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Current time:0:00Total duration:10:18

We're on problem
47 on page 281. If the two floors in a certain
building are 9 feet apart, how many steps are there in a set
of stairs that extends from the first floor to the second
floor of the building? So I think I can draw that. So this is the second floor,
this is the first floor. They want to know how many
there's-- they're 9 feet apart, that's what they tell us
already-- 9 feet, we need to figure out how many steps
there are to go from the first to the second floor. So piece of information one,
they give us, each step is 3/4 foot high. Well, this by itself should
give us enough information, right? Because each of these
distances-- I'll do it in green-- each of these
are 3/4 foot high. So to figure out how many total
steps to get to the top you just say 3/4 times the
number of steps is going to be equal to 9. And then you could just solve
this by multiplying both sides by 4/3. But we don't have to solve it. We just have to know that
we can solve for it. So one, alone, is
enough, right? And you could, what? If you multiply both sides
of this you get 12 steps. But anyway, we want to avoid
having to actually do math. We just want to figure out
whether we can do the math. Piece of information two, each
step is 1 foot wide. Well, that's clearly useless
because it doesn't tell us how much altitude we're making on
each step and that's what you have to figure out to say how
many steps you need to go up 9 feet in the air. So this is useless. So the answer is A: statement
one alone is sufficient. Next problem. 48. If xy is equal to 0, oh,
does not equal 0. So that tells us that neither
x nor y is 0, right? Is x divided by y going
to be less than 0? In order for this to be true
one of these numbers-- so I mean if x divided by y is
negative, that means that either x or y, but not both of
them, are negative, right? That's the only way you can
get a negative number when you're dividing. If both of them are negative
this would be a positive number. So let's see what we can do
with their information. Number one. x is equal to, and this is the
question, is xy less than 0? And one says x is equal
to minus y. Well, immediately let's just
substitute that back in. If x is equal to minus
y then what's x/y? x is equal to minus y, so you
have minus y/y and that will equal-- and we know
y doesn't equal 0. So for any other value other
than 0-- if this was 0 this would be undefined-- but then
this is equal to negative 1. So in this case x divided by y
is equal to negative 1, which is definitely less than 0. So that proves our statement. So statement one alone
is all we need. Now let's see what statement
two gives us. I'll do it here. Statement two. Minus x is equal to minus y. So that tells us that minus x is
equal to y and then we can just substitute the same
thing in again. Well, now we could substitute
for x, or we could just multiply both sides by negative,
you get x is equal to minus y, which is the same
thing as this here. So statement two, alone,
is also sufficient to solve this problem. And so the answer is D: each
statement, alone, is sufficient. Switch colors, problem 49. How many people are directors
of both company k and company r? OK. Directors of k and r. Statement number one. There were 17 directors present
at a joint meeting of the directors of companies
k and r and no directors were absent. At a joint meeting of the
directors, so that's all the directors of k and r. So me draw some Venn Diagrams.
So if that is k and then that is r. So they're saying that when
you add both of these together, because this is a
joint meeting of all the directors of both, we got 17. So there's 17 in this entire
universe of directors. That's what statement
one tells us. If you take this whole circle
and then you add up the extra-- don't double count the
intersection-- you add up the extra, you get 17. But that alone doesn't tell
us how many joint directors there are. Joint directors are
these people. People who are on the board
of both k and r. So this is k and this is r. So statement one alone
doesn't help us. Although, I'm suspecting, maybe
in conjunction with something else, it could. Statement number two is company
k has 12 directors and company r has 8 directors. OK. k is equal to 12 directors
and r is equal to 8 directors. OK, so everything in the k
circle combined is 8, right? Sorry, everything in the
k circle is 12, right? k directors. Everything in the
r circle is 8. Now, if we wanted to get the
total of the k and the r-- so by itself that doesn't tell
me how many overlap. So when you take statement one
or statement two independently that doesn't help us. But what if we were to
figure out if we would use them together? So how many total directors
are there going to be? There's going to be the total
directors in k-- so k's directors-- plus the
director's in r. But if you were to just add
those two up, you would double count the people who are
in both k and r, right? You would count them twice. You would count them when
they're in k and you would count them when they're in r. So if you wanted to figure out
how many total directors there are you would then subtract out
the people who are in k and r, right? You don't want to subtract them
twice, you just don't want to double count
them, right? Because when you we do k plus
r you're counting them for k and then you're counting
them for r. So let's subtract them out once
so that you only count them for k or that you only
count them for r. You only count them once. So minus kr-- so this notation
that's people who are in k and r. And what does that equal? Well, statement one
told us that. It told us that they're
total of 17 directors. And so statement two told us
there are 12 directors in k plus 8 directors in
r minus the joint directors is equal to 17. And, by the way, we don't
even have to do this. We could've just recognized
that if we know the total number of directors and we know
how many are in each of the groups that we can
figure it out. And we would just answer the
question that both statements together are sufficient. But I'll just show you that we
can figure it out just so that you're happy with it. Let's see, we get 20 minus
kr is equal to 17. And so you get kr
is equal to 3. So there are 3 directors that
overlap with both and we were only able to answer that
question by using statement one and statement two. Next problem, 50. If x and y are positive
is xy greater than 1? Statement number one tells
us xy is greater then 1. So let's see, does that
help us at all? And that's not obvious. That just tells us that x is
greater than 1/y or y is greater than 1/x. So let's just think about it,
this statement is equivalent, if we multiply both
sides by y. And we can do that without
changing the inequality because we know y is
greater than 0. So if we do that, we get
x is greater than y. If we can show this,
we can show that. And remember, the only reason
why I didn't have to change the inequality is because I knew
that when I'm multiplying both sides by y, that
y is greater than 0. If y was less than 0 I'd have to
switch the sign right here when I multiply both
sides by it. But anyway, if I can prove
x is greater than y, we're all set. This doesn't help me. Let's see, statement number
two is x minus y is greater than 0. Well, if I add y to
both sides of this equation what do I get? I get x is greater than y. So this proves exactly what I
need to prove, and if you want to go all the way to what
originally they asked, divide both sides by y-- and we don't
have to change the sign because y is positive-- you
get x divided by y is greater than 1. Which is exactly what
we needed to prove. So statement two, alone,
is sufficient. See you in the next video.