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Current time:0:00Total duration:11:03

We're on problem 138. In a certain business production
index, p is directly proportional to
efficiency index e. So we could say that p is equal
to some proportionality constant times e. Because they say p is directly
proportional to e. Fair enough. Which is, in turn, directly
proportional to investment index i. So e is proportional to i. So we could say it's equal
to some proportionality constant times i. And actually, we could go even
a step further and say, well, if p is proportional to e and
e is proportional to i, we could also say that p is
proportional to i. Directly proportional to i. So it might be some other
constant times i. And you could prove that
mathematically, right? You take this and substitute
it for e and you get p is equal to k times e, e is equal
to some constant times i, and so p is equal to kn times i. But k and n were both arbitrary
constants, so we could just call that m. So this is the information that
the problem gives us. p is proportional to e, e is
proportional to i, so p is also proportional to i. So what are they asking us? What is p if i is equal to 70? So if we knew this proportional constent, we'd be all set. We could actually solve
this problem. Maybe they're going to go
through e or something, I don't know. So statement number one. e is equal to 0.5 when
e is equal to 60. Well, we can definitely use
this information to figure out what n is. But that by itself isn't going
to help us, because we can figure out what n is and then
once we know what n is, we can figure out what e
is when i is 70. We'll put 70 here times whatever
we figure out n is. And we'll get e. But without knowing what k is,
that still won't help us. So statement one, by itself,
if I'm right, isn't right. That won't help us. And I'll show it to you. Oh, this would be waste
of time if you understood that logic. Because you can say 0.5 is equal
to n times 60-- this is so we can solve for n-- so n
is equal to 0.5 over 60. That's the same thing
as 1/120. So then that equation
boils down to e is equal to i over 120. 1/120 times i. So when i is equal to 70, e
would be equal to 70/120, which is equal to 7/12. But that still doesn't
help us. E is equal to 7/12, but we don't
know what k is, right? When i is equal to 70, e is
7/12, so we still don't know what k is based on just the
information in statement one. So as far as I can tell
right now, one by itself-- not so useful. Statement two. p is equal to 2 when
i is equal to 50. Well, this is useful. Because we already said,
p is proportional to i. P is equal to some constant
m times i. So we could use this information
to solve for m. So p is equal to 2 when
i is equal to 50. Divide both sides by 50, you
get m is equal to 1/25. So the question was, what is
p when i is equal to 70? So we could just say p is
equal to m times i. Well, i now is 70. And whatever 70 divided
by 25 is, that's p. So statement two alone
was sufficient to solve this problem. And statement one didn't
help us much. Next problem. 139. If x does not equal minus y--
that's interesting-- is x minus y over x plus
y greater than 1. So why do they say x does
not equal minus 1? I'm sorry. x does now
equal minus y. Well, if x were equal to minus
y, if this were minus y, then you'd have minus y plus y, the
denominator would be 0, and you'd be undefined. So maybe that's why they
put that out there. Let's see, I think we can
simplify this statement. So we could multiply both
sides by x plus y. You get x minus y is greater
than x plus y. And so this holds
true if what? Let's see, we could subtract
x from both sides. Subtracting x from both sides
you get-- just get rid of the x's-- you get minus y
is greater than y. Let's see, we could add y to
both sides of this equation. So the left side, if you
add y here, you get 0. If you add a y on this
side, you get 2y. And then you could divide both
sides of this equation by 2. And you get 0 is greater than
y, or y is less than 0. Either way. This statement is true
if, and only if, this statement is true. So this is really all
we need to test for. Is y less than 0? If we know that y is less than
0, we know that this is true. Or if we know that this
is false, then we know this is false. So statement one. Statement one tells us
x is greater than 0. Well, this is useless. This actually has no bearing on
whether y is less than 0. x can be anything. It doesn't change this. So this by itself is useless. Statement two is-- well,
there you go. y is less than 0. So statement two tells us this
is true and this is true if, and only if, this is true. So therefore, statement two
alone is sufficient to answer this question. Very seldom do you boil down the
statement to actually one of the statements that
they provide you. But that was interesting. Next problem. 140. In the rectangular coordinate
system, are the points r,s and u,v equidistant from
the origin. OK, so they're essentially
saying, is the distance of the point r,s-- distance from the
origin-- equal to the distance-- I'll call it d sub
o-- equal to the distance of the origin of point u,v. And here, just to get the
intuition of distance-- I always find it silly that they
teach something called a distance formula in high
schools, because it's really just the Pythagorean theorem. And by calling it something
different and making you memorize a different formula,
it'll just clutter your head. So this is the x-axis,
this is the y-axis. The point r,s will be here. This is r. This is s. This is the point r,s. What's its distance
from the origin? Well, its distance from origin
is the length of this line right there. What's the length of that? Well, we can use Pythagorean
theorem. The height right there is s. The base here is r. And if we call this distance,
we use Pythagorean theorum. r squared plus s squared is
equal to the distance squared. Or we can say that the distance
is equal to the square root of r squared
plus s squared. The distance to the origin. So this statement up here boils
down to that the square root of r squared plus s squared
needs to be equal to the square root of u squared
plus v squared. Well, just to simplify things,
let's just square both sides of this equation. The question they ask is, is r
squared plus s squared equal to u squared plus v squared? Is this true? That's what they ask us. And that kind of simplifies
things. Makes them more concrete. So let's see what the
statements give us. Statement one. r plus s is equal to 1. Just off the cuff, I don't see
where that's going to be actually useful. It's not like you can just
square this. r plus s squared is r squared plus 2rs
plus s squared. A lot of people make the mistake
that thinking, oh, r plus s squared is r squared
plus s squared. No, that's not true. You have to distribute all the
terms and you end up with three terms. So that's
not right. I don't see an r plus
s anywhere up here. Let's try statement
number two. Statement number two. u is equal to 1 minus r, and
v is equal to 1 minus s. So this seems like it could
be interesting. Because it allows us to,
essentially, reduce this question, which is a question of
four variables, and turn it into a question of two variables
by substituting u and v with these things. So let's do that. Let's turn this statement into
a statement of two variables. I'll switch colors just
to ease the monotony. So the left-hand side is r
squared plus s squared is equal to u squared. Well, now they're telling us
that u squared is 1 minus r squared-- plus v squared. Well, v is 1 minus s. 1 minus s squared. Let's just keep simplifying. r squared plus s squared is
equal to 1 minus 2, r plus r squared plus 1 minus
2s plus s squared. Let's see, we could subtract
r squared from both sides. We can subtract s squared
from both sides. And we're left with 0 is equal
to 2 minus 2r minus 2s. And let's see, we can bring the
r and the s terms over to the other side. So add 2r plus 2s
to both sides. So bring them over, you get
2r plus 2s is equal to 2. Divide the whole thing
by 2-- both sides. You get r plus s
is equal to 1. Interesting. So when you apply these two
constraints on our original question, the question
gets reduced to this. If we know that statement two is
correct, the question that the problem was asking gets
reduced to this: It's saying, if we know that this happens,
then the question is true if this is true. Well, just from statement
two alone, we don't know that this is true. But, as you see, statement one
tells us that that is true. So if you use both statements
together, you know that this question is correct. This question is true. And that was actually pretty
interesting and a little bit hairier than normal. Normally, you can identify
immediately just by eyeballing it. But anyway, I'll see you
in the next video.