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Current time:0:00Total duration:10:34

We're on problem 84. On a company-sponsored cruise,
2/3 of the passengers were company employees, and the
remaining passengers were their guests. So we could say, employees is
equal to 2/3 of passengers. And I guess we could also say
that guests are equal to 1/3 of passengers. The remainder are guests. If 3/4 of the company employee
passengers were managers, what was the number of company
employee passengers who were not managers? OK. So the second statement, in at
least the problem statement, it says if 3/4 of the company
employee passengers were managers-- so 3/4 of E were
managers, if I read that statement-- what was the number
of company employee passengers who were
not managers? So essentially, they want to
know what 1/4 of E is, right? This is what they're asking. Because they want to know,
what were the number of company employee passengers
that were not managers? And 3/4 were managers, 1/4
were not managers. So if we could figure out P, we
would be able to figure out E, and we'd be able to
figure out 1/4 E. If we were able to figure out
guests, we could then use that to figure out P, which we could
use to figure out E, which we could then
figure out 1/4 E. So that's how we should be
thinking about it, not that we should be spending that
much time on it. Problem number one. There were 690 passengers. 690 is equal to P. Well, we'd be done, right? Because E is equal to 2/3
P, and then we want to know what 1/4 E is. So the non-manager employees
would be 1/4 times E, which is just 2/3 times P. Times 690. And that would be our answer. So one alone is sufficient. And then statement number two. There were 230 passengers
who were guests of the company employees. So they're essentially saying
that guests are equal to 230. And then we could use this
information, that guests are just 1/3 of P. And then, of course, from
there you could solve. You could multiply 3 times both
sides of the equation to figure out that P
is equal to 690. Which is the same information
that they gave in one, which was enough to solve
the problem. So two alone is also
sufficient. So they are each independently
sufficient to solve the problem. That problem is confusing,
just with the language. Company, employee, manager,
passenger, guest. It was very confusing. 85. In the x-y plane, does
the point 4 comma 12 lie on line k? OK, this is interesting. So let me draw a line k, and
I'll worry about the point 4 comma 12 later. So let's see. Let me look at the points that
they do say lie on k. So if this is the x-axis,
that's the y-axis. Statement one tells us that
the point 1, 7 lies on k. So the point 1 comma
7 lies on k. And then they also tell us that
the point--- that alone isn't going let me know if the
point 4 comma 12 lies on it. So 1, 2, 3, 4. 4 comma 12. It might be up here. This is the point that
we care about. If you can see that. Let me scroll down. 4 comma 12. So if the line looks like that,
maybe it goes through, but the line could
be like that. So statement number one
alone isn't enough. Statement number two says the
point minus 2 comma 2. x is minus 2, y is positive 2. So they're saying
that lies on it. Eyeballing this line, k, it
seems like 4 comma 12 could very well lie on it. But we have to do a little bit
of math to figure it out. So first of all, just statement
two by itself doesn't help us, because this
line could go anywhere. You need two points to just
know what the line is. And so how do we figure out
if the point 4 comma 12 is on this line? Well, the easiest way to do it
is to figure out the slope-- well, I don't know if it's the
easiest way, but it's the way I would think about doing it--
is to figure out the slope of the line, and then extend that
slope here and see if the point 4 comma 12 lies on it. So let's see, what's the
slope of this line? So when we went up 7 minus 2. So the rise is 5. When you go up 5, we
went over how much? 1 minus minus 2. We went over 3, right? So for every 3 that you move
over, you go up 5. That's the easiest way
to think about it. So we're going from x is equal
to 1 to x is equal to 4, so we're going to the right by 3. And so we should
be going up 5. So 5 plus 7, sure enough,
is equal to 12. So if you just extend the
slope, it does hit the point 4 comma 12. Well, actually you know what? I just wasted a lot of your
time, because we just have to know whether the data
is sufficient to answer the question. We don't have to prove that
the question is true. I've just proven to you
that 4 common 12 does lie on the line. But you could have just said,
oh, well, you know what? Statement number one gives
me a point on the line. That alone isn't enough
to solve the problem. Statement number two gives
me another point. And then at this point, if
you're really taking the GMAT and time matters, you just
say, hey, I got two points for this line. Two points is everything I need
to know about a line. Once I know two points on a
line, I can figure out if any other point is on that line. I'm done. Both statements combined
are sufficient. You wouldn't have to do all of
this stuff that I did, which would waste your time. But just to prove to
you that you could figure it, I did that. But anyway. I have to admit I did waste
your time a little bit. You should just immediately
say, oh, I got two points on a line. Once I have two points, I can
completely figure out if any other point lies on that line. 86. The length of the edging that
surrounds circular garden K-- OK, so we have two gardens,
it looks like. So we have circular garden K,
and then we have circular garden-- let's see,
what does it say? It is 1/2 the length of the
edge that surrounds circular garden G. So this is circular garden G. This is K. This is G. So they're saying the length of
the edging that surrounds circular garden K is 1/2 the
length of the edging that surrounds circular garden G. So we could say circumference
of K-- right, that's the edging that surrounds it--
is equal to 1/2 the circumference of G. Fair enough. What is the area of garden K? Assume that the edging
has negligible width. Well, a couple of things. If we know the circumference,
we know the area, right? Because the circumference is 2
pi r, and once we know r, area is pi r squared. So if we can figure out the
circumference of K, then we know its area. And likewise, if we know
anything about the radius, or the diameter, or the area of
G, we can figure out its circumference. We know its area. We can use the formula, area
equals pi r squared to figure out what r is. And then once you figure out
what r is, you could just figure out that circumference
is equal to 2 pi r. So if you have any of this
information, you're able to figure out any of
the rest of it. So statement number one says,
the area of G is equal to 25 pi square meters. So let me write it. So immediately you should say,
hey, if I know the area of G, I can figure out the
radius of G. And if I know the radius of
G, I can figure out the circumference of G. If I know the circumference
of G, I can figure out the circumference of K. If I know the circumference of
K I can figure out the radius of K, and if I know the radius
of K, I know the area of K. You shouldn't have to calculate
any of that. So this is enough. This is sufficient. Statement two. And maybe I'll do that for you,
just to show you that it can be done. But hopefully it's making
some sense. The edging around G is
10 pi meters long. So they're actually telling us
that the circumference of G is equal to 10 pi. So this is even easier. They're giving us this. That means the circumference of
K is 5 pi, which means that we can figure out the radius
of K and we could figure out the area. So this is also sufficient
to solve the problem. And just so you don't take my
word for it, let me just show you how quickly you could
figure it out. If the area of G is 25 pi, that
means that pi times the radius squared of G
is equal to 25 pi. That means that the radius
is equal to 5, right? Cancel out the pi
on both sides. r squared is equal to 5. That means the radius is 5. That means the circumference
is 2 pi times this. So that means that the
circumference of G is equal to 2 pi times 5, which is 10 pi,
which is exactly what statement two told us. And then from there, we could
figure out the circumference of K, which is 1/2 of that. So circumference of K
is equal to 5 pi. And then we know that 2 pi
times the radius of K is equal to 5 pi. Divide both sides by pi. We know that the radius
is equal to 5/2. And then the area of K would
be pi times this squared. So it would be pi times 25/4,
and we'd be done. But all of this was
a waste of time. I just wanted to show you that
once you have the radius or the circumference, or the area
of either of these, you're able to figure out
everything else. And I'm out of time. See you in the next video.