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Welcome back.
We are on problem number 13.
It's on page 491.
The price of a telephone was first increased by 10%, and
then the new price was decreased by 25%.
The final price was what price of the initial price?
The price final is equal to the initial
price-- price initial.
So the price of a telephone, the initial price, was
increased by 10%.
So when you increase something by 10%, that's like
multiplying it by 1.1.
Or you could say 1 plus 10%.
That's 1.1.
And then it was decreased by 25%.
Well, decreasing something by 25% is the same thing as
multiplying by 0.75.
Why is that?
Because that's 1 minus 0.25.
And so this sets up our relationship, and so we just
have to do some math.
What is 0.75 times 1.1?
That's 75.
0, 75, 5, 8.
And how many points do we have in the decimal point?
One, two, three.
Three points.
So we have price final is equal to price
initial times 0.825.
And what do they say?
The final price was what percent of the initial price?
So that's it.
The final price is 82.5% of the initial price.
And the main thing to just realize, when you increase
something by 10%, that's the equivalent of
multiplying it by 1.1.
And then when you decrease something by 25%, that's the
same thing as multiplying it by 0.75.
And why?
Because it's 1 minus 0.25.
You're decreasing it by 25%.
And that is choice C.
Next problem.
Number 14.
This is what I do every day in my job.
I work for a hedge fund.
I'm always doing percentages.
Image.
Invert colors.
I do other things.
Don't think my entire livelihood is based on just
multiplying percentages, but a surprising fraction of it is.
Anyway, all right.
Problem number 14.
When the number w is multiplied by 4, the result is
the same as 4 is added to w.
OK, so it's saying when w is multiplied by 4, so 4w, is the
same thing as the result as when 4 is added to w.
So it's the same thing as w plus 4.
Fair enough.
What is the value of 3w?
3w is equal to what?
Well, subtract w from both sides of this equation.
You get 3w equals 4.
That might have been the fastest problem
we had to do, right?
We just solved it.
3w is equal to 4.
Choice E.
Problem 15.
The lengths of the sides of a right triangle are consecutive
even integers.
Fascinating.
And the length of the shortest side is x.
It looks like I'm going to have to draw a
triangle, in yellow.
So it's a right triangle.
That looks like a right triangle to me.
And they say they're consecutive even integers, and
the shortest side is x.
So let's say this is x.
Consecutive even integers.
So the next even integer is going to be x plus 2.
And then the last integer, it's going to be the
hypotenuse, the next one.
And that's going to have to be x plus 4.
And they want to know what is x?
Or what equation can be used to solve for x?
Well, this is the Pythagorean theorem.
So we just say this squared plus this squared is equal to
this squared.
So x squared plus x plus 2 squared is
equal to x plus 4 squared.
Now let me see if this is already there
in any of the choices.
x squared plus x plus 2 squared is
equal to x plus 4 squared.
Yeah, it's already there.
We didn't do any simplifying.
This is choice C.
And we are done.
These problems are extremely fast to do.
And the important realization is that they're consecutive
even integers.
So you add 2 each time, and you just use
the Pythagorean theorem.
Next problem.
Problem 16.
If x is an integer greater than 1, and if y is equal to x
plus 1/x, which of the following must be true?
I like these.
OK, I'll write all the choices down.
II, and III.
OK, this first one says y does not equal x.
The second one says y is an integer.
The last one says xy is greater than x squared.
Well, let's think about it.
First of all, do we know that y is an integer?
Well, no, not really.
I could immediately find an x.
If x is equal to 2, then y is what?
It's 2 plus 1/2.
It's equal to 2 and 1/2.
So we know y isn't an integer.
If x is an integer, y doesn't have to be one.
So we know this is definitely not the case.
Now, do we know for a fact that y does not equal x?
Well, let's play with this equation to see if we can get
it to a form that makes a little bit more sense to us.
So we know that x is greater than 1, or that x
does not equal 0.
And obviously, this would be undefined if x equals 0.
So let's multiply both sides of this equation by x.
So we get xy is equal to x squared plus 1.
This, all of a sudden, looks fascinating.
xy is equal to x squared plus 1.
So do we know immediately that xy is greater than x squared?
Well, sure.
No matter what x squared is, xy is going to be that number
plus 1, right?
And this term is always going to be positive.
This is always going to be positive.
There's no way you can get x squared plus 1.
If you're not dealing with imaginary numbers, it's going
to be negative, so this whole thing's going to be positive.
x is positive.
So if this is positive, x is positive, we know also that y
is going to be positive.
I don't know if that helps, but anyway.
But we know that xy is always 1 more than x squared, so we
do know that xy is greater than x
squared, so this is true.
Now, let's see if we can show that y has to not equal x.
Well, let's think about it.
No matter what x is, y is going to be equal to some
fraction larger than x.
And as you get really, really, really large x's, that
fraction, the difference, is going to get smaller.
You could try it on a calculator.
If you put in x is 5, then y is 5 and 1/5.
If x is 500, then y is 500 and 1/500.
But no matter what number you put in that's a larger number
than 1, you're going to get a little bit left over.
y is going to be just that little bit
fraction larger than x.
So we know for a fact that y is going to be not equal to x.
And so this is also true.
So choices I and III, and that is choice D.
We are done with this section, because there's only 16
problems as far as I can tell.
Yep, only 16 problems. Actually, I think we're done
with all the math sections in this test
number 2 in the book.
I will see you in the next test. See you soon.