SAT Prep: Test 7 Section 8 Part 3

WEBVTT
Kind: captions
Language: en

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Welcome back.

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I'm sorry.

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I actually had to take a
five-second pause because

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something came up, but
let me continue.

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Problem number 10.

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So let's see, they say that
this is x degrees.

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So there's a formula for the
angles inside of a polygon,

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but what I'm going to do is
derive it for you, because

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frankly, on an exam you can--
well, later in life, it's very

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easy to forget everything.

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So what I do is, we know that
all of the angles are

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congruent, right?

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So if we know that this is--
let's call this angle y, then

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this angle is going to be y,
this angle going to be y, that

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is y, that's y, that's y, that's
y, that's y, that's y,

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that's y, right?

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All of these angles
are the same.

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And what do we know about
these angles here?

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Let's called these
z, z, z, z, z.

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There are five of these z angles
and what are they going

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to add up to be?

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Well, they go around
in a circle, right?

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So these five z angles
are going to have to

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add up to 360 degrees.

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So we could say 5z is equal
to 360 degrees.

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z is equal to-- let's see, 5
goes into 360, 7, 35, 10, 72.

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So z is equal to 72 degrees.

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So each of these z's
is 72 degrees.

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That angle of 72 degrees,
what are the y's?

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Well, y plus y plus z has
to be 180, right?

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So you have y plus y plus 72
is equal to 180 degrees.

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2y plus 72 is equal to 180.

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2y is equal to what?

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108.

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And actually we could
solve for y.

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y is 54.

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But really we just want
to know what 2y is

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because look at this.

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We're trying to solve for x.

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x plus 2y is 180, right?

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How did I get that?

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x is supplementary
with y and y.

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This is 2y right here and
altogether they are

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supplementary.

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So that's where I get
x plus 2y is 180.

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We know what 2y is.

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It's 108.

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x plus 108 is equal to 180.

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x is equal to-- and actually,
it's interesting. x is going

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to be equal to z, right?

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x is 72 degrees.

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And that is choice C.

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And that actually is
interesting that x

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is the same as z.

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And actually, when you think
about it, it makes sense

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because y plus y plus z is 180,
and here, y plus y plus x

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has to be 180.

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So x is going to
be equal to z.

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You say there are 360 degrees
in the circle.

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Divide by the number of z's
there are, one, two, three,

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four, five, you get 72.

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Next problem.

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Problem 11.

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The length of a drawing of a
tool is 3/8 of the length of

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the actual tool.

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OK, if the length of the drawing
of the tool is 6

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inches, what is the length in
inches of the actual tool?

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OK, 6 inches is equal to 3/8
times the actual tool.

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Let's multiply both sides
of this equation by the

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reciprocal of this right here.

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So let's say 8/3 times 6 is
equal to 8/3 times 3/8a.

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This, of course, cancels out.

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That's why I multiplied
by the reciprocal.

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And this becomes a 2,
this becomes a 1.

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8 times 2 is 16.

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So the actual tool
is 16 inches.

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And that's choice C.

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Next problem.

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Problem 12.

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I might have space
to do it here.

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If x plus 3/2 is an integer-- so
they're saying x plus 3 is

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divisible by 2, right?

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Because in order for this to be
an integer-- and so what do

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we know about x?

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Well, x plus 3 has to be
even number, right?

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Because it's divisible by 2.

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So that must mean that x has
to be an odd number.

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How do I know that?

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So I know that x
plus 3 is even.

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If x plus 3 is even, then
I know that x is odd.

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Why is that?

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Because 3 is odd.

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The only way, if I had one
number plus an odd and I get

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an even, this number has
to be odd as well.

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2 odds added together equal
an even, right?

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For example, 5 plus 3 is
8, or 11 plus 3 is 14.

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So an odd plus an
odd is an even.

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So x has to be odd.

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So when we look at the choices,
we have choice E.

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x is an odd integer.

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That's the only thing that we
can really assume based on

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this, and that's because x
plus 3 has to be even.

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And anything plus an odd and
you get an even, that thing

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you added to the odd has
to be odd as well.

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And you can try it
out with numbers.

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Next problem.

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Problem 13.

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All right.

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In the x, y plane above, points
q and s are the centers

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of the circle.

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Which are tangent
to the x-axis?

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Let me see if I can draw this.

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So this is the y-axis.

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This is the x-axis.

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And now they draw a
couple of circles.

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A small one and a big one.

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Let's see if I can draw it.

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The small circle looks
something like that.

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And the big circle doesn't
touch the small circle.

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So the big circle looks
something like that.

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Fair enough.

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Let's see, this is point Q.

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This is point S and those
are the centers.

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And they tell us that this point
right here is 3 comma 6,

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and this point up here
is 11 comma 10.

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And what else?

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This is point R.

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this is point P, QS, oh, and
then one last thing, I'm

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drawing this line here
and a line here.

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In both cases, they say this
is a perpendicular.

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Let's do the problem.

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In the x, y plane above, points
Q and S are the centers

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of the circle, which are tangent
to the x-axis, right?

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They just barely touch the
x-axis right here.

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That's the x-axis.

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What is the slope of line QS?

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So what is the slope
of this line?

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Line QS?

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And it could go further out like
that, go further in this

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direction like that.

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So we really just have
to figure out the

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coordinates Q and S.

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So Q is the center of this, so
it's tangent to the x-axis.

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This entire diameter has what?

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Has height what?

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Height 6.

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So this the point 3.

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This is 3, 6, so what's
this Q going to be?

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The x is definitely going to
still be 3 and the y is going

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to be halfway to
this top point.

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Well, what's halfway to 6?

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3.

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Similarly, this is point 11
comma 10, so the x-coordinate

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is 11 and this y-coordinate
is to 10.

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So what's the coordinate
of S going to be?

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The x-coordinate is, of course,
still going to be 11.

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And the y-coordinate is going
to be halfway to 10.

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What's halfway to 10?

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It's 5.

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So what's the slope?

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Slope is change in y
over change in x.

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So let's take the y's.

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It's 5 minus 3 over
11 minus 3.

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That equals 2/9.

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No, sorry, 2/8.

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I always mess up
on that stuff.

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11 minus 3 is 8.

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So 2/8 and that equals 1/4.

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And that is choice D.

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I'll see you in the
next video.


SAT Prep: Test 7 Section 5 Part 2

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Kind: captions
Language: en

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We're in problem number seven.

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Let me switch colors.

00:00:05.580 --> 00:00:10.330
The average of the weights
of 14 books is p pounds.

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In terms of p, what is
the total weight of

00:00:12.080 --> 00:00:13.470
the books in pounds?

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So this comes up all the
time in the SAT.

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The average of weights of
14 books is p pounds.

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So if you had weight 1, plus
weight 2, plus-- if I were to

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add all 14 books, that's just
the weights of all the 14

00:00:30.300 --> 00:00:37.210
books, and I were to divide
by 14, that is equal to p.

00:00:37.210 --> 00:00:38.310
This is just the definition
of average.

00:00:38.310 --> 00:00:40.690
You add up the weights of all
the books, divide by the

00:00:40.690 --> 00:00:42.680
number of books, and you
get the average.

00:00:42.680 --> 00:00:46.500
Well, if you multiply both sides
by 14 you get w1-- you

00:00:46.500 --> 00:00:51.390
get the sum of all of the
weights of the books is equal

00:00:51.390 --> 00:00:54.100
to what? p times 14.

00:00:54.100 --> 00:00:55.760
And what are they asking?

00:00:55.760 --> 00:00:58.080
In terms of p, what is
the total weight

00:00:58.080 --> 00:00:58.970
of all of the books?

00:00:58.970 --> 00:01:01.260
Well, this is the total weight
of all of the books, right?

00:01:01.260 --> 00:01:03.350
I just take the weight of the
first one, plus the weight of

00:01:03.350 --> 00:01:04.980
the second one, and keep adding
until I get to the

00:01:04.980 --> 00:01:08.380
weight of the 14th, and that
is equal to p times 14.

00:01:08.380 --> 00:01:10.700
And that is choice E.

00:01:10.700 --> 00:01:13.000
14p.

00:01:13.000 --> 00:01:14.770
Choice E.

00:01:14.770 --> 00:01:15.860
Next problem.

00:01:15.860 --> 00:01:17.110
Problem eight.

00:01:23.036 --> 00:01:24.286
Draw the axes.

00:01:34.990 --> 00:01:37.820
This is x and y.

00:01:37.820 --> 00:01:41.190
And then they draw-- let me do
it in a different color.

00:01:41.190 --> 00:01:47.460
They have point A, which
is 2, negative 1.

00:01:47.460 --> 00:01:49.010
So it's 2 comma negative 1.

00:01:49.010 --> 00:01:51.970
So this is the point
x is equal to 2.

00:01:51.970 --> 00:01:54.560
If we were to go down, this
is y equal negative 1.

00:01:54.560 --> 00:02:01.330
And then they have point
B, which is k,t.

00:02:01.330 --> 00:02:07.450
And then they have point C
up here-- C is 2 comma 5.

00:02:07.450 --> 00:02:11.080
So it's 2 comma and this is 5.

00:02:11.080 --> 00:02:13.530
Point B is the midpoint of AC.

00:02:13.530 --> 00:02:16.830
So point B is this
midpoint of AC.

00:02:16.830 --> 00:02:19.670
So this distance is the
same as that distance.

00:02:19.670 --> 00:02:22.250
What is the value of t?

00:02:22.250 --> 00:02:23.190
So what's its y value?

00:02:23.190 --> 00:02:26.540
We know what k is, k's going
to have to be 2.

00:02:26.540 --> 00:02:27.790
So what's the y value?

00:02:27.790 --> 00:02:29.780
So the y value's essentially
just going to be the average

00:02:29.780 --> 00:02:31.980
of this y value and that y
value, because it's right in

00:02:31.980 --> 00:02:33.380
the middle of the two.

00:02:33.380 --> 00:02:39.680
So it's going to be 5
plus minus 1 over 2,

00:02:39.680 --> 00:02:41.070
which is equal to what?

00:02:41.070 --> 00:02:44.060
That's equal to 4 over 2,
which is equal to 2.

00:02:44.060 --> 00:02:47.090
So this point, right here,
is going to be 2 comma 2.

00:02:47.090 --> 00:02:49.910
And all I did is I took the
average of the y values.

00:02:49.910 --> 00:02:52.610
The other way you could have
thought about it is, what is

00:02:52.610 --> 00:02:54.000
this total distance?

00:02:54.000 --> 00:02:55.600
What's the distance from
5 to negative 1?

00:02:55.600 --> 00:02:57.010
Well, that's 6.

00:02:57.010 --> 00:02:58.435
So this distance would be 3.

00:02:58.435 --> 00:03:00.330
And what's 5 minus 3?

00:03:00.330 --> 00:03:01.690
It's 2 as well.

00:03:01.690 --> 00:03:04.340
But the easiest way is just to
average this y value and this

00:03:04.340 --> 00:03:07.330
y value, and you'll get the
y value for the midpoint.

00:03:07.330 --> 00:03:10.920
And that is choice C.

00:03:10.920 --> 00:03:12.170
Next problem.

00:03:17.160 --> 00:03:19.100
Problem nine.

00:03:19.100 --> 00:03:22.130
So far I didn't have to draw
anything, this is pretty good.

00:03:22.130 --> 00:03:33.130
If k times 2x plus 3, times x
minus 1 is equal to 0, and

00:03:33.130 --> 00:03:35.960
they also tell us that x is
greater than 1, what is the

00:03:35.960 --> 00:03:39.370
value of k?

00:03:39.370 --> 00:03:43.250
So let's see, if for this times
this, times this to

00:03:43.250 --> 00:03:46.770
equal 0, at least one of them
has to be equal to 0.

00:03:46.770 --> 00:03:50.530
Maybe more than one,
maybe all of them.

00:03:50.530 --> 00:03:54.670
In order for this term to be 0,
what does x have to equal?

00:03:54.670 --> 00:04:00.225
Well, for x to be 0 here-- for
this term to be 0, sorry-- x

00:04:00.225 --> 00:04:01.470
would have to be equal to 1.

00:04:01.470 --> 00:04:01.980
How did I get that?

00:04:01.980 --> 00:04:03.580
I said x minus 1
is equal to 0.

00:04:03.580 --> 00:04:05.640
Add 1 to both sides,
x is equal to 1.

00:04:05.640 --> 00:04:08.030
Well, they told us that x
is greater than 1, so x

00:04:08.030 --> 00:04:09.490
cannot equal 1.

00:04:09.490 --> 00:04:11.460
So this cannot happen.

00:04:11.460 --> 00:04:13.910
That cannot happen, because
x is greater than 1.

00:04:13.910 --> 00:04:16.050
So we know that this is not 0.

00:04:16.050 --> 00:04:18.850
This term is not-- this might
not equal-- that's

00:04:18.850 --> 00:04:20.480
not equal to 0.

00:04:20.480 --> 00:04:24.830
What has to be a value for
x for this term to be 0?

00:04:24.830 --> 00:04:27.630
2x plus 3 equals 0.

00:04:27.630 --> 00:04:29.350
Subtract 3 from both sides.

00:04:29.350 --> 00:04:30.960
2x is equal to minus 3.

00:04:30.960 --> 00:04:33.500
x is equal to minus 3/2.

00:04:33.500 --> 00:04:37.650
Once again, they tell us that
x has to be greater than 1.

00:04:37.650 --> 00:04:40.950
If x is greater than 1, we know
that x cannot be minus

00:04:40.950 --> 00:04:43.760
3/2, because obviously
this is less than 1.

00:04:43.760 --> 00:04:47.750
So once again, we know that this
term does not equal 0.

00:04:47.750 --> 00:04:51.650
So if this term doesn't equal 0,
this term doesn't equal 0.

00:04:51.650 --> 00:04:54.770
But when I multiply all three
of these terms I get 0.

00:04:54.770 --> 00:04:58.420
This k has to equal 0.

00:04:58.420 --> 00:05:00.360
This is non-zero, this
is non-zero.

00:05:00.360 --> 00:05:03.310
If I'm getting 0 when I multiply
them, k has to be 0,

00:05:03.310 --> 00:05:07.150
and that's choice B.

00:05:07.150 --> 00:05:09.540
Next problem.

00:05:09.540 --> 00:05:10.390
Problem number 10.

00:05:10.390 --> 00:05:13.505
Let's see if I have enough
space to do it.

00:05:13.505 --> 00:05:14.755
I should just clear
it every time.

00:05:17.510 --> 00:05:21.650
If all men in the Williams
family are over 6 feet tall--

00:05:21.650 --> 00:05:24.880
that's not true of my family, of
the Khan family; maybe one

00:05:24.880 --> 00:05:26.840
day if we eat right-- which
of the following

00:05:26.840 --> 00:05:28.060
statements must be true?

00:05:28.060 --> 00:05:30.570
If all of the men in the
Williams' family are over 6

00:05:30.570 --> 00:05:31.920
feet tall, which of
the following

00:05:31.920 --> 00:05:33.490
statements must be true?

00:05:33.490 --> 00:05:38.000
Choice A, no man under 6 foot
tall is a member of the

00:05:38.000 --> 00:05:40.180
Williams family.

00:05:40.180 --> 00:05:41.650
Sure.

00:05:41.650 --> 00:05:43.610
That sounds good to me.

00:05:43.610 --> 00:05:47.700
Choice A, no man under 6 feet
tall is a member of the

00:05:47.700 --> 00:05:48.850
Williams family.

00:05:48.850 --> 00:05:51.730
For example, I am 5 foot 9.

00:05:51.730 --> 00:05:54.210
If I was a member of the
Williams family, then the

00:05:54.210 --> 00:05:57.030
first statement could not be
have been said, that all men

00:05:57.030 --> 00:06:00.795
in the Williams family
are over 6 feet tall.

00:06:00.795 --> 00:06:03.190
So that makes sense, all men
in the Williams family are

00:06:03.190 --> 00:06:04.920
over 6 feet tall.

00:06:04.920 --> 00:06:07.750
So that means that no man under
6 foot tall is a member

00:06:07.750 --> 00:06:10.010
of the Williams family, because
if there were then you

00:06:10.010 --> 00:06:11.930
couldn't have said that all men
in the Williams family are

00:06:11.930 --> 00:06:13.810
over 6 feet tall.

00:06:13.810 --> 00:06:15.880
That probably required the most
talking and the least

00:06:15.880 --> 00:06:17.680
writing on my behalf,
but hopefully that's

00:06:17.680 --> 00:06:20.000
a convincing answer.

00:06:20.000 --> 00:06:20.720
Problem 11.

00:06:20.720 --> 00:06:23.240
Let me know if it wasn't.

00:06:23.240 --> 00:06:28.410
What is the radius of a circle
that has circumference of pi.

00:06:28.410 --> 00:06:33.820
So we know that circumference
is equal 2 pi r.

00:06:33.820 --> 00:06:36.440
And they're telling us that
the circumference is pi.

00:06:36.440 --> 00:06:39.870
So pi is equal to 2 pi r.

00:06:39.870 --> 00:06:43.660
Divide both sides by 2
pi, you get pi over 2

00:06:43.660 --> 00:06:45.980
pi is equal to radius.

00:06:45.980 --> 00:06:47.510
Divide the numerator and
the denominator by

00:06:47.510 --> 00:06:49.550
pi, you get 1, 1.

00:06:49.550 --> 00:06:55.380
So you get 1/2 is equal to the
radius, and that is choice B.

00:06:55.380 --> 00:06:57.590
Pretty straightforward, eh?

00:06:57.590 --> 00:07:05.270
Problem 12.

00:07:05.270 --> 00:07:09.950
If y is directly proportional
to x squared and y equals--

00:07:09.950 --> 00:07:12.440
OK, so when I say directly
proportional, that means y is

00:07:12.440 --> 00:07:15.975
equal to some constant, we don't
know what it is, y is

00:07:15.975 --> 00:07:18.620
equal to some constant
times x squared.

00:07:18.620 --> 00:07:20.150
That's what directly
proportional means.

00:07:20.150 --> 00:07:22.540
It means it's some constant
times x squared.

00:07:22.540 --> 00:07:25.740
y is directly proportional to
x squared, and y is equal to

00:07:25.740 --> 00:07:27.690
1/8 when x is equal to 1/2.

00:07:27.690 --> 00:07:35.110
So y is equal to 1/8 is equal
to k times when x is 1/2.

00:07:35.110 --> 00:07:37.910
So when x is 1/2, that's
all we're saying.

00:07:37.910 --> 00:07:38.760
So what does this tell us?

00:07:38.760 --> 00:07:43.340
That means that 1/8 is
equal to k times 1/4.

00:07:43.340 --> 00:07:49.590
Multiply both sides by 4, times
4, you get 4/8 is equal

00:07:49.590 --> 00:07:51.570
to k, because this
cancels out.

00:07:51.570 --> 00:07:53.890
And 4/8 is the same
thing as 1/2.

00:07:53.890 --> 00:07:59.880
So the relationship is, y is
equal to 1/2 x squared, and

00:07:59.880 --> 00:08:01.720
now what are they asking us?

00:08:01.720 --> 00:08:06.200
What is the positive value of
x when y is equal to 9/2?

00:08:06.200 --> 00:08:08.786
So they're saying when y equals
9/2, let's solve for x

00:08:08.786 --> 00:08:10.150
in the positive value.

00:08:10.150 --> 00:08:13.130
9/2 is equal to 1/2 x squared.

00:08:13.130 --> 00:08:16.870
Multiply both sides by 2,
this 2 cancels here,

00:08:16.870 --> 00:08:18.006
this 2 cancels here.

00:08:18.006 --> 00:08:19.410
I was multiplying.

00:08:19.410 --> 00:08:22.090
So you get 9 is equal
to x squared.

00:08:22.090 --> 00:08:23.670
So they want the positive
value of x.

00:08:23.670 --> 00:08:27.320
So x is equal to plus
or minus 3.

00:08:27.320 --> 00:08:30.100
And they want the positive
value, so x is equal to

00:08:30.100 --> 00:08:31.880
positive 3.

00:08:31.880 --> 00:08:35.260
And that's choice D.

00:08:35.260 --> 00:08:36.510
Choice D.

00:08:39.710 --> 00:08:41.120
Let me do the next problem
in the next video.

00:08:41.120 --> 00:08:42.789
I'll see you soon.


SAT Prep: Test 7 Section 5 Part 1

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.370
Problem number 1.

00:00:03.370 --> 00:00:09.400
It gives a series--
2, 6, 14, 30.

00:00:09.400 --> 00:00:12.770
And they say in the sequence
above the first term is 2.

00:00:12.770 --> 00:00:15.540
Each number after the first if
obtained by adding 1 to the

00:00:15.540 --> 00:00:18.150
preceding number and then
doubling the result.

00:00:18.150 --> 00:00:20.840
So you add 1, so you
get 3, and then you

00:00:20.840 --> 00:00:22.240
double it you get 6.

00:00:22.240 --> 00:00:25.230
You add 1 to 6 to 7, right, this
is plus 1 and then you

00:00:25.230 --> 00:00:26.500
multiply it by 2.

00:00:26.500 --> 00:00:29.030
You get 7 times 2 is 14.

00:00:29.030 --> 00:00:32.770
14 plus 1 is 15 times 2 is 30.

00:00:32.770 --> 00:00:34.380
And they want the sixth term.

00:00:34.380 --> 00:00:42.730
So then 30 plus 1 is 31 times
2 is 62 and then you add 1.

00:00:42.730 --> 00:00:45.290
We add 1 first. So
you get to 63.

00:00:45.290 --> 00:00:46.875
And then you multiply
that times 2.

00:00:46.875 --> 00:00:49.140
That's what, 126.

00:00:49.140 --> 00:00:52.660
So that is choice E.

00:00:52.660 --> 00:00:53.910
Problem 2.

00:00:56.250 --> 00:01:06.540
If a times x plus y is equal to
45, and ax is equal to 15,

00:01:06.540 --> 00:01:09.290
what is ay equal?

00:01:09.290 --> 00:01:10.630
So let's just distribute
this a here.

00:01:10.630 --> 00:01:15.640
You get ax plus ay
is equal to 45.

00:01:15.640 --> 00:01:18.350
They tell us ax is
equal to 15.

00:01:18.350 --> 00:01:20.730
So that is equal to 15--
ax is equal to 15.

00:01:20.730 --> 00:01:24.790
So you get 15 plus ay
is equal to 45.

00:01:24.790 --> 00:01:30.590
Subtract 15 from both sides and
you get ay is equal to 30.

00:01:30.590 --> 00:01:31.210
That's our answer.

00:01:31.210 --> 00:01:32.200
They want to know what ay is.

00:01:32.200 --> 00:01:34.820
You don't have to solve for
a or y, just ay, so

00:01:34.820 --> 00:01:37.640
that's choice E.

00:01:37.640 --> 00:01:38.890
Next problem.

00:01:40.890 --> 00:01:43.460
Before drawing this let me read
it so I can figure out

00:01:43.460 --> 00:01:44.560
what I have to draw.

00:01:44.560 --> 00:01:47.780
On the speedometer above what is
the speed in miles per hour

00:01:47.780 --> 00:01:50.580
indicated by the needle
position?

00:01:50.580 --> 00:01:54.010
So let me see if I
can draw this.

00:01:54.010 --> 00:01:59.860
So they give the bottom one
like that, that's given in

00:01:59.860 --> 00:02:02.580
feet per second.

00:02:02.580 --> 00:02:06.540
And then the next one
up is this one,

00:02:06.540 --> 00:02:09.960
that's miles per hour.

00:02:09.960 --> 00:02:14.170
And then the top one is
kilometers per hour.

00:02:14.170 --> 00:02:16.340
I'm going to write it on this
side, although they write it

00:02:16.340 --> 00:02:18.640
up here-- kilometers per hour.

00:02:18.640 --> 00:02:24.800
And then they indicate-- and on
the miles per hour thing--

00:02:24.800 --> 00:02:27.700
I don't know what you actually
call this measurement.

00:02:27.700 --> 00:02:32.290
So there's 30 and then there's
one slash, two slash, three

00:02:32.290 --> 00:02:35.350
slash, and then they go to 60.

00:02:35.350 --> 00:02:39.720
And then the line that they
draw is right here.

00:02:39.720 --> 00:02:41.010
It goes like that.

00:02:45.450 --> 00:02:48.550
And they say in the speedometer
above what is the

00:02:48.550 --> 00:02:50.460
speed in miles per
hour indicated

00:02:50.460 --> 00:02:51.510
by the needle position.

00:02:51.510 --> 00:02:55.290
And the reason why I didn't do
much detail on this line or on

00:02:55.290 --> 00:02:58.090
this line is because this one's
kilometers per hour and

00:02:58.090 --> 00:02:58.880
this is feet per second.

00:02:58.880 --> 00:03:00.890
So I don't care about those, I
just care about the miles per

00:03:00.890 --> 00:03:03.970
hour, and that is this one.

00:03:03.970 --> 00:03:05.520
And let's see, they
drew it here.

00:03:05.520 --> 00:03:07.690
So how big is each
of these slashes?

00:03:07.690 --> 00:03:09.510
1, 2, 3, 4.

00:03:09.510 --> 00:03:16.460
So each of these go 1/4 of the
distance between 30 and 60.

00:03:16.460 --> 00:03:18.200
Each of them goes 1/4
of the distance.

00:03:18.200 --> 00:03:21.200
So you could say, for example,
this is halfway between 30 and

00:03:21.200 --> 00:03:24.540
60, so this is going to be 45.

00:03:24.540 --> 00:03:26.940
And so what's halfway between
30 and 45, which is

00:03:26.940 --> 00:03:28.490
where the thing is?

00:03:28.490 --> 00:03:31.920
Well 45 is 15 more than 30, so
this is going to be 7 and 1/2

00:03:31.920 --> 00:03:33.050
more than 30.

00:03:33.050 --> 00:03:37.040
So this point right here
is going to be 37.5.

00:03:37.040 --> 00:03:37.820
And how do I know that?

00:03:37.820 --> 00:03:42.060
If you just say, well if I just
take 30 divided by 4-- 1,

00:03:42.060 --> 00:03:46.050
2, 3, 4-- each of these are
going to be 7.5-- right, 30

00:03:46.050 --> 00:03:48.940
divided by 4 is equal to 7.5.

00:03:48.940 --> 00:03:50.520
Or you could say well this
middle one's definitely going

00:03:50.520 --> 00:03:54.120
to be 45 because 45 is right
in between 30 and 60.

00:03:54.120 --> 00:03:57.360
And then what's right in
between 30 and 45?

00:03:57.360 --> 00:03:59.760
Well that's 37.5 again.

00:03:59.760 --> 00:04:02.320
So the speedometer's indicating
37.5 miles per

00:04:02.320 --> 00:04:05.160
hour, that's choice B.

00:04:05.160 --> 00:04:08.560
Next problem, problem 4.

00:04:08.560 --> 00:04:12.390
How many different positive
three-digit integers can be

00:04:12.390 --> 00:04:17.910
formed if the three digits, 4,
5 and 6, must be used in each

00:04:17.910 --> 00:04:19.160
of the integers.

00:04:22.580 --> 00:04:25.220
So they're essentially saying
how many combinations can you

00:04:25.220 --> 00:04:27.480
get of the number 456?

00:04:27.480 --> 00:04:31.310
So I could write out all those
combinations if you like.

00:04:31.310 --> 00:04:35.870
I mean you could 456, 465--
right, those are all of the

00:04:35.870 --> 00:04:37.650
ones that if you have
4 in front.

00:04:37.650 --> 00:04:41.670
Then if you have 5 in front,
you have 546 or

00:04:41.670 --> 00:04:44.180
you can have 564.

00:04:44.180 --> 00:04:45.210
I just switched them.

00:04:45.210 --> 00:04:50.920
And if you have 6 in front you
could have 645 or 654.

00:04:50.920 --> 00:04:52.170
So those are 6.

00:04:52.170 --> 00:04:53.880
The other way I think
about it is I have

00:04:53.880 --> 00:04:55.270
three numbers, right?

00:04:55.270 --> 00:04:57.860
I can put one of three in
the first position.

00:04:57.860 --> 00:04:59.880
I can put one of two in
the second position.

00:04:59.880 --> 00:05:03.250
And then I would have one left
over in the last position.

00:05:03.250 --> 00:05:04.790
I'd have three choices
to put in the first

00:05:04.790 --> 00:05:06.680
number, 4, 5 or 6.

00:05:06.680 --> 00:05:08.890
Then after I put one in that
first position I have two

00:05:08.890 --> 00:05:12.300
left, so there's two more
possibilities, and then I have

00:05:12.300 --> 00:05:14.290
one left over for the
third position.

00:05:14.290 --> 00:05:15.410
So you could also think
of it this way.

00:05:15.410 --> 00:05:16.520
You could just write
them out-- 1,

00:05:16.520 --> 00:05:18.440
2, 3, 4, 5, 6 numbers.

00:05:18.440 --> 00:05:21.360
Or you could use that latter
technique I just talked about

00:05:21.360 --> 00:05:23.180
and it would be 3 times
2 times 1, which is

00:05:23.180 --> 00:05:25.720
also equal to 6.

00:05:25.720 --> 00:05:26.970
Next problem.

00:05:29.660 --> 00:05:31.120
Let me draw what they
have drawn.

00:05:36.400 --> 00:05:41.570
Well I'll try my best. This
line goes like this.

00:05:48.160 --> 00:05:50.270
And then there's
a dotted line.

00:05:50.270 --> 00:05:51.610
Let me just see if
I can do that.

00:06:00.190 --> 00:06:02.490
The hard part of these problems
is drawing them.

00:06:02.490 --> 00:06:04.380
Good thing you don't have
to do that on the SAT.

00:06:04.380 --> 00:06:07.490
The three-dimensional figure
represented above consists of

00:06:07.490 --> 00:06:09.520
rectangular and triangle
faces.

00:06:09.520 --> 00:06:11.270
Fair enough, triangular faces.

00:06:11.270 --> 00:06:14.720
Each rectangular face
has area r.

00:06:14.720 --> 00:06:17.430
So this is a rectangular face
here, there's three

00:06:17.430 --> 00:06:19.110
rectangular faces.

00:06:19.110 --> 00:06:24.190
So area of the rectangle
is equal to r.

00:06:24.190 --> 00:06:25.980
And each triangular
face is area t.

00:06:25.980 --> 00:06:29.130
So area of each of the triangle
is equal to t.

00:06:29.130 --> 00:06:31.350
What is a total surface
area of the figure in

00:06:31.350 --> 00:06:32.410
terms of r and t?

00:06:32.410 --> 00:06:35.510
So how many rectangular
surfaces are there?

00:06:35.510 --> 00:06:37.470
Well there's this one in back.

00:06:40.660 --> 00:06:42.350
It's kind of behind the thing.

00:06:42.350 --> 00:06:46.550
You have this base that
I'll drawn in magenta.

00:06:46.550 --> 00:06:48.610
This is the base rectangular
surface.

00:06:48.610 --> 00:06:51.510
And then you have this one
that's closest to us, which I

00:06:51.510 --> 00:06:54.760
could-- this one-- that's the
closest. So you have three

00:06:54.760 --> 00:06:56.560
rectangular surfaces.

00:06:56.560 --> 00:06:59.895
So it's 3 times r, because each
rectangular surface has

00:06:59.895 --> 00:07:01.110
an area or r.

00:07:01.110 --> 00:07:02.230
And then how many of
the triangular

00:07:02.230 --> 00:07:03.130
surfaces do you have?

00:07:03.130 --> 00:07:06.020
You have the one in the back,
back here, and then you have

00:07:06.020 --> 00:07:08.210
the one of the front
right here.

00:07:08.210 --> 00:07:10.220
So you have two triangular
surfaces.

00:07:10.220 --> 00:07:15.010
So it's 3 times the rectangular
areas plus 2 times

00:07:15.010 --> 00:07:17.770
the triangular areas because
there's two triangular areas.

00:07:17.770 --> 00:07:21.980
So 3r plus 2t, and
that's choice B.

00:07:21.980 --> 00:07:23.230
Next problem.

00:07:25.730 --> 00:07:28.150
Problem 6.

00:07:28.150 --> 00:07:35.100
If n is a positive integer and
n plus 1 over 2 to the n is

00:07:35.100 --> 00:07:42.380
equal to 1/2, n equals what?

00:07:42.380 --> 00:07:44.530
So let's think about
how we can do this.

00:07:44.530 --> 00:07:49.110
Let's multiply both sides of
this equation by 2 to the n.

00:07:49.110 --> 00:07:51.130
So if you multiply both sides of
the equation times 2 to the

00:07:51.130 --> 00:07:59.090
n, you get n plus 1 is equal
to 1/2 times 2 to the n.

00:07:59.090 --> 00:08:01.940
Let me think about this
a little bit.

00:08:01.940 --> 00:08:03.210
Let's see, 1/2 times
2 to the n.

00:08:03.210 --> 00:08:07.150
1/2 is the same thing as
2 to the negative 1.

00:08:07.150 --> 00:08:14.000
So this is n plus 1 is equal
to-- let me think about this.

00:08:14.000 --> 00:08:21.920
n plus 1 is equal to 2 to the
negative 1 times 2 to the n.

00:08:21.920 --> 00:08:25.680
So n plus 1 is equal
to 2-- sorry.

00:08:25.680 --> 00:08:28.260
2 to the n minus 1.

00:08:28.260 --> 00:08:29.430
Huh.

00:08:29.430 --> 00:08:30.950
So that's where I can get it.

00:08:30.950 --> 00:08:33.090
Really at this point, the best
thing I can think of is just

00:08:33.090 --> 00:08:35.600
trying out the choices and
see which one works.

00:08:35.600 --> 00:08:40.080
If you look at choice A, 1--
actually, we could just do it

00:08:40.080 --> 00:08:41.679
from the original,
that's faster.

00:08:41.679 --> 00:08:45.850
1 plus 1 over 2 to the n is
equal to 2 over 2 to the 1,

00:08:45.850 --> 00:08:46.820
which is equal to 1.

00:08:46.820 --> 00:08:49.790
So it's not 1/2, so it's
not a equals 1.

00:08:49.790 --> 00:08:51.970
Choice B is 2.

00:08:51.970 --> 00:08:57.950
So it would be 2 plus 1, which
is 3 over 2 squared, which

00:08:57.950 --> 00:08:59.510
equals 3/4.

00:08:59.510 --> 00:09:00.700
So that's not the answer.

00:09:00.700 --> 00:09:04.230
If you look at choice
C, choice C is 3.

00:09:04.230 --> 00:09:10.450
So 3 plus 1 is 4 over
2 to the third.

00:09:10.450 --> 00:09:13.670
That's 4 over 8 equals 1/2.

00:09:13.670 --> 00:09:15.260
So it's choice C.

00:09:15.260 --> 00:09:16.780
And I'm trying to figure out
if there's a way that you

00:09:16.780 --> 00:09:18.900
could solve for that easier
other than trying out the

00:09:18.900 --> 00:09:22.500
numbers, but for some reason
it's not popping into my head.

00:09:22.500 --> 00:09:25.010
Let me know a message if
you figure it out.

00:09:25.010 --> 00:09:27.170
But I also do these under
time pressure.

00:09:27.170 --> 00:09:30.450
But I will see you in
the next video.


SAT Prep: Test 7 Section 5 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.830
We're on problem 13.

00:00:03.830 --> 00:00:11.520
And they say if 4x is equal to
6u, which is equal to 5v,

00:00:11.520 --> 00:00:17.670
which is equal to 7w, all of
which is greater than 0, which

00:00:17.670 --> 00:00:20.210
of the following is true.

00:00:20.210 --> 00:00:24.243
So they're essentially wanting
you to order w to-- they want

00:00:24.243 --> 00:00:27.670
you to order these terms in
order from least to greatest.

00:00:27.670 --> 00:00:30.710
So one thing I'm going to do,
let me order these terms, the

00:00:30.710 --> 00:00:32.900
coefficients from least to
greatest. That's the same

00:00:32.900 --> 00:00:37.330
thing as 4x is equal to 5v,
which is equal to 6u, which is

00:00:37.330 --> 00:00:38.840
equal to 7w.

00:00:38.840 --> 00:00:41.290
They're all greater than 0.

00:00:41.290 --> 00:00:45.290
So what's going to be
greater, x or w?

00:00:45.290 --> 00:00:48.350
So if I have 4 of something
and that equals 7 of this

00:00:48.350 --> 00:00:52.090
thing, this is going to be
the smallest, right?

00:00:52.090 --> 00:00:53.340
Think of it this way.

00:00:55.590 --> 00:01:00.990
If x was 7, then w would be 4.

00:01:00.990 --> 00:01:01.330
Try that out.

00:01:01.330 --> 00:01:04.660
If x was 7 and this term would
be 28, the w would be 4.

00:01:04.660 --> 00:01:08.920
So w is the smallest. And by
that same principle, u is

00:01:08.920 --> 00:01:09.800
going to be next.

00:01:09.800 --> 00:01:13.780
So w is less than u, which
is less than v, which

00:01:13.780 --> 00:01:14.990
is less than x.

00:01:14.990 --> 00:01:19.790
And it's just on the principle,
if it takes me 7 of

00:01:19.790 --> 00:01:23.080
something to make up the 4 of
something else, this thing has

00:01:23.080 --> 00:01:24.900
to be bigger because I
need fewer of it to

00:01:24.900 --> 00:01:27.000
make 7 of this thing.

00:01:27.000 --> 00:01:30.430
And by the same logic u is less
than v, right, because

00:01:30.430 --> 00:01:33.640
you need 6 of that thing
to make one of v.

00:01:33.640 --> 00:01:39.550
I mean if you really wanted to
kind of try it out you could

00:01:39.550 --> 00:01:44.100
divide all of these by 4 and
you'd get x is equal to 5/4v,

00:01:44.100 --> 00:01:47.710
which is equal to 6/4u, which
is equal to 7/4w.

00:01:47.710 --> 00:01:49.800
All of these terms are
greater than 1.

00:01:49.800 --> 00:01:53.560
So x is more than
all of these.

00:01:53.560 --> 00:01:55.800
And you could do it for all of
these and you would see that

00:01:55.800 --> 00:01:56.380
same order.

00:01:56.380 --> 00:01:58.655
That whichever one has the
highest coefficient is going

00:01:58.655 --> 00:02:01.320
to be the smallest number, and
then the next smallest is

00:02:01.320 --> 00:02:03.700
going to be u, then v, then x.

00:02:03.700 --> 00:02:06.210
So let's see what choice that
is. w is less than u, which is

00:02:06.210 --> 00:02:07.780
less than u, which
is less than x.

00:02:07.780 --> 00:02:10.930
That's choice D.

00:02:10.930 --> 00:02:14.800
Problem 14.

00:02:14.800 --> 00:02:22.210
Let the function h be defined by
h of t is equal to 2 times

00:02:22.210 --> 00:02:26.910
t cubed minus 3.

00:02:26.910 --> 00:02:33.280
When h of t is equal to minus
60, so h of t is equal to

00:02:33.280 --> 00:02:38.730
minus 60, what is 2 minus 3t?

00:02:38.730 --> 00:02:41.280
What is 2 minus 3t.

00:02:41.280 --> 00:02:45.260
So let's see what
we can do here.

00:02:45.260 --> 00:02:47.452
2 minus 3t.

00:02:47.452 --> 00:02:53.710
So h of t is equal to minus 60,
so minus 60 is equal to 2

00:02:53.710 --> 00:02:59.210
times t cubed minus 3.

00:02:59.210 --> 00:03:00.810
Fair enough.

00:03:00.810 --> 00:03:02.410
And then divide both
sides by 2.

00:03:02.410 --> 00:03:04.460
I'm just trying to see where
I can go with this.

00:03:04.460 --> 00:03:06.650
Divide both sides by 2.

00:03:06.650 --> 00:03:13.350
So you get minus 30 is equal
to t cubed minus 3.

00:03:13.350 --> 00:03:20.050
Add 3 to both sides you get
minus 27 is equal to t cubed.

00:03:20.050 --> 00:03:21.450
This is pretty straightforward.

00:03:21.450 --> 00:03:23.960
Something to the third power
is equal to minus 27.

00:03:23.960 --> 00:03:25.920
Well what's the cubed
root of 27?

00:03:25.920 --> 00:03:26.680
It's 3.

00:03:26.680 --> 00:03:29.830
So the cubed root of minus
27 is minus 3.

00:03:29.830 --> 00:03:32.360
So t is equal to minus 3.

00:03:32.360 --> 00:03:33.100
And you could try that out.

00:03:33.100 --> 00:03:35.430
What's minus 3 to
the third power?

00:03:35.430 --> 00:03:39.140
Minus 3 times minus 3 is 9 times
minus 3 is minus 27.

00:03:39.140 --> 00:03:44.340
So when h of t is minus 60,
t is equal to minus 3.

00:03:44.340 --> 00:03:46.810
And so what is 2 minus 3t, which
is what they asked us

00:03:46.810 --> 00:03:47.285
originally?

00:03:47.285 --> 00:03:51.670
It's going to be 2 minus
3 times minus 3.

00:03:51.670 --> 00:03:58.790
So it's 2 minus minus 9 or 2
plus 9, which is equal to 11.

00:03:58.790 --> 00:04:01.910
And that is choice B.

00:04:01.910 --> 00:04:03.160
Next problem.

00:04:07.410 --> 00:04:08.840
I'll switch colors.

00:04:08.840 --> 00:04:11.100
Where was I?

00:04:11.100 --> 00:04:13.770
My brain is fried from doing all
of these SAT problems. But

00:04:13.770 --> 00:04:16.430
I figure I do it once an it'll
be there forever for students

00:04:16.430 --> 00:04:19.589
for the rest of eternity to
learn SAT problems and compete

00:04:19.589 --> 00:04:22.220
with my future children and
they'll be [UNINTELLIGIBLE].

00:04:22.220 --> 00:04:23.630
Problem 15.

00:04:23.630 --> 00:04:28.410
If x is divisible by 3 and y
is divisible by 5, which of

00:04:28.410 --> 00:04:31.630
the following must be
divisible by 15?

00:04:31.630 --> 00:04:37.040
So x divisible by 3, so x is a
multiple of 3-- you can view

00:04:37.040 --> 00:04:38.130
it that way.

00:04:38.130 --> 00:04:42.130
y is the divisible by 5.

00:04:42.130 --> 00:04:45.970
Which of the following must
also be divisible by 15?

00:04:45.970 --> 00:04:46.710
Do you know how I'm
going to do it?

00:04:46.710 --> 00:04:50.400
I'm going to say look, you could
write x is equal to 3k

00:04:50.400 --> 00:04:51.660
where k is some integer.

00:04:51.660 --> 00:04:56.120
We don't know what multiple x
is of 3, but we know 3 times

00:04:56.120 --> 00:04:58.270
some integer, k,
is equal to x.

00:04:58.270 --> 00:05:01.550
And similarly, you could say y
is equal to 5, I don't know,

00:05:01.550 --> 00:05:04.690
5m where you could say
some integer m times

00:05:04.690 --> 00:05:06.550
5 is equal to y.

00:05:06.550 --> 00:05:08.960
Because we know that 5 times
something's equal to y.

00:05:08.960 --> 00:05:10.510
That's what divisible means.

00:05:10.510 --> 00:05:12.520
So let's look at the choices.

00:05:12.520 --> 00:05:13.890
1.

00:05:13.890 --> 00:05:15.090
They want to know which
are the ones that are

00:05:15.090 --> 00:05:16.350
divisible by 15.

00:05:16.350 --> 00:05:19.170
So choice 1 is x times y.

00:05:19.170 --> 00:05:25.540
Well x times y is the same
thing as 3k times 5m.

00:05:25.540 --> 00:05:29.530
And that's the same
thing as 15km.

00:05:29.530 --> 00:05:34.430
So xy is the same thing as 15
times a product of integers,

00:05:34.430 --> 00:05:36.040
and the product of two integers
is going to be an

00:05:36.040 --> 00:05:37.330
integer, right?

00:05:37.330 --> 00:05:39.560
So this is definitely
divisible by 15.

00:05:39.560 --> 00:05:42.720
So choice 1 is divisible
by 15.

00:05:42.720 --> 00:05:43.970
Problem 2.

00:05:47.750 --> 00:05:52.930
3x plus 5y.

00:05:52.930 --> 00:05:55.260
So we could do the same thing.

00:05:55.260 --> 00:06:03.610
x is 3k, so that's 3 times 3k
plus 5y plus 5 times 5m.

00:06:07.350 --> 00:06:12.210
That equals 9k plus 25m.

00:06:12.210 --> 00:06:14.270
This isn't necessarily
divisible by 15.

00:06:14.270 --> 00:06:16.810
I mean what if k and
m are both 1?

00:06:16.810 --> 00:06:20.680
Then you get 34 and-- this
is not divisible by 15.

00:06:20.680 --> 00:06:21.930
Choice 3.

00:06:24.730 --> 00:06:28.700
5x plus 3y.

00:06:28.700 --> 00:06:29.510
Doing the same thing.

00:06:29.510 --> 00:06:33.600
That equals 5 times 3k,
right, this is x.

00:06:33.600 --> 00:06:38.360
5 times 3k plus 3 times 5m.

00:06:38.360 --> 00:06:44.480
And this equals 15k plus 15m.

00:06:44.480 --> 00:06:46.670
An you can factor out the 15.

00:06:46.670 --> 00:06:50.020
That's 15 times k plus m.

00:06:50.020 --> 00:06:54.270
And once again, so this thing is
equal to 15 times k plus m.

00:06:54.270 --> 00:06:55.900
k plus m is going to
be some integer.

00:06:55.900 --> 00:06:57.760
We said k and m are
both integers.

00:06:57.760 --> 00:07:00.420
So this term right here is
definitely divisible by 15.

00:07:00.420 --> 00:07:02.720
So one and three
are our answer.

00:07:02.720 --> 00:07:05.940
And that is choice D.

00:07:05.940 --> 00:07:08.070
And you know, if you didn't want
to do this fancy stuff

00:07:08.070 --> 00:07:09.720
where you say it's multiplying
by some integer,

00:07:09.720 --> 00:07:11.160
et cetera, et cetera.

00:07:11.160 --> 00:07:14.090
Just x is divisible by 3, y is
divisible by 5, well let's

00:07:14.090 --> 00:07:17.810
just say x is equal to 3
and y is equal to 5 and

00:07:17.810 --> 00:07:18.920
then try them out.

00:07:18.920 --> 00:07:22.090
Then xy is going to be 15,
which is, of course,

00:07:22.090 --> 00:07:23.810
divisible by 15.

00:07:23.810 --> 00:07:26.670
This would be, let's see, 3
times 3 plus 5 times 5, this

00:07:26.670 --> 00:07:27.620
would be 34.

00:07:27.620 --> 00:07:29.380
Not divisible by 15.

00:07:29.380 --> 00:07:33.205
And then the final one, 3 times
5 is 15 plus 3 times 5

00:07:33.205 --> 00:07:35.970
is 15, this would be 30, which
is divisible by 15.

00:07:35.970 --> 00:07:37.950
So that would have been the
quick and dirty, not

00:07:37.950 --> 00:07:39.920
necessarily the most
mathematically rigorous way of

00:07:39.920 --> 00:07:42.170
doing it, but it would have
gotten you the right answer.

00:07:42.170 --> 00:07:45.400
And that really is what matters,
I guess, on the SAT.

00:07:49.610 --> 00:07:51.455
That would have been the quick
and dirty solution.

00:07:51.455 --> 00:07:55.900
Let me see if I can
draw this thing.

00:07:55.900 --> 00:07:57.920
One line like that.

00:07:57.920 --> 00:08:00.270
One line like that.

00:08:00.270 --> 00:08:02.900
One line something like that.

00:08:02.900 --> 00:08:06.460
And what do they tell
us about this?

00:08:06.460 --> 00:08:07.750
I'll use yellow.

00:08:07.750 --> 00:08:14.020
They tell us that this angle
right here is 115 degrees.

00:08:14.020 --> 00:08:16.910
This is line l.

00:08:16.910 --> 00:08:20.290
This is z degrees.

00:08:20.290 --> 00:08:24.190
Line m, line n, and
this is y degrees.

00:08:24.190 --> 00:08:26.730
Looks like we're going to have
to play the angle game here.

00:08:26.730 --> 00:08:27.920
This is y degrees.

00:08:27.920 --> 00:08:30.370
In the figure above
what is y plus z?

00:08:34.039 --> 00:08:36.370
So when I play the angle game
I just try to figure out as

00:08:36.370 --> 00:08:37.750
many sides as I can.

00:08:37.750 --> 00:08:41.330
So if this is 115 what
is this going to be?

00:08:41.330 --> 00:08:44.320
Well 115 plus this
supplementary, right?

00:08:44.320 --> 00:08:47.440
So these are going to add up to
be 180, so this is going to

00:08:47.440 --> 00:08:49.380
be 180 minus 115.

00:08:49.380 --> 00:08:51.070
What's 80 minus 15?

00:08:51.070 --> 00:08:54.910
It's 65, right, because
80 minus 10 is 70.

00:08:54.910 --> 00:08:56.010
So this is 65 degrees.

00:08:56.010 --> 00:08:57.720
And I just said that because
this angle plus

00:08:57.720 --> 00:09:00.390
this has to be 180.

00:09:00.390 --> 00:09:03.370
Now what is this angle
right here?

00:09:03.370 --> 00:09:05.760
This angle right here is going
to be 180 minus y for the same

00:09:05.760 --> 00:09:08.650
reason because it's
supplementary to y.

00:09:08.650 --> 00:09:15.330
This angle here, same reason,
it's going to be 180 minus z.

00:09:15.330 --> 00:09:17.690
And we know that this angle
plus this angle plus this

00:09:17.690 --> 00:09:19.680
angle has to add up to 180.

00:09:19.680 --> 00:09:29.060
So 65 plus 180 minus y plus 180
minus z is equal to 180.

00:09:29.060 --> 00:09:35.150
Let's see, so that's-- you could
subtract 180 from both

00:09:35.150 --> 00:09:36.770
sides, you get a 0 here.

00:09:36.770 --> 00:09:38.740
What's 65 plus 180?

00:09:38.740 --> 00:09:47.160
65 plus 180 is 245 minus y
minus z, is equal to 0.

00:09:47.160 --> 00:09:53.110
Add y and z to both sides, you
get 245 is equal to y plus z.

00:09:53.110 --> 00:09:55.780
And that is choice E.

00:09:55.780 --> 00:09:57.600
See you in the next video.


SAT Prep: Test 7 Section 5 Part 4

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:05.590
We're on problem 17.

00:00:05.590 --> 00:00:09.240
The sum of three consecutive
odd integers is 111.

00:00:09.240 --> 00:00:10.920
That's what they tell
us at the top.

00:00:10.920 --> 00:00:11.810
I won't write that.

00:00:11.810 --> 00:00:14.440
If n represents the least of the
three integers, which of

00:00:14.440 --> 00:00:15.900
the following equations
represents

00:00:15.900 --> 00:00:17.010
the statement above?

00:00:17.010 --> 00:00:19.610
So n is the least of the
three integers, so

00:00:19.610 --> 00:00:21.040
let's say that's n.

00:00:21.040 --> 00:00:25.440
So it's the least of three
consecutive odd integers.

00:00:25.440 --> 00:00:27.970
So what's the next odd
integer going to be?

00:00:27.970 --> 00:00:30.080
It's not going to be n plus 1,
because n plus 1 is going to

00:00:30.080 --> 00:00:31.110
be an even integer, right?

00:00:31.110 --> 00:00:34.430
If this was 3 then the next odd
integer's going to be 5,

00:00:34.430 --> 00:00:36.870
which is going to be n plus 2.

00:00:36.870 --> 00:00:41.470
And then the next one above that
is going to be n plus 4.

00:00:41.470 --> 00:00:42.240
And you can try it out.

00:00:42.240 --> 00:00:45.000
Let's say that n was 3, and
this is 5, and this is 7.

00:00:45.000 --> 00:00:47.460
That's three consecutive
odd integers.

00:00:47.460 --> 00:00:50.820
And they're telling us
that the sum is 111.

00:00:50.820 --> 00:00:51.900
So the sum is 111.

00:00:51.900 --> 00:00:54.870
What's the sum of these
three terms?

00:00:54.870 --> 00:00:56.400
Well, you add up
the three n's.

00:00:56.400 --> 00:00:59.110
n plus n plus n is 3n.

00:00:59.110 --> 00:01:02.380
And then 2 plus 4, 3n plus 6.

00:01:02.380 --> 00:01:05.500
And that's going to
be equal to 111.

00:01:05.500 --> 00:01:09.060
And that is choice D.

00:01:09.060 --> 00:01:10.640
Maybe I just confused you
how I wrote this.

00:01:10.640 --> 00:01:13.880
All I did is I said that's the
same thing as n plus n plus 2

00:01:13.880 --> 00:01:19.640
plus n plus 4, which is
equal to 3n plus 6.

00:01:19.640 --> 00:01:20.890
Next problem.

00:01:30.300 --> 00:01:31.160
Problem 8.

00:01:31.160 --> 00:01:32.150
And they do this arc.

00:01:32.150 --> 00:01:33.720
It looks something like that.

00:01:33.720 --> 00:01:35.920
It actually looks better
than what I just drew,

00:01:35.920 --> 00:01:39.190
but you get the point.

00:01:39.190 --> 00:01:42.030
So there's like a distance of 2,
and then there's a distance

00:01:42.030 --> 00:01:43.700
of b, it looks like.

00:01:43.700 --> 00:01:53.800
So it's 2 then b then 2 then
b then 2 then b again.

00:01:53.800 --> 00:01:55.630
The figure above shows part
of a circle whose

00:01:55.630 --> 00:01:57.660
circumference is 45.

00:01:57.660 --> 00:02:01.260
So the circumference of
the circle is 45.

00:02:01.260 --> 00:02:05.600
If arcs of length 2 and length b
continue to alternate around

00:02:05.600 --> 00:02:08.509
the entire circle, so this is
just part of the circle and

00:02:08.509 --> 00:02:10.889
they just keep going
around the circle.

00:02:10.889 --> 00:02:12.470
They just keep going around the
circle so that there are

00:02:12.470 --> 00:02:16.020
18 arcs of each length.

00:02:16.020 --> 00:02:19.990
What is the degree measure of
each of the arcs of length b?

00:02:19.990 --> 00:02:23.690
So they say there's 18
arcs of each length.

00:02:23.690 --> 00:02:27.880
They say that there are 18 of
these arcs of lengths 2, and

00:02:27.880 --> 00:02:31.690
there's 18 of these b segments
because they're saying this

00:02:31.690 --> 00:02:33.620
is-- well, they drew
this as part of the

00:02:33.620 --> 00:02:35.300
whole circle, right?

00:02:35.300 --> 00:02:40.050
So that means that if the entire
circumference is 45,

00:02:40.050 --> 00:02:43.385
but in terms of 2's and b's, we
know that there's 18 2's ,

00:02:43.385 --> 00:02:45.660
so it's 18 times 2.

00:02:45.660 --> 00:02:49.990
And then we also know that there
are 18 b's, plus 18b.

00:02:49.990 --> 00:02:51.940
This is another way of writing
the circumference of the

00:02:51.940 --> 00:02:54.540
circle. because they say that
the circumference is made of

00:02:54.540 --> 00:02:59.660
18 of these arcs and 18 of these
b arcs here, so this is

00:02:59.660 --> 00:03:01.270
also equal to the circumference
of the circle,

00:03:01.270 --> 00:03:04.890
which they told us is 45.

00:03:04.890 --> 00:03:07.500
Now let's see, this is 36.

00:03:07.500 --> 00:03:13.800
18 times 2 plus 18b
is equal to 45.

00:03:13.800 --> 00:03:15.260
Subtract 36 from both sides.

00:03:15.260 --> 00:03:18.900
You get 18b is equal to 9.

00:03:18.900 --> 00:03:26.710
b is equal to 9/18, which
is equal to 1/2.

00:03:26.710 --> 00:03:28.070
Now what are they asking?

00:03:28.070 --> 00:03:33.310
What is the degree measure of
each of the arcs of length b?

00:03:33.310 --> 00:03:40.190
So we know that each b is 1/2.

00:03:40.190 --> 00:03:40.760
And actually, you know what?

00:03:40.760 --> 00:03:42.720
We could have gone
to this step.

00:03:42.720 --> 00:03:47.180
We could have said that 18 b's--
well, no, actually, this

00:03:47.180 --> 00:03:51.680
is-- each b is of length
1/2, right?

00:03:51.680 --> 00:03:55.410
So what fraction is that of
the entire circumference?

00:03:55.410 --> 00:03:57.120
This length right here is 1/2.

00:03:57.120 --> 00:03:58.370
That's what we just
solved, right?

00:03:58.370 --> 00:04:02.840
You saying there's going to be
18 b's and 18 arcs of length 2

00:04:02.840 --> 00:04:04.120
and they all add up to 45.

00:04:04.120 --> 00:04:04.715
We solve for b.

00:04:04.715 --> 00:04:05.770
It's 1/2.

00:04:05.770 --> 00:04:07.910
We want to figure out what's
the degree measure.

00:04:07.910 --> 00:04:12.600
So to figure out the degree
measure, we say, well, 1/2 is

00:04:12.600 --> 00:04:18.230
to the circumference of the
entire circle, which is 45, is

00:04:18.230 --> 00:04:22.000
equal to the degree measure over
the total degrees in the

00:04:22.000 --> 00:04:25.460
entire circle, because the
fraction in terms of the

00:04:25.460 --> 00:04:27.520
circumference is the same thing
it's going to be in

00:04:27.520 --> 00:04:30.430
terms of degrees as a fraction
of 360 degrees.

00:04:30.430 --> 00:04:30.680
Why?

00:04:30.680 --> 00:04:33.320
Because there's 360 degrees
in the circle.

00:04:33.320 --> 00:04:35.010
So let's cross-multiply this.

00:04:35.010 --> 00:04:39.690
1/2 times 360 degrees, so you
get 180 degrees-- that's 1/2

00:04:39.690 --> 00:04:44.020
times 360-- is equal to 45x.

00:04:44.020 --> 00:04:49.130
So x is equal to 180
divided by 45.

00:04:49.130 --> 00:04:51.780
And what's 180 divided by 45?

00:04:51.780 --> 00:04:55.740
Let's see, 40 goes into 160,
it's 4 times, right?

00:04:55.740 --> 00:04:56.730
x is equal to 4.

00:04:56.730 --> 00:04:58.540
4 times 45, right?

00:04:58.540 --> 00:05:00.450
So x is equal to 4 degrees.

00:05:00.450 --> 00:05:01.690
And that's choice A.

00:05:01.690 --> 00:05:03.230
That also makes sense
intuitively.

00:05:03.230 --> 00:05:06.970
4 degrees is a very small amount
of degrees, so that

00:05:06.970 --> 00:05:09.190
makes sense if you were
going from the center.

00:05:09.190 --> 00:05:12.270
I know what I just drew just
probably confused you.

00:05:12.270 --> 00:05:14.290
Hopefully, you get the idea.

00:05:14.290 --> 00:05:16.620
Figure out how long the b's
are and then what fraction

00:05:16.620 --> 00:05:19.430
that is of the entire
circumference and then that's

00:05:19.430 --> 00:05:23.120
what fraction it is
of 360 degrees.

00:05:23.120 --> 00:05:25.830
Problem 19.

00:05:25.830 --> 00:05:29.090
The cost of maintenance on
an automobile increases

00:05:29.090 --> 00:05:31.190
each year by 10%.

00:05:31.190 --> 00:05:34.340
And Andrew paid $300 this year
for maintenance on his

00:05:34.340 --> 00:05:35.080
automobile.

00:05:35.080 --> 00:05:36.110
Fine.

00:05:36.110 --> 00:05:40.620
If the cost c for maintenance on
Andrew's automobile n years

00:05:40.620 --> 00:05:44.680
from now is given by the
function, so c of n is equal

00:05:44.680 --> 00:05:52.230
to 300x to the n, what
is the value of x?

00:05:52.230 --> 00:05:55.860
So they say every year the
cost increases by 10%.

00:05:55.860 --> 00:05:59.130
So what does it mean
to increase by 10%.

00:05:59.130 --> 00:06:03.150
Let me say the cost in year 1,
let's say the cost in year 1

00:06:03.150 --> 00:06:06.350
is going to be-- what's
the cost in year 2?

00:06:06.350 --> 00:06:11.040
It's going to be the cost of
year 1 increased by 10%.

00:06:11.040 --> 00:06:17.710
So plus 0.1 times the cost in
year 1, which is equal to 1.1

00:06:17.710 --> 00:06:20.410
times the cost in year 1.

00:06:20.410 --> 00:06:26.320
The cost in year 3 is equal to
1.1 times the cost in year 2,

00:06:26.320 --> 00:06:28.830
which is the same
thing as what?

00:06:28.830 --> 00:06:35.630
1.1 times the cost of year
2, which is this.

00:06:35.630 --> 00:06:40.620
1.1 times 1.1, cost of year
1, which is the same

00:06:40.620 --> 00:06:45.210
thing as 1.1 squared.

00:06:45.210 --> 00:06:49.900
So every year that you go out,
you're essentially just taking

00:06:49.900 --> 00:06:53.070
1.1 to that power of how many
years you're going out.

00:06:53.070 --> 00:06:55.240
Actually, I should probably make
this cost of year 0, make

00:06:55.240 --> 00:06:58.620
this cost of year 1, 0, 0, 0.

00:06:58.620 --> 00:07:00.956
This is probably confusing you,
but cost of year 2, so

00:07:00.956 --> 00:07:05.020
then the exponents at least
match up: 1 and then 0.

00:07:05.020 --> 00:07:06.500
Then the exponents match up.

00:07:06.500 --> 00:07:11.450
The cost of year 2 is 1.1
squared times cost of year 0.

00:07:11.450 --> 00:07:13.790
So as you can see, all we're
doing when we're doing this,

00:07:13.790 --> 00:07:15.290
this is kind of an exponential
growth problem.

00:07:15.290 --> 00:07:17.540
But they're just saying
what is this base?

00:07:17.540 --> 00:07:19.440
But when you increase something
by 10%, you're

00:07:19.440 --> 00:07:22.080
essentially multiplying
it by 1., so

00:07:22.080 --> 00:07:23.510
that's all they're asking.

00:07:23.510 --> 00:07:26.330
So that's choice C.

00:07:26.330 --> 00:07:27.580
Next problem.

00:07:33.490 --> 00:07:40.341
They drew this parallelogram,
which I have not drawn well.

00:07:40.341 --> 00:07:41.670
It gets the point.

00:07:49.910 --> 00:07:57.060
A, B, C, D.

00:07:57.060 --> 00:07:59.720
If the five line segments in
the figure above are all

00:07:59.720 --> 00:08:02.930
congruent-- OK, so the five
lines, these are all

00:08:02.930 --> 00:08:05.940
congruent, so they're all
equal to each other.

00:08:05.940 --> 00:08:08.530
all of those sides, what
is the ratio of the

00:08:08.530 --> 00:08:10.990
length of AC, not drawn?

00:08:10.990 --> 00:08:14.870
So they want to know AC,
so that's this length.

00:08:14.870 --> 00:08:20.560
They want to know
AC to length BD.

00:08:20.560 --> 00:08:21.160
All right.

00:08:21.160 --> 00:08:24.270
So all of those lines
are congruent.

00:08:24.270 --> 00:08:24.480
You know what?

00:08:24.480 --> 00:08:25.590
Let's just pick a number.

00:08:25.590 --> 00:08:27.290
That's what I find to
be easiest when

00:08:27.290 --> 00:08:28.260
they tell us it all.

00:08:28.260 --> 00:08:32.669
So let's just say that this
is 1, this is 1, this

00:08:32.669 --> 00:08:34.990
is 1, this is 1.

00:08:34.990 --> 00:08:37.510
This whole thing is
going to be 1.

00:08:37.510 --> 00:08:41.590
So how long is this and this?

00:08:41.590 --> 00:08:44.830
This will bisect this
line, right?

00:08:44.830 --> 00:08:48.230
I think we know that about
parallelograms that the two

00:08:48.230 --> 00:08:51.450
lines bisect each other,
the two diagonals.

00:08:51.450 --> 00:08:56.466
So this is going to be 1/2, and
this is going to be 1/2.

00:08:56.466 --> 00:08:58.730
And there's also something
else we know.

00:08:58.730 --> 00:09:05.900
We know that this is 60 degrees,
this is 60 degrees,

00:09:05.900 --> 00:09:11.710
that this whole angle is 60
degrees, this whole angle is

00:09:11.710 --> 00:09:12.280
60 degrees.

00:09:12.280 --> 00:09:13.010
And how do we know that?

00:09:13.010 --> 00:09:14.770
Before we drew the green line,
we knew that both of those

00:09:14.770 --> 00:09:17.980
triangles are equilateral
triangles.

00:09:17.980 --> 00:09:21.610
So if this whole angle was 60
degrees, then this angle right

00:09:21.610 --> 00:09:24.410
here is going to
be 30 degrees.

00:09:24.410 --> 00:09:26.870
And I think you see where
I'm going with this.

00:09:26.870 --> 00:09:30.980
This is a 30-60-90 triangle
where this is 1-- that's the

00:09:30.980 --> 00:09:32.610
hypotenuse-- this is 1/2.

00:09:32.610 --> 00:09:35.960
And then what is the side
opposite the 60-degree side,

00:09:35.960 --> 00:09:36.915
which is this?

00:09:36.915 --> 00:09:41.896
The square root of 3 times
this side right here.

00:09:41.896 --> 00:09:45.520
The square root of 3/2.

00:09:45.520 --> 00:09:46.250
And so would this one.

00:09:46.250 --> 00:09:50.500
This one would also be
square root of 3/2.

00:09:50.500 --> 00:09:52.350
I know maybe it's a
little confusing.

00:09:52.350 --> 00:09:54.360
Actually, I'm about to run out
of time, so I will continue

00:09:54.360 --> 00:09:56.110
this in the next video.


SAT Prep: Test 7 Section 5 Part 5

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:01.510
Welcome back.

00:00:01.510 --> 00:00:03.130
I was running out of time in
the last video and I was

00:00:03.130 --> 00:00:05.630
afraid that I was jumbling up
the problem just to get in

00:00:05.630 --> 00:00:06.700
under the wire.

00:00:06.700 --> 00:00:08.780
So I just drew the triangle.

00:00:08.780 --> 00:00:10.950
Actually, let me do it over
since I have all the time in

00:00:10.950 --> 00:00:13.380
the world now.

00:00:13.380 --> 00:00:16.660
Because I really want
you to get this.

00:00:16.660 --> 00:00:19.720
So the original parallelogram
looks like this.

00:00:28.560 --> 00:00:29.820
This is a diagonal.

00:00:29.820 --> 00:00:35.450
And they tell us that all of
these sides are congruent, so

00:00:35.450 --> 00:00:38.070
that means that they're
all the same length.

00:00:38.070 --> 00:00:40.500
So this side is equal to the
side is equal to this side,

00:00:40.500 --> 00:00:42.050
this side is equal
to this side.

00:00:42.050 --> 00:00:43.930
So that also tells you these
are both equilateral

00:00:43.930 --> 00:00:46.730
triangles, all the sides are
equals so all these angles,

00:00:46.730 --> 00:00:52.830
this must be 60, this must be
60, this must be 60, this must

00:00:52.830 --> 00:00:59.390
be 60, 60 and 60.

00:00:59.390 --> 00:01:01.770
Now what else do we know?

00:01:01.770 --> 00:01:12.180
They want us to figure out the
ratio of this diagonal, the

00:01:12.180 --> 00:01:17.020
long diagonal to this shorter
diagonal right here.

00:01:17.020 --> 00:01:19.300
We know the shorter diagonals
of length 1.

00:01:19.300 --> 00:01:23.260
And with parallelograms, you
might already kind of know

00:01:23.260 --> 00:01:25.830
this, and actually we could
prove this to you.

00:01:25.830 --> 00:01:29.490
Since all of these lines
are congruent, we know

00:01:29.490 --> 00:01:32.010
that-- we know what?

00:01:32.010 --> 00:01:34.130
What is this angle right here?

00:01:34.130 --> 00:01:38.130
What is half of-- this line
is going to bisect

00:01:38.130 --> 00:01:39.990
both of these angles.

00:01:39.990 --> 00:01:40.470
How do we know?

00:01:40.470 --> 00:01:44.040
Well if it didn't bisect
it, all the

00:01:44.040 --> 00:01:45.420
sides wouldn't be congruent.

00:01:45.420 --> 00:01:50.670
So we know that this side is 30,
this side is 30, this is

00:01:50.670 --> 00:01:53.080
30, this is 30.

00:01:53.080 --> 00:01:58.500
And we also know that for any
parallelogram-- actually for

00:01:58.500 --> 00:02:03.940
any rhombus-- the two soon.
diagonals are going to be

00:02:03.940 --> 00:02:05.980
perpendicular bisectors
of each other.

00:02:05.980 --> 00:02:08.940
It's not true for any
parallelogram, it's only true

00:02:08.940 --> 00:02:11.940
when the sides are
all congruent.

00:02:11.940 --> 00:02:14.750
In this situation, and we
already know this because this

00:02:14.750 --> 00:02:16.830
is a 30 degree angle, this is
a 60 degree angle, so what

00:02:16.830 --> 00:02:18.200
does this have to be?

00:02:18.200 --> 00:02:20.496
This has to be a 90 degree
angle because they have

00:02:20.496 --> 00:02:22.470
to add up to 180.

00:02:22.470 --> 00:02:24.100
So let's just pick numbers.

00:02:24.100 --> 00:02:26.570
Let's say that this side is 1,
this side is 1, this side is

00:02:26.570 --> 00:02:27.990
1, this side is 1.

00:02:27.990 --> 00:02:30.770
Then this entire diagonal here
would be 1, half of it would

00:02:30.770 --> 00:02:33.540
be 1/2, this would be 1/2,
this would be 1/2.

00:02:33.540 --> 00:02:35.490
If we didn't know it was 1/2, we
could just say well this is

00:02:35.490 --> 00:02:36.770
a 30, 60, 90 triangle.

00:02:36.770 --> 00:02:40.230
So the 30 degree side is
1/2 of the hypotenuse.

00:02:40.230 --> 00:02:42.730
And then what else do we know
about 30, 60, 90 triangles?

00:02:42.730 --> 00:02:48.030
The 60 degree side is squares
of 3 times the shorter side.

00:02:48.030 --> 00:02:52.180
So this side is going to be
square root of 3 over 2.

00:02:52.180 --> 00:02:54.990
And all we did is we realized
that this is

00:02:54.990 --> 00:02:56.310
a 30, 60, 90 triangle.

00:02:56.310 --> 00:02:58.320
And then we were using the
principle of 30, 60, 90

00:02:58.320 --> 00:03:01.240
triangle to figure out that
the side opposite the 60

00:03:01.240 --> 00:03:04.880
degree side is square
root of 3 over 2.

00:03:04.880 --> 00:03:07.770
If that side is square root of
3 over 2, then this is also

00:03:07.770 --> 00:03:09.810
square root of 3 over 2.

00:03:09.810 --> 00:03:11.650
So what's this entire length?

00:03:11.650 --> 00:03:16.310
It'd be square root of 3 over
2 plus square root of 3 over

00:03:16.310 --> 00:03:20.300
2, which is equal to
square root of 3.

00:03:20.300 --> 00:03:22.260
You add two halves of anything
you get the whole.

00:03:22.260 --> 00:03:23.760
So the square roots of 3.

00:03:23.760 --> 00:03:27.020
So the purple diagonal is square
roots of 3, and what is

00:03:27.020 --> 00:03:30.640
this-- this was d and c.

00:03:30.640 --> 00:03:31.890
This is ac.

00:03:34.880 --> 00:03:38.540
So we know that ac is equal
to the square root of 3.

00:03:38.540 --> 00:03:45.120
And we also know that d-- I'm
sorry, this is a b-- db is

00:03:45.120 --> 00:03:45.700
equal to 1.

00:03:45.700 --> 00:03:46.920
We just defined it as 1.

00:03:46.920 --> 00:03:51.220
So the ratio of ac to bd or
db is going to be square

00:03:51.220 --> 00:03:53.420
root of 3 to 1.

00:03:53.420 --> 00:03:55.580
And that is choice B.

00:03:55.580 --> 00:03:57.910
I hope I didn't confuse you.

00:03:57.910 --> 00:04:00.360
Next section-- actually, we just
finished that section.

00:04:00.360 --> 00:04:01.820
I'll see you


SAT Prep: Test 7 Section 2 Part 1

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:03.050
We're now starting test 7.

00:00:03.050 --> 00:00:04.280
And we're on section two.

00:00:04.280 --> 00:00:08.039
Let's see, problem
1 is on page 774.

00:00:08.039 --> 00:00:19.450
So they say set x-- x is equal
to this-- 30, 31, 32, 33.

00:00:19.450 --> 00:00:21.320
I haven't read the problem
yet, I'm just

00:00:21.320 --> 00:00:22.900
drawing what they drew.

00:00:22.900 --> 00:00:33.000
This is 32, 33, 34, 35 and 36.

00:00:33.000 --> 00:00:34.600
Sets x and y are shown above.

00:00:34.600 --> 00:00:37.850
How many numbers in set
x are also in set y?

00:00:37.850 --> 00:00:41.140
Well hopefully it's fairly
straightforward.

00:00:41.140 --> 00:00:42.650
Is 30 in set y?

00:00:42.650 --> 00:00:43.356
No.

00:00:43.356 --> 00:00:44.840
Then is 31 in set y?

00:00:44.840 --> 00:00:45.380
No.

00:00:45.380 --> 00:00:47.040
32 is, right?

00:00:47.040 --> 00:00:48.660
And so is 33.

00:00:48.660 --> 00:00:50.320
1, 2, the answer is 2.

00:00:50.320 --> 00:00:53.090
That's choice A.

00:00:53.090 --> 00:00:54.370
Not too tough, huh?

00:00:54.370 --> 00:00:56.550
All right, problem number 2.

00:00:56.550 --> 00:01:03.630
If Peg traveled 10 miles in 2
hours, and Linda traveled

00:01:03.630 --> 00:01:07.880
twice as far in 1/2 the
time-- So Linda

00:01:07.880 --> 00:01:09.280
traveled twice as far.

00:01:09.280 --> 00:01:10.950
So twice as far is 10 miles.

00:01:10.950 --> 00:01:13.170
So she traveled 20
miles, right?

00:01:13.170 --> 00:01:15.760
Times 2, that's all I did.

00:01:15.760 --> 00:01:18.810
And they say, in 1/2 the time,
so she traveled 20 miles in 1

00:01:18.810 --> 00:01:20.050
hour, right?

00:01:20.050 --> 00:01:21.700
I just do the 1/2 here.

00:01:21.700 --> 00:01:22.320
20 miles in 1 hour.

00:01:22.320 --> 00:01:24.220
What was Linda's average speed
in miles per hour?

00:01:24.220 --> 00:01:26.270
She did 20 miles in 1 hour.

00:01:26.270 --> 00:01:31.170
Well that's 20 miles per hour.

00:01:31.170 --> 00:01:33.230
And that's choice C.

00:01:33.230 --> 00:01:36.320
Also not a difficult problem.

00:01:36.320 --> 00:01:38.880
Not that you should feel bad if
you had it wrong, it's just

00:01:38.880 --> 00:01:41.860
all about practice.

00:01:41.860 --> 00:01:44.600
It's all about actually
reading the problem.

00:01:44.600 --> 00:01:46.190
I know that might seem
a bit obvious,

00:01:46.190 --> 00:01:47.860
but you'd be surprised.

00:01:47.860 --> 00:01:50.110
You've even seen me make a
couple of mistakes by not

00:01:50.110 --> 00:01:51.910
reading the problem properly.

00:01:51.910 --> 00:01:53.530
OK, problem 3.

00:01:53.530 --> 00:01:56.670
My cousin marked this up with a
black marker, so I only hope

00:01:56.670 --> 00:01:57.870
that what I see is
what you see.

00:01:57.870 --> 00:02:01.610
So let's see, x is equal to--
she did this when she was in

00:02:01.610 --> 00:02:02.680
sixth grade.

00:02:02.680 --> 00:02:05.400
So I'm very proud of her that
she was taking the SAT in

00:02:05.400 --> 00:02:05.985
sixth grade.

00:02:05.985 --> 00:02:10.684
But anyway, x is equal
to k times k minus 2.

00:02:10.684 --> 00:02:13.550
Then x plus 1 is
equal to what?

00:02:13.550 --> 00:02:14.400
[COUGH]

00:02:14.400 --> 00:02:18.300
Excuse me. x plus 1
is equal to what?

00:02:18.300 --> 00:02:21.200
So let's just add 1 to both
sides of this equation.

00:02:21.200 --> 00:02:27.560
So x plus 1 is equal to k times
k minus 2 plus 1, right?

00:02:27.560 --> 00:02:30.110
I just added 1 to both sides
of that equation, and that

00:02:30.110 --> 00:02:36.330
equals k squared minus
2k plus 1.

00:02:36.330 --> 00:02:40.195
And that is choice-- if I
can read this properly--

00:02:40.195 --> 00:02:43.390
that is choice C.

00:02:43.390 --> 00:02:45.730
k squared minus 2k plus 1.

00:02:45.730 --> 00:02:47.580
Problem number 4.

00:02:47.580 --> 00:02:49.240
I should have drank water before
starting this video,

00:02:49.240 --> 00:02:54.732
but I will move forward and
complete what I have begun.

00:02:54.732 --> 00:02:59.160
So they draw a graph that looks
something like that.

00:02:59.160 --> 00:03:05.370
And then there's a line that
goes something like this.

00:03:05.370 --> 00:03:07.450
And then they tell us--
what do they tell us?

00:03:07.450 --> 00:03:11.070
They tell us, well it looks
like, it intersects at this

00:03:11.070 --> 00:03:12.270
point, at y equals 1.

00:03:12.270 --> 00:03:14.520
So this is the point,
0,1, right?

00:03:14.520 --> 00:03:16.750
x is 0, so this is the x-axis.

00:03:16.750 --> 00:03:18.590
This is the y-axis.

00:03:18.590 --> 00:03:21.420
The figure above shows the graph
of a line of y equals ax

00:03:21.420 --> 00:03:27.040
plus b where a and
b are constants.

00:03:27.040 --> 00:03:29.400
Which of the following best
represents the graph of the

00:03:29.400 --> 00:03:34.910
line 2a x plus b?

00:03:34.910 --> 00:03:36.500
So let's draw it.

00:03:36.500 --> 00:03:37.580
I'm not even going to look
at all the choices.

00:03:37.580 --> 00:03:38.580
What would this look like?

00:03:38.580 --> 00:03:40.420
So it's going to have the
same y intercept, right?

00:03:40.420 --> 00:03:42.740
It's going to intersect
at the same point.

00:03:42.740 --> 00:03:43.630
That's the y intercept.

00:03:43.630 --> 00:03:45.960
Let me draw in this
peach color.

00:03:45.960 --> 00:03:48.530
So it's going to intersect
right here.

00:03:48.530 --> 00:03:51.650
And its slope is 2 times the
slope of the previous one.

00:03:51.650 --> 00:03:53.860
So it's going to be even
a steeper slope.

00:03:53.860 --> 00:03:55.750
So the line's going to look
something like this.

00:03:55.750 --> 00:03:57.960
It's going to be twice as
steep, so it's going

00:03:57.960 --> 00:03:59.210
to look like this.

00:04:02.090 --> 00:04:02.230
All right?

00:04:02.230 --> 00:04:05.060
And all I just have is that
same y intercept, right?

00:04:05.060 --> 00:04:08.500
Because the b is still d, but
they've doubled the slope, so

00:04:08.500 --> 00:04:10.070
it's going to go
up even faster.

00:04:12.840 --> 00:04:16.130
And so, if you look at all the
choices, we can immediately

00:04:16.130 --> 00:04:18.660
rule out all of the
choices that don't

00:04:18.660 --> 00:04:19.720
have the same y intercept.

00:04:19.720 --> 00:04:22.110
And the only choices that have
the same y intercept are

00:04:22.110 --> 00:04:24.140
choices B and C, right?

00:04:24.140 --> 00:04:26.060
Only those that intersect
at 0, 1.

00:04:26.060 --> 00:04:32.100
And then between B and C, B is
steeper so B is going to be

00:04:32.100 --> 00:04:33.080
our answer, right?

00:04:33.080 --> 00:04:35.230
C is actually less steep
than this green line.

00:04:39.100 --> 00:04:42.040
Choice C looks something
like this.

00:04:42.040 --> 00:04:45.080
So its slope is less than A.

00:04:45.080 --> 00:04:47.150
So it's not going to be choice
C, so it has to be choice B.

00:04:47.150 --> 00:04:50.370
B goes through the same y
intercept and it is steeper

00:04:50.370 --> 00:04:53.050
than our original green line.

00:04:53.050 --> 00:04:56.450
So that's the answer,
choice B.

00:04:56.450 --> 00:05:01.350
Next problem, problem 5.

00:05:04.770 --> 00:05:07.460
OK, they drew us a triangle.

00:05:07.460 --> 00:05:09.302
That's one side.

00:05:09.302 --> 00:05:10.552
That's another side.

00:05:12.790 --> 00:05:13.990
They have drawn this-- whoops.

00:05:13.990 --> 00:05:15.930
I can't-- OK, there you go.

00:05:15.930 --> 00:05:18.516
That's not as nice looking as
I would have liked, but you

00:05:18.516 --> 00:05:19.090
get the point.

00:05:19.090 --> 00:05:20.040
That's a right triangle.

00:05:20.040 --> 00:05:23.400
They say that this is x.

00:05:23.400 --> 00:05:26.940
And the figure above the
perimeter of the triangle is 4

00:05:26.940 --> 00:05:32.400
plus 2 root 2.

00:05:32.400 --> 00:05:34.260
What is the value of x?

00:05:34.260 --> 00:05:36.250
So what is the value
of this side?

00:05:36.250 --> 00:05:39.000
Well Pythagorean theorem, the
value of that side is the

00:05:39.000 --> 00:05:44.010
square root of x squared
plus x squared.

00:05:44.010 --> 00:05:45.600
So that equals, what?

00:05:45.600 --> 00:05:49.450
The square root of 2x squared.

00:05:49.450 --> 00:05:50.760
You can take the x out, right?

00:05:50.760 --> 00:05:52.065
Square root of x squared
is just x.

00:05:52.065 --> 00:05:54.120
So that equals x square
root of 2.

00:05:54.120 --> 00:05:56.060
And they actually even tell
you that on the first page

00:05:56.060 --> 00:05:59.630
of-- if you go back to page 774,
they tell you, you have a

00:05:59.630 --> 00:06:01.850
45, 45, 90 triangle, it's
s square roots of 2.

00:06:01.850 --> 00:06:04.280
But I like to solve
it every time.

00:06:04.280 --> 00:06:05.910
So this is x, this is x, this
is x square root of 2.

00:06:05.910 --> 00:06:06.490
So what's the perimeter?

00:06:06.490 --> 00:06:12.330
It's x plus x plus x root 2.

00:06:12.330 --> 00:06:16.080
So it's 2x plus x root 2.

00:06:16.080 --> 00:06:17.650
That's the perimeter, right?

00:06:17.650 --> 00:06:20.910
And they also tell us that,
that equals 4 plus

00:06:20.910 --> 00:06:22.830
2 root to 2, right?

00:06:22.830 --> 00:06:24.090
That's for the perimeter.

00:06:24.090 --> 00:06:25.355
So you could just do
pattern matching.

00:06:25.355 --> 00:06:28.200
You'd say well, this looks a
lot like x is equal to 2.

00:06:28.200 --> 00:06:31.450
And it is, because if x is equal
to 2, this becomes a 2,

00:06:31.450 --> 00:06:34.070
and then this becomes
a 4, right?

00:06:34.070 --> 00:06:36.560
So that's our answer,
x is equal to 2.

00:06:36.560 --> 00:06:37.910
And that is choice A.

00:06:40.840 --> 00:06:42.090
Next problem.

00:06:50.480 --> 00:06:51.490
Let me switch colors.

00:06:51.490 --> 00:06:51.960
OK.

00:06:51.960 --> 00:06:53.640
There's always something
for me to draw.

00:06:53.640 --> 00:06:54.760
Well before I draw it, let
me read the question.

00:06:54.760 --> 00:06:57.040
The scores on Tuesday's history
test for 16 students

00:06:57.040 --> 00:06:58.710
are shown on the table above.

00:06:58.710 --> 00:07:03.180
Sam was the only student
absent on Tuesday.

00:07:03.180 --> 00:07:04.640
He'll take the test next week.

00:07:04.640 --> 00:07:08.070
If Sam receives a 95 on the
test, what will be the median

00:07:08.070 --> 00:07:10.690
score for the test?

00:07:10.690 --> 00:07:11.570
OK.

00:07:11.570 --> 00:07:13.990
So they give us a score and
number of students.

00:07:13.990 --> 00:07:14.850
So I'm actually going
to write all this,

00:07:14.850 --> 00:07:19.800
score and then number.

00:07:19.800 --> 00:07:30.000
And so 1 kid got 100; two kids
got a 95; 90, four kids got

00:07:30.000 --> 00:07:44.810
it; 85, 1 kid; 80, 3 kids; 75,
2 kids; 70, 2 kids; 65, no

00:07:44.810 --> 00:07:49.250
kids; and then 60, 1 kid.

00:07:49.250 --> 00:07:52.670
And then Sam comes along and
gets a 95 on the exam, right?

00:07:52.670 --> 00:07:54.680
So before, there's only
2 kids who got a 95.

00:07:54.680 --> 00:07:56.950
Now Sam comes along and there's
going to be 3 kids who

00:07:56.950 --> 00:07:58.640
got a 95 on the exam.

00:07:58.640 --> 00:08:01.250
So to figure out the median, all
I do is I list out all the

00:08:01.250 --> 00:08:03.080
numbers, and then I pick
the middle number.

00:08:03.080 --> 00:08:04.600
That's all the median is.

00:08:04.600 --> 00:08:08.480
So one kid got 100.

00:08:08.480 --> 00:08:13.380
3 kids got a 95: 95, 95, 95.

00:08:13.380 --> 00:08:19.160
Four kids got a 90:
1, 2, 3, 4.

00:08:19.160 --> 00:08:21.260
One kid got an 85.

00:08:21.260 --> 00:08:25.320
Three kids got an
80: 80, 80, 80.

00:08:25.320 --> 00:08:29.370
Two kids got a 75.

00:08:29.370 --> 00:08:31.860
Two kids got a 70.

00:08:31.860 --> 00:08:33.590
And one kid got a 60.

00:08:33.590 --> 00:08:34.350
So how many does that total?

00:08:34.350 --> 00:08:35.340
I could have just added this.

00:08:35.340 --> 00:08:36.750
That's actually what
I should have done.

00:08:36.750 --> 00:08:42.010
But it's 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12,

00:08:42.010 --> 00:08:44.760
13, 14, 15, 16, 17.

00:08:44.760 --> 00:08:45.990
So there's 17 kids.

00:08:45.990 --> 00:08:49.230
So the median is going to be
the number that has eight

00:08:49.230 --> 00:08:50.370
above it and eight below it.

00:08:50.370 --> 00:08:56.220
So 1, 2, 3, 4, 5,
6, 7, 8, Bam!

00:08:56.220 --> 00:08:59.840
That's the median, because
there's 1, 2, 3, 4, 5, 6-- oh,

00:08:59.840 --> 00:09:00.810
no, no, that's not the median.

00:09:00.810 --> 00:09:03.340
Sorry, sorry, sorry, this
is the median: 85.

00:09:03.340 --> 00:09:07.130
Because there's 1, 2, 3,
4, 5, 6, 7, 8 above it.

00:09:07.130 --> 00:09:12.410
And there are 1, 2, 3, 4,
5, 6, 7, 8 below it.

00:09:12.410 --> 00:09:14.240
So the answer is 85.

00:09:14.240 --> 00:09:19.770
Before Sam came in there, then
the median actually would be--

00:09:19.770 --> 00:09:21.650
you'd have one number
less here, so you'd

00:09:21.650 --> 00:09:22.290
have to move it up.

00:09:22.290 --> 00:09:24.230
So it'd probably be 90,
or it'd be an average.

00:09:24.230 --> 00:09:26.410
But anyway, we know that
the median is 85.

00:09:26.410 --> 00:09:28.950
And that is choice C.

00:09:28.950 --> 00:09:30.850
See you in the next video.


SAT Prep: Test 7 Section 2 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:02.360
We're on problem number 7.

00:00:02.360 --> 00:00:05.100
Ahmed has two containers of--
oh, look at that, they're

00:00:05.100 --> 00:00:07.070
doing ethnic names now.

00:00:07.070 --> 00:00:08.930
For the first time, someone on
the SAT sounds like they might

00:00:08.930 --> 00:00:09.730
be related to me.

00:00:09.730 --> 00:00:13.860
Anyway, Ahmed has containers
of two different sizes.

00:00:13.860 --> 00:00:17.210
The total capacity of
16 containers of

00:00:17.210 --> 00:00:20.760
one size is x gallons.

00:00:20.760 --> 00:00:24.250
So let's see, so it's
container 1 size

00:00:24.250 --> 00:00:25.700
and container 2 size.

00:00:25.700 --> 00:00:31.990
So it's 16 containers of size 1,
so 16 times size 1 is equal

00:00:31.990 --> 00:00:33.240
to x gallons.

00:00:37.680 --> 00:00:40.940
And the total capacity of 8
containers on the other side

00:00:40.940 --> 00:00:43.520
is also x gallons.

00:00:43.520 --> 00:00:47.170
So 8 times c2 do is
also x gallons.

00:00:50.350 --> 00:00:51.770
And x is greater than
0, of course.

00:00:51.770 --> 00:00:55.480
You can't have negative
gallons.

00:00:55.480 --> 00:01:00.030
In terms of x, what is the
capacity, in gallons, of each

00:01:00.030 --> 00:01:03.939
of the larger containers?

00:01:03.939 --> 00:01:05.010
Let me see what it is.

00:01:05.010 --> 00:01:07.160
Ahmed has containers of
two different sizes.

00:01:07.160 --> 00:01:10.700
The total capacity of 16
containers of 1 size is x

00:01:10.700 --> 00:01:12.240
gallons, right.

00:01:12.240 --> 00:01:16.060
And the total capacity of 8
containers of the other size

00:01:16.060 --> 00:01:17.740
is also x gallons.

00:01:17.740 --> 00:01:19.980
And x is greater than 0.

00:01:19.980 --> 00:01:25.070
In terms of x, what is the
capacity, in gallons, of each

00:01:25.070 --> 00:01:26.470
of the larger containers?

00:01:26.470 --> 00:01:26.800
OK.

00:01:26.800 --> 00:01:29.200
So all we have to know is about
the larger containers.

00:01:29.200 --> 00:01:32.450
Oh, the trick here is which one
is the larger container?

00:01:32.450 --> 00:01:34.630
Right, that is the trick
to this question.

00:01:34.630 --> 00:01:39.140
So you really need 8 of the
second containers to get x.

00:01:39.140 --> 00:01:42.050
We need 16 of this container
to get to x.

00:01:42.050 --> 00:01:45.010
So this is a larger
container, right?

00:01:45.010 --> 00:01:47.310
And if we wanted to know the
exact size, you divide both

00:01:47.310 --> 00:01:52.190
sides by 8, you get container
2 is equal to x/8 gallons.

00:01:52.190 --> 00:01:54.430
And you could have figured
out container 1 is

00:01:54.430 --> 00:01:59.220
equal to x/16 gallons.

00:01:59.220 --> 00:02:01.090
And something divided by 8 is
going to be bigger than

00:02:01.090 --> 00:02:01.935
something divided by 16.

00:02:01.935 --> 00:02:02.770
So this is our answer.

00:02:02.770 --> 00:02:05.450
This is the larger container,
x divided by 8,

00:02:05.450 --> 00:02:07.180
that's choice D.

00:02:07.180 --> 00:02:08.850
Got to read these problems
carefully.

00:02:08.850 --> 00:02:11.260
Next question.

00:02:11.260 --> 00:02:12.510
Whoops.

00:02:15.090 --> 00:02:18.170
Problem 8.

00:02:18.170 --> 00:02:22.300
Rectangle a, b, c, d.

00:02:22.300 --> 00:02:24.160
I'm still getting over the fact
that they had the name

00:02:24.160 --> 00:02:25.740
Ahmed in the SAT.

00:02:25.740 --> 00:02:29.660
Anyway, rectangle A, B, C, D.

00:02:29.660 --> 00:02:32.330
One day, they'll have a Salman
in there too and I will be

00:02:32.330 --> 00:02:34.910
very proud.

00:02:34.910 --> 00:02:35.830
That's my full name.

00:02:35.830 --> 00:02:37.680
I go by Sal.

00:02:37.680 --> 00:02:39.970
Some people think that I'm not
proud of my ethnicity, but I

00:02:39.970 --> 00:02:41.470
explained that's what
my mother called me.

00:02:41.470 --> 00:02:43.020
But my full name is Salman.

00:02:43.020 --> 00:02:46.300
It's actually Salman Khan, which
is the name of an Indian

00:02:46.300 --> 00:02:49.360
actor that gets more
web hits than I do.

00:02:49.360 --> 00:02:50.490
But that's life.

00:02:50.490 --> 00:02:52.380
OK, problem eight.

00:02:52.380 --> 00:02:55.490
Rectangle A, B, C, D, lies in
the x, y coordinate plane, so

00:02:55.490 --> 00:02:57.360
the sides are not parallel
to the axes.

00:02:57.360 --> 00:03:00.615
Well, let me clear this.

00:03:00.615 --> 00:03:02.450
I have to focus.

00:03:02.450 --> 00:03:03.040
OK.

00:03:03.040 --> 00:03:04.560
It's not parallel to the axes.

00:03:04.560 --> 00:03:07.130
What is the product of the
slopes of all four

00:03:07.130 --> 00:03:09.410
sides of A, B, C, D?

00:03:09.410 --> 00:03:10.250
Oh, this is fun.

00:03:10.250 --> 00:03:10.922
OK.

00:03:10.922 --> 00:03:12.040
So let me draw it.

00:03:12.040 --> 00:03:17.300
So it looks like that,
like that, like

00:03:17.300 --> 00:03:22.610
that, like that, roughly.

00:03:22.610 --> 00:03:23.970
I didn't draw it perfectly,
but these are

00:03:23.970 --> 00:03:25.220
right angles, right?

00:03:27.730 --> 00:03:28.020
OK.

00:03:28.020 --> 00:03:33.860
So let's say the slope of this
line, right here, is m, right?

00:03:33.860 --> 00:03:35.910
Well this line's going to be
parallel to this line, so this

00:03:35.910 --> 00:03:38.140
is also going to have
a slope of m.

00:03:38.140 --> 00:03:40.670
And this is something you
should just memorize.

00:03:40.670 --> 00:03:43.980
The slope of a perpendicular
line-- so the slope

00:03:43.980 --> 00:03:46.320
perpendicular to this--
is a negative inverse.

00:03:46.320 --> 00:03:49.090
So the inverse of m is 1/m,
and it's going to be the

00:03:49.090 --> 00:03:49.630
negative inverse.

00:03:49.630 --> 00:03:53.060
So the slope of this line right
here is negative 1/m.

00:03:53.060 --> 00:03:54.020
And that's something that
you should just

00:03:54.020 --> 00:03:55.820
memorize, for life.

00:03:55.820 --> 00:03:57.450
And, of course, this one is
parallel to this one.

00:03:57.450 --> 00:03:58.735
It's also perpendicular
to these two, so

00:03:58.735 --> 00:04:01.220
it's also minus 1/m.

00:04:01.220 --> 00:04:03.130
So they want to know the product
of all of these.

00:04:03.130 --> 00:04:12.330
So you get m times minus 1/m
times m times minus 1/m.

00:04:12.330 --> 00:04:16.930
m times minus 1/m, this is
minus 1 times minus 1,

00:04:16.930 --> 00:04:18.810
which equals 1.

00:04:18.810 --> 00:04:22.430
And that is choice D.

00:04:22.430 --> 00:04:23.780
Next problem.

00:04:23.780 --> 00:04:26.315
This problem is all based on,
did you know that the slope of

00:04:26.315 --> 00:04:28.505
a perpendicular line is
a negative inverse?

00:04:33.510 --> 00:04:34.250
Invert colors.

00:04:34.250 --> 00:04:35.600
All right.

00:04:35.600 --> 00:04:38.080
Problem number 9.

00:04:38.080 --> 00:04:41.210
An hour long television program
includes 20 minutes of

00:04:41.210 --> 00:04:42.390
commercials.

00:04:42.390 --> 00:04:46.230
What fraction of the hour long
program was not commercials?

00:04:46.230 --> 00:04:50.110
So if it had 20 minutes of
commercials, it must have had

00:04:50.110 --> 00:04:55.700
40 minutes not commercials,
right?

00:04:55.700 --> 00:05:01.140
So 40 minutes over an hour is
the fraction that's not

00:05:01.140 --> 00:05:01.640
commercials.

00:05:01.640 --> 00:05:04.850
So that's 4/6, which
equals 2/3.

00:05:04.850 --> 00:05:05.690
That's the answer.

00:05:05.690 --> 00:05:06.720
That's it.

00:05:06.720 --> 00:05:08.930
Problem 10.

00:05:08.930 --> 00:05:14.740
If the product of 0.3 and a
number is equal to 1-- 0.3 and

00:05:14.740 --> 00:05:17.055
a number, the product of 0.3
and a number, I'll say x is

00:05:17.055 --> 00:05:17.555
equal to 1.

00:05:17.555 --> 00:05:19.170
So the product of 0.3
and x is equal to

00:05:19.170 --> 00:05:21.050
1-- what is the number?

00:05:21.050 --> 00:05:22.635
Divide both sides by 0.3.

00:05:22.635 --> 00:05:26.800
x is equal to 1 divided
by 0.3.

00:05:26.800 --> 00:05:28.890
And what is that equal to?

00:05:28.890 --> 00:05:30.500
I don't know, can you use
a calculator on this?

00:05:33.730 --> 00:05:36.770
You could multiply the top
and the bottom by 10.

00:05:36.770 --> 00:05:42.510
That is equal to 10/3, which is
equal to 3 and 1/3, which

00:05:42.510 --> 00:05:45.310
is equal to-- well, you could
just write 10/3, actually.

00:05:45.310 --> 00:05:46.520
Multiply the top and
the bottom of it.

00:05:46.520 --> 00:05:49.560
Well, that's also
3.333 repeating

00:05:49.560 --> 00:05:50.490
over, and over again.

00:05:50.490 --> 00:05:53.470
So you could do it
that way too.

00:05:53.470 --> 00:05:56.050
Next problem.

00:05:56.050 --> 00:05:56.930
Did I do that right?

00:05:56.930 --> 00:05:57.670
Is equal to what?

00:05:57.670 --> 00:05:57.980
Right.

00:05:57.980 --> 00:06:00.630
That's right.

00:06:00.630 --> 00:06:06.570
Problem 11.

00:06:06.570 --> 00:06:08.565
Let-- oh, I like when they
define these new operations.

00:06:11.120 --> 00:06:18.990
OK, this is a fun one, x y, z
can be defined as-- so that

00:06:18.990 --> 00:06:26.150
means x to the y minus
z to the y.

00:06:26.150 --> 00:06:29.600
For all positive integers x, y,
and z, what is the value of

00:06:29.600 --> 00:06:35.200
triangle 10, 3, 5?

00:06:35.200 --> 00:06:35.540
OK.

00:06:35.540 --> 00:06:36.590
So what's x?

00:06:36.590 --> 00:06:38.400
x is this term.

00:06:38.400 --> 00:06:43.220
So it's 10 to the y-- y is this
term, 10 to the third--

00:06:43.220 --> 00:06:45.580
minus z, z is this term.

00:06:45.580 --> 00:06:47.110
I'm just pattern matching.

00:06:47.110 --> 00:06:48.920
So that z is this.

00:06:48.920 --> 00:06:52.270
And then my z to
the y, y is 3.

00:06:52.270 --> 00:06:53.030
That's all it is.

00:06:53.030 --> 00:06:55.110
So it's 10 to the third
minus 5 to the third.

00:06:55.110 --> 00:06:58.650
10 to the third is 10 times 10
times 10, which is 1,000,

00:06:58.650 --> 00:07:02.100
minus 5 times 5 times
5, which is 125.

00:07:02.100 --> 00:07:02.890
And so that's what?

00:07:02.890 --> 00:07:04.730
That's 875.

00:07:04.730 --> 00:07:06.680
And we are done.

00:07:06.680 --> 00:07:07.930
Next problem.

00:07:15.890 --> 00:07:17.400
OK, draw a rectangle.

00:07:17.400 --> 00:07:19.040
Let's see, they have a
rectangle that looks

00:07:19.040 --> 00:07:19.740
something like that.

00:07:19.740 --> 00:07:27.485
And then there's a line there.

00:07:27.485 --> 00:07:40.940
And these are points P,
Q, R, S, T, and U.

00:07:40.940 --> 00:07:46.430
And the figure above P, Q,
S, T is a rectangle.

00:07:46.430 --> 00:07:49.200
And U, R, S, T is a square.

00:07:49.200 --> 00:07:50.100
So this is a square.

00:07:50.100 --> 00:07:50.650
Fair enough.

00:07:50.650 --> 00:07:52.150
So that means this
side's equal.

00:07:52.150 --> 00:07:54.720
P, U is equal to 5.

00:07:54.720 --> 00:07:56.420
So this is equal to 5.

00:07:56.420 --> 00:07:59.110
And U, T is a positive
integer.

00:07:59.110 --> 00:08:01.090
U, T is an integer, So
that's interesting.

00:08:01.090 --> 00:08:02.790
We know it's an integer, and we
know it can't be negative.

00:08:02.790 --> 00:08:07.800
You can't have a negative side
of a square in our universe.

00:08:07.800 --> 00:08:14.420
If the area of P, Q, S, T must
be more than 10-- so area is

00:08:14.420 --> 00:08:17.600
greater than 10 and
less than 30.

00:08:17.600 --> 00:08:20.610
Area is less than 30, so
we can write this way.

00:08:20.610 --> 00:08:30.710
Area less than 30-- what is one
possible value of U, T?

00:08:30.710 --> 00:08:34.090
Well let's just say that
U, T is x, right?

00:08:34.090 --> 00:08:36.309
And this is a square, so
this is x, and this

00:08:36.309 --> 00:08:38.429
is x as well, right?

00:08:38.429 --> 00:08:42.490
And so what is the area
of this big thing?

00:08:42.490 --> 00:08:45.130
It's going to be this
side which is what?

00:08:45.130 --> 00:08:50.880
5 plus x times this
side, which is x.

00:08:50.880 --> 00:08:52.430
So that's going to
be the area.

00:08:52.430 --> 00:08:57.050
So it's going to be
5x plus x squared.

00:08:57.050 --> 00:08:58.500
And so that's the area.

00:08:58.500 --> 00:09:03.860
And it has to be between these
two spaces, right?

00:09:03.860 --> 00:09:05.090
So we could try out
some numbers.

00:09:05.090 --> 00:09:09.130
Let's see, if x is 1
what's the area?

00:09:09.130 --> 00:09:11.240
It's 5 plus 1.

00:09:11.240 --> 00:09:12.650
So then the area is 7.

00:09:12.650 --> 00:09:14.850
No, that's not greater
than 10.

00:09:14.850 --> 00:09:17.130
What happens if x
is equal to 2?

00:09:17.130 --> 00:09:20.150
Then we have 5 times
of x is 2.

00:09:20.150 --> 00:09:23.760
You have 7 times
2, which is 14.

00:09:23.760 --> 00:09:25.690
And that satisfies
these conditions.

00:09:25.690 --> 00:09:28.230
14 is greater than 10
and less than 30.

00:09:28.230 --> 00:09:29.190
So that's it.

00:09:29.190 --> 00:09:30.100
You didn't even have to do x.

00:09:30.100 --> 00:09:31.610
You could have just said well,
what if x is, you could have

00:09:31.610 --> 00:09:32.950
tried 1, and just tried 2.

00:09:32.950 --> 00:09:36.020
And you could say 5 plus
2 is 7 times 2 is 14.

00:09:36.020 --> 00:09:37.900
That meets my conditions,
and I am done.

00:09:37.900 --> 00:09:39.240
That's all you've got to do.

00:09:39.240 --> 00:09:41.150
I'll see you in the
next video.