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Current time:0:00Total duration:10:30

We're on problem 99. And I like these that don't have
a lot of words in them. m does not equal n. We have to prove that. Is it true that m does
not equal n? Statement number 1 says that
m plus n is less than 0. Well, let's see, what
does this tell us? This just tells us that m
is less than negative n. It still doesn't tell
us anything. Maybe negative n is-- well,
let's say, if m is equal to n, which is equal to negative
10, then this would hold true, right? Because negative 10 plus
negative 10 is minus 20, which is less than 0. So this still doesn't
prove anything. Or, of course, I could have
two different numbers. I could have m being minus
5, and n being minus 1. And when you add them together
you get minus 6, both less than 0. So it still doesn't tell us
anything about whether m does not equal n. Statement two. m times n is less than 0. Now, this is interesting. m times n is less than 0. So that tells us that one is
positive and one is negative. The only way you can multiply
two numbers and get something less than 0 is one positive
and one negative. And it doesn't say equal 0, so
they both can't equal 0. And so, when you multiply two
numbers and you get a negative number, they have to be
different numbers, because one has to be positive and one
has to be negative. Try it out with as many
things as you want. But you know from your basic
negative multiplication rules that the only way to get a
negative as a product of two numbers is one positive,
one negative. So this is all we need to know
to know that m and n are different, because one's
positive and one's negative. I said that about five times. Problem 100. And you didn't need
statement one. Problem 100. When a player in a certain game
tossed a coin a number of times, four more heads
than tails resulted. So heads is equal
to tails plus 4. Heads or tails resulted
each time the player tossed the coin. Fair enough. How many times did
heads result? So essentially the problem
description just tells us that heads is equal to
tails plus 4. And that heads plus tails is
just all of the tosses. So statement number one
says, the player tossed the coin 24 times. So that's what I just said. Heads plus tails
is equal to 24. And so this is enough
information by itself to know how many times did
heads result. Because we could rewrite this
top equation as heads minus tails is equal to 4. And if you wanted to solve
it, you'd get 2 heads is equal to 28. Heads is equal to 14. All of that would have been a
waste of time as soon as you saw that you had two linear
equations and two unknowns. You could have said, oh, I have
enough information to solve for any of
the variables. This is a 4. So statement one by
itself is enough. Statement two. The players received 3 points
each time heads resulted and 1 point each time tails
resulted, for a total of 52 points. So he got 3 points for every
head-- that's how many points he got for all the heads-- plus
1 point for each tail. So that's how many points he
got for all the tails. And that equals 52. So once again, this is providing
us with another linear equation. So now we have two linear
equations with two unknowns. I'm not even looking at what
they gave us in statement one. So just using these two, we
have enough information to solve the problem. You can do it the
exact same way. This is what you learned
in algebra 1. So, each statement independently
is sufficient to solve the problem. Problem 101. We've already done 100
problems. 101. Once you get going, these
get kind of fun. If s is the infinite sequence--
OK, so it's like s1 is equal to 9. s2 is equal to 99. s3 is equal to three 9's. I get it. And the kth s is equal to
10 to the k minus 1. This is the third one. 10 to the third is
1000 minus 1. 10 squared is 100 minus 1. Right. So it all works. All right, that's what
they told us. Is every term in s divisible
by the prime number, p? Well, it depends what p is. And frankly, if just the first
term is divisible by p, all of them are going to be
divisible by p. Why is that? Because all of the terms
are divisible by the first term, or 9. So if p goes into 9, it's going
to go into all of these terms, because 9 goes
into all of them. So we just have to say,
does p go into 9? It's a much simpler way of
thinking about the problem. Statement number one says,
p is greater than 2. So does every prime number
greater than 2 go into 9? Well, no. 5, 7. These don't go into 9. 3 goes into 9. So we still don't know whether
p goes into 9. Statement two. At least one term in
the sequence, s, is divisible by p. If we knew that it was this
term, if we knew that the first term was divisible by
p, we would be all set. But we don't know whether it's
just the first term. For example, p could be 11. What if p is 11? If p is 11, then it goes
into this term. It goes into the 99. But it won't go into
the first term. It won't go into 9. So 11 is an example where it
holds for case two, but it doesn't hold for the
whole question. It's not divisible
into every term. Or we could say that if p was
equal to 3-- well, of course, that'll go into everyone. So statement two by itself
actually doesn't help us any. And actually, both of these
numbers satisfy both statements. And so, if you pick 11,
it doesn't work. If you pick 3, it does work. So both statements combined
still do not give us enough information. Problem 102. I have got to do some drawing. OK, so I have a quadrilateral
here. Let's see. That's RU. And then go up there. And then go flat to there. And then come down like that. And then they draw
an altitude here. All righty. OK, so this an R. This is a W. This is a U. A T. And an S. They're saying the height
of this is 60 meters. This length is 45 meters. And they're saying this
right here, this length is 15 meters. Fair enough. Quadrilateral RSTU, shown above,
is a site plan for a parking lot, in which side RU is
parallel to side ST. OK, so this and this are parallel. What is the area of
the parking lot? So, immediately I can figure
out the area of this triangle, right? Base times height, times 1/2. So I can figure out this
area immediately. I can immediately figure out
this square region right here. Because I know it's 60 by 45. And so the question is, can I
figure out this triangular region right here? Well, I don't know what
this distance is. If I knew what this distance
is, then I could say base times 60, times 1/2, and
figure this out. So this right here
is what matters. That's the crux of the
issue right there. So statement number one tells
us that RU is 80 meters. RU is equal to 80. I think this gets
us there, right? Because think about it, if this
whole thing is 80 meters, I can figure out that. How can I figure out that? Because I have 15 here. I have this length is 45. So this length is going to
be 80 minus 45, minus 15. And what is that? 60. So 80 minus 60. So this will be 20. And so just with statement one,
I can figure out this length right here. And I'm done because this
triangle is this base times this height, times 1/2. This rectangle is this base
times this height. And then this triangle is
20 times 60, times 1/2. So I have enough information
to figure out the area just with statement one. Now statement two. TU is equal to 20 square
roots of 10 meters. Well, this is enough
information, too. Think about it. This is a right triangle
right here. This height is 60. And remember, if we can figure
out-- let me switch colors-- if we figure out the base,
we're going to be set. So we could do Pythagorean
Theorem. So we could say, the based
squared plus 60 squared is equal to this number squared. And that number squared
is what? 400 times 10. So it's 4,000. So you get B squared is equal
to 4,000 minus 3,600, which equals 400. So B is equal to 20 again. So this actually also gives you
the information that this base right here is
equal to 20. And then, like I showed in the
previous statement, that's all you need to figure out
the area of the entire parking lot. So statement two alone
is also sufficient. So either of these statements
alone are sufficient to tell us the area of the
parking lot. See you in the next video.