We're on problem 73.

How much did a certain telephone call cost?

All right.

Well, we know nothing so far.

Statement one.

The call lasted 53 minutes.

Well, unless they tell us how much it costs per minute, that

still doesn't tell us how much the call costs.

Statement two tells us the cost for the first 3 minutes

was 5 times the cost for each additional minute.

But that still doesn't help us.

The cost for the first 3 minutes might have been $0.05

a minute, and then we might have been $0.01 a

minute after that.

Or it could have been $0.10 a minute, and then-- or let me

make it a multiple of 3-- $0.12 a minute, and then $0.04

after that.

Both of these satisfy that statement.

And actually, if I have to be honest, this statement is

slightly ambiguous.

Because they're saying the cost for the first 3 minutes.

Do they mean that the total cost for the first 3 minutes

was 5 times each additional minute?

Or was it the cost per minute of the first 3 minutes was 5

times additional minutes?

Well, either way, both statements combined don't help

us, because we're not able to figure out the cost per

minute, so that we could apply it to this.

So they are both useless.

74.

In a certain office, 50% of the

employees are college graduates.

So let's say 50% college.

I'll call it C.

Or I could say 50% of the employees are college

graduate, right?

60% of the employees are over 40 years old.

So it says 60% of the employees equal, we'll call it

O for over 40.

I don't know where this is going, I'm

just making up numbers.

And it said if 30% of those over 40 have Master's degrees,

how many of the employees over 40 have Master's degrees?

OK.

So 30% of the over 40 have Master's degrees.

I don't know, I'm just trying to make up some notation.

How many of the employees over 40 have Master's degrees?

So we're actually trying to come up with this number.

So if we knew how many employees over 40 there were,

we would be set, because we'd just multiply 30% times that.

If we knew how many employees there were, we would be all

set, because we could figure out the number over 40 and

then multiply 30 times that.

So let's see what they give us.

Statement one.

Exactly 100 of the employees are college graduates.

So essentially, C is equal to 100, right?

And they told us that 50% of the

employees are college graduates.

So that means how many total employees are there?

That means that there are 200 employees.

100 is 50% of 200.

So that told us there are 200 employees.

If there are 200 employees, how many

employees over 40 are there?

Well, that's O, right?

O for over 40.

Well, that's just 60% of 200.

That's 120.

And then how many over 40 with Master's degrees?

Let's that OMD.

How many of these are there?

Over 40 with Master's degrees.

Well, that's 30% of over 40, so that's 30%-- this is an O--

this is 30% of this, which is equal to-- so over 40 with

Master's degrees is equal to 36.

And we would be done.

So statement one alone is sufficient.

I should call it over 40 instead of a 0.

Don't want to get confused.

Over 40 with Mater's degrees.

So statement two.

Of the employees 40 years old or less, 25%

have Master's degrees.

So that doesn't tell us anything, really.

It doesn't tell us how many employees there are over 40,

which was one of the things that we realized we have to

figure out.

It doesn't help us figure out how many total employees there

are, so we really can't do this type of logic.

So it seems fairly useless.

So yeah, employees 40 years old or less.

25% of-- no.

This is a useless piece of information.

So the answer is-- what is that?

I always forget.

A.

Statement alone is sufficient and statement 2 is useless.

75.

They ask is rst equal to 1?

Statement one tells us that rs is equal to 1.

This by itself doesn't help us, right?

If rs is equal to 1, the only way that this whole thing is

going to be equal to 1 is if t is also equal 1.

But we don't know that, so-- I mean, t could be equal to 10.

So statement one alone doesn't help us.

Statement two tells us, st is equal to 1.

Well, I can show that even if we have both of these

statements, that we still can't prove this.

For example, I can come up with a scenario that meets

both statements where this is true.

Because I can say what if r is equal to 1, s is equal to 1,

and t is equal to 1?

Then we can make both of these statements true.

So then, definitely, rst is equal to 1.

But let me see if I can construct something where this

doesn't hold up.

So what if r is equal to 6.

s is equal to 1/6, and then t is equal to 6.

So then r times s would still be equal to 1, right?

s times t would still be equal to 1.

But what is rst now?

It's 6 times 1/6, times 6, which is equal to 6.

So I can satisfy both of these, but still end up not

knowing whether it equals 1 or something different.

I just picked this arbitrarily.

You can actually kind of pick an arbitrary number here and

set this up.

So anyway, this is E.

I think that's E, right?

Both statements combined are useless.

Right.

They're not sufficient.

Next problem.

Turn the page.

It's problem 76.

And they've drawn a nice pie chart for

us, which I'll reproduce.

76.

Total expenses for the 5 divisions of company h.

Let me draw this pie chart.

OK.

And then they have a bunch of-- let's see, how

many pies are there?

There are 5 pies.

I'll try to draw it the way they do it.

So that's one.

Two.

Three.

And they have one more.

There we go.

So that looks pretty close to what they've drawn.

And then they write, this is Q, R, S, T, and P.

O is the center.

They call this angle, x.

Fair enough.

So this is the total expenses for the 5 divisions.

So each of these are divisions for company h.

76.

The figure above represents a circle graph of company h's

totally expenses broken down by the expenses for each of

it's 5 divisions.

If O is the center of the circle-- fair enough, so this

is the center, although I didn't draw it quite at the

center-- if O is the center of the circle, and if company h's

total expenses are $5.4 million, what are the expenses

for division R?

So the whole thing is $5.4 million.

So the whole pie is $5.4 million.

And we need to figure out what R is equal to.

So the whole expenses are 360 degrees.

So whatever fraction x is of 360 degrees, that's going to

be the fraction that R is of the total expenses.

So let's see if they give us that.

If we can figure out what angle x is.

One, x is equal to 94.

Well, we're done.

Because x over the total degrees in the circle-- so 94

over 360 degrees-- that's going to be equal to the

expenses of division R divided by the total

expenses, $5.4 million.

And this is a trivial equation to solve.

So one alone is sufficient.

Let's see what they give us for statement number two.

The total expenses for divisions S and T are twice as

much as the expenses for division R.

So they're saying that this right here is twice as much as

what we're trying to figure out.

It's this right here.

Well, that doesn't help us much, because P and Q could be

arbitrarily small or large.

These two combined could be $2 and this could be $1, right?

In which case, P and Q are the bulk of the

expenses for the company.

Or maybe these two are $4 and R is $2,

which are all possible.

And these would represent the rest of the-- whatever.

It gets you to $5.4 million.

So statement two is useless.

So statement one alone is sufficient and statement two

does us no good.

And I'm out of time.

I'll see you in the next video.