:06:16.130 --> 00:06:18.740
If the square has sides of
length 3, what is the length

00:06:18.740 --> 00:06:21.190
of one side of the triangle?

00:06:21.190 --> 00:06:29.340
So a square and a triangle
have equal perimeters.

00:06:29.340 --> 00:06:32.240
And the square has--
what does it say?

00:06:32.240 --> 00:06:34.050
It has sides of length 3.

00:06:34.050 --> 00:06:36.740
So 3, 3, 3, 3.

00:06:36.740 --> 00:06:38.300
So what's this perimeter?

00:06:38.300 --> 00:06:40.020
It's 3 plus 3 plus 3 plus 3.

00:06:40.020 --> 00:06:41.010
That's 12.

00:06:41.010 --> 00:06:42.500
Perimeter is 12.

00:06:42.500 --> 00:06:44.420
Well, this triangle is also
going to have a perimeter of

00:06:44.420 --> 00:06:50.050
12, but it's going to have it
amongst 3 equal sides, right?

00:06:50.050 --> 00:06:52.140
Because there's only 3
sides of a triangle.

00:06:52.140 --> 00:06:55.315
So it's going to have 12 divided
by 3, so it's going to

00:06:55.315 --> 00:06:58.640
have to be 4, 4, and 4,
so one side of the

00:06:58.640 --> 00:07:00.750
triangle has length 4.

00:07:00.750 --> 00:07:02.100
That's choice C.

00:07:05.400 --> 00:07:07.134
Problem six.

00:07:07.134 --> 00:07:10.930
We'll do it here.

00:07:10.930 --> 00:07:18.540
If x is equal to negative 1 and
k is greater than 0, which

00:07:18.540 --> 00:07:22.110
of the following has
the greatest value?

00:07:22.110 --> 00:07:29.080
OK, so choice A is 2
times k times x.

00:07:29.080 --> 00:07:32.550
So k is positive, x is minus
1, so this is going to

00:07:32.550 --> 00:07:34.230
be less than 0.

00:07:34.230 --> 00:07:36.890
So this is going to be a
negative number, right?

00:07:36.890 --> 00:07:39.860
And so we know that's probably
not going to be the answer.

00:07:39.860 --> 00:07:42.740
So if we look at all of the
choices, the only positive

00:07:42.740 --> 00:07:45.650
answers, the only positive
choices, are going to be the

00:07:45.650 --> 00:07:50.280
ones where I raise x to an
even exponent, right?

00:07:50.280 --> 00:07:53.840
Because if I raised x to an odd
exponent, like let's see,

00:07:53.840 --> 00:07:59.195
choice C, choice C
is 6k x cubed.

00:07:59.195 --> 00:08:02.600
And that's equal to 6k times
minus 1 cubed, right?

00:08:02.600 --> 00:08:03.980
Because x is minus 1.

00:08:03.980 --> 00:08:06.530
So that's going to
be minus 6k.

00:08:06.530 --> 00:08:08.220
And we know k is a
positive number.

00:08:08.220 --> 00:08:12.440
I mean, really, you could just
ignore the k's, really.

00:08:12.440 --> 00:08:13.580
k is just a positive number.

00:08:13.580 --> 00:08:14.700
It's just kind of a
scaling factor.

00:08:14.700 --> 00:08:16.050
And they all have k in them.

00:08:16.050 --> 00:08:17.660
So you just have to worry about
the coefficient in the x

00:08:17.660 --> 00:08:21.850
terms. So one thing we can--
if x has an odd exponent on

00:08:21.850 --> 00:08:24.440
it, it's going to be a negative
number, so that's not

00:08:24.440 --> 00:08:25.990
going to be the greatest
value.

00:08:25.990 --> 00:08:28.560
So there are only two that
have x with a-- so

00:08:28.560 --> 00:08:31.260
you have 4k x squared.

00:08:31.260 --> 00:08:32.690
That's choice B.

00:08:32.690 --> 00:08:40.900
And then you have choice D,
which is 8k x to the fourth.

00:08:40.900 --> 00:08:44.230
Well, in both of these cases,
x squared is equal to 1.

00:08:44.230 --> 00:08:45.950
Negative 1 squared is 1.

00:08:45.950 --> 00:08:49.170
Negative 1 to the fourth is
also equal to 1, right?

00:08:49.170 --> 00:08:51.020
And if k is positive,
what's bigger?

00:08:51.020 --> 00:08:52.710
8k or 4k?

00:08:52.710 --> 00:08:54.930
Well, 8k is going
to be bigger.

00:08:54.930 --> 00:08:56.420
As I said, they all have k in
them so you could kind of

00:08:56.420 --> 00:08:58.310
ignore k because it's
a positive number.

00:08:58.310 --> 00:09:01.000
So the answer is choice D.

00:09:01.000 --> 00:09:02.860
I'll see you in the
next video.


SAT Prep: Test 7 Section 8 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:01.640
In the home stretch.

00:00:01.640 --> 00:00:05.910
Problem 14.

00:00:05.910 --> 00:00:13.200
If n and p are integers greater
than 1-- I didn't

00:00:13.200 --> 00:00:17.420
write that they're integers--
and if p is a factor of both n

00:00:17.420 --> 00:00:33.130
plus 3 and n plus 10, what
is the value of p?

00:00:33.130 --> 00:00:36.610
So n and p are integers greater
than 1 and p is a

00:00:36.610 --> 00:00:42.400
factor of both n plus
3 and n plus 10.

00:00:42.400 --> 00:00:43.860
This is interesting.

00:00:43.860 --> 00:00:52.890
So that means that n plus 3 is
equal to some number times p,

00:00:52.890 --> 00:00:55.710
where k is just some
random number.

00:00:55.710 --> 00:01:00.560
And we also know that n plus
10 is also equal to some--

00:01:00.560 --> 00:01:05.700
maybe, probably, definitely
some other number times p.

00:01:05.700 --> 00:01:09.720
And so we could subtract
3 from both

00:01:09.720 --> 00:01:10.730
sides in this equation.

00:01:10.730 --> 00:01:15.330
We'd get n is equal to k
times p, some number

00:01:15.330 --> 00:01:17.330
times p minus 3.

00:01:17.330 --> 00:01:19.300
And we also could do it here.

00:01:19.300 --> 00:01:20.640
We can subtract 10
from both sides.

00:01:20.640 --> 00:01:27.520
We say n is equal to some other
number times p minus 10.

00:01:27.520 --> 00:01:28.660
And I don't know where
this is going.

00:01:28.660 --> 00:01:30.550
I'm just really playing
around with this.

00:01:30.550 --> 00:01:31.630
So let's see.

00:01:31.630 --> 00:01:34.130
So both of these things
are equal to n.

00:01:34.130 --> 00:01:35.560
What are we trying
to solve for?

00:01:35.560 --> 00:01:37.460
We're trying to solve for p.

00:01:37.460 --> 00:01:40.560
So some number times p minus 3
is equal to some other number

00:01:40.560 --> 00:01:42.480
times p minus 10.

00:01:42.480 --> 00:01:44.360
Let me see where that gets me.

00:01:44.360 --> 00:01:52.300
So kp minus 3 is equal to
m times p minus 10.

00:01:52.300 --> 00:01:57.870
Let's add 10 to both sides
of this equation.

00:01:57.870 --> 00:02:05.340
You'll get kp plus 7 is
equal to m times p.

00:02:05.340 --> 00:02:07.460
Now, this is interesting.

00:02:07.460 --> 00:02:12.460
So if I multiply some integer
times p, right?

00:02:12.460 --> 00:02:15.590
So this is some multiple
of p, right?

00:02:15.590 --> 00:02:17.860
This is actually n plus 3.

00:02:17.860 --> 00:02:22.910
But if I have some multiple of
p right here and I add 7 to

00:02:22.910 --> 00:02:26.610
it, I get another
multiple of p.

00:02:26.610 --> 00:02:31.210
So that means that 7 has
to be divisible by p.

00:02:31.210 --> 00:02:38.060
There are only two numbers
that 7 is divisible by.

00:02:38.060 --> 00:02:38.885
1 and 7.

00:02:38.885 --> 00:02:42.210
And it's not going to be 1,
because it tells us that n and

00:02:42.210 --> 00:02:44.030
p are both greater
than 1, right?

00:02:44.030 --> 00:02:47.120
So p has to be 7.

00:02:47.120 --> 00:02:51.670
p is equal to 7, which
is choice B.

00:02:51.670 --> 00:02:59.960
And so the way I think about it
is just when you add 10 to

00:02:59.960 --> 00:03:01.590
a number-- think
of it this way.

00:03:01.590 --> 00:03:04.090
Ignore this whole n plus 3.

00:03:04.090 --> 00:03:08.390
Let's just say n plus 3, let's
just say that we'll

00:03:08.390 --> 00:03:11.030
call this Q, right?

00:03:11.030 --> 00:03:15.020
We're saying p is a factor
of Q, right?

00:03:15.020 --> 00:03:19.110
And then if Q is n plus 3, then
this would be Q plus 7.

00:03:19.110 --> 00:03:21.590
I hope I'm not confusing you.

00:03:21.590 --> 00:03:24.540
So this is divisible by p.

00:03:24.540 --> 00:03:28.040
And then when you add 7 to it,
it's also divisible by p.

00:03:28.040 --> 00:03:33.490
So p has to be 7, because when
I add 7 to-- think of it this

00:03:33.490 --> 00:03:43.020
way: if n plus 3 was-- let's
say this was 21.

00:03:43.020 --> 00:03:48.000
Not n, n plus 3 was 21, then
n would be what, 18?

00:03:48.000 --> 00:03:56.220
n is 18, n plus 3 is 21, and
n plus 10 would be 28.

00:03:56.220 --> 00:03:58.330
And these are both
divisible by 7.

00:03:58.330 --> 00:03:59.340
So that's another way
you could do it.

00:03:59.340 --> 00:04:01.350
You could actually just
try out the numbers.

00:04:01.350 --> 00:04:02.500
But hopefully, that'll give
you the intuition.

00:04:02.500 --> 00:04:06.590
But the big clue is some
multiple times p plus 7 is

00:04:06.590 --> 00:04:08.800
equal to another multiple
times p.

00:04:08.800 --> 00:04:13.140
So we know that p has to be 7
because p has to be divisible

00:04:13.140 --> 00:04:14.830
into 7 and greater than 1.

00:04:14.830 --> 00:04:17.570
And there's only one number
that's greater than 1 and that

00:04:17.570 --> 00:04:18.510
is a factor of 7.

00:04:18.510 --> 00:04:20.360
And that's B.

00:04:20.360 --> 00:04:21.950
Or that's 7, sorry.

00:04:21.950 --> 00:04:25.160
Next problem.

00:04:25.160 --> 00:04:26.950
That was tricky, I think.

00:04:26.950 --> 00:04:29.860
Problem 15.

00:04:29.860 --> 00:04:32.680
In the queue shown above,
points A, B and C are

00:04:32.680 --> 00:04:34.840
midpoints of the three edges.

00:04:34.840 --> 00:04:39.110
Which of the following angles
has the least measure?

00:04:39.110 --> 00:04:40.980
Oh man, this is going to
be a lot of drawing.

00:04:40.980 --> 00:04:42.670
This might take the whole
time, but I'll

00:04:42.670 --> 00:04:45.430
try my best. OK.

00:04:45.430 --> 00:04:46.760
B, C and E.

00:04:46.760 --> 00:04:54.080
I'm going to draw it big
just so-- actually, I'm

00:04:54.080 --> 00:04:55.330
drawing it too big.

00:04:59.930 --> 00:05:01.540
I'll do it the best I can.

00:05:17.130 --> 00:05:17.590
OK.

00:05:17.590 --> 00:05:19.315
And then they have these dotted
lines in the back.

00:05:30.440 --> 00:05:32.570
OK, now they have--
let me draw this

00:05:32.570 --> 00:05:33.510
in a different color.

00:05:33.510 --> 00:05:34.760
This is point x.

00:05:37.800 --> 00:05:40.620
This is point y.

00:05:40.620 --> 00:05:41.970
They draw a bunch of points.

00:05:41.970 --> 00:05:43.850
B, C, and E are the midpoints.

00:05:43.850 --> 00:05:47.170
So this is point B.

00:05:47.170 --> 00:05:51.100
This is point A.

00:05:51.100 --> 00:05:54.430
This is point C.

00:05:54.430 --> 00:05:57.700
Point D.

00:05:57.700 --> 00:06:00.720
Point E.

00:06:00.720 --> 00:06:04.020
And everything they drew
essentially starts at x, goes

00:06:04.020 --> 00:06:07.785
to one of these points, and then
goes back to y, right?

00:06:07.785 --> 00:06:12.590
It goes from x to B
to y, x to A to y.

00:06:12.590 --> 00:06:14.520
All the choices go to each
of these points and

00:06:14.520 --> 00:06:15.290
then go back to y.

00:06:15.290 --> 00:06:17.510
And what they want to
know is, which angle

00:06:17.510 --> 00:06:21.740
has the least measure?

00:06:21.740 --> 00:06:24.130
So the way we could think about
this is, all of these

00:06:24.130 --> 00:06:31.136
angles-- we could just draw this
part-- sorry, if this was

00:06:31.136 --> 00:06:33.680
x, this is one of the messier
things I've ever drawn, this

00:06:33.680 --> 00:06:36.020
is y and then the other angles,
you know it's going to

00:06:36.020 --> 00:06:37.955
go to some point and
then come back.

00:06:37.955 --> 00:06:40.800
And what we want to know
is, when do we

00:06:40.800 --> 00:06:44.540
get the least angle?

00:06:44.540 --> 00:06:47.540
Think of it this way, the longer
this base angle is, the

00:06:47.540 --> 00:06:50.000
longer this is, the
smaller the angle.

00:06:50.000 --> 00:06:51.730
Because this is the angle
we're measuring.

00:06:51.730 --> 00:06:54.950
So we want to make
this angle, this

00:06:54.950 --> 00:06:56.235
length as long as possible.

00:06:56.235 --> 00:06:59.900
The length from whatever letter
we are to y, right?

00:06:59.900 --> 00:07:02.150
So B to y is very short,
so this angle

00:07:02.150 --> 00:07:02.980
would be pretty big.

00:07:02.980 --> 00:07:05.420
A to y is even bigger.

00:07:05.420 --> 00:07:07.195
C to y is even bigger
than that.

00:07:07.195 --> 00:07:10.530
D to y is actually the biggest.
This is the biggest

00:07:10.530 --> 00:07:12.900
distance, right?

00:07:12.900 --> 00:07:16.746
So this will be the
smallest angle.

00:07:16.746 --> 00:07:23.282
So that's ydx or xdy,
so that's choice D.

00:07:23.282 --> 00:07:26.750
And remember, the underlying
intuition is, if I were to--

00:07:26.750 --> 00:07:30.500
let me do the colors so you know
what I'm saying-- this

00:07:30.500 --> 00:07:34.440
line, if I were to draw that,
that would be here.

00:07:34.440 --> 00:07:36.520
I'm saying that corresponds
to that right there.

00:07:36.520 --> 00:07:37.155
That's x.

00:07:37.155 --> 00:07:38.130
That's y.

00:07:38.130 --> 00:07:40.610
And then we're taking a
point from x to one of

00:07:40.610 --> 00:07:41.980
these points, right?

00:07:41.980 --> 00:07:47.770
xby, xay, xcy, and whatever
point it is, the way we get

00:07:47.770 --> 00:07:49.460
the smallest angle is
if we make this

00:07:49.460 --> 00:07:53.330
line as long as possible.

00:07:53.330 --> 00:07:54.830
And you can visualize that.

00:07:54.830 --> 00:07:57.090
Take this point further and
further out and this angle

00:07:57.090 --> 00:08:00.040
gets smaller and smaller
and smaller.

00:08:00.040 --> 00:08:04.130
So if you look at it that way,
what is going to be the

00:08:04.130 --> 00:08:07.110
longest distance y to which
of these points?

00:08:07.110 --> 00:08:10.480
y to C is the same as y to E.

00:08:10.480 --> 00:08:11.905
y to B is very short.

00:08:11.905 --> 00:08:12.990
y to A id a little longer.

00:08:12.990 --> 00:08:14.260
y to C is a little longer.

00:08:14.260 --> 00:08:16.320
y to D is the longest
distance.

00:08:16.320 --> 00:08:20.365
So the least angle will be,
if this point is D, xdy.

00:08:20.365 --> 00:08:23.428
And that's choice D.

00:08:23.428 --> 00:08:26.510
Let me see if I can squeeze
problem 16 in here.

00:08:26.510 --> 00:08:28.360
Otherwise, I'll do it
in the next video.

00:08:28.360 --> 00:08:29.930
Problem 16.

00:08:29.930 --> 00:08:41.130
if xy is equal to 7, x minus y
is equal to 5 then x squared y

00:08:41.130 --> 00:08:46.370
minus xy squared is
equal to what?

00:08:46.370 --> 00:08:48.430
All right, so we know what
xy is and we know

00:08:48.430 --> 00:08:49.410
what x minus y is.

00:08:49.410 --> 00:08:51.220
So let's watch this.

00:08:51.220 --> 00:08:52.360
Factor an x out of here.

00:08:52.360 --> 00:08:56.830
You get the same thing as x
times xy, minus-- let's factor

00:08:56.830 --> 00:08:57.890
a y out of here.

00:08:57.890 --> 00:09:02.410
This is the same thing
as y times xy, right?

00:09:02.410 --> 00:09:04.430
All I did is I factored
a y out.

00:09:04.430 --> 00:09:06.790
And now let's factor
the x minus y out.

00:09:06.790 --> 00:09:11.680
This is the same thing as xy
times x minus y, right?

00:09:11.680 --> 00:09:13.500
I'm just factoring the xy out.

00:09:13.500 --> 00:09:14.460
You could do it in reverse.

00:09:14.460 --> 00:09:17.840
You can distribute the xy and
you'd get this up here.

00:09:17.840 --> 00:09:19.070
And this equals what?

00:09:19.070 --> 00:09:21.730
xy is 7.

00:09:21.730 --> 00:09:25.190
And x minus y, it gives
us right there, is 5.

00:09:25.190 --> 00:09:28.420
7 times 5 is 35.

00:09:28.420 --> 00:09:29.885
That's choice D.

00:09:29.885 --> 00:09:32.600
And we are done with Test 7.

00:09:32.600 --> 00:09:35.260
I will see you in Test 8.


SAT Prep: Test 7 Section 8 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.200
I'm on problem number seven.

00:00:04.200 --> 00:00:07.250
Josephine's daily exercise
routine consists of swimming,

00:00:07.250 --> 00:00:08.720
cycling, and running
in that order.

00:00:08.720 --> 00:00:14.660
So swimming, cycling, and
running in that order.

00:00:14.660 --> 00:00:23.340
She runs faster than she swims,
and cycles faster than

00:00:23.340 --> 00:00:24.990
she runs, like most people.

00:00:27.640 --> 00:00:29.900
If she does not rest between the
activities, which of the

00:00:29.900 --> 00:00:32.580
following could be the graph
of the distance she covers

00:00:32.580 --> 00:00:35.650
during the entire time of
her exercise routine?

00:00:35.650 --> 00:00:37.460
So I'm not going to even
look at the choices.

00:00:37.460 --> 00:00:39.060
And I'll just give you an
intuition and then you can

00:00:39.060 --> 00:00:42.150
look for the choices
that match it.

00:00:42.150 --> 00:00:52.030
So if I were to graph distance
versus time, the slope of my

00:00:52.030 --> 00:00:53.980
line is going to be
my speed, right?

00:00:53.980 --> 00:00:54.830
Because what is speed?

00:00:54.830 --> 00:00:56.570
It's distance per time.

00:00:56.570 --> 00:00:58.550
And that's the same
thing as slope.

00:00:58.550 --> 00:01:00.320
It's rise over run.

00:01:00.320 --> 00:01:02.700
So in this context, that would
be the slope of the line.

00:01:02.700 --> 00:01:05.180
So what you want is a situation
where she swims

00:01:05.180 --> 00:01:08.460
first-- which is her slowest--
then she cycles-- which is her

00:01:08.460 --> 00:01:11.640
fastest-- then she runs-- which
is kind of in between.

00:01:11.640 --> 00:01:14.020
So what you're going to have
is a low slope, then a very

00:01:14.020 --> 00:01:16.680
high slope, and then you want
kind of a medium slope.

00:01:16.680 --> 00:01:18.550
So it's going to look
something like this.

00:01:18.550 --> 00:01:21.990
Because she swims first, so
that's the slowest exercise,

00:01:21.990 --> 00:01:24.050
so maybe it'll be like that.

00:01:24.050 --> 00:01:26.693
Then she cycles, which is her
fastest exercise, so maybe it

00:01:26.693 --> 00:01:28.180
will look something like that.

00:01:28.180 --> 00:01:30.250
And then she runs, which is
kind of an in-between

00:01:30.250 --> 00:01:33.390
exercise, so maybe it looks
something like this.

00:01:33.390 --> 00:01:36.460
So which of the choices
looks like that?

00:01:36.460 --> 00:01:39.630
Where you start relatively
slow, you peak out pretty

00:01:39.630 --> 00:01:43.090
fast, and then you do something
that looks a little

00:01:43.090 --> 00:01:45.300
bit faster-- that has a higher
slope than this.

00:01:45.300 --> 00:01:47.730
And if I look at all
of the choices, it

00:01:47.730 --> 00:01:51.040
looks like choice E.

00:01:51.040 --> 00:01:55.230
Choice E is actually pretty
close to what I drew.

00:01:55.230 --> 00:01:56.880
If you look at the other
choices, choice A she goes

00:01:56.880 --> 00:02:00.960
slow, then medium, then fast.
So that's like if she cycled

00:02:00.960 --> 00:02:02.630
last, right?

00:02:02.630 --> 00:02:04.920
In choice B she does her slow
thing last. That looks like

00:02:04.920 --> 00:02:07.690
she's swimming last. Choice C
looks like she's swimming last

00:02:07.690 --> 00:02:10.780
as well, because that's the
slowest. And then choice D

00:02:10.780 --> 00:02:13.620
looks like she cycles, then
she swims, then she runs.

00:02:13.620 --> 00:02:15.390
So it's definitely choice E.

00:02:15.390 --> 00:02:18.150
You go slow, fast,
and then medium.

00:02:18.150 --> 00:02:19.400
Next problem.

00:02:25.900 --> 00:02:28.510
Problem eight.

00:02:28.510 --> 00:02:34.000
In the xy-coordinate systems,
square root of 6-- sorry, the

00:02:34.000 --> 00:02:38.600
point is square root of 6 comma
k-- is one of the points

00:02:38.600 --> 00:02:40.075
of intersection of the graphs.

00:02:40.075 --> 00:02:41.820
So these are the graphs.

00:02:41.820 --> 00:02:49.560
y is equal to x squared minus
7, and y is equal to minus x

00:02:49.560 --> 00:02:54.890
squared plus j.

00:02:54.890 --> 00:02:55.640
Fair enough.

00:02:55.640 --> 00:02:56.860
Where j is a constant.

00:02:56.860 --> 00:02:58.930
What is the value of j?

00:02:58.930 --> 00:03:01.490
So we know the x-coordinate
where it intersects, right?

00:03:01.490 --> 00:03:04.200
We know it intersects
at square root of 6.

00:03:04.200 --> 00:03:07.590
So intersection means that the
y values are the same for a

00:03:07.590 --> 00:03:08.670
given x value.

00:03:08.670 --> 00:03:10.210
That's what intersects.

00:03:10.210 --> 00:03:12.810
So we can set these two
equations equal to each other.

00:03:12.810 --> 00:03:16.080
So we could say, x squared--
we could set the y values

00:03:16.080 --> 00:03:19.640
equal to each other-- x squared
minus 7 is equal to

00:03:19.640 --> 00:03:24.860
this: minus x squared plus j.

00:03:24.860 --> 00:03:29.940
Now we could keep things in
terms of x or we could-- well

00:03:29.940 --> 00:03:31.690
let's just algebraically play
with this a little bit.

00:03:31.690 --> 00:03:35.300
And I hope you understand that
this is what we would do to

00:03:35.300 --> 00:03:37.700
solve for the intersection of
these two graphs, if we wanted

00:03:37.700 --> 00:03:39.495
to solve for the x value.

00:03:39.495 --> 00:03:40.960
And we could play with
this a little bit.

00:03:40.960 --> 00:03:43.010
We could add x squared
to both sides.

00:03:43.010 --> 00:03:49.530
You get 2x squared minus
7 is equal to j.

00:03:49.530 --> 00:03:50.340
And what's x?

00:03:50.340 --> 00:03:51.640
We know where these intersect.

00:03:51.640 --> 00:03:53.570
We know one of the points
where they intersect is

00:03:53.570 --> 00:03:54.930
square root of 6.

00:03:54.930 --> 00:04:00.150
So let's input-- let's replace
x with square root of 6.

00:04:00.150 --> 00:04:06.610
So 2 times square root of 6
squared minus 7 is equal to j.

00:04:06.610 --> 00:04:08.185
What's the square root
of 6 squared?

00:04:08.185 --> 00:04:10.660
It's just 6, right?

00:04:10.660 --> 00:04:15.540
So it's 2 times 6 minus
7 is equal to j.

00:04:15.540 --> 00:04:19.079
So 12 minus 7 is equal to j.

00:04:19.079 --> 00:04:20.810
5 is equal to j.

00:04:20.810 --> 00:04:22.250
And that is choice A.

00:04:24.960 --> 00:04:26.210
Problem nine.

00:04:31.920 --> 00:04:38.850
If the absolute value of 2 minus
x is less than 3, which

00:04:38.850 --> 00:04:41.270
of the following is a
possible value of x?

00:04:41.270 --> 00:04:51.030
So this means that 2 minus
x is less than 3.

00:04:51.030 --> 00:04:51.840
Right?

00:04:51.840 --> 00:04:53.950
So I guess the way you can
think about is that the

00:04:53.950 --> 00:04:56.070
difference between 2 and
x is going to be

00:04:56.070 --> 00:04:58.750
less than 3, right?

00:04:58.750 --> 00:05:04.030
The other way you could view
it is that the negative of

00:05:04.030 --> 00:05:10.920
this-- x minus 2-- is going
to be less than 3.

00:05:10.920 --> 00:05:11.520
Right?

00:05:11.520 --> 00:05:15.140
We could have also said 2
minus x is greater than

00:05:15.140 --> 00:05:16.140
negative 3.

00:05:16.140 --> 00:05:17.470
That's the other thing we
could have written.

00:05:17.470 --> 00:05:21.080
And what I did is I just-- this
is either going to be a

00:05:21.080 --> 00:05:22.970
positive or a negative, right?

00:05:22.970 --> 00:05:26.350
So if it's a positive, we just
say 2 minus x is less than 3.

00:05:26.350 --> 00:05:30.030
If x is going to be greater than
2, then we could say that

00:05:30.030 --> 00:05:31.930
x minus 2 is less than 3.

00:05:31.930 --> 00:05:33.390
Because then it would be a
negative number and then it

00:05:33.390 --> 00:05:34.636
would become positive.

00:05:34.636 --> 00:05:36.420
Hope I didn't confuse you.

00:05:36.420 --> 00:05:40.220
So if this is true you would
get-- let's see-- you would

00:05:40.220 --> 00:05:44.840
get, if I were to add x to both
sides I'd get 2 is less

00:05:44.840 --> 00:05:47.050
than 3 plus x.

00:05:47.050 --> 00:05:49.020
Subtract 3 from both sides.

00:05:49.020 --> 00:05:52.040
You get negative 1
is less than x.

00:05:52.040 --> 00:05:53.360
This one, add 2 to both sides.

00:05:53.360 --> 00:05:57.110
You get x is less than 5.

00:05:57.110 --> 00:06:02.670
So x is greater than negative
1 and it is less than 5.

00:06:02.670 --> 00:06:11.840
So if you look at all of the
choices-- it has to be less

00:06:11.840 --> 00:06:18.750
than 5 and greater than negative
1-- so out of all of

00:06:18.750 --> 00:06:22.170
those choices, the only choice
that is less than 5 is choice

00:06:22.170 --> 00:06:25.340
A, which is 4.

00:06:25.340 --> 00:06:26.410
And you could look at it.

00:06:26.410 --> 00:06:27.990
You could actually just
try out the numbers.

00:06:27.990 --> 00:06:30.770
And frankly, even if you got
confused by absolute value,

00:06:30.770 --> 00:06:33.250
you could have just tried
each of the choices.

00:06:33.250 --> 00:06:34.890
2 minus 4 is minus 2.

00:06:34.890 --> 00:06:39.280
The absolute value of negative
2 is just 2, which

00:06:39.280 --> 00:06:40.070
is less than 3.

00:06:40.070 --> 00:06:41.150
So that's right.

00:06:41.150 --> 00:06:42.970
So you could have actually just
tried out the choices if

00:06:42.970 --> 00:06:43.620
this confused you.

00:06:43.620 --> 00:06:46.160
And that actually might have
been faster because in the

00:06:46.160 --> 00:06:48.240
worst case, you'd have to try
out five choices and each of

00:06:48.240 --> 00:06:51.770
them would have taken a few
seconds to evaluate.

00:06:51.770 --> 00:06:53.700
But sometimes it's good to
do it mathematically.

00:06:53.700 --> 00:06:57.840
But I wouldn't wrong you for--
I wouldn't blame you for just

00:06:57.840 --> 00:07:01.040
trying things out as well.

00:07:01.040 --> 00:07:01.840
Ten.

00:07:01.840 --> 00:07:03.560
OK.

00:07:03.560 --> 00:07:08.410
If all interior angles of the
polygon are congruent, then x

00:07:08.410 --> 00:07:11.450
is equal to-- OK so this
is interesting.

00:07:11.450 --> 00:07:13.120
All of the angles of this
polygon are congruent.

00:07:13.120 --> 00:07:13.985
So it's a pentagon.

00:07:13.985 --> 00:07:24.790
So it's 1, 2-- like that--
3, 4, and 5.

00:07:24.790 --> 00:07:27.260
I didn't draw really
that congruent.

00:07:27.260 --> 00:07:29.395
And actually this line keeps
going like this.

00:07:29.395 --> 00:07:31.490
And the important thing to
realize with any of these

00:07:31.490 --> 00:07:37.460
triangles is, if I were to
draw-- there's a formula for

00:07:37.460 --> 00:07:42.750
the angles of a polygon, but I
always-- so the way to think

00:07:42.750 --> 00:07:49.920
about it is that these are all
going to be equal angles.

00:07:49.920 --> 00:07:52.490
So think of it this way.

00:07:52.490 --> 00:07:55.840
These are five equal
triangles.

00:07:55.840 --> 00:07:56.990
Let me think of it
the best way.

00:07:56.990 --> 00:08:00.880
So they're telling us-- we
want to figure out what x

00:08:00.880 --> 00:08:03.050
degrees is, right?

00:08:03.050 --> 00:08:04.720
Well this x degrees
is going to be the

00:08:04.720 --> 00:08:06.000
same as this x degrees.

00:08:09.990 --> 00:08:12.080
And how do I know that?