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.