SAT Prep: Test 3 Section 2 Part 1

WEBVTT
Kind: captions
Language: en

00:00:01.210 --> 00:00:02.310
Good morning.

00:00:02.310 --> 00:00:04.890
It's the morning after
Thanksgiving and I'm still

00:00:04.890 --> 00:00:08.600
feeling sick with food,
but I will move

00:00:08.600 --> 00:00:10.470
forward and do SAT videos.

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So we are on the third test in
The Official SAT Study Guide

00:00:19.250 --> 00:00:20.990
on page 518.

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This is section two, so it's the
first math section in this

00:00:23.830 --> 00:00:24.840
third test.

00:00:24.840 --> 00:00:30.690
And let's get started with the
problems. The first problem is

00:00:30.690 --> 00:00:41.100
3x plus 9 is equal to 5x plus
1, and they want to know x.

00:00:41.100 --> 00:00:44.270
So this just a straightforward
algebra problem.

00:00:44.270 --> 00:00:46.930
Well, what we can do is, since
we don't have negative

00:00:46.930 --> 00:00:49.370
numbers, let's subtract
1 from both sides.

00:00:49.370 --> 00:00:55.040
So you get 3x plus
8 is equal to 5x.

00:00:55.040 --> 00:01:00.250
Subtract 3x from both sides,
you get 8 is equal to 2x.

00:01:00.250 --> 00:01:05.250
Divide both sides by 2, you get
4 is equal to x, or x is

00:01:05.250 --> 00:01:08.260
equal to 4, choice D.

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Problem number two.

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That was just a straightforward
problem.

00:01:13.680 --> 00:01:14.180
Problem number two.

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They give us a little sequence
here, so let's see, 7, 15, 31,

00:01:25.280 --> 00:01:28.912
63, and it just keeps going.

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And it says, the first term in
the sequence was 7, and each

00:01:32.600 --> 00:01:35.880
term after the first is
determined by multiplying the

00:01:35.880 --> 00:01:40.780
preceding term by m,
and then adding n.

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So we multiplied by m
and then added n.

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OK, what is the value of m?

00:01:47.900 --> 00:01:50.800
Well, you can just look at what
they're doing, right?

00:01:50.800 --> 00:01:53.710
What do you have to multiply
7 by and add to get to 15?

00:01:53.710 --> 00:01:58.430
Well, 7 times 2, plus 1
is equal to 15, right?

00:01:58.430 --> 00:02:03.635
7 times 2, plus 1
is equal to 15.

00:02:03.635 --> 00:02:07.140
And then if you do 15 times
2, plus 1, you get 31.

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15 times 2, plus 1 is equal
to 31, and so forth.

00:02:12.710 --> 00:02:18.950
31 times 2, plus 1 is
equal to 63, right?

00:02:18.950 --> 00:02:23.960
So in general, you're taking--
they're saying, each term

00:02:23.960 --> 00:02:25.625
after the first is determined
by multiplying the preceding

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term by m, and then adding p.

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So we're multiplying by
2 and then adding 1.

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And they want to know what m is,
so we're multiplying by 2.

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So the choice is B.

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Next problem, number three.

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And I'll switch colors
for this one.

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On number three, I went
to a bolder yellow.

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OK, so that was two
tables, they're

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T-shirts, so there's color.

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They have red, white,
and blue.

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And then under size they have
small, medium, large, and

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extra large.

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The table above shows the
different colors and sizes of

00:03:26.310 --> 00:03:29.200
T-shirts that are available at
Independence High School.

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How many different combinations
of color and size

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are possible?

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Well, for each color there
are four sizes, right?

00:03:36.570 --> 00:03:38.820
So you just multiply the number
of colors times the

00:03:38.820 --> 00:03:39.980
number of sizes, right?

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There's four reds, four
whites, four blues.

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So it's 3 times 4 is equal
to 12, choice B.

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Once again, that was just a very
straightforward problem,

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no real tricks there.

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Problem number four.

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We're just going through these
problems. For which of the

00:04:00.610 --> 00:04:04.440
following functions is-- and
they write-- let me clear this

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because this might
take some space.

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OK, so they say, for which
of these functions is the

00:04:10.360 --> 00:04:20.250
following true: f of negative
3 is greater than f of 3?

00:04:20.250 --> 00:04:22.140
And here are their choices.

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So choice a: f of x is
equal to 4x squared.

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Well, in this situation, we
can just look at this.

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f of negative 3 and f of 3 is
going to be the same thing,

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because when you square the x
term, f of negative 3 is going

00:04:42.610 --> 00:04:48.720
to be 4 times 9, which is the
same thing as f of 3, right?

00:04:48.720 --> 00:04:49.810
So that's not going
to help us.

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I got the 9 because I squared
the negative 3, or squared the

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3, so this isn't the
right choice.

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f of 3 is equal to f of
negative 3, choice a.

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Choice b: f of x is going
to be equal to 4.

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Well, once again, this isn't
going to be our choice,

00:05:04.220 --> 00:05:07.300
because f of x is the same
number no matter what, right?

00:05:07.300 --> 00:05:09.770
It's 4, so there's no way that f
of negative 3 is going to be

00:05:09.770 --> 00:05:11.700
greater than f of 3.

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So b is not our answer.

00:05:13.830 --> 00:05:21.880
Choice c says f of x is
equal to 4 over x.

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Well, this has possibility.

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So f of negative 3 is what?

00:05:28.620 --> 00:05:31.620
Minus 4/3.

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And f of 3-- well, no, this
isn't going to work.

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It's going to be positive 4/3,
so actually f of 3 is larger

00:05:37.710 --> 00:05:40.150
than f of minus 3.

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So that doesn't work.

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Choice e: f of x is equal to
4 minus x to the third.

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This has potential, because
f of negative 3 is what?

00:05:56.040 --> 00:06:05.900
It's 4 minus minus 27, which
equals 31, while f of 3 is 4

00:06:05.900 --> 00:06:08.820
minus 27, which equals
minus 24.

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So f of negative 3 is
larger than f of 3.

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So our answer is e.

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Next problem.

00:06:17.280 --> 00:06:19.460
We don't have to do the last
one, because we already found

00:06:19.460 --> 00:06:25.216
one that fits our answer and
that solves our problem.

00:06:25.216 --> 00:06:26.530
Different colors.

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OK, I will switch
to this color.

00:06:28.886 --> 00:06:30.136
OK, five.

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The force required to stretch
a spring beyond its natural

00:06:36.550 --> 00:06:38.580
length is proportional
to how far the

00:06:38.580 --> 00:06:42.470
spring is being stretched.

00:06:42.470 --> 00:06:45.090
And if you've taken physics,
you'd probably see this.

00:06:45.090 --> 00:06:50.050
The force necessary is equal
to some k, right?

00:06:50.050 --> 00:06:53.600
Which is the proportionality
constant times how far you're

00:06:53.600 --> 00:06:53.980
stretching.

00:06:53.980 --> 00:06:55.330
That's all they're
saying, right?

00:06:55.330 --> 00:06:58.490
The force required to stretch
a spring beyond its natural

00:06:58.490 --> 00:07:01.770
strength is proportional--
so that's what k means,

00:07:01.770 --> 00:07:04.670
proportional-- to how far the
spring is being stretched. x

00:07:04.670 --> 00:07:06.240
is how far its being
stretched.

00:07:06.240 --> 00:07:13.475
If a force of 15 pounds-- so
when F-- if a force of 15

00:07:13.475 --> 00:07:19.560
pounds stretches a spring 8
centimeters-- so 15 pounds

00:07:19.560 --> 00:07:23.415
stretches 8 centimeters--
we've got to [? solve ?]

00:07:23.415 --> 00:07:27.420
for length-- what force, in
pounds, is needed to stretch

00:07:27.420 --> 00:07:32.010
the spring 20 centimeters beyond
its natural length?

00:07:32.010 --> 00:07:34.895
So we can just solve for k in
this equation and then apply

00:07:34.895 --> 00:07:36.560
it here, right?

00:07:36.560 --> 00:07:38.270
This is x.

00:07:38.270 --> 00:07:45.840
So 15 is equal to k times
8, or we could say k is

00:07:45.840 --> 00:07:50.970
equal to 15 over 8.

00:07:50.970 --> 00:07:52.700
So we know that k is
equal to 15 over 8.

00:07:52.700 --> 00:07:54.670
And then here we have-- well,
what happens when x

00:07:54.670 --> 00:07:56.150
is 20, what is F?

00:07:56.150 --> 00:08:03.750
Well then, F is equal to k, 15
over 8, times how far this

00:08:03.750 --> 00:08:09.350
thing needs to be stretched
beyond its natural length.

00:08:09.350 --> 00:08:12.540
So 15 over 8 times 20.

00:08:12.540 --> 00:08:13.050
What is this?

00:08:13.050 --> 00:08:16.230
So we can divide the top
and the bottom by 4.

00:08:16.230 --> 00:08:19.380
So this goes 5, 2.

00:08:19.380 --> 00:08:21.436
And what's 5?

00:08:21.436 --> 00:08:23.140
What's 5 times 15?

00:08:23.140 --> 00:08:24.670
5 times 15, 75.

00:08:24.670 --> 00:08:29.250
So it's 75 over 2.

00:08:29.250 --> 00:08:31.560
And that's a strange number, but
it looks like we do have

00:08:31.560 --> 00:08:32.650
choices that match that.

00:08:32.650 --> 00:08:36.990
And that's 37.5, which
is choice E.

00:08:40.090 --> 00:08:41.030
Next problem.

00:08:41.030 --> 00:08:43.205
Actually, I've probably run out
of time, so I will do the

00:08:43.205 --> 00:08:44.520
next problem in the
next video.

00:08:44.520 --> 00:08:45.770
See you soon.


SAT Prep: Test 3 Section 2 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:07.220
We are on number 11.

00:00:07.220 --> 00:00:11.570
Sonia customized her bicycle by
exchanging the front wheel

00:00:11.570 --> 00:00:17.330
for a wheel that had 1/2 the
diameter of the back wheel.

00:00:17.330 --> 00:00:19.140
So this'll be a weird
looking bicycle.

00:00:19.140 --> 00:00:22.510
We're going to have a small
wheel in front, and then--

00:00:22.510 --> 00:00:24.830
let's see if I can
draw a bicycle.

00:00:24.830 --> 00:00:25.610
Oh, it looks like it.

00:00:25.610 --> 00:00:26.510
I don't know.

00:00:26.510 --> 00:00:28.300
Then you have a huge wheel
in the back, right?

00:00:28.300 --> 00:00:30.490
Because the diameter of this
wheel is 1/2 of the diameter

00:00:30.490 --> 00:00:31.940
of that wheel.

00:00:31.940 --> 00:00:35.030
Now when Sonia rides the
bicycle, how many revolutions

00:00:35.030 --> 00:00:37.550
does the front wheel
make for each

00:00:37.550 --> 00:00:40.210
revolution of the back wheel?

00:00:40.210 --> 00:00:42.030
All right.

00:00:42.030 --> 00:00:45.550
When you do this problem, how
many revolutions does the

00:00:45.550 --> 00:00:47.920
front wheel make for each
revolution of the back wheel?

00:00:47.920 --> 00:00:52.140
And let me just make sure it
says that the front wheel has

00:00:52.140 --> 00:00:54.402
1.2 the diameter of
the back wheel.

00:00:54.402 --> 00:00:56.410
So let me ask you a question.

00:00:56.410 --> 00:00:59.620
If the back wheel goes
around-- well, that's

00:00:59.620 --> 00:01:01.550
essentially what it
was just asking.

00:01:01.550 --> 00:01:02.260
Let's make up numbers.

00:01:02.260 --> 00:01:07.260
Let's say that the back wheel's
diameter-- or let's

00:01:07.260 --> 00:01:09.900
say it's radius.

00:01:09.900 --> 00:01:12.380
Let's say the back wheel
has a radius of 2.

00:01:12.380 --> 00:01:13.666
So then, what's its
circumference?

00:01:13.666 --> 00:01:17.670
Its circumference, for the back
wheel, would be 2 pi r.

00:01:17.670 --> 00:01:21.000
So it would be, 2 times
pi times 2, which is

00:01:21.000 --> 00:01:25.220
equal to 4 pi, right?

00:01:25.220 --> 00:01:27.920
The circumference of this has
1/2 the diameter, so it would

00:01:27.920 --> 00:01:28.930
also have 1/2 the radius.

00:01:28.930 --> 00:01:31.010
So the circumference will
just be 2 pi r,

00:01:31.010 --> 00:01:32.420
which would be 2 pi.

00:01:32.420 --> 00:01:34.980
So this one, if you have 1/2
the diameter, you also have

00:01:34.980 --> 00:01:36.540
1/2 the circumference.

00:01:36.540 --> 00:01:44.840
And so when this wheel does 1
revolution, how far has the

00:01:44.840 --> 00:01:47.350
bicycle traveled, assuming
the wheel

00:01:47.350 --> 00:01:49.080
hasn't skidded or anything?

00:01:49.080 --> 00:01:52.355
Well, when this back wheel does
1 revolution, the bicycle

00:01:52.355 --> 00:01:56.850
would have traveled 4 pi,
whatever our units.

00:01:56.850 --> 00:02:01.480
4 pi feet, if that's
our units, right?

00:02:01.480 --> 00:02:04.900
So if the bicycle's traveled
the 4 pi units.

00:02:04.900 --> 00:02:06.460
It's going to travel 4 pi units
on the front of the

00:02:06.460 --> 00:02:07.720
bicycle as well.

00:02:07.720 --> 00:02:11.600
So if we've traveled 4 pi units,
how many times does

00:02:11.600 --> 00:02:15.290
this wheel have to turn around
to travel 4 pi units?

00:02:15.290 --> 00:02:19.440
Well, in one revolution,
this wheel, it's 2 pi.

00:02:19.440 --> 00:02:22.760
So this wheel actually has
to turn around twice.

00:02:22.760 --> 00:02:24.770
It turns around twice
for every 1

00:02:24.770 --> 00:02:27.170
revolution of the big wheel.

00:02:27.170 --> 00:02:28.130
Hopefully that makes sense.

00:02:28.130 --> 00:02:30.340
You just realize that 1
revolution of the big wheel is

00:02:30.340 --> 00:02:31.800
going to be twice the
circumference

00:02:31.800 --> 00:02:32.830
of the front wheel.

00:02:32.830 --> 00:02:36.130
And so the front wheel has to
turn around twice when the

00:02:36.130 --> 00:02:38.660
bike goes over that
same distance.

00:02:38.660 --> 00:02:40.510
So the answer to their
question is, how many

00:02:40.510 --> 00:02:42.490
revolutions does the front wheel
make for each revolution

00:02:42.490 --> 00:02:43.410
of the back wheel?

00:02:43.410 --> 00:02:46.090
Well, the answer is, C.

00:02:46.090 --> 00:02:50.270
That looks like an E, but
the answer is C, 2.

00:02:50.270 --> 00:02:57.090
Problem number 12.

00:02:57.090 --> 00:02:58.270
All right.

00:02:58.270 --> 00:03:00.940
A list of numbers contains
p positive

00:03:00.940 --> 00:03:02.290
and n negative numbers.

00:03:02.290 --> 00:03:07.870
So p pos, n negative.

00:03:07.870 --> 00:03:10.970
If a number is picked at random
from the list, the

00:03:10.970 --> 00:03:17.090
probability that the number
is positive is 3/5.

00:03:17.090 --> 00:03:21.180
What is the value of n/p?

00:03:21.180 --> 00:03:31.920
OK, so the probability of
getting a positive is 3/5.

00:03:31.920 --> 00:03:34.360
So what is the probability
of getting a positive?

00:03:34.360 --> 00:03:36.190
Our probability of getting
a positive is going

00:03:36.190 --> 00:03:37.440
to be equal to p/n?

00:03:41.950 --> 00:03:45.150
That's the probability of the
number of positive numbers we

00:03:45.150 --> 00:03:47.630
have over the total numbers
we have, right?

00:03:47.630 --> 00:03:48.720
That's the probability
of getting a

00:03:48.720 --> 00:03:50.050
positive number, right?

00:03:50.050 --> 00:03:52.460
If this was red marbles and blue
marbles, the probability

00:03:52.460 --> 00:03:55.850
of getting a red marble would be
the number of red over the

00:03:55.850 --> 00:03:57.890
total number of marbles,
right?

00:03:57.890 --> 00:03:58.620
So similar here.

00:03:58.620 --> 00:04:01.540
It's the number of
positive numbers

00:04:01.540 --> 00:04:02.700
over the total numbers.

00:04:02.700 --> 00:04:06.630
And the total numbers are the
positive numbers plus the

00:04:06.630 --> 00:04:08.130
negative numbers.

00:04:08.130 --> 00:04:12.770
And now, they are asking us for
what is the value-- Well

00:04:12.770 --> 00:04:14.070
OK, I just want to make sure
I'm doing it right.

00:04:14.070 --> 00:04:15.725
If the number is picked at
random from the list, the

00:04:15.725 --> 00:04:17.900
probability that the number is
positive-- right, this is the

00:04:17.900 --> 00:04:20.959
probability that's positive
is 3/5-- what is

00:04:20.959 --> 00:04:22.640
the value of n/p?

00:04:22.640 --> 00:04:25.000
So now I'll solve for n/p.

00:04:25.000 --> 00:04:27.510
So we could cross multiply
here, right?

00:04:27.510 --> 00:04:33.820
So we could say 5p is equal
to 3 times this, is

00:04:33.820 --> 00:04:37.790
equal to 3p plus 3n.

00:04:37.790 --> 00:04:40.610
I just cross multiplied.

00:04:40.610 --> 00:04:43.320
And then, let's see, subtract
p from both sides.

00:04:43.320 --> 00:04:47.050
You get 2p is equal to 3n.

00:04:47.050 --> 00:04:51.560
And we want to figure
out n/p, right?

00:04:51.560 --> 00:04:54.320
So let's divide both
sides by p.

00:04:54.320 --> 00:04:58.530
So you get 2 is equal to 3n/p.

00:04:58.530 --> 00:05:00.750
And let's divide both
sides by 3.

00:05:00.750 --> 00:05:05.160
And you get 2/3 is
equal to n/p.

00:05:05.160 --> 00:05:06.550
And we are done.

00:05:06.550 --> 00:05:14.390
n/p is equal to 2/3,
which is choice C.

00:05:14.390 --> 00:05:15.840
Wasn't too bad.

00:05:15.840 --> 00:05:21.390
Problem 13.

00:05:21.390 --> 00:05:23.420
Clear image, invert colors.

00:05:23.420 --> 00:05:26.640
And I will do it in
this tacky green.

00:05:26.640 --> 00:05:30.220
Problem 13.

00:05:30.220 --> 00:05:35.060
The total daily cost, c, in
dollars, of producing x units

00:05:35.060 --> 00:05:36.920
of a certain product is
given by the function.

00:05:36.920 --> 00:05:41.380
All right, we've got a cost
function right here. c of x is

00:05:41.380 --> 00:05:56.150
equal to 600x minus
200 over x plus k.

00:05:56.150 --> 00:05:58.640
And they're telling us
k is a constant.

00:05:58.640 --> 00:06:03.710
And x is less than,
or equal to, 100.

00:06:03.710 --> 00:06:04.520
Fair enough.

00:06:04.520 --> 00:06:08.790
If 20 units were produced
yesterday for a total cost of

00:06:08.790 --> 00:06:10.970
$640, what is the value of k?

00:06:10.970 --> 00:06:12.430
So 20 units.

00:06:12.430 --> 00:06:16.910
So they're essentially saying
that c of 20, the cost of

00:06:16.910 --> 00:06:21.930
producing 20 units,
is equal to $640.

00:06:21.930 --> 00:06:23.660
So we can just use this
information to

00:06:23.660 --> 00:06:24.660
now solve for k.

00:06:24.660 --> 00:06:30.950
So c of 20 is going to be
equal-- put in 20 to this

00:06:30.950 --> 00:06:37.520
equation-- so it's 600
times 20 minus 200.

00:06:37.520 --> 00:06:39.870
All of that over 20, right?

00:06:39.870 --> 00:06:41.150
C of 20, right?

00:06:41.150 --> 00:06:44.120
All of that over 20.

00:06:44.120 --> 00:06:44.810
Plus k.

00:06:44.810 --> 00:06:48.400
I can't forget the plus k-- plus
k is equal to-- we know

00:06:48.400 --> 00:06:50.870
that equals to 640, right?

00:06:50.870 --> 00:06:53.150
And what is this equal to?

00:06:53.150 --> 00:06:54.920
Well, we actually don't even
have to multiply it, right?

00:06:54.920 --> 00:06:56.880
Because this is 600 times 20.

00:06:56.880 --> 00:06:58.060
And what's 200?

00:06:58.060 --> 00:07:01.880
200 is 10 times 20, right?

00:07:01.880 --> 00:07:04.390
So you can divide the top
and the bottom by 20.

00:07:04.390 --> 00:07:06.950
So you have 20, 20, 20.

00:07:06.950 --> 00:07:09.850
So you're just left with 600.

00:07:09.850 --> 00:07:11.290
I mean, you could multiply it
out if you don't understand

00:07:11.290 --> 00:07:12.710
what I just did.

00:07:12.710 --> 00:07:20.770
But you get 600 minus 10
plus k is equal to 640.

00:07:20.770 --> 00:07:25.600
So you get 590 plus
k is equal to 640.

00:07:25.600 --> 00:07:26.970
Subtract 590 from both sides.

00:07:26.970 --> 00:07:29.732
You get k is equal to 50.

00:07:29.732 --> 00:07:33.476
And that is choice B.

00:07:33.476 --> 00:07:41.210
So you really just have
to substitute the x

00:07:41.210 --> 00:07:42.850
into our cost function.

00:07:42.850 --> 00:07:45.120
And then set that
equal to $640.

00:07:45.120 --> 00:07:46.930
And you've solved for k.

00:07:46.930 --> 00:07:54.830
Problem 14.

00:07:54.830 --> 00:08:04.740
For how many ordered pairs of
positive integers x, y is 2x

00:08:04.740 --> 00:08:11.270
plus 3y less than 6?

00:08:11.270 --> 00:08:15.340
So these are positive
integers, right?

00:08:15.340 --> 00:08:16.160
We could figure them out.

00:08:16.160 --> 00:08:17.640
The 0, we can't use 0.

00:08:23.300 --> 00:08:28.060
If x is 1, what can y equal?

00:08:28.060 --> 00:08:32.970
So then we have 2 plus
3y is less than 6.

00:08:32.970 --> 00:08:34.990
This is a case of
x is equal to 1.

00:08:34.990 --> 00:08:39.460
So then we'd have 3y
is less than 4.

00:08:39.460 --> 00:08:42.470
So we'd have y would have
to be less than 4/3.

00:08:42.470 --> 00:08:45.200
So the only situation where y
is less than 4/3 is y would

00:08:45.200 --> 00:08:46.510
have to be 1, right?

00:08:46.510 --> 00:08:47.440
Because that's the
only positive

00:08:47.440 --> 00:08:48.900
integer less than 4/3.

00:08:48.900 --> 00:08:54.440
If x is 2, then we have
4 plus 3y, right?

00:08:54.440 --> 00:08:58.040
2 times 2 is less than 6.

00:08:58.040 --> 00:09:02.770
And so you have 3y
is less than 2.

00:09:02.770 --> 00:09:05.330
y is less than 2/3.

00:09:05.330 --> 00:09:10.170
Well, I don't see any positive
integers that

00:09:10.170 --> 00:09:11.030
are less than 2/3.

00:09:11.030 --> 00:09:15.910
So x can't even be equal
to 2, right?

00:09:15.910 --> 00:09:18.040
So neither of these numbers
can get any

00:09:18.040 --> 00:09:19.410
bigger than 1 or 1.

00:09:19.410 --> 00:09:22.800
So there's only 1 ordered pair
of positive integers that--

00:09:22.800 --> 00:09:25.680
positive integers, and that's
the trick-- that satisfy this

00:09:25.680 --> 00:09:27.280
equation, 1 and 1.

00:09:27.280 --> 00:09:28.660
And we know, because
immediately, when we just

00:09:28.660 --> 00:09:31.920
raise one of them above 2, we
get above 6, no matter what.

00:09:31.920 --> 00:09:34.340
We can't find a positive integer
that satisfies it.

00:09:34.340 --> 00:09:34.890
So that's it.

00:09:34.890 --> 00:09:36.210
It's just choice A.

00:09:36.210 --> 00:09:38.320
Not too bad.

00:09:38.320 --> 00:09:41.000
And I will see you in
the next video.


SAT Prep: Test 3 Section 2 Part 4

WEBVTT
Kind: captions
Language: en

00:00:01.220 --> 00:00:08.285
Problem number 15, and we
have two triangles.

00:00:08.285 --> 00:00:13.350
So let's see, we have one
triangle here, one

00:00:13.350 --> 00:00:14.600
triangle like this.

00:00:17.200 --> 00:00:20.490
We have another triangle that
looks fairly similar.

00:00:20.490 --> 00:00:21.760
I don't know if they actually
are similar.

00:00:21.760 --> 00:00:24.350
I haven't read the
problem yet.

00:00:24.350 --> 00:00:26.390
At least in the drawing, they've
drawn them to look

00:00:26.390 --> 00:00:28.710
fairly similar.

00:00:28.710 --> 00:00:32.360
Then they tell us that
this is triangle ABC.

00:00:35.580 --> 00:00:36.830
This is triangle DEF.

00:00:42.470 --> 00:00:44.400
We know that this is 8.

00:00:44.400 --> 00:00:45.390
This is 5.

00:00:45.390 --> 00:00:47.780
This is x degrees.

00:00:47.780 --> 00:00:51.720
This is also x degrees.

00:00:51.720 --> 00:00:55.860
This is 5.

00:00:55.860 --> 00:01:00.000
Then they tell us that this
is y degrees, and

00:01:00.000 --> 00:01:03.020
that this is y degrees.

00:01:03.020 --> 00:01:03.860
Now let's read the problem.

00:01:03.860 --> 00:01:11.680
If y is equal to 60 in triangle
DEF above, how much

00:01:11.680 --> 00:01:17.310
greater is the perimeter of ABC
than the perimeter of DEF?

00:01:17.310 --> 00:01:29.370
So y is equal to 60, right?

00:01:29.370 --> 00:01:32.770
So what is this angle up here?

00:01:32.770 --> 00:01:38.060
Well if we know that these two
are the base angles-- well

00:01:38.060 --> 00:01:40.240
actually, we know 60, 60.

00:01:40.240 --> 00:01:41.770
So what's this top angle
going to be?

00:01:41.770 --> 00:01:43.320
Well this is an equilateral
triangle, right?

00:01:43.320 --> 00:01:45.250
This also has to
be equal to 60.

00:01:45.250 --> 00:01:47.010
My brain's malfunctioning.

00:01:47.010 --> 00:01:49.490
So if all the angles are the
same, then all the sides are

00:01:49.490 --> 00:01:51.890
the same as well, right?

00:01:51.890 --> 00:01:53.700
It's 5 and 5.

00:01:53.700 --> 00:01:55.235
That's an equilateral
triangle.

00:01:55.235 --> 00:02:00.650
Then the perimeter is 5
plus 5 plus 5, is 15.

00:02:00.650 --> 00:02:04.550
This problem, we don't know the
x angles, but we know that

00:02:04.550 --> 00:02:05.180
they're equal.

00:02:05.180 --> 00:02:07.770
So if these two base angles
are equal-- we know from

00:02:07.770 --> 00:02:09.669
geometry class that,
of course-- the

00:02:09.669 --> 00:02:11.700
sides are also equal.

00:02:11.700 --> 00:02:14.380
So if this side is 8,
this side is also 8.

00:02:14.380 --> 00:02:16.520
So the perimeter here is what?

00:02:16.520 --> 00:02:20.220
8 plus 8 is 16 plus 5 is 21.

00:02:20.220 --> 00:02:23.190
They want to know how much
greater is the perimeter of

00:02:23.190 --> 00:02:24.770
this triangle than
this triangle?

00:02:24.770 --> 00:02:28.230
So 21 minus 15 is 6.

00:02:28.230 --> 00:02:32.100
That is choice C.

00:02:32.100 --> 00:02:34.900
Next problem.

00:02:34.900 --> 00:02:35.360
Right?

00:02:35.360 --> 00:02:36.610
Yeah, choice C.

00:02:39.560 --> 00:02:46.330
Problem number 16, if x and y
are positive, consecutive, odd

00:02:46.330 --> 00:02:55.630
integers where y is greater than
x, which of the following

00:02:55.630 --> 00:03:03.200
is equal to y squared
minus x squared?

00:03:03.200 --> 00:03:08.510
So x and y are positive,
consecutive, odd integers, and

00:03:08.510 --> 00:03:09.690
y is greater than x.

00:03:09.690 --> 00:03:13.480
So that means that y is
equal to x plus what?

00:03:13.480 --> 00:03:16.020
It's equal to x plus 2.

00:03:16.020 --> 00:03:19.800
If they were consecutive just
integers, then y would be

00:03:19.800 --> 00:03:20.910
equal to x plus 1.

00:03:20.910 --> 00:03:22.340
It would be 1 more than x.

00:03:22.340 --> 00:03:26.010
But since we know that they're
both odd, and y is the next

00:03:26.010 --> 00:03:28.640
larger odd number-- the
difference between any two odd

00:03:28.640 --> 00:03:32.220
numbers is 2-- so y is going
to be 2 more than x.

00:03:32.220 --> 00:03:35.540
If x is 5, then y is 7.

00:03:35.540 --> 00:03:39.600
Consecutive, odd integers,
if x is 3, y is 5.

00:03:39.600 --> 00:03:41.280
y is 2 more than x.

00:03:41.280 --> 00:03:43.790
So we know y is equal
to x plus 2.

00:03:43.790 --> 00:03:46.240
So they want us to kind of
rewrite this expression.

00:03:46.240 --> 00:03:50.370
Well we know from algebra class
we can factor this as y

00:03:50.370 --> 00:03:58.810
plus x times y minus x, right?

00:03:58.810 --> 00:04:01.970
Then what is y plus x?

00:04:01.970 --> 00:04:03.030
Actually, we don't even
have to do this.

00:04:03.030 --> 00:04:04.440
We don't even have
to factor it.

00:04:04.440 --> 00:04:06.620
We know what y is
equal to, right?

00:04:06.620 --> 00:04:07.660
y is equal to x plus 2.

00:04:07.660 --> 00:04:13.410
So that is equal to x plus 2
squared minus x squared.

00:04:13.410 --> 00:04:16.000
This is equal to-- we just
multiply it out-- x squared

00:04:16.000 --> 00:04:19.110
plus 4x plus 4.

00:04:19.110 --> 00:04:20.170
That's this term.

00:04:20.170 --> 00:04:22.150
Now we minus x squared.

00:04:22.150 --> 00:04:27.350
So this cancels out with this,
and we're left with 4x plus 4,

00:04:27.350 --> 00:04:30.810
which is choice E.

00:04:30.810 --> 00:04:32.740
Whenever I see something like
this, my first temptation is

00:04:32.740 --> 00:04:34.610
to factor it out.

00:04:34.610 --> 00:04:36.010
You could have done it, and
then you could have

00:04:36.010 --> 00:04:36.770
substituted it back in.

00:04:36.770 --> 00:04:38.700
But you didn't even
have to do that.

00:04:38.700 --> 00:04:44.200
Image: clear image,
invert colors.

00:04:44.200 --> 00:04:47.220
It's actually 6:30 in the
morning after Thanksgiving.

00:04:47.220 --> 00:04:49.870
You've got to appreciate
my discipline.

00:04:49.870 --> 00:04:55.660
All right, problem number 17.

00:04:55.660 --> 00:04:58.110
My wife's actually a medical
resident and she had to work

00:04:58.110 --> 00:05:00.760
36 hours yesterday.

00:05:00.760 --> 00:05:03.170
Then she came home, essentially
passed out, missed

00:05:03.170 --> 00:05:05.730
Thanksgiving, everything,
and then she just

00:05:05.730 --> 00:05:06.500
went to work again.

00:05:06.500 --> 00:05:08.710
She has like a 12-hour
shift, it's crazy.

00:05:08.710 --> 00:05:11.690
She only has a year left and
I look forward to that.

00:05:11.690 --> 00:05:14.460
But there might be fewer
math videos once I

00:05:14.460 --> 00:05:15.550
have my wife back.

00:05:15.550 --> 00:05:17.050
But anyway, back to Math.

00:05:17.050 --> 00:05:21.210
Problem number 17, in the xy
plane, line l passes through

00:05:21.210 --> 00:05:24.440
the origin and is perpendicular
to the line.

00:05:24.440 --> 00:05:33.895
So l is perpendicular to the
line 4x plus y is equal to k,

00:05:33.895 --> 00:05:41.190
and l passes through the origin,
through 0, 0-- that's

00:05:41.190 --> 00:05:45.340
the origin-- where
k is a constant.

00:05:45.340 --> 00:05:53.220
If the two lines intersect at
the point t, t plus 1, what is

00:05:53.220 --> 00:05:54.720
the value of t?

00:05:54.720 --> 00:05:56.920
So first of all, we
know this line is

00:05:56.920 --> 00:06:00.120
perpendicular to this line.

00:06:00.120 --> 00:06:02.390
So what does perpendicular
tell us?

00:06:02.390 --> 00:06:09.240
Perpendicular says whatever the
slope was of this line,

00:06:09.240 --> 00:06:12.680
the slope of this line would
be the negative inverse.

00:06:12.680 --> 00:06:15.360
So let's figure out what the
slope of this line is.

00:06:15.360 --> 00:06:20.730
So we know that 4x plus
y is equal to k.

00:06:20.730 --> 00:06:24.540
So y is equal to minus
4x plus k.

00:06:24.540 --> 00:06:31.160
Now this is kind of familiar,
y equals mx plus b format.

00:06:31.160 --> 00:06:33.150
So the slope of this
line is minus 4.

00:06:33.150 --> 00:06:38.360
The slope equals minus 4.

00:06:38.360 --> 00:06:40.730
So the slope of this
line is minus 4.

00:06:40.730 --> 00:06:42.170
So what is the slope of l?

00:06:47.080 --> 00:06:51.870
Well l is perpendicular
to this line.

00:06:51.870 --> 00:06:55.270
So we know that l is going to
have the negative inverse of

00:06:55.270 --> 00:06:56.130
this slope.

00:06:56.130 --> 00:07:00.330
So the negative inverse of minus
4 is-- well it'll become

00:07:00.330 --> 00:07:01.840
positive, and you take
the inverse.

00:07:01.840 --> 00:07:04.520
The slope of l is 1/4.

00:07:04.520 --> 00:07:07.710
We also know that l goes
through the origin.

00:07:07.710 --> 00:07:11.450
So its y-intercept is 0.

00:07:11.450 --> 00:07:13.630
So we know the equation of l.

00:07:13.630 --> 00:07:21.030
We know that the equation of
l is y is equal to 1/4 x.

00:07:21.030 --> 00:07:24.930
We know there's no plus b,
because it goes through 0, 0.

00:07:24.930 --> 00:07:28.700
If you didn't believe me, you
can put 0, 0 in here and then

00:07:28.700 --> 00:07:29.160
solve for b.

00:07:29.160 --> 00:07:30.960
You'll get 0 equals 0 plus b.

00:07:30.960 --> 00:07:32.180
So b is 0.

00:07:32.180 --> 00:07:34.500
But anyway, so this is l.

00:07:34.500 --> 00:07:37.070
We've solved for the
equation for l.

00:07:37.070 --> 00:07:41.710
Now they tell us that
we intersect--

00:07:41.710 --> 00:07:42.490
where was the problem?

00:07:42.490 --> 00:07:44.030
I lost track of the page--
we intersect at the

00:07:44.030 --> 00:07:45.580
point t, t plus 1.

00:07:45.580 --> 00:07:47.490
They want to figure
out what t is.

00:07:47.490 --> 00:07:49.720
So when x is t, y is t plus 1.

00:07:49.720 --> 00:07:54.890
So let's see, so y is
t plus 1, so t plus

00:07:54.890 --> 00:07:58.920
1 is equal to 1/4.

00:07:58.920 --> 00:08:00.170
Then, x is t.

00:08:03.180 --> 00:08:07.840
I just substituted t and
t plus 1 for x and y.

00:08:07.840 --> 00:08:09.240
Now we just solve.

00:08:09.240 --> 00:08:13.390
Let's see, if we subtract 1/4
t from both sides-- let me

00:08:13.390 --> 00:08:16.750
finish this problem up
here-- we subtract

00:08:16.750 --> 00:08:18.730
1/4 t from both sides.

00:08:18.730 --> 00:08:24.290
We get 3/4 t plus
1 is equal to 0.

00:08:24.290 --> 00:08:27.330
I just subtracted 1/4
t from this 1t.

00:08:27.330 --> 00:08:29.070
So you get 3/4 t.

00:08:29.070 --> 00:08:32.130
Subtract 1 from both sides,
you get 3/4 t is

00:08:32.130 --> 00:08:33.850
equal to minus 1.

00:08:33.850 --> 00:08:38.650
Then you just multiply both
sides times the inverse, 4/3.

00:08:38.650 --> 00:08:40.929
So t-- this cancels out,
of course-- t is

00:08:40.929 --> 00:08:44.870
equal to minus 4/3.

00:08:44.870 --> 00:08:46.620
What is my choice there?

00:08:46.620 --> 00:08:47.450
Oh, there I go.

00:08:47.450 --> 00:08:49.200
Choice number A.

00:08:49.200 --> 00:08:49.540
I got scared.

00:08:49.540 --> 00:08:51.240
I looked at the wrong problem,
I didn't see my choice.

00:08:51.240 --> 00:08:52.630
I thought maybe I
did it wrong.

00:08:52.630 --> 00:08:54.120
Choice number A.

00:08:54.120 --> 00:08:57.490
And I will do problem 18
in the next video.

00:08:57.490 --> 00:08:59.130
I'll see you soon.


Test 3 Section 2 Part 5

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:07.440
Problem number 18.

00:00:07.440 --> 00:00:11.610
The average arithmetic
mean of x and y is k.

00:00:11.610 --> 00:00:16.140
So the average, so x plus
y/2 is k, right?

00:00:16.140 --> 00:00:19.000
That's the average of x
and y is equal to k.

00:00:19.000 --> 00:00:22.200
Which of the following is the
average of x, y and z?

00:00:22.200 --> 00:00:27.400
So they want to know what x plus
y-- so the average of x,

00:00:27.400 --> 00:00:29.115
y, and z is, you add up
all the numbers and

00:00:29.115 --> 00:00:31.550
you divide by 3.

00:00:31.550 --> 00:00:33.640
They want to know in
terms of k and z.

00:00:33.640 --> 00:00:37.240
So if we could substitute for
x plus y, then we'd be done.

00:00:37.240 --> 00:00:42.580
So multiplying this equation by
2, we get x plus y is equal

00:00:42.580 --> 00:00:43.750
to 2k, right?

00:00:43.750 --> 00:00:45.950
Just multiply both sides by 2.

00:00:45.950 --> 00:00:47.660
And then x plus y is 2k.

00:00:47.660 --> 00:00:52.610
So let's just substitute
2k here for x plus y.

00:00:52.610 --> 00:00:57.410
So then you just get
2k plus z/3.

00:00:57.410 --> 00:01:06.230
That's the average of x, y
and z: 2k plus z/3, and

00:01:06.230 --> 00:01:07.480
that is choice A.

00:01:10.610 --> 00:01:17.870
Problem 19.

00:01:17.870 --> 00:01:20.070
We have a neat drawing here.

00:01:20.070 --> 00:01:23.386
Let me see if I can draw this.

00:01:23.386 --> 00:01:31.000
Let's see, we have a circle,
and then we have a triangle

00:01:31.000 --> 00:01:32.440
inside of that.

00:01:32.440 --> 00:01:35.070
Well it's not really inside of
it, it kind of does something

00:01:35.070 --> 00:01:37.980
like-- let's see, the triangle
goes like that.

00:01:37.980 --> 00:01:44.800
And the other line kind of
goes-- Oh, oh, close enough.

00:01:44.800 --> 00:01:45.410
Close enough.

00:01:45.410 --> 00:01:46.725
I think you get the point.

00:01:46.725 --> 00:01:49.260
It actually kind of touches
at the bottom there.

00:01:49.260 --> 00:01:51.930
This should be touching
at the bottom.

00:01:51.930 --> 00:01:53.800
And then we have-- oh, we
have another line that

00:01:53.800 --> 00:01:56.940
goes down like this.

00:01:56.940 --> 00:01:58.340
All right.

00:01:58.340 --> 00:01:59.590
OK.

00:02:01.430 --> 00:02:10.580
This is O, Y, X, W, Z.

00:02:10.580 --> 00:02:11.460
This is W.

00:02:11.460 --> 00:02:14.480
We can do a better
color on W, OK?

00:02:14.480 --> 00:02:16.860
That's O, V, all right.

00:02:16.860 --> 00:02:20.350
In the figure above, triangle
X, Y, Z is equilateral--

00:02:20.350 --> 00:02:24.130
that's good to know-- with
side of length 2.

00:02:24.130 --> 00:02:26.490
So this side is 2, this
side is 2, and this

00:02:26.490 --> 00:02:28.710
whole side is 2.

00:02:28.710 --> 00:02:33.500
If w, y is the diameter of the
circle with center O-- so w,

00:02:33.500 --> 00:02:37.360
y, this is the diameter of the
circle with center O-- then

00:02:37.360 --> 00:02:41.570
the area of the circle
is-- fascinating.

00:02:41.570 --> 00:02:45.610
So essentially, we just need to
figure out what the height

00:02:45.610 --> 00:02:48.810
of this line is, because if we
know the length of this, we

00:02:48.810 --> 00:02:49.520
can figure out the area.

00:02:49.520 --> 00:02:52.290
Because that's the diameter,
we'd know the radius, and we'd

00:02:52.290 --> 00:02:53.270
use pi r squared.

00:02:53.270 --> 00:02:55.090
So how do we figure that out?

00:02:55.090 --> 00:02:59.210
Well, this is an equilateral
triangle, and so that means

00:02:59.210 --> 00:03:00.500
all of the angles
are 60 degrees.

00:03:00.500 --> 00:03:05.200
So that's 60, this
is 60 degrees.

00:03:05.200 --> 00:03:08.450
And now this whole angle up here
is going to be 60, but we

00:03:08.450 --> 00:03:11.710
split it exactly
in two, right?

00:03:11.710 --> 00:03:16.100
So this angle is going to be 30
degrees and this angle is

00:03:16.100 --> 00:03:16.770
30 degrees.

00:03:16.770 --> 00:03:18.840
And if you don't believe me,
how else do we know that?

00:03:18.840 --> 00:03:21.450
Well they tell us that this
is a right angle, right?

00:03:21.450 --> 00:03:23.610
This is a right angle, this
is a 90 degree angle.

00:03:23.610 --> 00:03:25.640
So that's how we know this was
actually split in two, because

00:03:25.640 --> 00:03:29.040
this is 90, this is 60, and they
have to add up to 180.

00:03:29.040 --> 00:03:31.410
So this angle up here is 30.

00:03:31.410 --> 00:03:34.430
So I think you are going to see
now that this is a classic

00:03:34.430 --> 00:03:39.710
30, 60, 90 triangle, right?

00:03:39.710 --> 00:03:43.470
If this side up here is 2, then
we know that the side

00:03:43.470 --> 00:03:47.100
opposite the 30 degree side
is 1/2 the hypotenuse.

00:03:47.100 --> 00:03:51.030
This side is the hypotenuse
right here.

00:03:51.030 --> 00:03:52.980
The corresponding side to the
30 degree side, or the

00:03:52.980 --> 00:03:55.580
opposite to the 30 degree side,
is 1/2 the hypotenuse.

00:03:55.580 --> 00:03:57.100
So this is 2, this is 1.

00:03:57.100 --> 00:04:00.120
And that also makes sense,
because this is a symmetric

00:04:00.120 --> 00:04:05.110
triangle, and this w is the
midpoint, because it's

00:04:05.110 --> 00:04:07.070
perpendicular and these
angles the same.

00:04:07.070 --> 00:04:08.110
So this would also be 1.

00:04:08.110 --> 00:04:10.650
So 1, and 1, and 2,
1 plus 1 is 2.

00:04:10.650 --> 00:04:11.870
So that makes sense as well.

00:04:11.870 --> 00:04:13.850
And then we know-- and if you
forget this, they actually

00:04:13.850 --> 00:04:17.730
give you the formula at the
beginning of every SAT exam--

00:04:17.730 --> 00:04:21.610
that the side opposite the 60
degree side-- which is the

00:04:21.610 --> 00:04:25.040
side we're trying to figure
out-- that square root of 3

00:04:25.040 --> 00:04:29.650
times this side-- so square root
of 3 times this is what?

00:04:29.650 --> 00:04:33.630
Well it's square root of
3 times 1, which is

00:04:33.630 --> 00:04:33.940
square root of 3.

00:04:33.940 --> 00:04:36.610
And if you don't believe this,
you could try the Pythagorean

00:04:36.610 --> 00:04:37.190
theorem, right?

00:04:37.190 --> 00:04:42.980
Square root of 3 squared plus 1
squared is equal to-- well 3

00:04:42.980 --> 00:04:47.330
plus 1, which is equal to 4,
which is equal to 2 squared.

00:04:47.330 --> 00:04:48.390
Which is equal to hypotenuse
squared.

00:04:48.390 --> 00:04:51.005
So we figured out this diameter
right here is

00:04:51.005 --> 00:04:51.660
square root of 3.

00:04:51.660 --> 00:04:55.320
The whole thing is
square root of 3.

00:04:55.320 --> 00:04:57.425
If the whole thing is square
root of 3, what is the radius?

00:04:57.425 --> 00:04:59.990
The radius is going
to be 1/2 of that.

00:04:59.990 --> 00:05:02.540
The radius is 1/2 of that,
so the radius is

00:05:02.540 --> 00:05:05.030
square root of 3/2.

00:05:05.030 --> 00:05:07.025
And now we're ready to
figure out the area.

00:05:07.025 --> 00:05:08.540
Area is equal to pi r squared.

00:05:08.540 --> 00:05:09.730
And they give you this
formula too at the

00:05:09.730 --> 00:05:10.970
beginning of the SAT.

00:05:10.970 --> 00:05:13.820
So area is equal
to pi times the

00:05:13.820 --> 00:05:16.495
square root of 3/2 squared.

00:05:16.495 --> 00:05:19.790
And that equals pi times square
root of 3 squared is

00:05:19.790 --> 00:05:24.830
3/-- and then 2 squared
is 4, so 3/4 pi.

00:05:24.830 --> 00:05:28.620
And 3/4 pi is choice C.

00:05:32.680 --> 00:05:36.190
Image, clear image, invert
colors, all right.

00:05:36.190 --> 00:05:38.270
Problem 20.

00:05:38.270 --> 00:05:39.950
We're already at the end
of this section.

00:05:39.950 --> 00:05:45.170
Problem 20, when 15 is divided
by the positive integer k, the

00:05:45.170 --> 00:05:46.130
remainder 3.

00:05:46.130 --> 00:05:54.020
So 15 divided by k has
a remainder of 3.

00:05:54.020 --> 00:05:55.270
I'm lazy.

00:05:57.800 --> 00:06:01.180
For how many different values
of k is this true?

00:06:01.180 --> 00:06:07.650
So when 15 is divided by k,
you have a remainder of 3.

00:06:07.650 --> 00:06:08.900
Interesting, OK.

00:06:13.900 --> 00:06:20.590
Actually the easiest thing to do
is say well, 15 divided by

00:06:20.590 --> 00:06:25.800
k is going to be, essentially--
Well the easiest

00:06:25.800 --> 00:06:27.660
way-- I'm just trying to think
of whether we should just do

00:06:27.660 --> 00:06:28.500
it by brute force.

00:06:28.500 --> 00:06:32.730
We can actually just do it by
brute force, because it says k

00:06:32.730 --> 00:06:33.950
is a positive integer, right?

00:06:33.950 --> 00:06:35.460
So what values of k work?

00:06:35.460 --> 00:06:38.230
Well does k equal 1 work?

00:06:38.230 --> 00:06:40.520
When 15 is divided by 1,
there's no remainder.

00:06:40.520 --> 00:06:43.030
So k 1 does not work.

00:06:43.030 --> 00:06:45.000
k equals 2?

00:06:45.000 --> 00:06:48.720
Actually, let's just do
it. k remainder 1, 2,

00:06:48.720 --> 00:06:52.500
3, 4, 5, 6, 7, right?

00:06:52.500 --> 00:06:54.930
It's going to go all
the way up to 15.

00:06:54.930 --> 00:06:56.000
You know, we could
keep trying.

00:06:56.000 --> 00:06:59.260
So when 1, the remainder is 0.

00:06:59.260 --> 00:07:03.090
2 goes into 15 7 times with
a remainder of 1.

00:07:03.090 --> 00:07:06.240
3 goes into 15 with
a remainder of 0.

00:07:06.240 --> 00:07:11.690
4 goes into 15 with a remainder
of 3, right?

00:07:11.690 --> 00:07:16.600
Because 4 gets to 12, and then
you have a remainder of 3.

00:07:16.600 --> 00:07:18.750
5 has a remainder of 0.

00:07:18.750 --> 00:07:22.535
6, once again, 6 gets to
12, and then it has a

00:07:22.535 --> 00:07:25.550
remainder of 3.

00:07:25.550 --> 00:07:27.070
So 6 works.

00:07:27.070 --> 00:07:30.260
So 4 and 6 work.

00:07:30.260 --> 00:07:35.210
7 has a remainder of 1, right?

00:07:35.210 --> 00:07:38.030
7, 14 remainder of 1.

00:07:38.030 --> 00:07:44.290
And then if you just did 8,
9, 10, 11, 12, 13, 8 has a

00:07:44.290 --> 00:07:45.780
remainder of 7.

00:07:45.780 --> 00:07:47.580
9 has a remainder of 6.

00:07:47.580 --> 00:07:48.760
10 has a remainder of 5.

00:07:48.760 --> 00:07:52.590
11 has a remainder
of 4, right?

00:07:52.590 --> 00:07:54.580
15 divided by 11 is
1, remainder of 4.

00:07:54.580 --> 00:07:58.870
12 has a remainder of
3, so 12 works.

00:07:58.870 --> 00:08:00.190
13 has a remainder of 2.

00:08:00.190 --> 00:08:01.400
14 has a remainder of 1.

00:08:01.400 --> 00:08:03.370
And then 15 has no
remainder again.

00:08:03.370 --> 00:08:09.570
So the numbers that work
for k are 12, 6, and 4.

00:08:09.570 --> 00:08:16.730
So it was essentially the
factors of-- well all right, I

00:08:16.730 --> 00:08:17.960
can't actually say that.

00:08:17.960 --> 00:08:21.960
Well, we just did it by brute
force and those are the 3 k's

00:08:21.960 --> 00:08:23.520
that work for this problem.

00:08:23.520 --> 00:08:26.620
So the answer is choice C.

00:08:26.620 --> 00:08:28.800
I'm trying to think if there's
a faster, elegant

00:08:28.800 --> 00:08:29.410
way of doing this.

00:08:29.410 --> 00:08:32.309
But this actually didn't
take too much time.

00:08:32.309 --> 00:08:34.390
But anyway, that's all
for this section.

00:08:34.390 --> 00:08:37.700
I'll see you in the
next SAT section.


SAT Prep: Test 2 Section 9 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:07.430
We are on problem number
10 on page 489.

00:00:07.430 --> 00:00:12.400
Stacy noted that she is both the
12th tallest and the 12th

00:00:12.400 --> 00:00:14.510
shortest student in her class.

00:00:14.510 --> 00:00:16.580
So let me write 12th down.

00:00:16.580 --> 00:00:20.050
She's the 12th tallest and
shortest. If everyone in the

00:00:20.050 --> 00:00:22.490
class is a different height,
how many students are

00:00:22.490 --> 00:00:23.700
there in the class?

00:00:23.700 --> 00:00:26.310
And these are the type of
questions that I just like to

00:00:26.310 --> 00:00:29.470
make sure that I'm
not miscounting.

00:00:29.470 --> 00:00:32.940
So she's the 12th tallest.
So what does that mean?

00:00:32.940 --> 00:00:39.110
That means that there are 11
people taller than her.

00:00:39.110 --> 00:00:47.460
So let's say from shortest to
tallest. So this is Stacy

00:00:47.460 --> 00:00:50.650
right here.

00:00:50.650 --> 00:00:54.150
And so, if she is the
12th tallest-- let

00:00:54.150 --> 00:00:54.890
me put it this way.

00:00:54.890 --> 00:00:58.220
If I said that she was the
first tallest, that means

00:00:58.220 --> 00:00:59.420
there's no one taller
than her.

00:00:59.420 --> 00:01:02.510
If she's the second tallest,
that means there's one person

00:01:02.510 --> 00:01:03.440
taller than her.

00:01:03.440 --> 00:01:06.560
If she's the third tallest, that
means there's two people

00:01:06.560 --> 00:01:07.660
taller than her.

00:01:07.660 --> 00:01:10.580
And likewise, if she's the 12th
tallest, that means that

00:01:10.580 --> 00:01:16.240
there are 11 people taller.

00:01:16.240 --> 00:01:20.930
Similarly, if she's the 12th
shortest, well, that must mean

00:01:20.930 --> 00:01:22.890
that there are 11 shorter
than her.

00:01:22.890 --> 00:01:26.250
So this is really just making
sure that you parse the words

00:01:26.250 --> 00:01:27.290
correctly and count properly.

00:01:27.290 --> 00:01:29.820
So if there's 11 shorter, 11
taller, and then there's

00:01:29.820 --> 00:01:32.900
Stacy, well, that's
22 plus her.

00:01:32.900 --> 00:01:36.780
There's 23 students
in the class.

00:01:36.780 --> 00:01:38.920
And this is just to make sure
that you don't say, oh, 12

00:01:38.920 --> 00:01:43.030
taller, 12 shorter, and say
25 or something like that.

00:01:43.030 --> 00:01:46.650
It's just to confuse you
really, I think.

00:01:46.650 --> 00:01:49.780
Problem number 11.

00:01:49.780 --> 00:01:50.510
Image.

00:01:50.510 --> 00:01:51.650
Clear image.

00:01:51.650 --> 00:01:52.680
Invert colors.

00:01:52.680 --> 00:01:54.200
Problem number 11.

00:01:54.200 --> 00:01:55.450
Switch colors.

00:01:57.620 --> 00:02:02.560
The quadratic function g is
given by-- this is just the

00:02:02.560 --> 00:02:04.970
general way people write
quadratic functions-- is equal

00:02:04.970 --> 00:02:12.280
to ax squared plus bx plus
c, where a and c

00:02:12.280 --> 00:02:13.530
are negative constants.

00:02:19.050 --> 00:02:21.000
Which of the following could
be the graph of g?

00:02:21.000 --> 00:02:22.310
So I'm not going to even
look at the choices.

00:02:22.310 --> 00:02:24.660
I'll just show you how I would--
if someone didn't even

00:02:24.660 --> 00:02:26.610
give me any choices, and say,
well, if a and c are less than

00:02:26.610 --> 00:02:28.280
0, what's this going
to look like?

00:02:28.280 --> 00:02:31.352
So let's draw the x and y.

00:02:31.352 --> 00:02:33.320
So let's say that that
is the x-axis.

00:02:36.520 --> 00:02:40.730
Actually, I drew this--
let me edit, undo.

00:02:40.730 --> 00:02:41.130
OK.

00:02:41.130 --> 00:02:44.720
So let me draw the x-axis.

00:02:44.720 --> 00:02:47.490
You might realize pretty soon
why I have to redo this.

00:02:47.490 --> 00:02:48.740
The y-axis.

00:02:53.790 --> 00:02:55.790
x, y.

00:02:55.790 --> 00:02:56.510
All right.

00:02:56.510 --> 00:02:59.060
So we know that a and
c are negative.

00:02:59.060 --> 00:03:01.030
So c is the y-intercept,
right?

00:03:01.030 --> 00:03:03.770
When x is equal to 0, we're
just left with g of

00:03:03.770 --> 00:03:04.710
x is equal to c.

00:03:04.710 --> 00:03:05.790
So c is the y-intercept.

00:03:05.790 --> 00:03:09.140
So if c is less than 0, we know
that the y-intercept is

00:03:09.140 --> 00:03:10.590
going to be below the x-axis.

00:03:10.590 --> 00:03:12.780
It's going to be a
negative number.

00:03:12.780 --> 00:03:17.360
We know that this is c.

00:03:17.360 --> 00:03:20.180
That's where the graph
intersects the y-axis.

00:03:20.180 --> 00:03:22.020
We also know it's going to be
symmetric around the y-axis.

00:03:25.750 --> 00:03:28.970
Well, actually, we don't know
for sure that it's going to be

00:03:28.970 --> 00:03:30.620
symmetrical around the y-axis.

00:03:30.620 --> 00:03:34.630
And I could go into that in
more detail, but I won't.

00:03:34.630 --> 00:03:35.880
But that's all we
need to know.

00:03:35.880 --> 00:03:38.710
It depends what the bx--
this term-- is.

00:03:38.710 --> 00:03:40.290
But let's just ignore
that for now.

00:03:40.290 --> 00:03:42.310
And then we know that
a is negative.

00:03:42.310 --> 00:03:45.260
So in general, you might want
to prove it to yourself, you

00:03:45.260 --> 00:03:47.560
know the graph of y
equals x squared.

00:03:47.560 --> 00:03:48.875
That opens up like this.

00:03:48.875 --> 00:03:52.650
And the graph of y is equal
to minus x squared.

00:03:52.650 --> 00:03:55.970
It goes down like that.

00:03:55.970 --> 00:03:59.590
So a possible graph for this
would look like this.

00:04:03.040 --> 00:04:06.300
The important thing is these
are the two things that you

00:04:06.300 --> 00:04:08.340
know when they tell you
a and c is negative.

00:04:08.340 --> 00:04:12.070
It's going to be a graph that
opens downwards, and it's

00:04:12.070 --> 00:04:21.180
going to intersect the y-axis
in the negative area, below

00:04:21.180 --> 00:04:22.520
the x-axis.

00:04:22.520 --> 00:04:25.660
A legitimate graph also would
have looked like this.

00:04:25.660 --> 00:04:28.890
It could have also
looked like this.

00:04:28.890 --> 00:04:30.310
It didn't actually have
to be symmetric.

00:04:30.310 --> 00:04:33.500
It depends what this
bx term does.

00:04:33.500 --> 00:04:37.450
But if we look at the choices,
the only choice that

00:04:37.450 --> 00:04:42.330
intercepts the y-axis below
the x-axis at a negative

00:04:42.330 --> 00:04:45.940
y-intercept and opens
down is choice A.

00:04:45.940 --> 00:04:48.680
There's only three choices
that open down, and only

00:04:48.680 --> 00:04:51.540
choice A intercepts in
the negative area, so

00:04:51.540 --> 00:04:52.150
the answer is A.

00:04:52.150 --> 00:04:54.520
It actually looks eerily
like what I drew,

00:04:54.520 --> 00:04:57.130
just out of the blue.

00:04:57.130 --> 00:04:58.380
Next question.

00:05:00.770 --> 00:05:03.930
Problem number 12.

00:05:03.930 --> 00:05:06.190
OK, so they drew a square.

00:05:06.190 --> 00:05:10.350
And I will draw a square,
because they drew a square.

00:05:10.350 --> 00:05:17.960
And they say this is A, this
is B, this is C, D.

00:05:17.960 --> 00:05:21.720
And this is P.

00:05:21.720 --> 00:05:24.110
This is Q.

00:05:24.110 --> 00:05:25.990
And out here is R.

00:05:25.990 --> 00:05:27.580
I don't know where this
problem's going, but it looks

00:05:27.580 --> 00:05:28.120
interesting.

00:05:28.120 --> 00:05:30.600
Figure not drawn to
scale, of course.

00:05:30.600 --> 00:05:33.510
In the figure above, ABCD
is a rectangle, with

00:05:33.510 --> 00:05:34.760
BC is equal to 4.

00:05:37.240 --> 00:05:40.940
AB is equal to 6.

00:05:40.940 --> 00:05:45.800
P, Q, and R are different points
on a line, not shown,

00:05:45.800 --> 00:05:47.850
that is parallel to AD.

00:05:47.850 --> 00:05:54.020
So P, Q, and R are apparently
all on the same line that is

00:05:54.020 --> 00:05:55.640
parallel to AD.

00:05:55.640 --> 00:05:57.500
Well, I guess that is the
way they drew it.

00:05:57.500 --> 00:06:00.060
So let me erase.

00:06:00.060 --> 00:06:05.400
Let's just say this is Q.

00:06:05.400 --> 00:06:06.610
That R is down here.

00:06:06.610 --> 00:06:08.130
This isn't R.

00:06:08.130 --> 00:06:09.320
So they're all on
the same line.

00:06:09.320 --> 00:06:10.970
And they say the line
isn't shown, but

00:06:10.970 --> 00:06:14.100
it's parallel to AD.

00:06:14.100 --> 00:06:15.200
So let me draw that line.

00:06:15.200 --> 00:06:18.116
So that line looks something
like this.

00:06:18.116 --> 00:06:21.520
Let me draw it straight.

00:06:21.520 --> 00:06:26.310
And on that line, you have
P, Q, and R, and it

00:06:26.310 --> 00:06:28.870
is parallel to AD.

00:06:28.870 --> 00:06:30.120
Good enough.

00:06:32.070 --> 00:06:37.220
Points P and Q are symmetric
about line AB.

00:06:41.790 --> 00:06:46.690
OK, so that means that P and
Q are equidistant from this

00:06:46.690 --> 00:06:52.450
line, so that means that
this distance is

00:06:52.450 --> 00:06:54.350
equal to this distance.

00:06:54.350 --> 00:06:57.300
When they say P and Q are
symmetric around AB, you can

00:06:57.300 --> 00:06:59.720
almost pretend like AB's a
mirror right here, and P and Q

00:06:59.720 --> 00:07:01.940
are the mirror images
of each other.

00:07:01.940 --> 00:07:04.380
Good enough.

00:07:04.380 --> 00:07:10.240
And then it also says
points Q and R are

00:07:10.240 --> 00:07:12.460
symmetric around line CD.

00:07:16.710 --> 00:07:21.650
So we also know that
this distance is

00:07:21.650 --> 00:07:24.440
equal to this distance.

00:07:24.440 --> 00:07:26.990
Good enough.

00:07:26.990 --> 00:07:31.680
What is the length of PR?

00:07:36.600 --> 00:07:38.100
This is fascinating.

00:07:38.100 --> 00:07:40.830
So let's say that this length
right here-- let me do it in

00:07:40.830 --> 00:07:45.840
yellow-- this length is x.

00:07:45.840 --> 00:07:48.990
Then what is this green
length going to be?

00:07:48.990 --> 00:07:51.800
This green length right here
is going to be 4 minus x.

00:07:54.600 --> 00:07:55.840
How did I know it
was 4 minus x?

00:07:55.840 --> 00:07:58.025
Well, this distance
up here is 4.

00:08:00.550 --> 00:08:01.520
So this is x.

00:08:01.520 --> 00:08:02.800
This is 4 minus x.

00:08:02.800 --> 00:08:04.860
Well, then this distance right
here is also going

00:08:04.860 --> 00:08:07.490
to be 4 minus x.

00:08:07.490 --> 00:08:11.070
And if this yellow distance is
x, then we also know that this

00:08:11.070 --> 00:08:12.940
distance is x.

00:08:12.940 --> 00:08:15.630
So the distance from P to R is
going to be the sum of all of

00:08:15.630 --> 00:08:19.240
these, right? x plus x plus
4 minus x plus 4 minus x.

00:08:19.240 --> 00:08:21.460
And I think you see where
this is going.

00:08:21.460 --> 00:08:30.920
It's x plus x plus 4 minus
x plus 4 minus x.

00:08:30.920 --> 00:08:32.360
These x's all cancel
out, right?

00:08:32.360 --> 00:08:34.520
2x minus 2x.

00:08:34.520 --> 00:08:35.990
And we're just left with 8.

00:08:35.990 --> 00:08:36.929
And that's our answer.

00:08:36.929 --> 00:08:37.919
B.

00:08:37.919 --> 00:08:38.929
And you could've tried it out.

00:08:38.929 --> 00:08:40.619
You could have assumed Q was
right in the middle, and it

00:08:40.619 --> 00:08:43.340
would have worked out the same,
but you know for sure

00:08:43.340 --> 00:08:45.360
that the answer is 8.

00:08:45.360 --> 00:08:47.990
Next problem.

00:08:47.990 --> 00:08:50.590
Image clear.

00:08:50.590 --> 00:08:51.680
Invert.

00:08:51.680 --> 00:08:54.010
I don't know if I'll
have time for this.

00:08:54.010 --> 00:08:56.200
The price-- I get nervous when
I'm running out of time.

00:08:56.200 --> 00:08:57.540
So I'm going to do this
in the next video.

00:08:57.540 --> 00:08:59.300
See you soon.


SAT Prep: Test 2 Section 9 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:01.330
Welcome back.

00:00:01.330 --> 00:00:02.440
Let me erase what I did.

00:00:02.440 --> 00:00:04.880
And I should have done this
before starting this.

00:00:04.880 --> 00:00:06.350
But it doesn't take too long.

00:00:06.350 --> 00:00:07.600
We're on problem number 6.

00:00:10.430 --> 00:00:13.070
And once again, a figure that
has not been drawn to scale.

00:00:13.070 --> 00:00:14.680
They do that to you
on purpose.

00:00:14.680 --> 00:00:17.070
Sometimes it is drawn to scale
and they say that it's not.

00:00:17.070 --> 00:00:19.040
You shouldn't assume anything.

00:00:23.330 --> 00:00:27.300
And they're saying that this is
3x degrees, they say that

00:00:27.300 --> 00:00:34.310
this is 2x degrees, and they say
that this is 4x degrees.

00:00:34.310 --> 00:00:37.380
In the figure above, three line
segments meet at a point

00:00:37.380 --> 00:00:38.710
to form three angles.

00:00:38.710 --> 00:00:40.350
What is the value of x?

00:00:40.350 --> 00:00:42.020
Well, what are all of these
angles going to

00:00:42.020 --> 00:00:43.250
have to add up to?

00:00:43.250 --> 00:00:50.410
If you add this angle plus this
angle plus this angle,

00:00:50.410 --> 00:00:50.820
what do you do?

00:00:50.820 --> 00:00:52.230
You went all around
the circle.

00:00:52.230 --> 00:00:55.650
So they have to add
up to 360 degrees.

00:00:55.650 --> 00:01:03.120
So 2x plus 4x plus 3x is going
to equal 360 degrees.

00:01:03.120 --> 00:01:04.640
And what's 2 plus 4 plus 3?

00:01:04.640 --> 00:01:06.282
That's 9x.

00:01:06.282 --> 00:01:07.020
Let me see.

00:01:07.020 --> 00:01:07.870
This is always where
I mess up.

00:01:07.870 --> 00:01:09.980
2 plus 4 is 6, 6 plus 3 is 9.

00:01:09.980 --> 00:01:13.555
9x is equal to 360.

00:01:13.555 --> 00:01:17.740
x is equal to 360
divided by 9.

00:01:17.740 --> 00:01:19.420
It equals 40.

00:01:19.420 --> 00:01:22.800
And that is choice C.

00:01:22.800 --> 00:01:24.780
Problem number 7.

00:01:31.910 --> 00:01:34.660
Positive integers x, y, and
z-- so they're integers--

00:01:34.660 --> 00:01:46.960
satisfy the equations x to the
minus 1/2 is equal to 1/3, y

00:01:46.960 --> 00:01:51.790
to the z is equal to 16.

00:01:51.790 --> 00:01:52.940
OK, those are the
two equations.

00:01:52.940 --> 00:02:01.090
If z is greater than y, what
is the value of x plus z?

00:02:04.640 --> 00:02:07.770
So the trick is-- because you're
saying, well, how do I

00:02:07.770 --> 00:02:08.350
solve for this?

00:02:08.350 --> 00:02:09.699
Well, the trick is that
you know that

00:02:09.699 --> 00:02:11.080
these are all integers.

00:02:11.080 --> 00:02:12.140
So first of all,
let's do this.

00:02:12.140 --> 00:02:15.020
So x to the minus 1/2
is equal to 1/3.

00:02:15.020 --> 00:02:20.210
That means that 1 over the
square root of x-- I just

00:02:20.210 --> 00:02:23.280
rewrote x to the minus 1/2.

00:02:23.280 --> 00:02:24.650
1/2 power is square
root, and the

00:02:24.650 --> 00:02:25.710
negative means the inverse.

00:02:25.710 --> 00:02:30.590
So 1 over the square root
of x is equal to 1/3.

00:02:30.590 --> 00:02:34.340
Or you could say the square
root of x is equal to 3.

00:02:34.340 --> 00:02:36.750
x is equal to 9.

00:02:36.750 --> 00:02:37.450
So that was easy.

00:02:37.450 --> 00:02:38.170
We just solve this.

00:02:38.170 --> 00:02:41.060
You just have to know that a
negative means to invert, and

00:02:41.060 --> 00:02:43.500
that 1/2 power is the same
thing as square root.

00:02:43.500 --> 00:02:45.670
Now y to the z equals 16.

00:02:45.670 --> 00:02:49.240
We've got to come up with
integers y and z where is z is

00:02:49.240 --> 00:02:52.070
greater than y that
end up at 16.

00:02:52.070 --> 00:02:54.640
Well, the number that
immediately pops out is 2 to

00:02:54.640 --> 00:02:56.460
the fourth power.

00:02:56.460 --> 00:02:58.250
2 to the fourth power
is equal to 16.

00:02:58.250 --> 00:03:01.480
The other option was 4 to the
second power, soon. right?

00:03:01.480 --> 00:03:05.060
But 4 to the second power
wouldn't have satisfied this,

00:03:05.060 --> 00:03:07.830
because 4 is greater than 2.

00:03:07.830 --> 00:03:09.530
So the exponent has to be
greater than the base.

00:03:09.530 --> 00:03:11.180
So 2 to the fourth.

00:03:11.180 --> 00:03:13.730
And we know that because
it has to be integers.

00:03:13.730 --> 00:03:17.230
So we know that z
is equal to 4.

00:03:17.230 --> 00:03:18.776
So what's 4 plus 9?

00:03:18.776 --> 00:03:20.570
It's 13.

00:03:20.570 --> 00:03:21.360
x plus z.

00:03:21.360 --> 00:03:22.700
13.

00:03:22.700 --> 00:03:25.540
And that is choice D.

00:03:25.540 --> 00:03:26.790
Next problem.

00:03:30.400 --> 00:03:33.670
I'm on problem number 8.

00:03:33.670 --> 00:03:38.570
And let me draw what
they have drawn.

00:03:38.570 --> 00:03:40.750
I don't even know if it's useful
for me to draw, but I

00:03:40.750 --> 00:03:42.980
will do it because
they've done it.

00:03:42.980 --> 00:03:44.530
And then they have
a semicircle.

00:03:44.530 --> 00:03:47.690
So I could do a semicircle,
something like this.

00:03:47.690 --> 00:03:53.920
I could draw it-- use the
circle tool like that.

00:03:53.920 --> 00:03:56.340
Roughly.

00:03:56.340 --> 00:03:57.830
That's good enough.

00:03:57.830 --> 00:04:01.090
I could erase the bottom if I
wanted to, but let's just

00:04:01.090 --> 00:04:03.140
assume that we only
see this top part.

00:04:03.140 --> 00:04:09.030
And they say-- whoops, I'm still
using the line tool.

00:04:09.030 --> 00:04:12.110
This point is 4 comma 0.

00:04:12.110 --> 00:04:17.029
This point is 8 comma 0.

00:04:17.029 --> 00:04:19.160
And they say, in the semicircle
above, the center

00:04:19.160 --> 00:04:20.800
is at 4 comma 0.

00:04:20.800 --> 00:04:23.450
Which of the following are
x-coordinates of two points on

00:04:23.450 --> 00:04:28.050
the semicircle whose
y-coordinates are equal?

00:04:34.710 --> 00:04:36.270
Well, this is an interesting
question.

00:04:36.270 --> 00:04:38.850
Because it's not so much math,
but it really is getting the

00:04:38.850 --> 00:04:41.410
intuition of kind of the
symmetry of the circle.

00:04:41.410 --> 00:04:43.520
So it wants to know two
points on the circle

00:04:43.520 --> 00:04:45.400
where the y is equal.

00:04:45.400 --> 00:04:51.430
So, for example, this point
and this point.

00:04:51.430 --> 00:04:54.165
The y is going to be equal.

00:04:54.165 --> 00:04:58.030
And so it would be this
point and this point.

00:04:58.030 --> 00:05:01.940
And similarly, we could say,
well, this point and this

00:05:01.940 --> 00:05:03.155
point, the y's are equal.

00:05:03.155 --> 00:05:07.620
If we take down, it would be
this point and this point.

00:05:07.620 --> 00:05:11.500
The key is any of these pairs of
points have to be the exact

00:05:11.500 --> 00:05:12.780
same distance from the center.

00:05:12.780 --> 00:05:13.410
And that makes sense.

00:05:13.410 --> 00:05:16.440
If we go the same distance to
the left and go up to the

00:05:16.440 --> 00:05:18.055
circle, and we go the same
distance to the right and go

00:05:18.055 --> 00:05:20.150
up to the circle, they're going
to intersect the y-axis

00:05:20.150 --> 00:05:21.350
at the same point.

00:05:21.350 --> 00:05:23.950
So we have to look at their
choices and see which of their

00:05:23.950 --> 00:05:26.980
pairs of choices are equal
distance from x equals 4.

00:05:30.840 --> 00:05:33.650
So the first choice
is 1 and 6.

00:05:33.650 --> 00:05:37.530
1 is 3 less than 4, and
6 is 2 more than 4, so

00:05:37.530 --> 00:05:39.190
that doesn't work.

00:05:39.190 --> 00:05:44.190
A is 1 and 6, and that doesn't
work because 6 is much closer

00:05:44.190 --> 00:05:46.560
to 4 than 1 is.

00:05:46.560 --> 00:05:51.000
B is choices 1 and 8.

00:05:51.000 --> 00:05:53.350
Well, 1 is 3 away from 4.

00:05:53.350 --> 00:05:53.570
Right?

00:05:53.570 --> 00:05:54.860
1 is here.

00:05:54.860 --> 00:05:56.480
Well, 8 is all the way here.

00:05:56.480 --> 00:05:59.040
So 8 is 4 more than 4.

00:05:59.040 --> 00:06:00.560
3 you less, 4 more.

00:06:00.560 --> 00:06:02.740
Doesn't work.

00:06:02.740 --> 00:06:05.190
C.

00:06:05.190 --> 00:06:06.300
2 and 6.

00:06:06.300 --> 00:06:07.140
This looks promising.

00:06:07.140 --> 00:06:09.150
2 is 2 less than 4.

00:06:09.150 --> 00:06:10.940
6 is 2 more than 4.

00:06:10.940 --> 00:06:13.800
So that's 2 less, bam, it's
going to be there.

00:06:13.800 --> 00:06:14.760
2 more, bam.

00:06:14.760 --> 00:06:15.310
Going to be there.

00:06:15.310 --> 00:06:16.250
So that's our answer.

00:06:16.250 --> 00:06:17.660
We don't even have to look
at the other ones.

00:06:17.660 --> 00:06:18.980
But if you look at the other
ones, you'll see that they're

00:06:18.980 --> 00:06:24.200
all different distances
away from x equals 4.

00:06:24.200 --> 00:06:31.440
Next problem, problem
number 9.

00:06:31.440 --> 00:06:36.590
If p is an integer, and 3 is
the remainder when-- OK, so

00:06:36.590 --> 00:06:40.890
let me write this. p integer.

00:06:40.890 --> 00:06:56.370
3 is remainder when 2p plus
7 is divided by 5.

00:06:56.370 --> 00:06:58.670
3 is the remainder.

00:06:58.670 --> 00:07:04.590
Then p could be-- well, there's
a bunch of ways you

00:07:04.590 --> 00:07:07.500
could do it, but really the
simplest way, the very

00:07:07.500 --> 00:07:10.765
simplest way, is to just
try out the numbers.

00:07:10.765 --> 00:07:12.710
And let me ask you a question.

00:07:12.710 --> 00:07:15.250
When you divide something by 5,
if you get a remainder of

00:07:15.250 --> 00:07:18.800
3, what is that number going
to have to end in?

00:07:18.800 --> 00:07:22.010
Well, it's going to end
in a 0 or 5, right?

00:07:22.010 --> 00:07:22.510
Oh, sorry.

00:07:22.510 --> 00:07:24.390
It's going to end
in a 3 or an 8.

00:07:24.390 --> 00:07:26.860
Because if something is
divisible by 5, it ends

00:07:26.860 --> 00:07:28.350
with a 0 or a 5.

00:07:28.350 --> 00:07:32.750
5, 10, 15, 20, 25.

00:07:32.750 --> 00:07:35.940
All of these are
divisible by 5.

00:07:35.940 --> 00:07:41.530
And if you add 3 to any of these
numbers-- 8, 13, 18, 23,

00:07:41.530 --> 00:07:45.500
28, and so on, 33-- all of these
numbers are going to

00:07:45.500 --> 00:07:49.440
have a remainder of 3 when
you divide by 5.

00:07:49.440 --> 00:07:53.140
And then you could also say,
well, any of these numbers is

00:07:53.140 --> 00:07:54.390
going to be the numerator.

00:08:01.050 --> 00:08:03.040
And then you can actually,
if you want, you

00:08:03.040 --> 00:08:04.720
could subtract 7.

00:08:04.720 --> 00:08:05.720
But actually, we
won't do that.

00:08:05.720 --> 00:08:08.160
What we could do is just try out
the different choices they

00:08:08.160 --> 00:08:12.670
give, and see if any of these
choices create a number that

00:08:12.670 --> 00:08:14.645
ends in an 8 or a 3.

00:08:14.645 --> 00:08:16.060
So let's see.

00:08:16.060 --> 00:08:18.990
Choice A is 2.

00:08:18.990 --> 00:08:21.460
What is 2 times 2 plus 7?

00:08:21.460 --> 00:08:23.500
It's 4 plus 7 which is 11.

00:08:23.500 --> 00:08:25.840
It's not that one because
it ends with a 1.

00:08:25.840 --> 00:08:27.060
B.

00:08:27.060 --> 00:08:27.910
3.

00:08:27.910 --> 00:08:30.260
3 times 2 is 6.

00:08:30.260 --> 00:08:34.260
6 plus 7 is 13.

00:08:34.260 --> 00:08:35.370
We're done.

00:08:35.370 --> 00:08:36.970
And you can try it out.

00:08:36.970 --> 00:08:40.169
If p is 3, 2 times 3 is 6.

00:08:40.169 --> 00:08:41.010
Plus 7, 13.

00:08:41.010 --> 00:08:42.400
Divided by 5.

00:08:42.400 --> 00:08:45.570
Well, the remainder is going to
be-- it's 2, remainder 3.

00:08:45.570 --> 00:08:46.550
So it works.

00:08:46.550 --> 00:08:47.595
And that was it.

00:08:47.595 --> 00:08:48.420
And it's good.

00:08:48.420 --> 00:08:51.540
We didn't have to waste time
figuring out C, D, and E.

00:08:51.540 --> 00:08:53.000
OK.

00:08:53.000 --> 00:08:54.680
And then-- oh, I only
have a minute left.

00:08:54.680 --> 00:08:58.340
So I will wait to do the next
problem in the next video.

00:08:58.340 --> 00:08:59.590
See


SAT Prep: Test 2 Section 9 Part 1

WEBVTT
Kind: captions
Language: en

00:00:01.250 --> 00:00:02.270
Welcome back.

00:00:02.270 --> 00:00:05.390
I will be eating turkey in a
little over an hour, and I

00:00:05.390 --> 00:00:06.380
want to work up an appetite.

00:00:06.380 --> 00:00:09.280
So I will move forward and
continue with the SAT

00:00:09.280 --> 00:00:11.690
problems. Today is Thanksgiving,
by the way.

00:00:11.690 --> 00:00:16.580
That's why I'm a little extra
excited about the turkey.

00:00:16.580 --> 00:00:18.210
Problem number 1.

00:00:18.210 --> 00:00:21.010
Oh, by the way, we're on
page 47, section 9, of

00:00:21.010 --> 00:00:22.260
the-- what's it called?

00:00:22.260 --> 00:00:25.210
The official SAT study guide.

00:00:25.210 --> 00:00:26.770
Problem number 1.

00:00:26.770 --> 00:00:30.850
There is the same number of boys
and girls on a school bus

00:00:30.850 --> 00:00:34.360
when it departs from school.

00:00:34.360 --> 00:00:38.230
So at the beginning,
boys equals girls.

00:00:38.230 --> 00:00:39.210
I don't know if this is
going to be useful

00:00:39.210 --> 00:00:40.990
for me to write that.

00:00:40.990 --> 00:00:43.390
At the first stop, 4 boys
get off the bus

00:00:43.390 --> 00:00:44.390
and nobody gets on.

00:00:44.390 --> 00:00:50.570
So stop one, minus 4 boys.

00:00:54.710 --> 00:00:59.030
After the first stop, there
are twice as many girls as

00:00:59.030 --> 00:01:01.580
boys on the bus.

00:01:01.580 --> 00:01:09.450
So now, girls is equal
to 2 times the boys.

00:01:09.450 --> 00:01:12.090
How many girls are
there on the bus?

00:01:12.090 --> 00:01:14.090
So you could almost do
this by inspection.

00:01:14.090 --> 00:01:19.920
But what you could say is, well,
here the boys is equal

00:01:19.920 --> 00:01:21.710
to the girls.

00:01:21.710 --> 00:01:24.080
Then we subtract 4
from the boys.

00:01:24.080 --> 00:01:25.360
So this b is actually
a different b.

00:01:25.360 --> 00:01:29.910
This is 2 times the new
one, b minus 4.

00:01:29.910 --> 00:01:31.160
Why does that make sense?

00:01:34.800 --> 00:01:36.320
We start off with b boys.

00:01:36.320 --> 00:01:38.400
At the first stop, 4 get off.

00:01:38.400 --> 00:01:40.730
So now there are
b minus 4 boys.

00:01:40.730 --> 00:01:43.440
And at the end, it says that
there are twice as many girls

00:01:43.440 --> 00:01:46.470
as there are boys now, and b
minus 4 are the number of boys

00:01:46.470 --> 00:01:47.630
there are now.

00:01:47.630 --> 00:01:50.275
So now girls is equal to
2 times b minus 4.

00:01:50.275 --> 00:01:53.970
But we also know that girls is
equal to this initial b.

00:01:53.970 --> 00:01:57.610
So you say b is equal
to 2b minus 8.

00:01:57.610 --> 00:01:59.770
I just distributed the 2.

00:01:59.770 --> 00:02:04.070
And then if we subtract b from
both sides and add 8 to both

00:02:04.070 --> 00:02:06.870
sides, you get 8
is equal to b.

00:02:06.870 --> 00:02:08.520
And you could have probably
thought about that.

00:02:08.520 --> 00:02:10.710
I mean, you didn't have to do
all this fancy algebra.

00:02:10.710 --> 00:02:15.720
You could have just said, well,
when 4 boys leave, I'm

00:02:15.720 --> 00:02:17.610
left with half as many
boys, right?

00:02:17.610 --> 00:02:20.290
Because before, I had just
as many girls as boys.

00:02:20.290 --> 00:02:23.230
Then 4 leave, and I
have half as many.

00:02:23.230 --> 00:02:25.680
Because now the number
of girls is doubled.

00:02:25.680 --> 00:02:28.830
So if I have some amount of
something and when 4 leave,

00:02:28.830 --> 00:02:31.190
I'm left with half, then we
know we started with 8.

00:02:31.190 --> 00:02:33.010
So you could have just done
it without the algebra.

00:02:33.010 --> 00:02:34.560
But this is one way you
could make sure you

00:02:34.560 --> 00:02:36.590
had the right answer.

00:02:36.590 --> 00:02:39.900
But it probably would've taken
you too long, so the best way

00:02:39.900 --> 00:02:43.190
might have been just to
do it in your head.

00:02:43.190 --> 00:02:46.470
Problem number 2.

00:02:46.470 --> 00:02:48.800
And I will do it in magenta.

00:02:48.800 --> 00:02:51.260
Which of the following is the
graph of a linear function

00:02:51.260 --> 00:02:54.000
with a negative slope and
a positive y-intercept?

00:02:54.000 --> 00:02:56.610
So I'm not even going to
look at their choices.

00:02:56.610 --> 00:02:58.160
It'll take me forever
to draw them.

00:02:58.160 --> 00:03:01.190
I'm just going to draw my own
graph that describes what

00:03:01.190 --> 00:03:02.542
they're saying.

00:03:02.542 --> 00:03:03.540
soon.

00:03:03.540 --> 00:03:05.560
That's the x and y-axes.

00:03:05.560 --> 00:03:07.730
And so they say, graph
of a linear function.

00:03:07.730 --> 00:03:10.620
So linear function means it's
going to be a line.

00:03:10.620 --> 00:03:11.670
Negative slope.

00:03:11.670 --> 00:03:14.880
So negative slope means that it
goes from the top left to

00:03:14.880 --> 00:03:15.830
the bottom right.

00:03:15.830 --> 00:03:17.970
Because we move to the right.

00:03:17.970 --> 00:03:20.040
Negative slope means that the
graph's going to move down.

00:03:20.040 --> 00:03:23.750
So it's going to be a negative
inclination.

00:03:23.750 --> 00:03:26.570
And a positive y-intercept, so
that means it intersects the

00:03:26.570 --> 00:03:30.060
y-axis in the positive y area.

00:03:30.060 --> 00:03:33.330
So a suitable line would look
something like this.

00:03:33.330 --> 00:03:34.770
It would intersect here
and move down.

00:03:34.770 --> 00:03:36.020
So it would look like this.

00:03:38.860 --> 00:03:42.160
And if we look at the choices of
which ones do that, well, D

00:03:42.160 --> 00:03:44.230
is the only one that looks
like what I just drew.

00:03:44.230 --> 00:03:45.920
So the choice is D.

00:03:48.940 --> 00:03:50.655
Problem number 3.

00:03:58.510 --> 00:04:00.690
Refer to the following price
list. So I guess for

00:04:00.690 --> 00:04:01.570
problems 3 and 4.

00:04:01.570 --> 00:04:05.690
So let me draw the price list.
So number of donuts.

00:04:08.730 --> 00:04:09.600
I can never spell donuts.

00:04:09.600 --> 00:04:12.500
I always want to write
out dough.

00:04:12.500 --> 00:04:14.660
And then it says, total price.

00:04:19.120 --> 00:04:23.570
It says if I get 1 donut, it's
going to cost me $0.40.

00:04:23.570 --> 00:04:28.990
If I get 6 donuts, a box of 6,
it's going to cost me $1.89.

00:04:28.990 --> 00:04:35.020
And if I get 12 donuts, it's
going to cost me $3.59.

00:04:35.020 --> 00:04:39.330
And the question is-- question
number 3-- of the following,

00:04:39.330 --> 00:04:43.630
which is the closest
approximation of the cost per

00:04:43.630 --> 00:04:45.620
donut when one purchases
a box of 6?

00:04:45.620 --> 00:04:48.610
Well, a box of 6 costs $1.89.

00:04:48.610 --> 00:04:53.410
So it's literally $1.89
divided by 6. $1.89/6.

00:04:53.410 --> 00:04:54.510
And you can eyeball that.

00:04:54.510 --> 00:04:59.760
It's a little over $0.30.

00:04:59.760 --> 00:05:01.480
If it was $1.80, it
would be $0.30.

00:05:01.480 --> 00:05:02.800
And then you have to have
9 divided by 6.

00:05:02.800 --> 00:05:05.050
So it's like 31 and 1/3 cents.

00:05:05.050 --> 00:05:07.410
It's approximately $0.31.

00:05:07.410 --> 00:05:09.630
And if you look at it, they give
you very rough numbers.

00:05:09.630 --> 00:05:11.280
So it's choice B.

00:05:11.280 --> 00:05:12.840
And what they wanted you to
do is do this quickly.

00:05:12.840 --> 00:05:14.210
They don't want you to actually
divide the numbers.

00:05:14.210 --> 00:05:17.330
They want to say, well, $1.89
divided by 6 is roughly $0.30.

00:05:17.330 --> 00:05:19.580
You do that problem really,
really fast.

00:05:19.580 --> 00:05:23.620
Now, problem number 4, we use
the same price list. What

00:05:23.620 --> 00:05:26.870
would be the least amount of
money needed to purchase

00:05:26.870 --> 00:05:30.340
exactly 21 donuts?

00:05:30.340 --> 00:05:32.840
Well, if we want to purchase 21
donuts, we want to get it

00:05:32.840 --> 00:05:36.280
in kind of as big chunks
as possible.

00:05:36.280 --> 00:05:37.640
So let's see.

00:05:37.640 --> 00:05:44.340
If we got 12-- exactly
21 donuts.

00:05:44.340 --> 00:05:45.200
It could've been tricky.

00:05:45.200 --> 00:05:46.960
Maybe 24 donuts would've been
cheaper or something.

00:05:46.960 --> 00:05:47.910
But we want exactly 21.

00:05:47.910 --> 00:05:51.130
So we want to get one dozen of
this, because this is going to

00:05:51.130 --> 00:05:53.450
be a little under $0.30
per donut, right?

00:05:53.450 --> 00:05:55.710
Because $3.60 would be $0.30.

00:05:55.710 --> 00:05:58.180
And then we want to
get 1 of these.

00:06:01.000 --> 00:06:02.410
And then we want to
get-- let's see.

00:06:02.410 --> 00:06:04.186
6-- that would get us to 18.

00:06:04.186 --> 00:06:04.840
you

00:06:04.840 --> 00:06:08.260
And then we want to
get 3 of these.

00:06:08.260 --> 00:06:09.930
Times 3.

00:06:09.930 --> 00:06:10.620
So let's see.

00:06:10.620 --> 00:06:22.220
It would be 3 times $0.40, which
is $1.20, plus $1.89,

00:06:22.220 --> 00:06:26.060
plus $3.59.

00:06:26.060 --> 00:06:29.440
So 9 plus 9 is 18.

00:06:29.440 --> 00:06:32.780
1 plus 2 plus 8 is 11, 16.

00:06:32.780 --> 00:06:35.500
And then $6.68.

00:06:35.500 --> 00:06:38.500
And that is choice B.

00:06:38.500 --> 00:06:42.620
This is for problem number
4, not problem number 3.

00:06:42.620 --> 00:06:43.870
Let's move on.

00:06:47.360 --> 00:06:51.190
And we'll switch colors.

00:06:51.190 --> 00:06:52.700
Number 5.

00:06:52.700 --> 00:06:54.805
I think I'm going to have
to do some drawing now.

00:06:54.805 --> 00:06:56.190
The figure above.

00:06:56.190 --> 00:06:58.970
Which of the following
is closest to h of 5?

00:06:58.970 --> 00:07:00.800
Yeah, I'll have to
draw that one.

00:07:00.800 --> 00:07:03.320
So let me draw the
coordinate axes.

00:07:03.320 --> 00:07:05.880
That's my x-axis.

00:07:05.880 --> 00:07:07.130
This is my y-axis.

00:07:10.560 --> 00:07:10.830
Let's see.

00:07:10.830 --> 00:07:18.380
The graph does something
like this.

00:07:18.380 --> 00:07:20.560
Let me draw the markers.

00:07:20.560 --> 00:07:30.040
So this is 2, 4, 6, and so on.

00:07:30.040 --> 00:07:35.760
And this is 2, 4,
6, and so on.

00:07:35.760 --> 00:07:40.600
This is minus 2,
minus 4, so on.

00:07:40.600 --> 00:07:42.640
And the graph looks something
like this.

00:07:42.640 --> 00:07:46.620
So it's minus 6.

00:07:46.620 --> 00:07:48.190
Let me do a different color.

00:07:48.190 --> 00:07:52.275
So if we start at minus 6,
and then it crosses.

00:07:52.275 --> 00:07:57.350
It goes someplace like
there, roughly.

00:07:57.350 --> 00:08:01.190
Then it dips down to here.

00:08:01.190 --> 00:08:03.960
Dips down something like that.

00:08:03.960 --> 00:08:12.140
And then it does something--
I'm trying to be exact,

00:08:12.140 --> 00:08:14.045
because this is actually all
dependent on the graph.

00:08:16.750 --> 00:08:19.850
It's going to be something
like that.

00:08:19.850 --> 00:08:23.080
And they want to know, after
I've drawn this graph, they

00:08:23.080 --> 00:08:28.610
want to know, what is the
closest to h of 5?

00:08:28.610 --> 00:08:30.890
Well, this is h of x.

00:08:30.890 --> 00:08:31.750
This is 5.

00:08:31.750 --> 00:08:37.929
So h of 5 is just going to
be this point right here.

00:08:37.929 --> 00:08:46.020
And if we go to the left, it
looks like right in between 2

00:08:46.020 --> 00:08:46.880
and 4, right?

00:08:46.880 --> 00:08:48.070
And what's right in
between 2 and 4?

00:08:48.070 --> 00:08:50.040
Well, 3.

00:08:50.040 --> 00:08:52.580
And it looks even more like that
when you look at it on

00:08:52.580 --> 00:08:54.020
the graph they drew.

00:08:54.020 --> 00:08:56.310
My graph, you could argue
it's close to 2.

00:08:56.310 --> 00:08:58.375
But when you look at the one in
the book, you can see that

00:08:58.375 --> 00:09:01.020
it's close to 3.

00:09:01.020 --> 00:09:04.280
And I have a minute
left, so I will do

00:09:04.280 --> 00:09:05.830
that in the next video.

00:09:05.830 --> 00:09:07.080
See


SAT Prep: Test 2 Section 9 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:01.380
Welcome back.

00:00:01.380 --> 00:00:07.160
We are on problem number 13.

00:00:07.160 --> 00:00:09.560
It's on page 491.

00:00:09.560 --> 00:00:14.070
The price of a telephone was
first increased by 10%, and

00:00:14.070 --> 00:00:17.300
then the new price was
decreased by 25%.

00:00:17.300 --> 00:00:22.860
The final price was what price
of the initial price?

00:00:22.860 --> 00:00:28.320
The price final is equal
to the initial

00:00:28.320 --> 00:00:31.540
price-- price initial.

00:00:31.540 --> 00:00:33.640
So the price of a telephone,
the initial price, was

00:00:33.640 --> 00:00:36.610
increased by 10%.

00:00:36.610 --> 00:00:39.970
So when you increase something
by 10%, that's like

00:00:39.970 --> 00:00:43.322
multiplying it by 1.1.

00:00:43.322 --> 00:00:46.040
Or you could say 1 plus 10%.

00:00:46.040 --> 00:00:48.080
That's 1.1.

00:00:48.080 --> 00:00:52.030
And then it was decreased
by 25%.

00:00:52.030 --> 00:00:55.270
Well, decreasing something by
25% is the same thing as

00:00:55.270 --> 00:00:57.810
multiplying by 0.75.

00:00:57.810 --> 00:00:58.340
Why is that?

00:00:58.340 --> 00:01:02.370
Because that's 1 minus 0.25.

00:01:02.370 --> 00:01:05.250
And so this sets up our
relationship, and so we just

00:01:05.250 --> 00:01:06.800
have to do some math.

00:01:06.800 --> 00:01:16.510
What is 0.75 times 1.1?

00:01:16.510 --> 00:01:18.540
That's 75.

00:01:18.540 --> 00:01:23.540
0, 75, 5, 8.

00:01:23.540 --> 00:01:24.930
And how many points do we have
in the decimal point?

00:01:24.930 --> 00:01:26.310
One, two, three.

00:01:26.310 --> 00:01:27.240
Three points.

00:01:27.240 --> 00:01:31.310
So we have price final
is equal to price

00:01:31.310 --> 00:01:36.220
initial times 0.825.

00:01:36.220 --> 00:01:37.420
And what do they say?

00:01:37.420 --> 00:01:42.690
The final price was what percent
of the initial price?

00:01:42.690 --> 00:01:43.440
So that's it.

00:01:43.440 --> 00:01:47.335
The final price is 82.5%
of the initial price.

00:01:51.290 --> 00:01:52.930
And the main thing to just
realize, when you increase

00:01:52.930 --> 00:01:54.980
something by 10%, that's
the equivalent of

00:01:54.980 --> 00:01:56.960
multiplying it by 1.1.

00:01:56.960 --> 00:01:59.240
And then when you decrease
something by 25%, that's the

00:01:59.240 --> 00:02:01.610
same thing as multiplying
it by 0.75.

00:02:01.610 --> 00:02:02.020
And why?

00:02:02.020 --> 00:02:04.640
Because it's 1 minus 0.25.

00:02:04.640 --> 00:02:07.990
You're decreasing it by 25%.

00:02:07.990 --> 00:02:10.690
And that is choice C.

00:02:10.690 --> 00:02:11.720
Next problem.

00:02:11.720 --> 00:02:12.425
Number 14.

00:02:12.425 --> 00:02:14.130
This is what I do every
day in my job.

00:02:14.130 --> 00:02:15.800
I work for a hedge fund.

00:02:15.800 --> 00:02:18.270
I'm always doing percentages.

00:02:18.270 --> 00:02:19.530
Image.

00:02:19.530 --> 00:02:20.780
Invert colors.

00:02:23.130 --> 00:02:23.960
I do other things.

00:02:23.960 --> 00:02:27.000
Don't think my entire livelihood
is based on just

00:02:27.000 --> 00:02:31.520
multiplying percentages, but a
surprising fraction of it is.

00:02:31.520 --> 00:02:32.970
Anyway, all right.

00:02:32.970 --> 00:02:35.200
Problem number 14.

00:02:35.200 --> 00:02:39.590
When the number w is multiplied
by 4, the result is

00:02:39.590 --> 00:02:42.120
the same as 4 is added to w.

00:02:42.120 --> 00:02:47.000
OK, so it's saying when w is
multiplied by 4, so 4w, is the

00:02:47.000 --> 00:02:51.200
same thing as the result as
when 4 is added to w.

00:02:51.200 --> 00:02:54.160
So it's the same thing
as w plus 4.

00:02:54.160 --> 00:02:55.850
Fair enough.

00:02:55.850 --> 00:02:58.580
What is the value of 3w?

00:02:58.580 --> 00:03:00.320
3w is equal to what?

00:03:00.320 --> 00:03:03.730
Well, subtract w from both
sides of this equation.

00:03:03.730 --> 00:03:06.460
You get 3w equals 4.

00:03:06.460 --> 00:03:08.730
That might have been
the fastest problem

00:03:08.730 --> 00:03:09.690
we had to do, right?

00:03:09.690 --> 00:03:10.420
We just solved it.

00:03:10.420 --> 00:03:11.770
3w is equal to 4.

00:03:11.770 --> 00:03:13.020
Choice E.

00:03:15.220 --> 00:03:19.560
Problem 15.

00:03:19.560 --> 00:03:23.330
The lengths of the sides of a
right triangle are consecutive

00:03:23.330 --> 00:03:25.530
even integers.

00:03:25.530 --> 00:03:26.520
Fascinating.

00:03:26.520 --> 00:03:29.120
And the length of the
shortest side is x.

00:03:29.120 --> 00:03:30.010
It looks like I'm going
to have to draw a

00:03:30.010 --> 00:03:33.450
triangle, in yellow.

00:03:33.450 --> 00:03:35.335
So it's a right triangle.

00:03:41.350 --> 00:03:43.470
That looks like a right
triangle to me.

00:03:43.470 --> 00:03:47.350
And they say they're consecutive
even integers, and

00:03:47.350 --> 00:03:49.100
the shortest side is x.

00:03:49.100 --> 00:03:50.940
So let's say this is x.

00:03:50.940 --> 00:03:53.340
Consecutive even integers.

00:03:53.340 --> 00:03:58.040
So the next even integer is
going to be x plus 2.

00:03:58.040 --> 00:04:02.320
And then the last integer,
it's going to be the

00:04:02.320 --> 00:04:03.400
hypotenuse, the next one.

00:04:03.400 --> 00:04:06.350
And that's going to have
to be x plus 4.

00:04:06.350 --> 00:04:08.770
And they want to
know what is x?

00:04:08.770 --> 00:04:11.410
Or what equation can be
used to solve for x?

00:04:11.410 --> 00:04:12.860
Well, this is the Pythagorean
theorem.

00:04:12.860 --> 00:04:14.650
So we just say this squared plus
this squared is equal to

00:04:14.650 --> 00:04:15.170
this squared.

00:04:15.170 --> 00:04:21.269
So x squared plus x
plus 2 squared is

00:04:21.269 --> 00:04:24.480
equal to x plus 4 squared.

00:04:24.480 --> 00:04:26.500
Now let me see if this
is already there

00:04:26.500 --> 00:04:27.790
in any of the choices.

00:04:27.790 --> 00:04:29.940
x squared plus x plus
2 squared is

00:04:29.940 --> 00:04:31.100
equal to x plus 4 squared.

00:04:31.100 --> 00:04:31.870
Yeah, it's already there.

00:04:31.870 --> 00:04:32.820
We didn't do any simplifying.

00:04:32.820 --> 00:04:34.500
This is choice C.

00:04:34.500 --> 00:04:36.390
And we are done.

00:04:36.390 --> 00:04:39.690
These problems are extremely
fast to do.

00:04:39.690 --> 00:04:41.710
And the important realization
is that they're consecutive

00:04:41.710 --> 00:04:43.370
even integers.

00:04:43.370 --> 00:04:45.800
So you add 2 each time,
and you just use

00:04:45.800 --> 00:04:47.770
the Pythagorean theorem.

00:04:47.770 --> 00:04:50.260
Next problem.

00:04:50.260 --> 00:04:52.670
Problem 16.

00:04:52.670 --> 00:05:05.990
If x is an integer greater than
1, and if y is equal to x

00:05:05.990 --> 00:05:11.900
plus 1/x, which of the following
must be true?

00:05:11.900 --> 00:05:13.450
I like these.

00:05:13.450 --> 00:05:15.460
OK, I'll write all
the choices down.

00:05:15.460 --> 00:05:18.500
II, and III.

00:05:18.500 --> 00:05:22.850
OK, this first one says
y does not equal x.

00:05:22.850 --> 00:05:25.870
The second one says
y is an integer.

00:05:29.110 --> 00:05:36.340
The last one says xy is greater
than x squared.

00:05:36.340 --> 00:05:37.660
Well, let's think about it.

00:05:40.240 --> 00:05:42.920
First of all, do we know
that y is an integer?

00:05:42.920 --> 00:05:44.530
Well, no, not really.

00:05:44.530 --> 00:05:46.710
I could immediately find an x.

00:05:46.710 --> 00:05:50.460
If x is equal to 2,
then y is what?

00:05:50.460 --> 00:05:51.840
It's 2 plus 1/2.

00:05:51.840 --> 00:05:53.450
It's equal to 2 and 1/2.

00:05:53.450 --> 00:05:55.050
So we know y isn't an integer.

00:05:55.050 --> 00:05:56.960
If x is an integer, y doesn't
have to be one.

00:05:56.960 --> 00:06:00.040
So we know this is definitely
not the case.

00:06:00.040 --> 00:06:03.880
Now, do we know for a fact
that y does not equal x?

00:06:03.880 --> 00:06:06.610
Well, let's play with this
equation to see if we can get

00:06:06.610 --> 00:06:09.720
it to a form that makes a little
bit more sense to us.

00:06:09.720 --> 00:06:11.990
So we know that x is greater
than 1, or that x

00:06:11.990 --> 00:06:13.650
does not equal 0.

00:06:13.650 --> 00:06:15.790
And obviously, this would be
undefined if x equals 0.

00:06:15.790 --> 00:06:19.200
So let's multiply both sides
of this equation by x.

00:06:19.200 --> 00:06:27.980
So we get xy is equal
to x squared plus 1.

00:06:27.980 --> 00:06:30.680
This, all of a sudden,
looks fascinating.

00:06:30.680 --> 00:06:33.850
xy is equal to x
squared plus 1.

00:06:33.850 --> 00:06:41.850
So do we know immediately that
xy is greater than x squared?

00:06:41.850 --> 00:06:42.960
Well, sure.

00:06:42.960 --> 00:06:47.970
No matter what x squared is, xy
is going to be that number

00:06:47.970 --> 00:06:50.030
plus 1, right?

00:06:50.030 --> 00:06:52.940
And this term is always
going to be positive.

00:06:52.940 --> 00:06:54.260
This is always going
to be positive.

00:06:54.260 --> 00:06:55.910
There's no way you can
get x squared plus 1.

00:06:55.910 --> 00:06:58.015
If you're not dealing with
imaginary numbers, it's going

00:06:58.015 --> 00:07:00.660
to be negative, so this whole
thing's going to be positive.

00:07:00.660 --> 00:07:01.580
x is positive.

00:07:01.580 --> 00:07:03.840
So if this is positive, x is
positive, we know also that y

00:07:03.840 --> 00:07:04.770
is going to be positive.

00:07:04.770 --> 00:07:06.770
I don't know if that
helps, but anyway.

00:07:06.770 --> 00:07:10.860
But we know that xy is always
1 more than x squared, so we

00:07:10.860 --> 00:07:13.080
do know that xy is
greater than x

00:07:13.080 --> 00:07:15.750
squared, so this is true.

00:07:15.750 --> 00:07:21.400
Now, let's see if we can show
that y has to not equal x.

00:07:21.400 --> 00:07:23.110
Well, let's think about it.

00:07:23.110 --> 00:07:31.740
No matter what x is, y is going
to be equal to some

00:07:31.740 --> 00:07:33.470
fraction larger than x.

00:07:33.470 --> 00:07:36.130
And as you get really, really,
really large x's, that

00:07:36.130 --> 00:07:39.420
fraction, the difference,
is going to get smaller.

00:07:39.420 --> 00:07:40.570
You could try it on
a calculator.

00:07:40.570 --> 00:07:47.390
If you put in x is 5,
then y is 5 and 1/5.

00:07:47.390 --> 00:07:55.240
If x is 500, then y
is 500 and 1/500.

00:07:55.240 --> 00:07:58.280
But no matter what number you
put in that's a larger number

00:07:58.280 --> 00:08:01.460
than 1, you're going to get
a little bit left over.

00:08:01.460 --> 00:08:03.080
y is going to be just
that little bit

00:08:03.080 --> 00:08:05.240
fraction larger than x.

00:08:05.240 --> 00:08:11.010
So we know for a fact that y is
going to be not equal to x.

00:08:11.010 --> 00:08:13.680
And so this is also true.

00:08:13.680 --> 00:08:19.230
So choices I and III, and
that is choice D.

00:08:19.230 --> 00:08:22.430
We are done with this section,
because there's only 16

00:08:22.430 --> 00:08:23.850
problems as far as I can tell.

00:08:23.850 --> 00:08:25.480
Yep, only 16 problems. Actually,
I think we're done

00:08:25.480 --> 00:08:27.890
with all the math sections
in this test

00:08:27.890 --> 00:08:29.240
number 2 in the book.

00:08:29.240 --> 00:08:32.120
I will see you in the next
test. See you soon.