Problem number one.
In the number 2,010, or 2,010, the value represented by the
digit 1 is what fraction of the value represented
by the digit 2?
We have the number 2,010, or we could rewrite this as 2,000
plus 0 hundreds plus 10 plus 0.
And they're saying the value represented by the digit 1--
well, the value represented by the digit 1 is 10-- is what
fraction of the value represented by the digit 2?
Well, the digit 2 represents 2,000.
So the fraction is going to be the 10 divided by what the 2
represents, 2,000.
The 1 represents 10, the 2 represents 2,000.
You can cancel out a 0 right there, we get 1 over 200,
which is choice B.
Problem two.
If P is a positive integer, which of the following must
also be a positive integer?
Let's see, choice A.
1 minus P.
Well, if P was 2, 2 is a positive integer, then 1 minus
2 would be equal to minus 1.
This is not a positive integer.
This is a negative integer.
So it can't be A.
1 over P.
Well, once again, I'm just picking 2 at random, a
positive integer.
If P is 2, then 1 over 2 is not a positive integer, so it
can't be B.
Let me cross that out.
Cross that out right there.
Well, I'm going to keep running with my number 2.
If it doesn't work for number 2, it's definitely not going
to work for any positive integer.
So if P is 2, right here, then the square root of 2, once
again, not a positive integer.
And then finally P squared.
Well, for 2, if I say P is equal to 2, then P squared is
equal to 4.
And in general, if I take any positive integer and I square
it, if I multiply it by itself, I'm just going to get
another-- it's definitely going to be positive, it's
definitely going to be an integer.
So my choice is D.
Next problem.
Problem three.
Let me switch colors, just for fun.
According to an article in a financial journal, a certain
company earned 3.85-- now why isn't it writing?
[UNINTELLIGIBLE]--
earned $3.85 million last year.
Based on this report of the company's yearly earnings, a
person reading the article estimates that the company
earned an average of approximately $30,000 a month,
and they're saying estimates.
So we don't have to do the exact math.
Which of the following statements best describes the
reasonableness of this estimate for the company's
average monthly earnings?
So $3.85 million last year, and then we want to be able to
estimate how many monthly earnings.
So we want to divide 3.85 million by 12 months.
That's 3.85 million a year, and we want to
divide it by 12.
We could do the exact number, but we could say,
hey, you know what?
This is pretty close to 3.6 million.
The reason why I'm picking 3.6 million is because
36 is 3 times 12.
So if we do 3.6 million, we could say that this is
approximately 3.6 million divided by 12,
which is equal to what?
If this was 36 million divided by 12, we
would have 3 million.
But it's not 36 million divided by 12,
it's 1/10 of that.
It's 3.6 million divided by 12.
So it would be 0.3-- and you could put a 0 out front-- 0.3
million per month.
Or another way to write 0.3 million is 300,000 per month.
Now, someone estimated that they made $30,000 per month.
So they were off, they were too low by a factor of 10.
You would have to divide by this, the actual estimate, the
correct estimate, by 10 to get their estimate.
So their estimate is too low by a factor of 10.
That is choice B.
Another way you could have thought about this is instead
of writing 3.85 million, you could have written that as
3,850K divided by 12.
Or, you could have said, well that's approximately 3,600K
divided by 12, and then you would have
said, OK, that's 300,000.
You would have gotten the same answer, which is 10 times more
than their estimate.
Or their estimate was too low by a factor of 10.
Problem four.
The mean distance from the earth to the sun is
approximately 93 million miles, or one
astronomical unit.
The mean distance from Neptune to the sun is approximately
2.794 times 10 to the ninth miles.
What is the approximate mean distance from Neptune to the
sun in astronomical units?
So essentially we want to divide the average distance of
Neptune-- 2.794 times 10 to the ninth miles-- divided by 1
astronomical unit, which is 93 million.
When you write it like this, it's kind of hard to digest,
or see what's going on.
But what we can do is write this guy, right here, in
scientific notation as well.
So 93 million looks like that.
We could say that this is 9.3 times 10 to the one, two,
three, four, five, six, seventh power.
So now this thing, up here, simplifies to 2.794 times 10
to the ninth over 9.3 times 10 to the seventh.
Now we could just divide this, but the way my brain works, I
would like, instead of this to be 2.7, I would
like this to be 27.
And the reason why I would like this to be 27 is because
we have a 9 down here.
Then we could say, hey, 27 divided by 9 is
roughly equal to 3.
So we can rewrite the top part of this.
We could rewrite this.
If we multiply this part by 10, we would have to divide
this part by 10 in order to not change the number.
We don't want to actually change the value of the
number, so we get 27.94 times 10 to the eighth.
All I did is I multiplied this part by 10, and to offset
that, I divided this part by 10.
10 to the ninth divided by 10 is 10 to the eighth divided by
9.3 times 10 to the seventh.
And now you could say, look, 27.94 divided by 9.3, that's
going to be approximately 3.
Remember, we're trying to find the approximate mean distance.
We can estimate a little bit.
And then 10 to the eighth divided by ten to the
seventh-- you could just subtract 7 from 8 of the
exponents, and so that's times 10 to the one.
10 to eighth divided by 10 to the seventh is exactly 10
times 10 to the seventh.
So our approximate answer is going to be 3
times 10 to the one.
Or 3 times 10.
Or approximately equal to 30 astronomical units.
And that is choice A.