SAT Prep: Test 6 Section 3 Part 4

WEBVTT
Kind: captions
Language: en

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We are on problem number 12.

00:00:03.040 --> 00:00:05.890
Problem 12.

00:00:05.890 --> 00:00:08.615
OK, what is the greatest of five
consecutive integers if

00:00:08.615 --> 00:00:10.750
the sum of these integers
is 185?

00:00:10.750 --> 00:00:11.630
So what is the greatest?

00:00:11.630 --> 00:00:15.160
Let's assume x is the greatest.
And they're

00:00:15.160 --> 00:00:17.220
consecutive integers, so what's
the second greatest?

00:00:17.220 --> 00:00:19.580
Well, that would be x minus 1.

00:00:19.580 --> 00:00:23.462
The third greatest would
be x minus 2.

00:00:23.462 --> 00:00:24.960
x minus 3.

00:00:24.960 --> 00:00:26.050
x minus 4, right?

00:00:26.050 --> 00:00:28.590
These are five consecutive
integers where x is the

00:00:28.590 --> 00:00:30.350
largest, right?

00:00:30.350 --> 00:00:35.460
If x was 150, x minus 4 would
be-- these numbers would be

00:00:35.460 --> 00:00:39.820
150, 149, 148, 147, 146
like that, right?

00:00:39.820 --> 00:00:42.140
So these are consecutive
integers, x is the largest.

00:00:42.140 --> 00:00:47.100
And they're telling us
that the sum is 185.

00:00:47.100 --> 00:00:49.720
So the sum is equal to 185.

00:00:49.720 --> 00:00:50.850
So what is this sum?

00:00:50.850 --> 00:00:52.220
How do sum these up?

00:00:52.220 --> 00:00:54.100
So we have five x's, right?

00:00:54.100 --> 00:00:56.210
So get 5x.

00:00:56.210 --> 00:00:59.180
And then what's minus 1, minus
2, minus 3, minus 4?

00:00:59.180 --> 00:01:02.700
So 1 plus 2 is 3, 3 plus 3 is--
this is where I always

00:01:02.700 --> 00:01:07.730
mess up-- 3 plus 3 6, 6
plus 4 is 10, right?

00:01:07.730 --> 00:01:11.060
So you get 5x minus 10
is equal to 185.

00:01:11.060 --> 00:01:12.770
Add 10 to both sides.

00:01:12.770 --> 00:01:17.160
5x is equal to 195.

00:01:17.160 --> 00:01:19.390
And you could eyeball this, you
could say well, how many

00:01:19.390 --> 00:01:26.230
times does 5 go into 200, and
it's going to be 1 less than

00:01:26.230 --> 00:01:27.480
that, right?

00:01:27.480 --> 00:01:30.740
Five goes into 200, what, if
it goes into it 40 times,

00:01:30.740 --> 00:01:34.470
it'll be 39 or you can divide.

00:01:34.470 --> 00:01:37.090
If you didn't want to do it that
way, you could say 5 goes

00:01:37.090 --> 00:01:41.680
into 195, 3 times 5 is 15.

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45, 39.

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So that's our answer, x is 39.

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It's the largest of the
consecutive numbers.

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So those numbers are
39, 38, 37, 36, 35.

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Those are our five consecutive
numbers.

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Next problem.

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Problem 13.

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A salesman's monthly gross pay
consists of $1,200 plus 20

00:02:12.770 --> 00:02:15.230
percent of the dollar
amount of the sales.

00:02:15.230 --> 00:02:19.070
So plus 20 percent
of the sales, s

00:02:19.070 --> 00:02:20.460
I'll say is for sales.

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If his gross pay for one month
was $2,500, what was the

00:02:29.430 --> 00:02:32.230
dollar amount of the sales
for that month?

00:02:32.230 --> 00:02:34.670
Well when I read it, I
essentially set up this

00:02:34.670 --> 00:02:35.420
equation, right?

00:02:35.420 --> 00:02:38.570
His compensation is $1,200 plus
20 percent of the sales.

00:02:38.570 --> 00:02:40.765
And we know that that equals
$2,500 in this month.

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So let's subtract $1,200
from both sides.

00:02:43.250 --> 00:02:51.890
So you get 0.2s is equal
to $1,300, right?

00:02:51.890 --> 00:02:57.410
And so s is equal to $1,300
divided by 0.2.

00:02:57.410 --> 00:03:00.620
And the way I like to think
about this, 0.2, instead of

00:03:00.620 --> 00:03:04.110
having to do this decimal, which
you could do, is I could

00:03:04.110 --> 00:03:05.310
have rewritten this.

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20 percent is the same
thing as 1/5, right?

00:03:07.740 --> 00:03:10.800
So it's also 1/5 s is
equal to $1,300.

00:03:10.800 --> 00:03:11.790
And now this is easy.

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Multiply both sides by 5, you
get s is equal to $1,300 times

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5, which equals what?

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Thirteen times 5 is 65,
and add the two 0's.

00:03:24.660 --> 00:03:26.430
So s is equal to $6,500.

00:03:26.430 --> 00:03:29.730
So that's the monthly
sales that month.

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I was a little redundant with
the word "monthly" just now.

00:03:32.260 --> 00:03:33.510
Next problem.

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Problem 14.

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OK, so they've drawn a circle.

00:03:46.570 --> 00:03:56.050
And then, let's see, they've
told us that

00:03:56.050 --> 00:04:00.850
this is, let's see.

00:04:00.850 --> 00:04:11.800
They tell us that this
is 40 degrees.

00:04:11.800 --> 00:04:14.880
Let's see, Naomi makes
silver jewelry.

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For one style of earring, she
cuts wedges from a silver

00:04:17.690 --> 00:04:20.860
disk, which I guess is depicted
here, as shown in the

00:04:20.860 --> 00:04:21.410
figure above.

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Each wedge makes a 40-degree
angle at the

00:04:23.990 --> 00:04:25.380
center of the disk.

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If the weight of each uncut
disk is a uniformly

00:04:28.910 --> 00:04:33.070
distributed 2.5 grams-- that's
how much this whole silver

00:04:33.070 --> 00:04:35.950
disk-- I should've done it in
grey instead of magenta

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because it's silver, but
anyway-- how many grams does

00:04:40.940 --> 00:04:43.220
each wedge weigh?

00:04:43.220 --> 00:04:46.220
So we essentially just have to
figure out what fraction of

00:04:46.220 --> 00:04:49.510
the entire circle
is each wedge?

00:04:49.510 --> 00:04:52.860
Well, each wedge
is 40 degrees.

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How many degrees are in
the entire circle?

00:04:55.500 --> 00:04:57.880
Hopefully it's second nature to
you right now, but there's

00:04:57.880 --> 00:05:01.160
360 degrees in an entire
circle, right?

00:05:01.160 --> 00:05:05.070
So the wedge is what fraction
of the entire circle?

00:05:05.070 --> 00:05:13.960
Each wedge is 40 over 360
of the entire circle.

00:05:13.960 --> 00:05:15.120
And that's what?

00:05:15.120 --> 00:05:17.460
That's equal to 1/9.

00:05:17.460 --> 00:05:19.600
Divide the top and the
bottom by 40, right?

00:05:19.600 --> 00:05:22.150
Four goes into 36,
right, 9 times.

00:05:22.150 --> 00:05:24.640
So each wedge is 1/9 of the
entire circle, and I just did

00:05:24.640 --> 00:05:25.780
that based on the fact
that it's 40

00:05:25.780 --> 00:05:28.890
degrees over 360 degrees.

00:05:28.890 --> 00:05:33.170
So if each wedge is 1/9 of the
entire silver piece and the

00:05:33.170 --> 00:05:36.536
silver piece weights 2.5 grams,
then each wedge will

00:05:36.536 --> 00:05:43.470
weigh 1/9 times 2.5
grams, right?

00:05:43.470 --> 00:05:51.120
And if we want to write 2.5 as a
fraction, how do we do that?

00:05:51.120 --> 00:05:53.740
That might be convenient
right now.

00:05:53.740 --> 00:05:55.190
Well, I'll tell you, that's
the same thing.

00:05:55.190 --> 00:05:57.580
And you could multiply it out,
and that's all fine, but you

00:05:57.580 --> 00:06:03.410
could also recognize that 2.5
can also be written as 9/4.

00:06:03.410 --> 00:06:05.860
Oh no, no it can't, sorry.

00:06:05.860 --> 00:06:09.410
2.5 can be written as 5/4.

00:06:09.410 --> 00:06:10.660
No.

00:06:12.350 --> 00:06:14.220
I'm pathetic.

00:06:14.220 --> 00:06:18.940
2.5 can be written
as 10/4, right?

00:06:18.940 --> 00:06:21.990
Because four goes into
10 two times.

00:06:21.990 --> 00:06:23.140
Why am I even writing 10/4?

00:06:23.140 --> 00:06:25.260
See, my brain is
malfunctioning.

00:06:25.260 --> 00:06:29.950
2.5 can be written as
5/2, which is 10/4.

00:06:29.950 --> 00:06:32.560
I was somehow thinking of the
number 9, and I'm not going to

00:06:32.560 --> 00:06:35.530
make excuses for my deficits.

00:06:35.530 --> 00:06:36.950
Anyway.

00:06:36.950 --> 00:06:38.990
So what's 1/9 times 5/2?

00:06:38.990 --> 00:06:41.440
It's 5/18.

00:06:41.440 --> 00:06:44.500
So each wedge is going to
weight 5/18 of a gram.

00:06:44.500 --> 00:06:48.930
The other option is you could
have divided 2.5 by 9 and you

00:06:48.930 --> 00:06:51.940
would have gotten another
answer with decimals.

00:06:51.940 --> 00:06:53.260
Either one would have
worked because these

00:06:53.260 --> 00:06:55.770
are free answer questions.

00:06:55.770 --> 00:06:57.870
Next problem.

00:06:57.870 --> 00:06:59.075
Can you actually use
a calculator on

00:06:59.075 --> 00:07:00.240
the SAT these days?

00:07:00.240 --> 00:07:04.250
If you can, that would've
been an option as well.

00:07:04.250 --> 00:07:07.940
Problem 15.

00:07:07.940 --> 00:07:14.280
If x squared minus y squared is
equal to 10 and x plus y is

00:07:14.280 --> 00:07:18.610
equal to 5, what is x minus y?

00:07:18.610 --> 00:07:21.700
Before I even read the rest of
the question, when I saw this,

00:07:21.700 --> 00:07:23.540
it should be like your knee-jerk
reaction when you

00:07:23.540 --> 00:07:24.080
take the SAT.

00:07:24.080 --> 00:07:26.500
When you see x squared minus y
squared equals 10, well that's

00:07:26.500 --> 00:07:29.590
the same thing as x plus
y times x minus

00:07:29.590 --> 00:07:31.100
y is equal to 10.

00:07:31.100 --> 00:07:32.850
You just factor this difference
of squares

00:07:32.850 --> 00:07:33.700
immediately.

00:07:33.700 --> 00:07:34.860
You almost can do that
before you read

00:07:34.860 --> 00:07:36.450
the rest of the question.

00:07:36.450 --> 00:07:38.430
And they tell us what's
x plus y?

00:07:38.430 --> 00:07:40.210
x plus y is 5.

00:07:40.210 --> 00:07:43.950
So 5 times x minus
y is equal to 10.

00:07:43.950 --> 00:07:49.320
Divide both sides by 5, you get
x minus y is equal to 2.

00:07:49.320 --> 00:07:51.860
That's it.

00:07:51.860 --> 00:07:53.110
Next problem.

00:07:56.800 --> 00:08:01.400
So they draw us--
let me color.

00:08:01.400 --> 00:08:06.150
So they draw us a rectangle.

00:08:06.150 --> 00:08:08.840
And then within that rectangle
there is another rectangle.

00:08:08.840 --> 00:08:10.030
It looks like this.

00:08:10.030 --> 00:08:12.180
Another square, really.

00:08:12.180 --> 00:08:15.340
And what I've drawn doesn't look
that much like a square,

00:08:15.340 --> 00:08:16.770
but it's close enough,
I think.

00:08:19.270 --> 00:08:21.000
You get the idea.

00:08:21.000 --> 00:08:24.050
And this is, I see, a right
angle, right angle, right

00:08:24.050 --> 00:08:26.380
angle, right angle.

00:08:26.380 --> 00:08:29.790
They tell us that this
distance is 2,

00:08:29.790 --> 00:08:32.110
this distance is 3.

00:08:32.110 --> 00:08:38.090
In the figure above, what is the
area of the shaded square?

00:08:38.090 --> 00:08:40.159
So this is a bit of a trick.

00:08:40.159 --> 00:08:42.350
And actually, let me do it in
the next video so I can kind

00:08:42.350 --> 00:08:44.610
of delve deeper into
the trick.

00:08:44.610 --> 00:08:46.490
I'll see you in the
next video.


SAT Prep: Test 5 Section 9 Part 3

WEBVTT
Kind: captions
Language: en

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Welcome back.

00:00:01.570 --> 00:00:06.680
I'm on problem number 10.

00:00:06.680 --> 00:00:10.980
Phillip used 4 pieces of masking
tape, each 6 inches

00:00:10.980 --> 00:00:25.020
long, to put up each
of his posters.

00:00:27.650 --> 00:00:30.150
So this is per poster.

00:00:33.080 --> 00:00:34.060
Good enough.

00:00:34.060 --> 00:00:37.610
Phillip had a 300-foot roll of
masking tape when he started.

00:00:37.610 --> 00:00:40.130
So he starts at 300 feet.

00:00:40.130 --> 00:00:42.760
I can already tell you that
some unit conversion will

00:00:42.760 --> 00:00:45.180
happen, because they're talking
about 6 inches here

00:00:45.180 --> 00:00:47.940
and they're talking about
300 feet here.

00:00:47.940 --> 00:00:51.850
If no tape was wasted, which
of the following represents

00:00:51.850 --> 00:00:55.330
the number of feet--and they
underline it-- of masking tape

00:00:55.330 --> 00:00:59.080
that was left on the roll after
he put up the n posters?

00:00:59.080 --> 00:01:02.330
And they actually tell us that
12 inches are equal to a foot

00:01:02.330 --> 00:01:06.140
in case you aren't
from this planet.

00:01:06.140 --> 00:01:07.715
So how do we do this?

00:01:12.520 --> 00:01:15.460
So he's going to put
up n posters.

00:01:15.460 --> 00:01:17.630
So he's going to start
off-- well, how much

00:01:17.630 --> 00:01:18.830
tape will he use?

00:01:18.830 --> 00:01:21.390
So for each of those n posters,
how much will he use?

00:01:21.390 --> 00:01:26.980
He uses 4 pieces
times 6 inches.

00:01:26.980 --> 00:01:28.300
But we want to go into feet.

00:01:28.300 --> 00:01:29.230
We want to know how many
feet are left.

00:01:29.230 --> 00:01:31.170
So let's just convert
immediately to feet.

00:01:31.170 --> 00:01:36.070
6 inches is equal to
how many feet?

00:01:36.070 --> 00:01:37.980
Well, it's half a foot.

00:01:37.980 --> 00:01:39.160
6/12 inches.

00:01:39.160 --> 00:01:42.000
So it equals 1/2 foot.

00:01:42.000 --> 00:01:46.790
So he does 4 pieces for each
poster, and each of those

00:01:46.790 --> 00:01:48.640
pieces is 1/2 foot.

00:01:48.640 --> 00:01:50.800
And now we're immediately
in feet length.

00:01:50.800 --> 00:01:52.290
So this is how much he uses.

00:01:52.290 --> 00:01:55.790
So he will use-- so 4
times 1/2 is just 2.

00:01:55.790 --> 00:01:57.640
So he uses 2n feet.

00:02:00.400 --> 00:02:06.210
So if he starts with 300, the
amount that he has left is

00:02:06.210 --> 00:02:08.485
what he started with
minus what he used.

00:02:08.485 --> 00:02:10.900
He used 2n.

00:02:10.900 --> 00:02:14.360
So he starts with 300 feet minus
2n feet, so that's the

00:02:14.360 --> 00:02:15.350
expression.

00:02:15.350 --> 00:02:18.600
That's choice B.

00:02:18.600 --> 00:02:20.680
Next problem.

00:02:20.680 --> 00:02:24.990
Problem 11.

00:02:24.990 --> 00:02:28.870
I'll switch to magenta.

00:02:28.870 --> 00:02:32.450
In the x, y coordinate plane,
line m is the reflection of

00:02:32.450 --> 00:02:35.320
line l about the x-axis.

00:02:35.320 --> 00:02:42.650
If the slope of m-- so m slope--
is equal to minus 4/5,

00:02:42.650 --> 00:02:45.300
what is the slope of l?

00:02:45.300 --> 00:02:48.000
So you should hopefully be able
to do this on the real

00:02:48.000 --> 00:02:50.980
exam without having
to draw it.

00:02:50.980 --> 00:02:53.330
Or you could actually just draw
a really quick and dirty

00:02:53.330 --> 00:02:55.740
one, and that actually
probably would

00:02:55.740 --> 00:02:56.990
do the job for you.

00:03:00.020 --> 00:03:01.450
So minus 4/5.

00:03:01.450 --> 00:03:05.460
That means for every
5-- and it's a

00:03:05.460 --> 00:03:06.510
reflection about the x-axis.

00:03:06.510 --> 00:03:08.805
So let's draw a line m.

00:03:12.870 --> 00:03:14.510
Now let's just assume that
the origin's here.

00:03:14.510 --> 00:03:16.350
They don't tell us that,
but they don't

00:03:16.350 --> 00:03:17.150
tell us it's not that.

00:03:17.150 --> 00:03:21.860
So that's zero, one, two,
three, four, five.

00:03:21.860 --> 00:03:24.900
And this is one, two,
three, four.

00:03:24.900 --> 00:03:25.850
Let me do it here.

00:03:25.850 --> 00:03:28.330
One, two, three, four.

00:03:28.330 --> 00:03:30.490
I just want to draw it
so you understand.

00:03:30.490 --> 00:03:34.410
So 4, minus 4, this is 5.

00:03:34.410 --> 00:03:36.120
So we know line m.

00:03:36.120 --> 00:03:42.530
For every line m, for every
5 it goes to the right,

00:03:42.530 --> 00:03:43.450
it goes down 4.

00:03:43.450 --> 00:03:46.590
So this could be a legitimate
line m right here.

00:03:46.590 --> 00:03:48.670
It could be like this.

00:03:48.670 --> 00:03:50.840
Line m could look like that.

00:03:54.170 --> 00:03:57.367
So a reflection about the
x-axis, if I were to reflect

00:03:57.367 --> 00:04:00.520
it about the x-axis.

00:04:00.520 --> 00:04:04.160
This is the x-axis right here,
so I just want to take its

00:04:04.160 --> 00:04:06.190
mirror image, or flip
it over the x-axis.

00:04:06.190 --> 00:04:07.440
It would look like this.

00:04:11.900 --> 00:04:14.960
Oh, I thought I was using
the line tool.

00:04:14.960 --> 00:04:16.210
It would look like this.

00:04:18.829 --> 00:04:20.950
I'm still not using
the line tool.

00:04:20.950 --> 00:04:23.430
Now, I'm using the line tool.

00:04:23.430 --> 00:04:25.440
It would look like
that, right?

00:04:25.440 --> 00:04:27.270
So what's the slope here?

00:04:27.270 --> 00:04:35.230
Well, for every 5 I go to
the right, I move up 4.

00:04:35.230 --> 00:04:37.940
So change in y.

00:04:37.940 --> 00:04:40.790
Let me make sure this is
line m, this is line l.

00:04:40.790 --> 00:04:47.350
Change in y over change in
x for line l is equal to

00:04:47.350 --> 00:04:48.600
positive 4/5.

00:04:52.260 --> 00:04:55.000
It shouldn't take you
that long to do it.

00:04:55.000 --> 00:04:57.720
One thing that you could just
do as well, you could just

00:04:57.720 --> 00:04:58.530
draw a quick and dirty one.

00:04:58.530 --> 00:05:00.160
It's like, well, if I have
something with a negative

00:05:00.160 --> 00:05:02.640
slope-- let's say I have
a negative, really

00:05:02.640 --> 00:05:04.590
shallow slope like that.

00:05:04.590 --> 00:05:07.040
If I flip it, it's going to have
the same slope, but it's

00:05:07.040 --> 00:05:08.620
going to be a positive slope,
but it's still going to be

00:05:08.620 --> 00:05:12.260
shallow, so it's going to
be the same magnitude.

00:05:12.260 --> 00:05:14.290
You'll flip the sign.

00:05:14.290 --> 00:05:16.100
So the answer is B, 4/5.

00:05:16.100 --> 00:05:18.920
But I did this just to give
you the intuition.

00:05:18.920 --> 00:05:21.940
The next problem.

00:05:21.940 --> 00:05:24.550
Or to let you know why you got
it wrong, if you got it wrong.

00:05:24.550 --> 00:05:24.960
Anyway.

00:05:24.960 --> 00:05:28.090
Problem number 12.

00:05:28.090 --> 00:05:37.900
If n is equal to 3p, for what
value of p is n equal to p?

00:05:51.630 --> 00:05:53.380
This is kind of crazy.

00:05:53.380 --> 00:05:56.240
And at first, I was like,
what are they saying?

00:05:56.240 --> 00:05:57.890
And then I read one
of the choices.

00:05:57.890 --> 00:06:02.810
Because no matter what,
n is equal to 3p.

00:06:02.810 --> 00:06:08.100
There's no circumstance--
well, oh, sorry.

00:06:08.100 --> 00:06:09.820
I was incorrect.

00:06:09.820 --> 00:06:12.350
There is a circumstance in
which n is equal to 3p.

00:06:15.850 --> 00:06:19.070
Well, what's the circumstance?

00:06:19.070 --> 00:06:25.290
Well, you might initially say,
well, as long as p is not 0, n

00:06:25.290 --> 00:06:28.580
is going to be exactly
3 times p.

00:06:28.580 --> 00:06:31.890
But then in our statement, I
just told you the answer.

00:06:31.890 --> 00:06:33.750
They both can be 0.

00:06:33.750 --> 00:06:37.680
If p is 0, then 3
times 0 is 0.

00:06:37.680 --> 00:06:39.540
So there's no real
algebra there.

00:06:39.540 --> 00:06:42.310
It's just kind of to realize
that 0 is a choice.

00:06:42.310 --> 00:06:43.970
And if you looked at the
choices, you'd immediately see

00:06:43.970 --> 00:06:45.160
choice A is 0.

00:06:45.160 --> 00:06:45.870
Try it out.

00:06:45.870 --> 00:06:49.180
You say, oh, well, if p is
0, then n is also 0.

00:06:49.180 --> 00:06:51.930
So then n would equal p.

00:06:51.930 --> 00:06:53.140
So next problem.

00:06:53.140 --> 00:06:56.310
Problem 13.

00:06:56.310 --> 00:06:58.920
That was one of those problems
that in some ways are so easy

00:06:58.920 --> 00:07:00.700
that you waste time
on it, making sure

00:07:00.700 --> 00:07:03.300
you didn't miss something.

00:07:03.300 --> 00:07:07.265
Let's draw what they drew.

00:07:07.265 --> 00:07:09.350
So we have a line like that.

00:07:09.350 --> 00:07:11.385
I have a line like that.

00:07:11.385 --> 00:07:15.030
Then I have a line like that.

00:07:15.030 --> 00:07:18.250
And this is line l.

00:07:18.250 --> 00:07:20.580
This is y degrees.

00:07:20.580 --> 00:07:23.430
This is line m.

00:07:23.430 --> 00:07:29.520
This is x degrees, and this
is line n right here.

00:07:29.520 --> 00:07:32.440
In the figure above, if z--
oh, they tell us this is z

00:07:32.440 --> 00:07:34.670
right here.

00:07:34.670 --> 00:07:42.155
In the figure above, if z is
equal to 30, what is the value

00:07:42.155 --> 00:07:44.670
of x plus y?

00:07:44.670 --> 00:07:46.720
x plus y is what?

00:07:46.720 --> 00:07:48.800
Well, what can we figure out?

00:07:48.800 --> 00:07:53.050
What do we know about this
angle right here?

00:07:53.050 --> 00:07:55.130
It's supplementary
to y, right?

00:07:55.130 --> 00:07:56.760
So this is kind of the angle
game, but we're going to have

00:07:56.760 --> 00:07:59.260
a little bit more variables
than normal.

00:07:59.260 --> 00:08:01.810
It's supplementary to y, so this
is going to be 180 minus

00:08:01.810 --> 00:08:03.880
y because y plus this
angle are going to

00:08:03.880 --> 00:08:05.610
have to equal 180.

00:08:05.610 --> 00:08:07.980
And for the exact same reason,
this angle right here

00:08:07.980 --> 00:08:12.870
is 180 minus x.

00:08:12.870 --> 00:08:13.530
And what do we know?

00:08:13.530 --> 00:08:17.115
We know this angle plus this
angle plus z is equal to 180.

00:08:17.115 --> 00:08:18.730
So let's write that down.

00:08:18.730 --> 00:08:29.230
This angle, 180 minus y, plus
this angle, plus 180 minus x,

00:08:29.230 --> 00:08:33.419
plus z is equal to 180
because they're

00:08:33.419 --> 00:08:35.390
all in the same triangle.

00:08:35.390 --> 00:08:37.380
So let's try our best
to simplify this.

00:08:37.380 --> 00:08:44.540
Well, we could immediately get
rid of one of the 180's on

00:08:44.540 --> 00:08:47.610
that side, and that becomes 0.

00:08:47.610 --> 00:08:50.100
z is 30, right?

00:08:50.100 --> 00:08:51.490
So let's simplify it.

00:08:51.490 --> 00:08:58.990
We get minus y minus x, and
then you have 180 plus 30,

00:08:58.990 --> 00:09:02.660
plus 210, is equal to 0.

00:09:02.660 --> 00:09:05.160
Now let's add x and
y to both sides.

00:09:05.160 --> 00:09:06.100
I'm kind of skipping a step.

00:09:06.100 --> 00:09:08.440
You could add x to both sides
and then add y to both sides.

00:09:08.440 --> 00:09:12.410
But if you add x and y to both
sides, you get 210 is

00:09:12.410 --> 00:09:14.190
equal to x plus y.

00:09:14.190 --> 00:09:15.680
And that's the answer.

00:09:15.680 --> 00:09:17.710
They want to know what
x plus y is.

00:09:17.710 --> 00:09:20.440
And that is choice D.

00:09:20.440 --> 00:09:22.060
And so the trick here is really
saying, well, they

00:09:22.060 --> 00:09:23.370
only give us z.

00:09:23.370 --> 00:09:26.440
The only thing I know is that
z is in a triangle with this

00:09:26.440 --> 00:09:28.220
angle and this angle.

00:09:28.220 --> 00:09:30.800
And let me express those two
angles in terms of x and y,

00:09:30.800 --> 00:09:34.800
because they are supplementary
to x and y.

00:09:34.800 --> 00:09:36.610
See you in the next video.


SAT Prep: Test 5 Section 9 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.840
We're on Test 5, Section
9, on page 679.

00:00:04.840 --> 00:00:09.530
Let's do some problems.
Problem number 1.

00:00:09.530 --> 00:00:13.560
If 6 cars out of 10 on an
assembly line are red, what is

00:00:13.560 --> 00:00:16.320
the probability that a car
selected at random from the

00:00:16.320 --> 00:00:18.700
assembly line will be red?

00:00:18.700 --> 00:00:20.390
Well, 6 out of 10 are red.

00:00:20.390 --> 00:00:24.640
So if I take a random one, I
have a 6 in 10 chance of

00:00:24.640 --> 00:00:26.640
getting a red car.

00:00:26.640 --> 00:00:30.310
If you look at the choices,
there is no 6/10, but you can

00:00:30.310 --> 00:00:32.080
see that this can be reduced.

00:00:32.080 --> 00:00:37.010
If you divide the top and the
bottom by 2, you get 3/5.

00:00:37.010 --> 00:00:40.330
So this is really just an
exercise in reducing a

00:00:40.330 --> 00:00:44.370
fraction to its lowest common
or simplified form, so the

00:00:44.370 --> 00:00:46.930
answer is B.

00:00:46.930 --> 00:00:48.965
Problem 2.

00:00:48.965 --> 00:00:51.240
Let me draw that diagram.

00:00:51.240 --> 00:00:52.520
So I have a triangle.

00:00:55.310 --> 00:00:57.460
Let me try my best
to draw this.

00:01:01.500 --> 00:01:03.830
There's an altitude that
comes down the center.

00:01:03.830 --> 00:01:06.380
Let's put a 2.

00:01:06.380 --> 00:01:08.000
And then what do they write?

00:01:08.000 --> 00:01:13.090
They write this angle
right here is w.

00:01:13.090 --> 00:01:20.320
This is x, y, z over here.

00:01:20.320 --> 00:01:23.170
And they say, note: figure
not drawn to scale.

00:01:23.170 --> 00:01:24.760
AB is equal to BC.

00:01:24.760 --> 00:01:26.395
Oh, I didn't draw
the letters in.

00:01:26.395 --> 00:01:32.740
So A, B, C, and then D.

00:01:32.740 --> 00:01:35.330
So they say that AB
is equal to BC.

00:01:40.050 --> 00:01:42.010
So immediately, a trigger should
go off in your head.

00:01:42.010 --> 00:01:45.490
If two sides are equal, what
does it say about the base

00:01:45.490 --> 00:01:47.180
angles or their corresponding
angles?

00:01:47.180 --> 00:01:49.800
That means that those are
also going to be equal.

00:01:49.800 --> 00:01:53.700
So if those two sides are equal,
then we also know that

00:01:53.700 --> 00:01:55.410
these two angles are equal.

00:01:55.410 --> 00:01:57.480
That should immediately be a
trigger in your head, that we

00:01:57.480 --> 00:01:59.330
know that w is equal to z.

00:01:59.330 --> 00:02:01.890
What's the next part
of the information?

00:02:01.890 --> 00:02:06.900
And BD, this is BD,
bisects AC.

00:02:06.900 --> 00:02:10.220
so bisects means it essentially
intersects it at

00:02:10.220 --> 00:02:11.039
its midpoint.

00:02:11.039 --> 00:02:16.380
So that also tells us that this
distance here from A to D

00:02:16.380 --> 00:02:20.080
is the same as the distance
from D to C.

00:02:20.080 --> 00:02:24.320
Which of the following
cannot be concluded?

00:02:24.320 --> 00:02:26.050
That w is equal to x.

00:02:31.110 --> 00:02:32.850
Actually, immediately, I
don't even know how I

00:02:32.850 --> 00:02:33.470
can conclude that.

00:02:33.470 --> 00:02:34.920
Let's look at the
other choices.

00:02:34.920 --> 00:02:36.190
w is equal to z.

00:02:36.190 --> 00:02:38.760
Well, w is equal to z can
definitely be concluded.

00:02:38.760 --> 00:02:39.990
That's choice B.

00:02:39.990 --> 00:02:42.410
So we know that choice
B is w is equal to z.

00:02:42.410 --> 00:02:44.050
We know that this isn't it
because we concluded that

00:02:44.050 --> 00:02:45.770
immediately.

00:02:45.770 --> 00:02:49.020
Choice C is that x
is equal to y.

00:02:51.640 --> 00:02:53.370
That actually can
be concluded.

00:02:53.370 --> 00:02:54.090
How do we know that?

00:02:54.090 --> 00:03:00.730
Well, since this bisects this
line and this is an isosceles

00:03:00.730 --> 00:03:03.560
triangle, even though that's
not how I drew it.

00:03:03.560 --> 00:03:06.390
This triangle is completely
symmetric.

00:03:06.390 --> 00:03:07.930
Maybe I should draw a
little bit better.

00:03:07.930 --> 00:03:11.240
How do I know it's completely
symmetric?

00:03:11.240 --> 00:03:14.860
Because this side is equal to
this side, that is equal to

00:03:14.860 --> 00:03:17.980
that, and this side is
equal to this side.

00:03:17.980 --> 00:03:20.290
And of course, this side
is equal to that side.

00:03:20.290 --> 00:03:21.930
And actually, we can also
conclude that this is a

00:03:21.930 --> 00:03:24.970
90-degree angle, because if this
was anything other than a

00:03:24.970 --> 00:03:28.870
90-degree angle, this wouldn't
be able to bisect at the base.

00:03:28.870 --> 00:03:34.340
So choice C, we actually can
deduce that x is equal to y.

00:03:34.340 --> 00:03:36.060
So that's not the answer.

00:03:36.060 --> 00:03:39.880
Choice D is AD is equal to DC.

00:03:39.880 --> 00:03:41.120
Well, they essentially
told us that one.

00:03:41.120 --> 00:03:45.460
They said BD bisects AC, so
that's not our answer.

00:03:45.460 --> 00:03:47.100
Choice E.

00:03:47.100 --> 00:03:49.100
DB is perpendicular to AC.

00:03:49.100 --> 00:03:52.130
And I just said the triangle
is completely symmetric.

00:03:52.130 --> 00:03:54.620
And given that it's an isosceles
triangle, the only

00:03:54.620 --> 00:03:57.410
way that BD can bisect this
bottom line is if it's

00:03:57.410 --> 00:03:59.220
perpendicular.

00:03:59.220 --> 00:04:01.570
So we can also rule out E.

00:04:01.570 --> 00:04:03.380
So our answer really is A.

00:04:03.380 --> 00:04:06.550
And really, there was no way
that we could figure out.

00:04:06.550 --> 00:04:09.780
And a good way of thinking
about that is actually we

00:04:09.780 --> 00:04:12.560
could make this point-- we could
make the length of this

00:04:12.560 --> 00:04:16.339
line, this length BD,
arbitrarily high because they

00:04:16.339 --> 00:04:17.670
gave us no information.

00:04:17.670 --> 00:04:20.600
The triangle could
look like this.

00:04:20.600 --> 00:04:25.792
It could look like that, or it
could look like this based on

00:04:25.792 --> 00:04:27.270
the information they gave us.

00:04:27.270 --> 00:04:32.610
So these angles could
change arbitrarily.

00:04:32.610 --> 00:04:35.770
So we really don't know
what those angles are.

00:04:35.770 --> 00:04:38.280
Next problem.

00:04:38.280 --> 00:04:39.530
Problem 3.

00:04:42.844 --> 00:04:44.420
Switch colors.

00:04:44.420 --> 00:04:46.214
About to sneeze.

00:04:46.214 --> 00:04:47.060
[SNEEZES]

00:04:47.060 --> 00:04:48.540
Excuse me.

00:04:48.540 --> 00:04:59.550
If 30% of m is 40, so 0.3
m, what is 15% of m?

00:04:59.550 --> 00:05:02.100
What is 0.15 of m?

00:05:02.100 --> 00:05:03.070
You could solve this.

00:05:03.070 --> 00:05:05.540
Divide both sides by 0.3,
et cetera, et cetera.

00:05:05.540 --> 00:05:09.010
But the important thing to
recognize is that 15%, or

00:05:09.010 --> 00:05:13.130
0.15, is half of 0.13.

00:05:13.130 --> 00:05:20.160
So if 15% is half of 0.3, or
30%, then 15% m is going to be

00:05:20.160 --> 00:05:24.460
half of whatever 30% m is, so
it's going to be equal to 20.

00:05:24.460 --> 00:05:27.060
Another thing you could have
done, if that's not completely

00:05:27.060 --> 00:05:28.770
obvious, take this
top equation and

00:05:28.770 --> 00:05:30.760
multiply it by 1/2.

00:05:30.760 --> 00:05:35.415
Then you'll get 0.15m
is equal to 20.

00:05:35.415 --> 00:05:37.560
And that's choice B.

00:05:37.560 --> 00:05:39.690
Problem 4.

00:05:39.690 --> 00:05:44.940
If n is any negative number, so
n is less than 0, which of

00:05:44.940 --> 00:05:48.390
the following must
be positive?

00:05:48.390 --> 00:05:51.110
So I can already tell you that
the only way you can take a

00:05:51.110 --> 00:05:54.120
negative number and make it
positive is either if you

00:05:54.120 --> 00:05:58.160
square it or if you multiply it
by another negative number.

00:05:58.160 --> 00:05:59.150
So let's see.

00:05:59.150 --> 00:06:01.715
Or if you subtract it, actually,
now that I look at

00:06:01.715 --> 00:06:02.670
the choices.

00:06:02.670 --> 00:06:05.335
So I could tell you an/2, that's
definitely going to be

00:06:05.335 --> 00:06:06.710
a negative number.

00:06:06.710 --> 00:06:08.800
B is definitely going to
be a negative number.

00:06:08.800 --> 00:06:10.890
You're multiplying
by a positive.

00:06:10.890 --> 00:06:13.130
C, you're just adding something
to it, so it still

00:06:13.130 --> 00:06:15.500
could be negative, if that's
like negative 10

00:06:15.500 --> 00:06:16.820
or something, right?

00:06:16.820 --> 00:06:18.350
D, you're just subtracting from
it, so you're just going

00:06:18.350 --> 00:06:19.780
to make it more negative.

00:06:19.780 --> 00:06:20.050
E.

00:06:20.050 --> 00:06:23.540
2 minus n.

00:06:23.540 --> 00:06:26.190
This will definitely result
in a positive number.

00:06:26.190 --> 00:06:27.360
Why?

00:06:27.360 --> 00:06:31.190
Because let's say that you could
express n as the product

00:06:31.190 --> 00:06:32.960
of a negative number.

00:06:32.960 --> 00:06:36.330
Let's say n is equal to
minus p, where p is

00:06:36.330 --> 00:06:37.520
some positive number.

00:06:37.520 --> 00:06:40.050
You can represent any negative
number like that.

00:06:40.050 --> 00:06:44.430
Then 2 minus n is the same
thing as 2 minus minus p,

00:06:44.430 --> 00:06:47.730
which is the same thing
as 2 plus p.

00:06:47.730 --> 00:06:50.730
And we said p is a
positive number.

00:06:50.730 --> 00:06:51.510
So there we go.

00:06:51.510 --> 00:06:55.670
So choice number 4
is definitely E.

00:06:55.670 --> 00:06:57.410
And you could try it
out with a number.

00:06:57.410 --> 00:07:00.400
Try it out with the number n
equals negative 10, and I

00:07:00.400 --> 00:07:02.560
think it should work out.

00:07:02.560 --> 00:07:03.520
Next problem.

00:07:03.520 --> 00:07:06.530
Problem number 5.

00:07:06.530 --> 00:07:15.640
The ratio 1.2:1 is equal to
which of the following ratios?

00:07:15.640 --> 00:07:17.990
So they give us a bunch
of choices.

00:07:17.990 --> 00:07:21.390
1.2:1, so it's a little
bit more than 1:1.

00:07:21.390 --> 00:07:23.610
So we could look at 1:2.

00:07:23.610 --> 00:07:27.750
No, that makes no sense, because
1 is smaller than 2.

00:07:27.750 --> 00:07:29.250
12:1.

00:07:29.250 --> 00:07:33.910
12 is 12 times 1,
not 1.2 times 1.

00:07:33.910 --> 00:07:36.585
So you're just looking for a
ratio where the first number

00:07:36.585 --> 00:07:38.660
is 1.2 times the
second number.

00:07:38.660 --> 00:07:40.450
5:6, that's choice C.

00:07:40.450 --> 00:07:43.080
That doesn't work, because
5 is less than 6.

00:07:43.080 --> 00:07:43.680
Choice D.

00:07:43.680 --> 00:07:44.800
6:5.

00:07:44.800 --> 00:07:48.170
I think we hit it because 6 is
just a little bit more than 5.

00:07:48.170 --> 00:07:51.840
And actually, if you wanted to
confirm that, why don't you

00:07:51.840 --> 00:07:53.630
divide the top and
the bottom by 5?

00:07:53.630 --> 00:07:56.900
So 5 goes into 6
how many times?

00:07:56.900 --> 00:08:00.490
5 goes into 6 one time.

00:08:00.490 --> 00:08:03.530
How many times does
5 go into 10?

00:08:03.530 --> 00:08:04.390
It's 2.

00:08:04.390 --> 00:08:06.200
5 goes into 6 1.2 times.

00:08:06.200 --> 00:08:09.280
So the answer is 6:5.

00:08:09.280 --> 00:08:11.080
Next problem.

00:08:11.080 --> 00:08:14.140
Let's see if I have
time to do this.

00:08:14.140 --> 00:08:15.180
Clear image.

00:08:15.180 --> 00:08:16.680
Invert colors.

00:08:16.680 --> 00:08:20.750
The legend of a certain
pictograph shows that this

00:08:20.750 --> 00:08:25.260
thing is equal to 5
million new homes.

00:08:25.260 --> 00:08:26.750
Approximately how many
new homes are

00:08:26.750 --> 00:08:28.360
represented by the symbols?

00:08:28.360 --> 00:08:31.180
So they drew that
3 and 1/2 times.

00:08:31.180 --> 00:08:40.110
One, two, three, and then
they draw half a house.

00:08:40.110 --> 00:08:41.900
That's half a house.

00:08:41.900 --> 00:08:43.270
So these three are
going to be what?

00:08:43.270 --> 00:08:50.280
This is 3 times 5, which is
equal to 15 million homes.

00:08:50.280 --> 00:08:51.560
And then this is half a house.

00:08:51.560 --> 00:08:53.430
That's what I tried to draw what
they drew in the diagram.

00:08:53.430 --> 00:08:54.290
So half a house is what?

00:08:54.290 --> 00:08:55.620
2.5 million houses.

00:08:55.620 --> 00:08:58.880
It's just half of
the 5 million.

00:08:58.880 --> 00:09:00.850
So it's 15 million plus 2.5.

00:09:00.850 --> 00:09:04.470
It equals 17.5 million homes.

00:09:04.470 --> 00:09:06.330
And that is choice D.

00:09:08.950 --> 00:09:11.380
You might be tempted to say that
this represents half a

00:09:11.380 --> 00:09:13.840
million homes, but remember,
this represents half of one of

00:09:13.840 --> 00:09:14.445
these pictures.

00:09:14.445 --> 00:09:17.600
And one of these pictures
represents 5 million homes.

00:09:17.600 --> 00:09:19.140
So this is 2 and 1/2 million.

00:09:19.140 --> 00:09:23.040
This is 2 and 1/2, not 0.5.

00:09:23.040 --> 00:09:24.810
See you in the next video.


SAT Prep: Test 5 Section 9 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:01.390
Welcome back.

00:00:01.390 --> 00:00:03.165
We're on problem number 7.

00:00:06.320 --> 00:00:10.110
If a and b are positive
integers, and a squared minus

00:00:10.110 --> 00:00:17.575
b squared is equal to 7,
what is the value of a?

00:00:27.450 --> 00:00:30.060
So there's a couple of ways that
I would think about this.

00:00:30.060 --> 00:00:32.470
You could say-- well, this
is actually the way I

00:00:32.470 --> 00:00:33.450
would think about it.

00:00:33.450 --> 00:00:35.700
Whenever I see something squared
minus something, I

00:00:35.700 --> 00:00:37.510
immediately just want to factor
it, and we'll see where

00:00:37.510 --> 00:00:38.910
that gets us.

00:00:38.910 --> 00:00:45.370
You get that a plus b times
a minus b is equal to 7.

00:00:45.370 --> 00:00:48.280
And we know that a and b
are positive integers.

00:00:52.280 --> 00:00:54.470
Well, we know this is going to
be a positive integer, right?

00:00:54.470 --> 00:00:55.870
You add two positive integers,
you definitely

00:00:55.870 --> 00:00:57.340
get a positive integer.

00:00:57.340 --> 00:00:58.610
And then we also know that
this is going to

00:00:58.610 --> 00:01:00.080
be a positive integer.

00:01:00.080 --> 00:01:01.180
How do I know that?

00:01:01.180 --> 00:01:04.200
Well, when I multiply a
positive integer times

00:01:04.200 --> 00:01:06.310
something and I get another
positive integer or I get

00:01:06.310 --> 00:01:08.430
another positive number,
I know that whatever I

00:01:08.430 --> 00:01:10.110
multiplied by is positive,
right?

00:01:10.110 --> 00:01:13.370
Because a positive times
a positive is positive.

00:01:13.370 --> 00:01:14.120
So what does this say?

00:01:14.120 --> 00:01:17.080
So these are both positive
integers, and when I multiply

00:01:17.080 --> 00:01:19.250
them, I get 7.

00:01:19.250 --> 00:01:20.740
So what could they be?

00:01:20.740 --> 00:01:23.260
Well, what numbers when you
multiply it can be 7?

00:01:23.260 --> 00:01:25.230
Well, there's only
two choices.

00:01:25.230 --> 00:01:28.230
1 times 7 because 7 is prime.

00:01:28.230 --> 00:01:31.750
7 has no other factors
other than 1 and 7.

00:01:31.750 --> 00:01:35.080
So which one's going to be 7 and
which one's going to be 1?

00:01:35.080 --> 00:01:38.730
Well, the larger one's
going to be 7.

00:01:38.730 --> 00:01:41.230
And the smaller one's
going to be 1.

00:01:41.230 --> 00:01:42.300
So you could eyeball this.

00:01:42.300 --> 00:01:43.950
You could say, well, what two
numbers when I add them

00:01:43.950 --> 00:01:48.130
together is 7 and when I
subtract them, I get 1?

00:01:48.130 --> 00:01:52.145
And you can say, well, what
if a is 4 and b is 3?

00:01:52.145 --> 00:01:53.620
And that works.

00:01:53.620 --> 00:01:55.380
4 plus 3 is 7.

00:01:55.380 --> 00:01:57.220
4 minus 3 is 1.

00:01:57.220 --> 00:02:00.480
And if that didn't just dawn
on you all of a sudden, you

00:02:00.480 --> 00:02:02.490
could do a system
of equations.

00:02:02.490 --> 00:02:05.290
a plus b is equal to 7.

00:02:05.290 --> 00:02:09.690
a minus b is equal to 1.

00:02:09.690 --> 00:02:10.740
Add these two equations.

00:02:10.740 --> 00:02:12.550
You get 2a.

00:02:12.550 --> 00:02:14.370
The b's cancel out.

00:02:14.370 --> 00:02:16.510
Plus 0 is equal to 8.

00:02:16.510 --> 00:02:17.940
7 plus 1 is 8.

00:02:17.940 --> 00:02:19.360
a is equal to 4.

00:02:19.360 --> 00:02:22.100
Then you can substitute back
in or just play around with

00:02:22.100 --> 00:02:25.150
the numbers, and you get
b is equal to 3.

00:02:25.150 --> 00:02:28.530
So that is choice-- oh, they
want to just know what a is.

00:02:28.530 --> 00:02:31.010
So a is 4, which is choice B.

00:02:35.330 --> 00:02:36.580
Next problem.

00:02:41.490 --> 00:02:42.986
You have a number line here.

00:02:42.986 --> 00:02:44.700
Let me draw it nice and big.

00:02:44.700 --> 00:02:46.940
I don't know how much
space I'll need.

00:02:53.280 --> 00:02:54.560
Have 0 in the middle.

00:02:59.100 --> 00:03:03.890
Negative 1, 1.

00:03:03.890 --> 00:03:13.820
Let me draw these points
now: t, u, v, and w.

00:03:22.560 --> 00:03:28.085
And then you have
x and y here.

00:03:30.830 --> 00:03:32.080
And then you have z.

00:03:36.050 --> 00:03:39.360
On the number line above,
which of the following

00:03:39.360 --> 00:03:44.130
corresponds to the absolute
value of u minus v?

00:03:44.130 --> 00:03:45.860
And this is just an
approximation problem.

00:03:45.860 --> 00:03:48.500
You might get intimidated, like,
how do I figure out u?

00:03:48.500 --> 00:03:49.440
How do I figure out v?

00:03:49.440 --> 00:03:51.020
Well, the answer is you
just kind of guess.

00:03:51.020 --> 00:03:51.800
Well, you don't guess.

00:03:51.800 --> 00:03:53.220
You approximate.

00:03:53.220 --> 00:03:53.840
So let's see.

00:03:53.840 --> 00:03:55.570
What does u look like?

00:03:55.570 --> 00:03:57.570
u looks like-- I don't know.

00:03:57.570 --> 00:04:02.250
If you're eyeballing it, it
looks like minus 0.8, if I

00:04:02.250 --> 00:04:02.960
were to eyeball it.

00:04:02.960 --> 00:04:03.670
Maybe a little more?

00:04:03.670 --> 00:04:05.820
Minus 0.85?

00:04:05.820 --> 00:04:07.420
Because it's almost
at negative 1, but

00:04:07.420 --> 00:04:08.240
not all the way there.

00:04:08.240 --> 00:04:11.870
I don't think it's 9/10 there.

00:04:11.870 --> 00:04:13.470
Yeah, I would say minus 0.85.

00:04:13.470 --> 00:04:14.690
I'm just eyeballing it.

00:04:14.690 --> 00:04:18.459
So let's just say u is
equal to minus 0.85.

00:04:18.459 --> 00:04:19.380
And what's v?

00:04:19.380 --> 00:04:23.840
v looks about 2/3
on the way to u.

00:04:23.840 --> 00:04:28.310
So let's say v is equal
to minus 0.6.

00:04:28.310 --> 00:04:32.530
Once again I'm just-- let's
say minus 0.66.

00:04:32.530 --> 00:04:34.970
I'm just approximating
once again.

00:04:34.970 --> 00:04:35.970
So it's minus, right?

00:04:35.970 --> 00:04:37.430
And they're both negative
numbers, and they're both

00:04:37.430 --> 00:04:38.980
almost at 1.

00:04:38.980 --> 00:04:41.020
So what is the absolute
value of u minus v?

00:04:41.020 --> 00:04:46.700
Well, that's the absolute value
of minus 0.85 minus v.

00:04:46.700 --> 00:04:51.790
So minus minus 0.66.

00:04:51.790 --> 00:04:54.570
So the minus minus
becomes a plus.

00:04:54.570 --> 00:04:57.970
So what's minus 0.85
plus 0.66?

00:04:57.970 --> 00:05:03.300
Well, that's roughly
minus 0.19.

00:05:03.300 --> 00:05:04.960
And if I'm taking the absolute
value of that,

00:05:04.960 --> 00:05:08.200
that becomes 0.19.

00:05:08.200 --> 00:05:12.360
0.19, roughly 1/5, and there's
no other number around.

00:05:12.360 --> 00:05:14.220
Roughly 1/5 is around here, and
there's no other number

00:05:14.220 --> 00:05:18.310
there even remotely close,
so that's going to be x.

00:05:18.310 --> 00:05:20.800
And another way you could have
thought about that, if you

00:05:20.800 --> 00:05:22.490
didn't want to approximate,
use numbers.

00:05:22.490 --> 00:05:23.900
I do that just to make sure.

00:05:23.900 --> 00:05:27.010
The absolute value, when you
subtract two numbers and you

00:05:27.010 --> 00:05:28.320
take the absolute value, you're
essentially saying,

00:05:28.320 --> 00:05:29.710
well, what's the distance
between the two?

00:05:29.710 --> 00:05:30.990
What's the positive distance?

00:05:30.990 --> 00:05:32.610
I don't care whether one's
larger or one's

00:05:32.610 --> 00:05:33.700
less than the other.

00:05:33.700 --> 00:05:36.250
So you're essentially saying
what's this distance, this

00:05:36.250 --> 00:05:41.510
distance here that's equal to
this, to this expression?

00:05:41.510 --> 00:05:43.670
So that distance, that
positive distance, is

00:05:43.670 --> 00:05:47.250
approximately equal to x, even
in my not-exact drawing.

00:05:47.250 --> 00:05:50.740
And if you look at how they drew
it-- actually, even in

00:05:50.740 --> 00:05:52.130
the way they drew it.

00:05:52.130 --> 00:05:53.220
Oh, sorry!

00:05:53.220 --> 00:05:56.070
They did u minus w.

00:05:56.070 --> 00:05:58.130
So then it actually makes even
more sense, because w

00:05:58.130 --> 00:05:59.380
is more like 0.5.

00:06:01.970 --> 00:06:03.140
I misread the problem.

00:06:03.140 --> 00:06:06.640
This w is equal to
more like 0.5.

00:06:06.640 --> 00:06:12.800
So the absolute value of u minus
w is going to be equal

00:06:12.800 --> 00:06:17.730
to the absolute value of 0.85--
sorry, w is minus 0.5.

00:06:17.730 --> 00:06:24.700
It's in the negative domain,
so minus 0.85 minus 0.5, so

00:06:24.700 --> 00:06:27.230
that equals minus 0.35.

00:06:27.230 --> 00:06:30.380
Absolute value is equal to 0.35,
still is equal to x.

00:06:30.380 --> 00:06:31.560
And they did that.

00:06:31.560 --> 00:06:34.550
I was just approximating
that x still works.

00:06:34.550 --> 00:06:37.180
But according to the drawing,
it makes even more sense

00:06:37.180 --> 00:06:39.570
because we're actually talking
about this distance.

00:06:39.570 --> 00:06:41.090
Sorry, to w.

00:06:41.090 --> 00:06:43.820
Let me do it in a different
better color.

00:06:43.820 --> 00:06:46.240
This is what u-- not this.

00:06:46.240 --> 00:06:48.260
This is what this represents.

00:06:48.260 --> 00:06:52.620
The absolute value of u minus
w represents this distance.

00:06:52.620 --> 00:06:57.250
And that distance is the
same as the value of x.

00:06:57.250 --> 00:06:59.082
And it really makes sense if you
just look at the distance

00:06:59.082 --> 00:07:01.230
between u and w in the book.

00:07:01.230 --> 00:07:02.670
It's almost exact.

00:07:02.670 --> 00:07:05.960
This actually looks like
it was drawn to scale.

00:07:05.960 --> 00:07:06.780
Next problem.

00:07:06.780 --> 00:07:08.160
And that's choice
C either way.

00:07:08.160 --> 00:07:10.080
Even when I did it wrong
the first way.

00:07:10.080 --> 00:07:12.830
I didn't do it wrong,
I misread.

00:07:12.830 --> 00:07:15.040
To give myself a little
bit of credit.

00:07:15.040 --> 00:07:16.160
Problem number 9.

00:07:16.160 --> 00:07:17.840
Although you won't get much
credit if you misread

00:07:17.840 --> 00:07:20.280
when you take it.

00:07:20.280 --> 00:07:24.200
A number n is increased by 5.

00:07:24.200 --> 00:07:27.210
I write these out as I read
them, is increased by 5, and

00:07:27.210 --> 00:07:29.130
the result is multiplied by 5.

00:07:33.860 --> 00:07:38.070
This result is decreased by 5.

00:07:38.070 --> 00:07:40.455
Finally, that result
is divided by 5.

00:07:43.140 --> 00:07:45.020
The whole result is
divided by 5.

00:07:45.020 --> 00:07:48.220
In terms of n, what is
the final result?

00:07:48.220 --> 00:07:50.850
Well, what can we do here?

00:07:50.850 --> 00:07:54.180
We can divide the numerator
and denominator by 5.

00:07:54.180 --> 00:07:56.470
So this 5 will cancel out with
this and this, right?

00:07:56.470 --> 00:07:57.670
Because you distribute.

00:07:57.670 --> 00:08:04.160
So you're left with n plus
5 times 1, right?

00:08:04.160 --> 00:08:06.170
Because this just becomes
a 1, this becomes a 1,

00:08:06.170 --> 00:08:08.240
this become a 1.

00:08:08.240 --> 00:08:11.850
n plus 5 times 1-- I won't
write the 1, because it

00:08:11.850 --> 00:08:13.730
doesn't matter anymore--
minus 1.

00:08:13.730 --> 00:08:15.064
All of that over 1.

00:08:15.064 --> 00:08:17.480
So we could just ignore that.

00:08:17.480 --> 00:08:20.030
So it's n plus 5 minus 1.

00:08:20.030 --> 00:08:22.410
Well, it's n plus 4.

00:08:22.410 --> 00:08:23.810
Just get rid of the
parentheses.

00:08:23.810 --> 00:08:25.980
Plus 5 minus 1 is n plus 4.

00:08:25.980 --> 00:08:29.530
And so that's choice D.

00:08:29.530 --> 00:08:32.669
Let's see if I have space
to do the next problem.

00:08:32.669 --> 00:08:35.990
Or time, actually,
more than space.

00:08:35.990 --> 00:08:38.840
I will actually wait to do it
in the next video, just so I

00:08:38.840 --> 00:08:40.570
don't rush things, which
I tend to do.

00:08:40.570 --> 00:08:42.090
See you soon.


SAT Prep: Test 5 Section 9 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.670
We're on problem 14.

00:00:04.670 --> 00:00:10.970
If the function f as defined
by f of x is equal to x

00:00:10.970 --> 00:00:16.160
squared plus bx plus c, where
b and c are positive

00:00:16.160 --> 00:00:19.110
constants, which of the
following could be

00:00:19.110 --> 00:00:21.440
the graph of f?

00:00:21.440 --> 00:00:24.330
So b and c are positive.

00:00:24.330 --> 00:00:25.470
So what does that tell us?

00:00:25.470 --> 00:00:28.560
So a couple of things you
immediately know.

00:00:28.560 --> 00:00:31.455
The x squared term-- whatever's
the coefficient of

00:00:31.455 --> 00:00:34.650
x squared, that tells us whether
the parabola opens up

00:00:34.650 --> 00:00:36.110
or opens down.

00:00:36.110 --> 00:00:39.820
It opens up if the coefficient
on the x term is positive,

00:00:39.820 --> 00:00:40.360
which it is.

00:00:40.360 --> 00:00:41.650
It's positive 1, right?

00:00:41.650 --> 00:00:43.480
So we know the graph is
going to open up.

00:00:43.480 --> 00:00:44.840
It's not going to be a

00:00:44.840 --> 00:00:46.870
downward-opening graph like that.

00:00:46.870 --> 00:00:49.080
So we know the graph is going
to look like a U, as opposed

00:00:49.080 --> 00:00:51.760
to-- I don't know what
this is like.

00:00:51.760 --> 00:00:52.800
An A without the line?

00:00:52.800 --> 00:00:53.710
Whatever.

00:00:53.710 --> 00:00:55.180
So we know it's going
to be a U.

00:00:55.180 --> 00:00:57.210
And what else do we
know about it?

00:00:57.210 --> 00:01:00.450
What do we know about
its y-intercept?

00:01:00.450 --> 00:01:02.320
Well, this is the y-intercept.

00:01:02.320 --> 00:01:06.340
When x is equal to 0, f
of 0 is equal to what?

00:01:06.340 --> 00:01:09.530
These two terms are going to be
0, so f of 0 is equal to c.

00:01:09.530 --> 00:01:12.280
And so we know that the
y-intercept is positive.

00:01:12.280 --> 00:01:15.450
So let's see if that
alone allows us

00:01:15.450 --> 00:01:16.700
to solve the problem.

00:01:20.040 --> 00:01:28.850
So we know that the y-intercept
is positive, so

00:01:28.850 --> 00:01:31.110
when f of 0 is going to
intersect someplace on the

00:01:31.110 --> 00:01:33.230
positive y-axis.

00:01:33.230 --> 00:01:35.660
And we know that it's an
upward-opening graph.

00:01:35.660 --> 00:01:38.800
So the graph could
look like this.

00:01:43.520 --> 00:01:47.820
It could also look like-- well,
this b term-- I won't

00:01:47.820 --> 00:01:49.480
get too much into the b term.

00:01:49.480 --> 00:01:52.350
So we actually know that it
is shifted to the left.

00:01:52.350 --> 00:01:57.070
But based on just what I said,
that it's opening upwards and

00:01:57.070 --> 00:02:01.350
that it has a positive
y-intercept-- this would've

00:02:01.350 --> 00:02:03.540
also been a legitimate graph.

00:02:03.540 --> 00:02:06.780
Anything that is opening upwards
like a U-- because

00:02:06.780 --> 00:02:10.630
this is a positive x squared
here-- and intersects the

00:02:10.630 --> 00:02:13.680
y-axis in the positive area
would be a correct answer.

00:02:13.680 --> 00:02:15.690
And if you look at the choices,
A opens down.

00:02:15.690 --> 00:02:16.440
That's not the right answer.

00:02:16.440 --> 00:02:17.250
B opens down.

00:02:17.250 --> 00:02:17.890
Not the right answer.

00:02:17.890 --> 00:02:21.440
C intercepts the y-axis
at 0, so that's

00:02:21.440 --> 00:02:22.430
not the right answer.

00:02:22.430 --> 00:02:27.330
D opens up, but it intersects
the y-axis in the negative y.

00:02:27.330 --> 00:02:29.450
And then finally E is
very similar to

00:02:29.450 --> 00:02:30.390
what I drew in yellow.

00:02:30.390 --> 00:02:32.270
So that is our answer.

00:02:32.270 --> 00:02:34.610
E.

00:02:34.610 --> 00:02:35.860
Next problem.

00:02:40.430 --> 00:02:42.690
Almost finished with
this section.

00:02:42.690 --> 00:02:45.670
OK, they drew us a cube.

00:02:45.670 --> 00:02:47.760
Let me draw it big, because
this looks like it might

00:02:47.760 --> 00:02:51.905
involve some fancy-- let
me draw the cube.

00:02:51.905 --> 00:02:55.642
So this is the front face.

00:02:55.642 --> 00:02:58.107
Whoops.

00:02:58.107 --> 00:02:59.810
Undo.

00:02:59.810 --> 00:03:03.890
All my time is spent drawing
diagrams. Cube.

00:03:10.860 --> 00:03:12.660
Drawing a diagram.

00:03:12.660 --> 00:03:14.690
And then it comes
back like that.

00:03:24.422 --> 00:03:28.130
Goes back like that.

00:03:28.130 --> 00:03:30.660
I should always draw the back
first. Good enough.

00:03:30.660 --> 00:03:31.530
I think you get the point.

00:03:31.530 --> 00:03:33.660
And it draws the dotted line.

00:03:33.660 --> 00:03:37.550
Maybe I'll draw that later,
if I have to.

00:03:37.550 --> 00:03:42.970
So they tell us this is point
A, and that this right

00:03:42.970 --> 00:03:45.790
here is point B.

00:03:45.790 --> 00:03:48.660
They say the cube shown above
has edges of length 2.

00:03:48.660 --> 00:03:51.670
So each of these sides are 2.

00:03:51.670 --> 00:03:54.450
2, 2, 2, 2, and so on.

00:03:54.450 --> 00:03:56.900
And A and B are midpoints
of the two edges.

00:03:56.900 --> 00:03:59.310
So A and B are midpoints.

00:03:59.310 --> 00:04:00.760
So this side is equal
to this side.

00:04:00.760 --> 00:04:04.410
So if the whole side is equal
to 2, and A is a midpoint,

00:04:04.410 --> 00:04:07.200
then we know that this
is 1, this is 1, this

00:04:07.200 --> 00:04:09.910
is 1, this is 1.

00:04:09.910 --> 00:04:12.540
What is the length of AB?

00:04:12.540 --> 00:04:16.362
So this is what they want
us to figure out.

00:04:16.362 --> 00:04:18.779
They want me to draw it in
kind of a darker color.

00:04:18.779 --> 00:04:22.089
No, that's not a darker--
that's that line.

00:04:22.089 --> 00:04:23.860
This is a pure visualization
problem.

00:04:23.860 --> 00:04:25.950
And at first, you're like, boy,
that's three dimensions.

00:04:25.950 --> 00:04:26.840
It's crazy.

00:04:26.840 --> 00:04:27.895
What am I going to do?

00:04:27.895 --> 00:04:30.470
What you're going to have to
do is do the Pythagorean

00:04:30.470 --> 00:04:32.000
Theorem twice.

00:04:32.000 --> 00:04:33.600
So what do we know?

00:04:33.600 --> 00:04:35.960
Well, we know everything we need
to solve this problem.

00:04:35.960 --> 00:04:37.410
Let's see if we can do it.

00:04:37.410 --> 00:04:42.530
So let me draw that bottom
surface of the cube.

00:04:48.360 --> 00:04:49.100
So I'll ask you a question.

00:04:49.100 --> 00:04:51.310
Can we figure out-- so
let me draw a line

00:04:51.310 --> 00:04:52.620
along the bottom surface.

00:04:52.620 --> 00:04:55.250
Let's say that this line I'm
drawing is along the bottom

00:04:55.250 --> 00:04:56.430
surface of the cube.

00:04:56.430 --> 00:04:59.570
So I'm going to draw
this line.

00:05:02.560 --> 00:05:04.640
That's along the bottom
surface of the cube.

00:05:04.640 --> 00:05:07.860
And let me label this
side like this.

00:05:07.860 --> 00:05:10.800
This is actually a really
fun problem.

00:05:10.800 --> 00:05:12.050
I'll do this in yellow.

00:05:14.460 --> 00:05:17.560
So can we figure out what
that magenta dotted

00:05:17.560 --> 00:05:19.260
line's length is?

00:05:19.260 --> 00:05:19.950
Well, sure.

00:05:19.950 --> 00:05:24.960
We know, first of all, that
is a right triangle,

00:05:24.960 --> 00:05:25.740
the way I drew it.

00:05:25.740 --> 00:05:32.770
I can redraw it like this, where
this is yellow, brown,

00:05:32.770 --> 00:05:34.020
and then dotted line magenta.

00:05:36.500 --> 00:05:41.450
All I did is I kind of
flipped it up so that

00:05:41.450 --> 00:05:42.820
you could see it.

00:05:42.820 --> 00:05:43.370
So what is that?

00:05:43.370 --> 00:05:44.640
We know the brown length.

00:05:44.640 --> 00:05:46.200
We know that's 1.

00:05:46.200 --> 00:05:48.270
We know the yellow
length is what?

00:05:48.270 --> 00:05:49.070
It's 2.

00:05:49.070 --> 00:05:51.460
Because all the sides
of the cube are 2.

00:05:51.460 --> 00:05:53.240
So we can use the Pythagorean
Theorem to figure out the

00:05:53.240 --> 00:05:54.040
magenta line.

00:05:54.040 --> 00:05:59.460
So it's 1 squared plus
2 squared is equal

00:05:59.460 --> 00:06:00.295
to this side squared.

00:06:00.295 --> 00:06:03.040
So we can just take the
square root of it.

00:06:03.040 --> 00:06:06.040
So that equals the square
root of 5, right?

00:06:06.040 --> 00:06:07.580
1 plus 4.

00:06:07.580 --> 00:06:09.420
Square root of 5 is
this magenta line.

00:06:12.340 --> 00:06:14.830
So now can we figure
out that gray line

00:06:14.830 --> 00:06:16.310
that I drew at first?

00:06:16.310 --> 00:06:16.770
Well, sure.

00:06:16.770 --> 00:06:19.350
Because now we have another
right triangle.

00:06:19.350 --> 00:06:21.490
That gray line is just the
hypotenuse of this right

00:06:21.490 --> 00:06:24.720
triangle, that's kind
of at an angle.

00:06:24.720 --> 00:06:28.350
So let me make sure you
understand what I'm saying.

00:06:28.350 --> 00:06:30.610
If I were to flatten
it out, I have this

00:06:30.610 --> 00:06:33.330
green side right here.

00:06:33.330 --> 00:06:41.180
I have this dotted line base,
and then I have the gray line,

00:06:41.180 --> 00:06:42.430
which is the hypotenuse.

00:06:44.170 --> 00:06:48.170
And just so you know, this point
right here would be A,

00:06:48.170 --> 00:06:51.250
and then this point right
here would be B.

00:06:51.250 --> 00:06:53.000
And do we know the sides?

00:06:53.000 --> 00:06:54.480
Yes, we do.

00:06:54.480 --> 00:06:57.540
We know this green line here is
1, so this side here is 1.

00:06:57.540 --> 00:06:59.080
I'm just redrawing it here.

00:06:59.080 --> 00:07:01.550
We know the magenta dotted line
at the bottom, which I

00:07:01.550 --> 00:07:03.320
kind of switched colors a bit.

00:07:03.320 --> 00:07:05.880
We just figured out that that
magenta bottom line is

00:07:05.880 --> 00:07:07.830
square root of 5.

00:07:07.830 --> 00:07:11.010
So now we can just use the
Pythagorean Theorem to figure

00:07:11.010 --> 00:07:12.950
out the length of this line.

00:07:12.950 --> 00:07:15.750
The length of this line squared
is going to be equal

00:07:15.750 --> 00:07:19.670
to the square root of 5,
squared, plus 1 squared.

00:07:19.670 --> 00:07:21.460
So the hypotenuse squared is
equal to this, so we could

00:07:21.460 --> 00:07:23.700
take the square root of that.

00:07:23.700 --> 00:07:26.940
So that equals the square
root of-- what's the

00:07:26.940 --> 00:07:27.995
square root of 5 squared?

00:07:27.995 --> 00:07:30.480
Well, it's just 5, right?

00:07:30.480 --> 00:07:31.010
Plus 1.

00:07:31.010 --> 00:07:34.440
So it's just equal to the
square root of 6.

00:07:34.440 --> 00:07:37.150
So this line is equal to
the square root of 6.

00:07:37.150 --> 00:07:38.530
And that is choice D.

00:07:38.530 --> 00:07:41.830
We luckily have not
made a mistake.

00:07:41.830 --> 00:07:43.080
Last problem.

00:07:46.610 --> 00:07:49.780
Problem 16.

00:07:49.780 --> 00:07:53.930
Let x-- with this kind of
oval-looking thing around it--

00:07:53.930 --> 00:08:01.830
be defined as x squared minus
x, for all values of x.

00:08:01.830 --> 00:08:11.540
If a-- so they're telling us
that a is equal to a minus 2,

00:08:11.540 --> 00:08:13.510
what is the value of a?

00:08:13.510 --> 00:08:16.320
So what does this mean? a
kind of with this oval?

00:08:16.320 --> 00:08:21.680
Well, that just means that
a squared minus a, right?

00:08:21.680 --> 00:08:23.700
That maps to that,
just like that.

00:08:23.700 --> 00:08:25.750
And what does a minus
2 map to?

00:08:25.750 --> 00:08:28.470
And I'm just bringing
the equal sign down.

00:08:28.470 --> 00:08:31.800
a minus 2 maps to-- every place
where I see an x, I put

00:08:31.800 --> 00:08:32.669
an a minus 2 in there.

00:08:32.669 --> 00:08:39.510
So that equals a minus 2
squared, minus a minus 2.

00:08:39.510 --> 00:08:41.115
And now we just keep solving.

00:08:41.115 --> 00:08:43.280
And most of the solving is
on the right-hand side.

00:08:43.280 --> 00:08:43.700
So let's see.

00:08:43.700 --> 00:08:57.850
That equals a squared minus
4a plus 4 minus a plus 2.

00:08:57.850 --> 00:09:04.190
So we get a squared minus
a is equal to a

00:09:04.190 --> 00:09:08.480
squared minus 5a, right?

00:09:08.480 --> 00:09:12.200
Minus 4a minus a plus 6.

00:09:12.200 --> 00:09:17.270
We can subtract a squared
from both sides.

00:09:17.270 --> 00:09:19.730
Let me add 5a to both sides.

00:09:19.730 --> 00:09:24.370
So 5a, so you get 4a
is equal to 6.

00:09:24.370 --> 00:09:27.040
Divide both sides by 4.

00:09:27.040 --> 00:09:33.732
You get a is equal to 6/4, which
equals 1 and 1/2 or 3/2.

00:09:33.732 --> 00:09:36.540
And that is choice C.

00:09:36.540 --> 00:09:38.260
And we are done this section.

00:09:38.260 --> 00:09:40.800
I'll see you in the next
practice test.


SAT Prep: Test 5 Section 7 Part 5

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.730
OK, we're on problem
number 16.

00:00:03.730 --> 00:00:06.490
Let's see, they drew
us something here.

00:00:06.490 --> 00:00:11.560
From point A to point
B, I have a line.

00:00:11.560 --> 00:00:15.930
Then I have another, B to C.

00:00:15.930 --> 00:00:19.750
Then I have another,
from here to here.

00:00:19.750 --> 00:00:21.060
OK, so what are they
telling us?

00:00:21.060 --> 00:00:24.450
They tell us, this is
a 90 degree angle.

00:00:24.450 --> 00:00:29.165
Whoops, not what I wanted
to do, edit, undo.

00:00:29.165 --> 00:00:31.000
I'm using the wrong tool.

00:00:31.000 --> 00:00:33.350
So they tell us, this is a 90
degree angle, this is 90

00:00:33.350 --> 00:00:35.280
degree angle.

00:00:35.280 --> 00:00:44.140
This is A, this is B, this
is C, this is D.

00:00:44.140 --> 00:00:46.890
And they say, the figure above
shows the route of Diane's

00:00:46.890 --> 00:00:48.810
trip from her house
to her job.

00:00:48.810 --> 00:00:53.620
Diane travels 16 miles from
A to B, so this is 16.

00:00:53.620 --> 00:00:57.700
15 miles from B to
C, so this is 15.

00:00:57.700 --> 00:01:01.286
And 4 miles from C to
D, so this is 4.

00:01:01.286 --> 00:01:04.019
If she were to travel from
A to D directly, how much

00:01:04.019 --> 00:01:06.420
shorter in miles would
the trip be?

00:01:06.420 --> 00:01:08.460
So what would be the
direct route?

00:01:08.460 --> 00:01:11.000
I'm assuming, they're kind of
talking if she could fly.

00:01:11.000 --> 00:01:12.560
What would be her
direct route?

00:01:12.560 --> 00:01:13.850
So that would be this route.

00:01:13.850 --> 00:01:16.600
I thought I was using
the line tool.

00:01:16.600 --> 00:01:17.290
Edit, undo.

00:01:17.290 --> 00:01:20.150
I'll do it in a better
color anyway.

00:01:20.150 --> 00:01:22.280
So this is the direct
route, right here.

00:01:22.280 --> 00:01:25.130
This is we want to figure out,
the distance of this.

00:01:25.130 --> 00:01:27.150
So how can we figure
out the distances?

00:01:27.150 --> 00:01:31.170
And the trick here is to just
realize that this is going to

00:01:31.170 --> 00:01:33.390
be the hypotenuse of
a right triangle.

00:01:33.390 --> 00:01:36.330
And you might-- seem confusing,
your initial

00:01:36.330 --> 00:01:38.780
impulse might have been to say
this is x, and this is 15

00:01:38.780 --> 00:01:40.730
minus x, and all of
this fancy stuff.

00:01:40.730 --> 00:01:46.260
Or what you could do, is you can
say, well, this side here,

00:01:46.260 --> 00:01:48.760
we know what this
is, this length.

00:01:48.760 --> 00:01:50.002
Whoops.

00:01:50.002 --> 00:01:51.700
Let me draw it a little
bit better.

00:01:54.200 --> 00:01:55.990
That length, what
is that length?

00:01:55.990 --> 00:01:58.020
Well, this length is going
to be the same as

00:01:58.020 --> 00:01:58.700
this length, right?

00:01:58.700 --> 00:02:00.820
If we assume this is
a 90 degree angle.

00:02:00.820 --> 00:02:03.110
It's going to be 15.

00:02:03.110 --> 00:02:04.770
And then what's this
length, right here?

00:02:08.880 --> 00:02:10.699
Well it's going to same as this
length up here, right?

00:02:10.699 --> 00:02:12.980
I just kind of extended
a rectangle.

00:02:12.980 --> 00:02:16.180
So this length right
here is 4.

00:02:16.180 --> 00:02:21.100
So what is this entire length?

00:02:21.100 --> 00:02:25.630
It's going to be 16 plus
4, so it's 20.

00:02:25.630 --> 00:02:29.880
So now, what we're trying to
figure out, her straight line

00:02:29.880 --> 00:02:34.200
distance, is nothing but the
hypotenuse of a rectangle-- of

00:02:34.200 --> 00:02:37.300
a right triangle, where we
know the other two sides.

00:02:37.300 --> 00:02:42.740
So we can say, if this is her
straight line distance, x, we

00:02:42.740 --> 00:02:47.980
could say x squared is equal to
20 squared, 20 squared plus

00:02:47.980 --> 00:02:51.500
15 squared.

00:02:51.500 --> 00:02:54.500
x squared is equal to-- let's
see, this is 400, plus 225, is

00:02:54.500 --> 00:02:56.590
equal to 625.

00:02:56.590 --> 00:03:02.610
And the square root of 625 is
25, so x is equal to 25.

00:03:02.610 --> 00:03:04.830
So her straight line
distance is 25.

00:03:04.830 --> 00:03:06.870
And how far was she
travelling before?

00:03:06.870 --> 00:03:16.220
Let's see, she was going
16 plus 15 plus 4.

00:03:16.220 --> 00:03:19.400
So that's 20-- she was going
35, what is that, miles?

00:03:19.400 --> 00:03:21.070
Right, miles.

00:03:21.070 --> 00:03:22.960
She was going 35 miles,
and if she goes

00:03:22.960 --> 00:03:25.590
straight, she goes 25 miles.

00:03:25.590 --> 00:03:30.310
So she would save 10 miles if
she went straight, if she was

00:03:30.310 --> 00:03:31.490
able to fly.

00:03:31.490 --> 00:03:34.690
So that's choice C.

00:03:34.690 --> 00:03:35.940
Next problem.

00:03:39.110 --> 00:03:45.510
Problem number 17.

00:03:45.510 --> 00:03:49.320
One circle has a radius of 1/2,
and another circle has a

00:03:49.320 --> 00:03:50.750
radius of 1.

00:03:50.750 --> 00:03:54.010
What is the ratio of the area
of the larger circle to the

00:03:54.010 --> 00:03:56.130
area of the smaller circle?

00:03:56.130 --> 00:04:04.530
Let's see, so the area of circle
1, area 1, is going to

00:04:04.530 --> 00:04:06.660
be equal to what?

00:04:06.660 --> 00:04:09.330
It has a radius of 1/2, right?

00:04:09.330 --> 00:04:12.380
So it's pi r squared,
is the radius.

00:04:12.380 --> 00:04:18.103
So it's pi 1/2 squared.

00:04:18.103 --> 00:04:20.740
And what's the area
of circle 2?

00:04:20.740 --> 00:04:26.580
The area of circle 2, it's
pi times its radius.

00:04:26.580 --> 00:04:28.340
And its radius is 1.

00:04:28.340 --> 00:04:31.360
So 1 squared.

00:04:31.360 --> 00:04:34.920
So the ratio of the area of
the larger circle to the

00:04:34.920 --> 00:04:38.290
smaller circle-- which one is
going to be the larger circle?

00:04:38.290 --> 00:04:40.690
This one is, because it has
a larger radius, right?

00:04:40.690 --> 00:04:42.310
It has a radius of 1.

00:04:42.310 --> 00:04:45.610
So the ratio of the larger
circle to the smaller circle--

00:04:45.610 --> 00:04:47.820
and that might be where you
messed up, if you flipped the

00:04:47.820 --> 00:04:50.135
two-- the ratio of the larger
circle to the smaller circle

00:04:50.135 --> 00:04:53.030
is equal to pi times 1, right?

00:04:53.030 --> 00:04:54.460
Well that's just pi.

00:04:54.460 --> 00:04:59.385
Over pi times 1/2 squared.

00:05:03.030 --> 00:05:08.860
Pi's cancel out, so we have,
it equals 1 over 1/4.

00:05:11.460 --> 00:05:13.650
Or what we could do is we could
multiply the numerator

00:05:13.650 --> 00:05:19.440
and the denominator by
4, which equals 4/1.

00:05:19.440 --> 00:05:25.590
So the ratio is 4:1, and
that's choice D.

00:05:25.590 --> 00:05:30.390
Problem 18.

00:05:30.390 --> 00:05:33.190
Clear image.

00:05:33.190 --> 00:05:34.930
Problem 18.

00:05:34.930 --> 00:05:39.290
If the sum of the consecutive
integers from minus 22 to x

00:05:39.290 --> 00:05:42.440
inclusive is 72, what
is the value of x?

00:05:42.440 --> 00:05:45.670
So this is the number-- we've
actually done a problem very,

00:05:45.670 --> 00:05:47.800
very similar to this, so it
shows you that people at the

00:05:47.800 --> 00:05:50.510
SAT, they think of a lot of
creative problems, but every

00:05:50.510 --> 00:05:52.690
now and then, they
have to repeat.

00:05:52.690 --> 00:05:54.900
So they're saying the sum of the
consecutive integers from

00:05:54.900 --> 00:05:56.670
minus 22 to x is 72.

00:05:56.670 --> 00:05:59.510
So let's think about this,
let's draw the sum.

00:05:59.510 --> 00:06:09.390
So it's minus 22, plus minus
21, plus minus 20, right?

00:06:09.390 --> 00:06:16.050
Plus-- it keeps going, until
it gets to 0-- plus 0.

00:06:16.050 --> 00:06:18.910
And then-- and I'm going to go
this way, again, I'm doing

00:06:18.910 --> 00:06:21.550
that for a reason.

00:06:21.550 --> 00:06:27.280
And then you can say, plus 1,
plus 2, plus-- you keep going,

00:06:27.280 --> 00:06:39.300
and then you'd have plus 20,
plus 21, plus 22, plus 23,

00:06:39.300 --> 00:06:43.920
plus 24, all the way until
we get to x, right?

00:06:43.920 --> 00:06:50.540
What you need to realize is all
the way from minus 22 to

00:06:50.540 --> 00:06:53.370
22, the sum is 0.

00:06:53.370 --> 00:06:55.000
And why is that?

00:06:55.000 --> 00:06:57.310
Well, look at the
way I drew it.

00:06:57.310 --> 00:06:59.570
So minus 22-- we're taking the
sum of everything, right?

00:06:59.570 --> 00:07:03.150
So minus 22 is going to cancel
out with positive 22.

00:07:03.150 --> 00:07:04.850
You add those together,
you get 0.

00:07:04.850 --> 00:07:08.230
Minus 21 is going to cancel
out with positive 21.

00:07:08.230 --> 00:07:10.760
Minus 20 is going to cancel
out with positive 20.

00:07:10.760 --> 00:07:11.750
And so forth, and so on.

00:07:11.750 --> 00:07:16.500
So everything-- so the sum from
minus 22 plus, bam, bam,

00:07:16.500 --> 00:07:22.060
bam, bam, all the way to plus
22, that equals 0, right?

00:07:22.060 --> 00:07:26.820
So we only get into the
positive sum once

00:07:26.820 --> 00:07:28.810
we get to 23, right?

00:07:28.810 --> 00:07:32.250
So 23 plus 24 plus something,
something,

00:07:32.250 --> 00:07:34.060
something, is x, right?

00:07:34.060 --> 00:07:37.520
And we have to get to 72.

00:07:37.520 --> 00:07:39.170
And here we can just
eyeball it.

00:07:39.170 --> 00:07:44.890
23 plus 24 plus 25,
what's that?

00:07:44.890 --> 00:07:47.070
Well their average is 24,
24 times 3 is 72.

00:07:47.070 --> 00:07:48.390
Or we can just add it up.

00:07:48.390 --> 00:07:55.930
So 23 plus 24 is 47, 47
plus 25, it equals 72.

00:07:55.930 --> 00:07:59.620
So the last number that we use
in this whole range is 25.

00:07:59.620 --> 00:08:02.040
So this is x.

00:08:02.040 --> 00:08:03.520
That is x.

00:08:03.520 --> 00:08:05.750
And all we did is we said--
think about it, take the sum

00:08:05.750 --> 00:08:09.360
from minus 22, all the
way to positive 25.

00:08:09.360 --> 00:08:13.620
Everything up to negative 22,
from minus 22 to positive 22,

00:08:13.620 --> 00:08:15.490
all of those cancel out
with each other.

00:08:15.490 --> 00:08:19.470
So the sum is just going to
be 23 plus 24 plus 25.

00:08:19.470 --> 00:08:23.600
That equals 72, so
then x is 25.

00:08:23.600 --> 00:08:26.910
Next problem-- I'm running out
of time, so I'd better do it

00:08:26.910 --> 00:08:28.400
in the next video.

00:08:28.400 --> 00:08:29.880
See you soon.


SAT Prep: Test 5 Section 7 Part 6

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:05.700
We're on problem 19.

00:00:05.700 --> 00:00:10.700
If k, n, x, and y are positive
numbers satisfying-- let's

00:00:10.700 --> 00:00:11.560
see, what did they write?

00:00:11.560 --> 00:00:22.340
They wrote x to the minus 4/3 is
equal to k to the minus 2.

00:00:22.340 --> 00:00:27.680
y to the positive 4/3 is
equal to n squared.

00:00:30.280 --> 00:00:42.760
What is xy to the minus 2/3
in terms of n and k?

00:00:42.760 --> 00:00:45.160
All right, so we just have to
rewrite this in terms of n and

00:00:45.160 --> 00:00:47.380
k, so let me write it big.

00:00:47.380 --> 00:01:03.000
So xy to the minus 2/3, that
equals x to the minus 2/3,

00:01:03.000 --> 00:01:08.690
times y to the minus 2/3.

00:01:08.690 --> 00:01:10.700
We know what x to the 4/3 is.

00:01:10.700 --> 00:01:13.820
This is what I wrote here, it
might not be that legible.

00:01:13.820 --> 00:01:15.000
Sorry.

00:01:15.000 --> 00:01:16.090
So this is a 4, right?

00:01:16.090 --> 00:01:18.740
We know what x to the
minus 4/3 is.

00:01:18.740 --> 00:01:24.300
So can we rewrite this as x to
the minus 4/3 to something?

00:01:24.300 --> 00:01:24.780
Well, sure.

00:01:24.780 --> 00:01:28.370
We can rewrite this term, right
here, we can write this

00:01:28.370 --> 00:01:30.310
as-- let me write it right below
it so you don't get too

00:01:30.310 --> 00:01:38.030
confused-- as x to the minus
4/3, and then you raise that

00:01:38.030 --> 00:01:40.590
to the 1/2 power.

00:01:40.590 --> 00:01:42.460
This is the same
thing as this.

00:01:42.460 --> 00:01:43.730
How do I know that?

00:01:43.730 --> 00:01:48.070
Well, if I multiply 4/3 times
1/2-- because I'm raising

00:01:48.070 --> 00:01:50.780
something to an exponent, and
then that to an exponent-- if

00:01:50.780 --> 00:01:56.420
I multiply minus 4/3 times 1/2,
I get minus 2/3, right?

00:01:56.420 --> 00:01:57.970
This 2 cancels out, this
becomes a 2, and

00:01:57.970 --> 00:01:59.450
this is what I get.

00:01:59.450 --> 00:02:00.870
You can try it out.

00:02:00.870 --> 00:02:05.886
Similarly, I want to write this
second term-- I'm going

00:02:05.886 --> 00:02:09.479
to do it in green-- this
y to the minus 2/3.

00:02:09.479 --> 00:02:12.040
I know what y to the
4/3 is, right?

00:02:12.040 --> 00:02:18.970
So I want to write it as y to
the 4/3 to some other power.

00:02:18.970 --> 00:02:23.340
So 4/3 times what is
equal to minus 2/3?

00:02:23.340 --> 00:02:27.080
Well, 4/3 times minus 1/2.

00:02:27.080 --> 00:02:28.510
And you can try it out.

00:02:28.510 --> 00:02:32.570
4/3 times minus 1/2 is equal
to minus 2/3, right?

00:02:32.570 --> 00:02:35.780
A positive times a negative
equals a negative.

00:02:35.780 --> 00:02:38.380
And then the 4 and the
2 end up just the

00:02:38.380 --> 00:02:40.690
2-- sorry, I'm burping.

00:02:40.690 --> 00:02:41.580
So what does this become?

00:02:41.580 --> 00:02:45.010
Well, we know what x to the
minus 4/3 is, right? x to the

00:02:45.010 --> 00:02:49.380
minus 4/3 is equal to
k to the minus 2.

00:02:49.380 --> 00:02:53.990
So that first term is equal to
this, is equal to k-- that's

00:02:53.990 --> 00:03:00.690
too dark-- k to the minus 2, and
then we're raising that to

00:03:00.690 --> 00:03:05.000
the 1/2-- right, that's the
same 1/2 right there-- and

00:03:05.000 --> 00:03:06.190
then y to the 4/3.

00:03:06.190 --> 00:03:07.150
What's y to the 4/3?

00:03:07.150 --> 00:03:08.400
Well, that's n squared.

00:03:13.680 --> 00:03:17.310
And then that's, of course,
the minus 1/2.

00:03:17.310 --> 00:03:18.820
And then we can multiply.

00:03:18.820 --> 00:03:20.980
What's minus 2 times 1/2?

00:03:20.980 --> 00:03:21.960
Well, it's just minus 1.

00:03:21.960 --> 00:03:24.050
So it's k to the minus 1.

00:03:24.050 --> 00:03:27.560
And then what's 2
times minus 1/2?

00:03:27.560 --> 00:03:28.473
It's minus 1 again.

00:03:28.473 --> 00:03:31.910
So k to the minus 1,
n to the minus 1.

00:03:31.910 --> 00:03:38.790
And that's the same thing
as 1/kn, or 1/nk,

00:03:38.790 --> 00:03:41.720
which is choice A.

00:03:41.720 --> 00:03:44.090
And I just realized that there's
actually an easier way

00:03:44.090 --> 00:03:44.850
to do that.

00:03:44.850 --> 00:03:48.290
We could've done it like this,
we could have just said-- let

00:03:48.290 --> 00:03:51.160
me see, let me erase this stuff,
because it's, I think,

00:03:51.160 --> 00:03:52.850
good for you to know the easier
way, instead of the way

00:03:52.850 --> 00:03:55.000
I just showed you
how to do it.

00:03:55.000 --> 00:03:57.000
Let me erase all of this.

00:03:57.000 --> 00:03:58.820
I think we would've
saved some steps,

00:03:58.820 --> 00:04:00.070
and been more intuitive.

00:04:02.770 --> 00:04:05.350
Let me erase all of this, I just
wanted to keep what I had

00:04:05.350 --> 00:04:08.320
written before.

00:04:08.320 --> 00:04:11.820
Erase, erase, erase.

00:04:11.820 --> 00:04:13.800
OK, I think that's
good enough.

00:04:13.800 --> 00:04:15.220
You can just ignore the
stuff on the right.

00:04:18.230 --> 00:04:20.310
So we could've simplified
these ahead of time.

00:04:20.310 --> 00:04:22.645
We could have said, x to the
minus 4/3 is equal to k

00:04:22.645 --> 00:04:24.590
to the minus 2.

00:04:24.590 --> 00:04:34.870
If we raised both of these
to the 1/2 power,

00:04:34.870 --> 00:04:36.520
we would get what?

00:04:36.520 --> 00:04:39.860
Minus 4/3 times 1/2 is-- we
would have gotten x to the

00:04:39.860 --> 00:04:44.740
minus 2/3 is equal to-- what's
negative 2 times 1/2?

00:04:44.740 --> 00:04:46.360
It's negative 1, right?

00:04:46.360 --> 00:04:48.270
It would've equaled k
to the negative 1.

00:04:48.270 --> 00:04:56.050
And similarly, if we had raised
y to the 4/3 equals n

00:04:56.050 --> 00:05:02.690
squared, if we'd raised both of
those to the minus 1/2-- if

00:05:02.690 --> 00:05:04.095
you do something on one side of
the equation, you just have

00:05:04.095 --> 00:05:07.090
to do it on the other-- then you
would have gotten y to the

00:05:07.090 --> 00:05:12.500
minus 2/3 is equal to
n to the minus 1.

00:05:12.500 --> 00:05:15.550
And then this would have been
really easy to solve.

00:05:15.550 --> 00:05:18.480
xy to the minus 2/3 is the same
thing as x to the minus

00:05:18.480 --> 00:05:22.430
2/3, times y to the minus 2/3.

00:05:22.430 --> 00:05:25.200
x to the the minus 2/3
is k to the minus 1.

00:05:25.200 --> 00:05:29.350
And y to the negative 2/3
is n to the minus 1.

00:05:29.350 --> 00:05:32.780
So it would have been k to the
minus 1, n to the minus 1.

00:05:32.780 --> 00:05:34.600
That's 1/nk.

00:05:34.600 --> 00:05:37.470
Well, I guess in hindsight I can
say this was easier, but

00:05:37.470 --> 00:05:39.260
the way we did it would've also
worked, and you really

00:05:39.260 --> 00:05:41.394
don't know sometimes which one
will be the shortcut when you

00:05:41.394 --> 00:05:42.490
do it the first time.

00:05:42.490 --> 00:05:43.580
But either way could've
worked.

00:05:43.580 --> 00:05:46.630
You could've simplified ahead of
time, if you had seen that

00:05:46.630 --> 00:05:49.130
you wanted to get it into
2/3, not 4/3, because

00:05:49.130 --> 00:05:50.390
they had a 2/3 here.

00:05:50.390 --> 00:05:52.390
Or you could just go the way
we did it the first time,

00:05:52.390 --> 00:05:54.450
either way works.

00:05:54.450 --> 00:06:01.780
Next problem, problem
number 20.

00:06:01.780 --> 00:06:04.140
OK, functions of f and g.

00:06:04.140 --> 00:06:06.100
So I'm going to have to do some
drawing here, if we have

00:06:06.100 --> 00:06:08.362
enough time.

00:06:08.362 --> 00:06:11.740
So I have to draw two of them.

00:06:11.740 --> 00:06:14.090
That one's going to be
for f, and this one's

00:06:14.090 --> 00:06:15.340
going to be for g.

00:06:20.060 --> 00:06:22.430
What points are there?

00:06:22.430 --> 00:06:27.090
So f of x-- let's see,
[UNINTELLIGIBLE]

00:06:27.090 --> 00:06:30.875
f of x is 2 points back, and
it intersects there.

00:06:30.875 --> 00:06:36.210
And it intersects at the point
minus 1 comma 1, 2, 3, so it

00:06:36.210 --> 00:06:38.400
intersects here.

00:06:38.400 --> 00:06:41.190
And then it goes through
the origin.

00:06:41.190 --> 00:06:43.530
And then it goes
to 1, minus 3.

00:06:43.530 --> 00:06:48.830
So that's 1, 2, 3, 1,
so it goes here.

00:06:48.830 --> 00:06:52.120
And then it goes at 2 comma
0, and it intersects here.

00:06:52.120 --> 00:06:54.010
So let me try to draw that.

00:06:54.010 --> 00:06:56.056
So the graph looks something
like this.

00:06:56.056 --> 00:06:59.920
It goes up, it intersects this
point, comes down through the

00:06:59.920 --> 00:07:04.410
origin, goes down here, and
then goes up like that.

00:07:04.410 --> 00:07:05.795
That is y equals f of x.

00:07:10.760 --> 00:07:12.295
Now what does g of x do?

00:07:15.150 --> 00:07:19.360
g of x goes to 2 comma 1.

00:07:19.360 --> 00:07:22.630
So 1, 2 comma 1.

00:07:25.450 --> 00:07:27.600
Then it's to 4, minus 5.

00:07:27.600 --> 00:07:33.030
So 2, 3, 4, 1, 2, 3,
4, 5-- 4, minus 5.

00:07:35.860 --> 00:07:36.920
And then it goes back up.

00:07:36.920 --> 00:07:38.370
So it looks something
like this.

00:07:38.370 --> 00:07:44.170
It goes down, goes up like that,
goes back up like that.

00:07:44.170 --> 00:07:46.240
The figures above show
the graph of the

00:07:46.240 --> 00:07:48.330
functions f and g.

00:07:48.330 --> 00:07:55.350
The function f is defined by--
so this one-- f of x is equal

00:07:55.350 --> 00:08:00.890
to x to the third minus 4x.

00:08:00.890 --> 00:08:09.150
The function g of x is equal
to f of x plus h, plus k,

00:08:09.150 --> 00:08:11.250
where h and k are constants.

00:08:11.250 --> 00:08:13.040
What is the value of k?

00:08:13.040 --> 00:08:16.570
So essentially, what they're
saying is, g of x is f of x

00:08:16.570 --> 00:08:20.190
shifted up or down by some
value, k, right?

00:08:20.190 --> 00:08:21.750
Because when you add a constant
to a function, you're

00:08:21.750 --> 00:08:27.000
just moving it up or down
by some value, k.

00:08:27.000 --> 00:08:34.590
And when you add h, that means
you're shifting it to-- let me

00:08:34.590 --> 00:08:38.380
make sure I get this right-- so
if it was x minus h, that

00:08:38.380 --> 00:08:41.570
means you're shifting it
to the right by h.

00:08:41.570 --> 00:08:46.690
So x plus h means you're
shifting it to the left by h.

00:08:50.610 --> 00:08:52.840
So k is the easy one.

00:08:52.840 --> 00:08:55.190
This point corresponds
to this point.

00:08:55.190 --> 00:08:56.360
So how much did we shift down?

00:08:56.360 --> 00:09:01.030
This is the point negative
1 comma 3.

00:09:01.030 --> 00:09:05.070
And this is the point
2 comma 1.

00:09:08.140 --> 00:09:12.020
So we went from y equals
3 to y equals 1, right?

00:09:12.020 --> 00:09:15.130
So we shifted down by 2.

00:09:15.130 --> 00:09:17.710
So k is going to be minus 2.

00:09:17.710 --> 00:09:19.730
And similarly you could
do it with this point.

00:09:19.730 --> 00:09:23.110
So k is equal to minus 2.

00:09:23.110 --> 00:09:25.120
Now we have to figure out how
much we shifted left to right.

00:09:25.120 --> 00:09:25.690
So [? similarly, ?]

00:09:25.690 --> 00:09:27.810
these are the corresponding
points, because you can look

00:09:27.810 --> 00:09:29.250
at them in the graph.

00:09:29.250 --> 00:09:32.575
So we went from minus
1 to positive 2.

00:09:35.760 --> 00:09:41.100
So we shifted to
the right by 3.

00:09:41.100 --> 00:09:42.860
And you could do the same thing
with this point here at

00:09:42.860 --> 00:09:43.380
the bottom.

00:09:43.380 --> 00:09:45.840
We shifted to the right by 3.

00:09:45.840 --> 00:09:48.990
So if I wanted to shift f of x
to the right by 3, I would

00:09:48.990 --> 00:09:54.730
write it like this,
f of x minus 3.

00:09:54.730 --> 00:09:58.500
That's how I would shift a
function to the right by 3.

00:09:58.500 --> 00:10:02.080
They're saying plus h, so
h has to be minus 3.

00:10:02.080 --> 00:10:03.870
h is minus 3.

00:10:03.870 --> 00:10:06.090
And minus 3 times minus 2?

00:10:06.090 --> 00:10:09.530
Well, that's positive 6,
which is choice E.


SAT Prep: Test 5 Section 7 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.930 --> 00:00:01.490
Welcome back.

00:00:01.490 --> 00:00:02.990
We're on problem number 13.

00:00:02.990 --> 00:00:06.340
And actually, I started problem
number 13 at the end

00:00:06.340 --> 00:00:08.360
of the last video, and then I
realized when I was going

00:00:08.360 --> 00:00:10.400
through the choices that
I wasn't explaining

00:00:10.400 --> 00:00:11.180
it perfectly correctly.

00:00:11.180 --> 00:00:14.310
And I have to admit, this is the
first problem that of any

00:00:14.310 --> 00:00:15.950
of the problems we've seen on
the SAT, this is probably the

00:00:15.950 --> 00:00:18.520
one that-- the wording
is probably the most

00:00:18.520 --> 00:00:20.110
difficult to interpret.

00:00:20.110 --> 00:00:21.940
But I think I've figured
it out now.

00:00:21.940 --> 00:00:24.570
So they're saying Kyle's lock
combination consists of three

00:00:24.570 --> 00:00:25.900
two digit numbers.

00:00:25.900 --> 00:00:29.420
The combination satisfies
three conditions below.

00:00:29.420 --> 00:00:35.710
One number is odd, one number
is a multiple of 5, so we'll

00:00:35.710 --> 00:00:39.540
just write that as times 5, and
then one number is a day

00:00:39.540 --> 00:00:40.805
of the month of Kyle's
birthday.

00:00:49.640 --> 00:00:51.790
And actually, this is what
confused me, it actually

00:00:51.790 --> 00:00:54.450
matters that it's his birthday,
not that it's just a

00:00:54.450 --> 00:00:56.940
day of the month, and
I'll show you why.

00:00:56.940 --> 00:00:58.840
So let's say we have choice A.

00:01:02.690 --> 00:01:05.930
And let me color code
these conditions.

00:01:05.930 --> 00:01:09.500
So odd I'll do in
this blue color.

00:01:09.500 --> 00:01:14.690
Let me see, and times 5
in this yellow color.

00:01:14.690 --> 00:01:19.570
And then-- let me pick another
good color-- brown, the day of

00:01:19.570 --> 00:01:20.820
the month of his birthday.

00:01:23.520 --> 00:01:25.560
So the first choice-- and now
let me write these down in a

00:01:25.560 --> 00:01:28.780
fairly neutral color,
in this magenta.

00:01:28.780 --> 00:01:37.340
So choice A says
14, 20, and 13.

00:01:37.340 --> 00:01:39.550
So what we can do is, let's see,
one of the numbers has to

00:01:39.550 --> 00:01:43.610
be odd, and only one of
the numbers, right?

00:01:43.610 --> 00:01:46.280
So let's say this one.

00:01:46.280 --> 00:01:48.620
This satisfies the odd
condition, and actually only

00:01:48.620 --> 00:01:51.350
satisfies the odd condition.

00:01:51.350 --> 00:01:53.670
Then let's see, what number
here is times 5,

00:01:53.670 --> 00:01:55.080
a multiple of 5?

00:01:55.080 --> 00:02:01.110
Well, this number-- let me do it
in the yellow-- this number

00:02:01.110 --> 00:02:03.510
is a multiple of 5.

00:02:03.510 --> 00:02:05.740
And then, finally, if that one
is the multiple of 5, which

00:02:05.740 --> 00:02:08.620
one has to be the day of the
month of his birthday?

00:02:08.620 --> 00:02:10.840
Well, it's this choice,
right here.

00:02:10.840 --> 00:02:13.060
Now what confused me, when I
first did this problem, and

00:02:13.060 --> 00:02:15.840
probably confused you as well,
it says if each number

00:02:15.840 --> 00:02:19.980
satisfies exactly one of the
conditions-- exactly one,

00:02:19.980 --> 00:02:21.820
exactly one, right?

00:02:21.820 --> 00:02:25.090
So to me that means that it can
only satisfy one of the

00:02:25.090 --> 00:02:25.690
conditions.

00:02:25.690 --> 00:02:31.780
And what confused me was that 20
satisfies multiple of five,

00:02:31.780 --> 00:02:35.480
but to me 20 could just
as easily be a

00:02:35.480 --> 00:02:39.390
day of a month, right?

00:02:39.390 --> 00:02:43.070
But the problem that this
encounters is here we have to

00:02:43.070 --> 00:02:45.740
kind of assume that the
14th is his birthday.

00:02:45.740 --> 00:02:49.480
Because-- let's put it this
way-- if the 20th was his

00:02:49.480 --> 00:02:54.970
birthday, this works
if, and only if,

00:02:54.970 --> 00:02:56.670
the 14th is his birthday.

00:02:56.670 --> 00:02:58.010
Which it could be.

00:02:58.010 --> 00:03:01.940
And if the 14th is his birthday,
then 20 doesn't

00:03:01.940 --> 00:03:05.440
satisfy the day of the month
of his birthday.

00:03:05.440 --> 00:03:07.870
It satisfies a day of the month,
it wouldn't satisfy the

00:03:07.870 --> 00:03:10.030
day of the month of
his birthday.

00:03:10.030 --> 00:03:11.700
That might seem a little
technical, but I think you'll

00:03:11.700 --> 00:03:13.290
see why I have to make that
distinction, as you go

00:03:13.290 --> 00:03:16.690
on to choice B.

00:03:16.690 --> 00:03:20.190
Choice B says-- let me do it in
that neutral color-- it's

00:03:20.190 --> 00:03:25.870
14, 25, and 13.

00:03:25.870 --> 00:03:31.610
So here, once again, 13
satisfies the odd criteria.

00:03:31.610 --> 00:03:35.950
We could say that 14-- we'll
assume that his birthday is

00:03:35.950 --> 00:03:38.050
the 14th, we don't know, they
haven't told us, but

00:03:38.050 --> 00:03:39.000
it could be, right?

00:03:39.000 --> 00:03:40.575
That's what they're asking, what
could be the combination

00:03:40.575 --> 00:03:41.790
of the lock.

00:03:41.790 --> 00:03:44.840
So we'll say that the 14th
could be his birthday.

00:03:44.840 --> 00:03:46.840
And if the 14th is
his birthday,

00:03:46.840 --> 00:03:48.800
what does 25 satisfy?

00:03:48.800 --> 00:03:53.620
Well, 25 now satisfies the odd
criteria, and it satisfies the

00:03:53.620 --> 00:03:56.450
multiple of 5 criteria.

00:03:56.450 --> 00:04:01.080
So this option, choice
B, does not work.

00:04:01.080 --> 00:04:04.175
Because 25 satisfies two
of the criteria.

00:04:04.175 --> 00:04:06.540
And I know what you're saying,
well look, in choice A,

00:04:06.540 --> 00:04:08.400
doesn't 20 satisfy
two criteria?

00:04:08.400 --> 00:04:10.660
Doesn't it satisfy that it could
be a day of the month?

00:04:10.660 --> 00:04:14.330
And what I'm saying is, if we
assume that the 14th is his

00:04:14.330 --> 00:04:17.579
birthday, than 20
does not satisfy

00:04:17.579 --> 00:04:19.630
the day of his birthday.

00:04:19.630 --> 00:04:19.910
Right?

00:04:19.910 --> 00:04:21.769
We're assuming the 14th
is his birthday.

00:04:21.769 --> 00:04:22.750
I want you to think
that about it.

00:04:22.750 --> 00:04:25.570
20 satisfies a potential day of
the month, but if we assume

00:04:25.570 --> 00:04:27.610
his birthday is the 14th,
then obviously

00:04:27.610 --> 00:04:29.800
20th is not the 14th.

00:04:29.800 --> 00:04:31.750
And we could go through
the other choices.

00:04:31.750 --> 00:04:36.920
Choice C says 15-- let
me do it in a neutral

00:04:36.920 --> 00:04:43.770
color-- 15, 18, 16.

00:04:43.770 --> 00:04:47.640
And here, let's say one
of them has to be odd.

00:04:47.640 --> 00:04:50.400
There's only one odd option
here, it's this one.

00:04:50.400 --> 00:04:51.730
One of them has to be
a multiple of 5.

00:04:51.730 --> 00:04:54.100
Well, there's no multiples of
5 here, so we know that C is

00:04:54.100 --> 00:04:56.990
not the right answer.

00:04:56.990 --> 00:05:04.370
D-- 20, 15, 20.

00:05:04.370 --> 00:05:06.760
Well here we have two of the
same number, so they're going

00:05:06.760 --> 00:05:08.570
to satisfy the same criteria.

00:05:08.570 --> 00:05:11.480
So they're each either going
to satisfy two criteria, or

00:05:11.480 --> 00:05:12.470
they're each going
to satisfy none.

00:05:12.470 --> 00:05:14.860
And they actually both satisfy--
you know, there's

00:05:14.860 --> 00:05:19.540
only one odd one here, so one of
the 20's has to satisfy the

00:05:19.540 --> 00:05:21.340
day of the month of
his birthday.

00:05:21.340 --> 00:05:25.980
But if the day of the month of
his birthday is 20th, and then

00:05:25.980 --> 00:05:30.600
we used the multiple of five
as being the 20th as well,

00:05:30.600 --> 00:05:33.470
then both of these numbers
satisfy the day of his

00:05:33.470 --> 00:05:37.120
birthday, and a multiple of 5.

00:05:37.120 --> 00:05:38.940
So that can't be the choice.

00:05:38.940 --> 00:05:43.240
Then E, choice E-- let me do it
in a neutral color again--

00:05:43.240 --> 00:05:47.630
is 34, 30, 21.

00:05:47.630 --> 00:05:50.620
And see 34, it's not odd, it's
not a multiple of 5, and it

00:05:50.620 --> 00:05:52.660
can't be a day of the month of
someone's birthday, so that

00:05:52.660 --> 00:05:54.160
one doesn't work.

00:05:54.160 --> 00:05:56.470
I think these three are
obvious to you,

00:05:56.470 --> 00:05:57.930
why they don't work.

00:05:57.930 --> 00:06:02.150
Here the subtlety is that
this criteria isn't just

00:06:02.150 --> 00:06:03.070
a day of the month.

00:06:03.070 --> 00:06:08.020
The criteria doesn't just say,
any number between 1 and 31.

00:06:08.020 --> 00:06:10.470
It has to be the day of the
month of his birthday.

00:06:10.470 --> 00:06:13.770
So we have to kind of pick, we
have to say, let's assume that

00:06:13.770 --> 00:06:15.250
his birthday is the 14th.

00:06:15.250 --> 00:06:17.630
If his birthday is the 14th,
and this is the day of the

00:06:17.630 --> 00:06:21.390
month of his birthday, then 20
is of course not the day of

00:06:21.390 --> 00:06:22.170
the month of his birthday.

00:06:22.170 --> 00:06:24.000
So it doesn't satisfy
this criterion.

00:06:24.000 --> 00:06:26.080
It only satisfies times 5.

00:06:26.080 --> 00:06:28.160
So that's how you have to think
about it, and I have to

00:06:28.160 --> 00:06:33.140
completely admit that I think
that this is a little bit of a

00:06:33.140 --> 00:06:34.830
too nuanced question.

00:06:34.830 --> 00:06:36.900
Out of the five tests we've done
so far, this is the first

00:06:36.900 --> 00:06:40.250
one that I had to take a little
bit of off time to

00:06:40.250 --> 00:06:41.730
figure out on my own.

00:06:41.730 --> 00:06:43.800
Let's move on to the next
problem, now that we have that

00:06:43.800 --> 00:06:45.050
one out of the way.

00:06:48.450 --> 00:06:50.930
Problem number 14.

00:06:50.930 --> 00:06:54.310
Just hope you don't face that
when you take the SAT.

00:06:54.310 --> 00:07:01.000
Square root of x plus 9
is equal to x minus 3.

00:07:01.000 --> 00:07:06.020
And they say, for all values
of x greater than 3, the

00:07:06.020 --> 00:07:11.770
equation above is equivalent
to which of the following?

00:07:11.770 --> 00:07:12.490
Let's see where this goes.

00:07:12.490 --> 00:07:16.201
So let's square both sides
of this equation.

00:07:16.201 --> 00:07:18.150
Let's square both sides,
just to get rid of

00:07:18.150 --> 00:07:19.580
the square root sign.

00:07:19.580 --> 00:07:21.650
So the left hand side-- I'm
going to switch colors--

00:07:21.650 --> 00:07:24.980
becomes x plus 9.

00:07:24.980 --> 00:07:26.700
And then we just have to
multiply that out, so that

00:07:26.700 --> 00:07:29.950
becomes x squared-- x
times x-- and then

00:07:29.950 --> 00:07:33.510
minus 6x plus 9, right?

00:07:33.510 --> 00:07:36.740
You have x times minus 3 is
minus 3x, and when you

00:07:36.740 --> 00:07:39.390
multiply it again, you
get minus 3x again.

00:07:39.390 --> 00:07:41.100
And if you don't remember how to
do this, you might want to

00:07:41.100 --> 00:07:45.010
just multiply it x minus
3 times x minus 3.

00:07:45.010 --> 00:07:48.050
Going back to this, so
this is the relation.

00:07:48.050 --> 00:07:50.590
And they want us to write
in terms of x.

00:07:50.590 --> 00:07:53.220
So let's subtract 9
from both sides.

00:07:53.220 --> 00:07:57.800
We get x is equal to
x squared minus 6x.

00:07:57.800 --> 00:08:02.100
And that is choice C.

00:08:02.100 --> 00:08:03.790
And we are done.

00:08:03.790 --> 00:08:08.950
And the reason why they said x
is greater than 3 is because

00:08:08.950 --> 00:08:12.090
they actually-- well, they
didn't want you to take the

00:08:12.090 --> 00:08:15.970
negative square root, right?

00:08:15.970 --> 00:08:17.820
They wanted you to make sure
that this is the positive

00:08:17.820 --> 00:08:19.850
square root.

00:08:19.850 --> 00:08:27.350
Next problem, problem 15.

00:08:27.350 --> 00:08:31.240
How many integers in the set of
all integers from 1 to 100

00:08:31.240 --> 00:08:36.140
inclusive are not the square
of an integer?

00:08:36.140 --> 00:08:40.309
So it's 1 to 100 inclusive,
so it includes 1 and 100.

00:08:40.309 --> 00:08:43.320
So what we have to think about
is how many squares are there

00:08:43.320 --> 00:08:45.140
between 1 and 100?

00:08:45.140 --> 00:08:52.560
Well, 100 is 10 squared, 1 is
1 squared, so every number

00:08:52.560 --> 00:08:54.930
between 1 and 10 squared
is going to be

00:08:54.930 --> 00:08:55.790
in this range, right?

00:08:55.790 --> 00:09:00.390
2 squared-- so these are the
numbers whose squares-- 4, 5,

00:09:00.390 --> 00:09:03.900
6, 7, 8, 9, 10-- and
of course, those

00:09:03.900 --> 00:09:05.420
are 10 numbers, right?

00:09:05.420 --> 00:09:08.540
So they're 10 numbers-- you
know, 1 squared is 1, 2

00:09:08.540 --> 00:09:11.030
squared is 4-- all of these
numbers are going to be

00:09:11.030 --> 00:09:13.400
between 1 and 100 inclusive.

00:09:13.400 --> 00:09:15.880
So all of these numbers are
there, there's 10 of them.

00:09:15.880 --> 00:09:17.770
So how many of them are
not a perfect square?

00:09:17.770 --> 00:09:22.340
It's going to be 100 minus 10--
minus these 10 numbers.

00:09:22.340 --> 00:09:25.040
So it's 90.

00:09:25.040 --> 00:09:28.000
Which is choice E.

00:09:28.000 --> 00:09:30.140
I'll see you in the
next video.