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Current time:0:00Total duration:13:04

We are on problem number 22. And their question is
straightforward enough. What is the value of
the integer x? And they tells us
it's an integer. x is equal to what? So the first statement,
x is prime. Well, there are a lot
of prime integers. There's actually an infinite
number of them. So that alone doesn't
tell us what x is. Two. Let's see, 31 is less than or
equal to x, which is less than or equal to 37. Well, immediately, this is
useless information because 31 and 37 are both prime numbers
as far as I can tell, right? They're both prime numbers. So this means that
x could be 31. It could be 37. Or let's see, are there any
prime numbers in between? 32, 33, 34, 35. Well, there are no prime numbers
in between, but it could be either 31 or 37, so it
doesn't help us much even used in combination. I mean, this alone is definitely
useless, because this is a range of
six numbers. And even if we know x is prime,
if we use this and this, all we know is that
x is either 31 or 37, so the answer is E. Together, they're not
sufficient, E Now, if this was a little bit
different, if this was x is greater than 31 and less than or
equal to 37, then we would know, oh, x has to be 37. Or if this wasn't here and this
was there, then we would know that x is 31. But anyway, they
didn't do that. They gave us the option,
so and the answer is E. Problem 23. If P, Q and R are three distinct
points, do line segments PQ and PR have
the same length? So P, Q and R. They want to know if these two
lines are the same length. This is line PQ and this
is the line PR. So are those the same length? Let's see if we can
figure this out. Statement one tells us
P is the midpoint of line segment QR. OK, P, midpoint, so I've
already drawn it wrong. Midpoint of QR. So let me redraw it, so all
three points actually end up being on the same line,
because P is the midpoint of QR. So this is the line. This is point Q. This is point R. So if P is the midpoint, P is
right there, and that tells us that this distance is the same
as this distance, which also tells us that QP, line segment
QP is equal to PR. And that's what I think what
they wanted us-- yeah, right. They want to say does PQ and
PR have same length? This is PQ So PQ and
PR definitely have the same length. So statement one is
definitely true, because P is the midpoint. Let's see about statement two. I mean, statement one implies
that they have to be on the same line; otherwise,
P couldn't have been the midpoint. Let's see if statement
two is interesting. Statement two: Q and R lie in
the same circle with center P. Interesting. So this is saying a circle with
center P, and Q and R both lie on this circle. So Q could be here. I'm just picking two
random points. R could be here. So we want to know whether
PQ is equal to PR, right? That was the question. Well, the definition of a circle
is all points that are equal distant from P. So R has to be the same
distance from P. It's going to be one radius of
the circle away from P as Q is because they're both on the
circle, and they're all equal distance from p. So PR is going to
be equal to PQ. They're going to be equal
to each other. So once again, two alone
is sufficient. So the answer is D, each
statement alone is sufficient. Problem 24. The problems are getting a
little bit more interesting. Is the number x between
0.2 and 0.7? So if I just write it 0.2 is
less than x, which less than 0.7, that's what they
want to know. Statement one tells us that
560x is less than 280. You know you don't even
have to solve this. This is a fairly simple
equation. You can easily solve for x, and
if you can solve for x, you can answer this question. So you actually don't have
to go to the trouble. If you want to, you can then
just divide both sides by 560, but that takes time,
valuable time. You're taking the GMAT. And that is equal
to 1/2, right? 28/56 is 1/2. And then you'd say, sure, x is
between the two numbers. But you didn't have
to do this. That's a waste of time. All you have to do say, oh, I
can solve for x, so I can test that, whether that's
true or not. Statement two says 700x
is equal to 280. Same thing. You just look at it. You're like, oh, I could
solve for x. If I can solve for x, I can tell
you whether x is between those two numbers, and
that's the end of it. And you say, oh, each of these
independently are sufficient to answer this question. But if you wanted to prove it
to yourself and see if this question is true or false, you
can just say, well, x is equal to 228/700. That is equal to 2.28/7. That is equal to 0.4. I just divide the top and the
bottom by 7, and you get 0.4. So once again, it
is between it. You answered the question. But you remember, the question
doesn't have to be affirmative. You just have to figure out if
the data is enough to answer the question, and we
didn't have to go through this and this. You could have just looked at
it and said, oh, I can solve for x in either way and
then just test it. Next, problem 25. If i and j are integers,
is i plus j even? So statement number one is
that i is less than 10. That tells me nothing. That to me seems kind
of useless. Statement number two
is i is equal to j. Well, that's interesting. So what's i plus j then
going to be equal to? Well, that's the same thing as--
since i is equal to j, that's the same thing as j
plus j, which equals 2j. Or you could say, if you
substitute j for i, or if you substitute i for j, that's the
same thing as i plus i, which is equal to 2i. In either case, you immediately
see that either of these numbers are a multiple of
two and therefore have to be even, because i and
j are integers. So statement two alone is
sufficient, and statement one is useless. Problem 26. n plus k is equal to m. What is the value of k? So we could solve for it, if we
could answer this, because k is equal to m minus n, and
that would solve our answer. Let's see if the statements
give us any help. n is equal to 10. Well, that'll just tells us that
k is equal to m minus 10. It still doesn't tell us
what k is, because we don't know what m is. Statement number two. m plus 10 is equal to n. This is interesting. So what does this tell us? So let's see, if we can figure
out from this statement what m minus n is, then we're
in business. Let's see, let's subtract
n from both sides. You get m minus n plus
10 is equal to 0. Subtract 10 from both sides,
you get m minus n is equal to minus 10. Well, we know that k is
equal to m minus n. m minus n is equal to minus
10, so this is equal to k. So we solved for k, using just
statement two alone. Just statement two alone
is sufficient. And statement one is-- I mean,
it kind of gets us halfway, but you really don't need it. Statement two is all you
need, so that is b. Let's see if we have
time for one more. Yeah, sure, why not. They have drawn a
triangle for us. I'll draw it quick and dirty. This is angle x. This is angle z, and
this is angle y. And they want to know is this
triangle equilateral, which means are all the sides
equal to each other? So statement one tells us
that x is equal to y. So we break out a little bit of
our eighth or ninth grade geometry here. So if x is equal to y, if these
angles are equal to each other, what that tells us is
that these sides are equal to each other, the corresponding
sides are the same. I could actually redraw this
triangle like this. Maybe it'll jog those neurons
from ninth grade. This is x. This is y, and this is z. And you could test it out,
if you don't believe me. And we've gone over this
in the geometry module. If x and y are equal to each
other, then these sides have to be equal to each other. And you could just try to
imagine drawing a triangle that has two base angles equal
to each other and the sides are different. It would be impossible. The sides would have
to be the same. This triangle would have to be
at least isosceles and maybe equilateral. Isosceles is two sides
being the same. But that by itself doesn't
tell us that this is an equilateral triangle. In order for this to be
equilateral, z has to be the same as x and y. They all have to be. And if they're all going to be
the same, they all add up to 180, they all have
to be 60 degrees. So you could have a situation
where x and y are both 20 degrees. This is 20, this is 20, then
they add up to 40. And then z would have to be 140
in which case, we would not have an equilateral
triangle, so one alone is not sufficient. Let's see was statement
two tells us. Statement two tells us z
is equal to 60 degrees. So that by itself-- so first
of all, we know if we know both of these pieces of
information, we are definitely dealing with an equilateral
triangle. Because think about it. If z is 60, and then we use this
information that these two are equal to each other,
what do we know? If z is 60, then these two added
together have to-- so x plus y plus z are going
to be equal to 180. If z is 60, then x plus y have
to be equal to-- subtract z from both sides, or subtract 60
from both sides-- has to be equal to 120. And since these are equal to
each other, x and y both would have to be equal to
60 degrees, right? So if you use both of this
information, all of the angles are going to be 60 degrees
and it's eqilateral. So we know that definitely
in combination, we can figure it out. But what about just if
z is equal to 60? Well, I can easily draw a
triangle that disproves that. So if I have a 0-degree angle
here-- in fact, it might be a 30-60-90 triangle like you
learned in school, so it could be 60 degrees, 30 degrees,
and 90 degrees. This could be z. And this definitely is not
an equilateral triangle. And so if just by z being 60
degrees, in no way are you saying that this is definitely
an equilateral triangle. You have to have the other
condition that x is equal to y. And so the answer is, let's
see, C, both statements together are sufficient, but
neither statement alone is sufficient. The answer is C. See you in the next video.