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Let's keep going. We're on problem number six
on page 278 of the data sufficiency sample questions. So let's see. They've drawn a little figure. And they say, in the
figure above, lines k and m are parallel. I guess I better draw
lines k and m. That's line k. That's line m. And then they have a
transversal line. And we've gone over all of this
in the geometry playlist. You might want to review
it if this looks completely foreign to you. And so this is line k. This is line m. And they're calling this angle
right here x degrees. This line here is z degrees. And then this angle here--
did I say line? That's an angle. This angle here is y degrees. And the question is-- question
number six-- in the figure above, if lines k and m are
parallel -- so they're both parallel -- what is
the value of x? OK. In statement number one
they tell us that y is equal to 120 degrees. So if y is equal to 120-- let me
do this one in magenta-- so this is based on this
is equal to 120. What can we figure out? Well, this y is supplementary
to z, right? So when you add these two angles
together, it's 180 degrees, right? Because you're completing
kind of a whole 1/2 arc, or 1/2 circle. So z would be 60 degrees. And then you could say that --
there's all these words that people use in geometry class. But if you have two parallel
lines and a transversal, then these opposite inside angles
are going to be the same. So you know that x is
equal to 60 degrees. Another way you could have done
it, you could have said, OK, if y is equal to 120
degrees, then y and this angle right here are also
supplementary, right? Because they complete
this whole arc. So y plus this angle have
to be equal to 180. So this angle would
also be 60. And then you use what you
learn in geometry class. That corresponding angles on a
transversal intersecting two parallel lines, that
they're also equal. So you'd also get to the
same conclusion. That x is equal to 60. So either way, statement number
one alone is enough to figure out x. Now, what did they give us
for statement number two? And I'll do that in
a different color. Statement number two.
z is equal to 60. Well, this actually gives us the
same information as that. Because if we know that z is
equal to 60, then we know that y is going to be equal to 120,
even if the book never told us this the first time. So z is equal to 60 is the
same information as y is equal to 120. And so you can make the exact
same argument as you did for the first one. So actually point number two
alone is also enough to figure out that x is equal to 60. You actually didn't even
have to figure it out. An important skill, eventually,
when you're when you're taking the GMAT, is to
be able just look at it and say, oh, I can figure that
out, and then move on. Instead of actually having
to figure out that x is equal to 60. But anyway. soon. So this one is, either of them
alone are sufficient. So that's d. OK. Problem number seven. What percentage of a
group of people are women with red hair? So women with red hair,
percentage. So statement number one tells
us, of the women in the group, 5% have red hair. 5% of women have red hair. That alone doesn't tell me what
percentage of the entire group are women with red hair,
because I don't know how large the whole group is. There could be 20 women, and
there could be 10 million men. Or there could be 20
women and no men. So that still doesn't help me
with what percentage of the group are women with red hair. Statement number two tells us,
of the men in the group, 10% have red hair. So 10% of men have red hair. That's really useless. Once again, I don't know
how big the group is. Think about it. If I have 20 women, then that
tells me that there's one woman with red hair. And I don't know I have
20 women, right? But I still don't know how
many men there are. If there are 20 women and one
has red hair, there could be a million men. There could be no man. In which case this answer would
turn out very different. What percentage of the group
are women with red hair? So both of these combined are
fairly useless questions. And actually, let me draw a Venn
diagram, because I think it's useful. So the entire group is
both women and men. I'll draw a Venn rectangle
instead. So that's women and men. So some percentage of-- we
don't know how many women there are, and how many men. So this area is women. That's the number of women. And this is the number of men. And this first point tells
us that 5% of the women have red hair. So it just tells us that 5% of
this area is red, right? Which is maybe, I don't
know, I'll eyeball it. It's like that. And then this says 10% of
the men have red hair. So maybe that area looks
something like that. So we know the ratio of
this to this box. And we know the ratio of this
to this box is 10%. But we don't know the ratio of
this to the entire universe, because we don't know how many--
we don't know what the total population sizes,
so we'll never be able to figure it out. Anyway, so that is e. All right. Problem number eight. Maybe I missed something. Problem eight. If r and s are positive
integers, you r is what percent of s? So r, s, positive integers. And we want to know r is
what percent of s? So essentially, we just want to
figure out what r over s is equal to, right? This'll give us some decimal. And then you multiply by 100,
and you know the percentage. So if you can figure out this,
you can figure out the percentage of r is what
percentage of s. So statement number one. They tell us that r
is equal to 3/4s. Well, let's just do a little
algebraic manipulation. We're trying to get r over
s, so let's divide both sides by s. So you get r over s is
equal to 3 over 4. So there we got it. We got the answer. That was a helpful data point. All we needed was that
data point, actually. Let's see what the second
data point gives us. Data point two. r divided by s-- well, they
wrote it like this; they wrote it the way you did in second
grade-- r divided by s is equal to 75 over 100. Well, that's just another way
of just writing r over s is equal to 75 over 100,
which is exactly the same thing as this. So these are actually equivalent
statements, almost. So each of them independently
are enough to figure out r over s, or what percentage
r is of s. All right. Problem number nine. I'll draw a line here, I don't
want to get too messy. Is it true that a is
greater than b? I sometimes find these
statements slightly humorous. Is it true that a is
greater than b? All right. The first statement is,
2a is greater than 2b. So I don't know if you remember
from algebra, but you can operate on inequalities
the exact same way you can operate on equalities, or I
guess you call them equations. And you just have to remember
that if you multiply or divide by a negative number, that
you have to swap the inequality sign. Well, luckily in this case, we
could divide both sides by a positive number. So if you multiplying or
dividing by a positive on both sides, you don't have to
change the inequality. So just divide both
sides by 2. And you can test that with
numbers, just to see why that make sense. So you divide both sides by 2. And you get a is
greater than b. So that's all we needed. We just needed statement one. Now let's see what statement
two does for us. Statement two tells us
that a plus c is greater than b plus c. Well, once again, we can
subtract c from both sides of this equation, or from both
sides of this inequality, without changing the
inequality sign. So you subtract c
from both sides. And once again, you get
a is greater than b. So each of these statements
independently are enough for us to figure out that it
is true, that a is greater than b. Let's do one more. Actually, I've run out of
chalkboard space, so I might as well just wait until
the next video. See