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Alright, we're on problem 59. 59 on page 282. If a real estate agent received
a commission of 6% of the selling price of a certain
house, what was the selling price of the house? So the price of the house. And they just told us that the
real estate agent received 6% percent but that alone
doesn't tell us much. So statement number one, they
tell us the selling price minus the real estate agent's
commission was $84,600. So let's see if we can write
that algebraically. The selling price, let's say
p for price, minus the real estate agent's commission--
well, the they told us at the beginning the real estate
agent received 6% of the selling price. So that's 0.06, 6%
of the price. So they're telling us that
that is equal to $84,600. Well, we're done! That's a linear equation
with one unknown. This is algebra one. You can solve for this. Let's see, you could say this
is, I mean if you had to, 0.94p is equal to 84,600 and
then you'd have p is equal to 84,600 divided by 0.94,
whatever that is. But we don't care. We just have to know that
we could solve it. Well, since we got so close,
let's just solve it. 84,600 divided by point-- this
is a bad habit when you're taking the GMAT, you want to
just know that you could solve it-- so the selling price
of the house is $90,000. We didn't have to solve that. That would be a waste of time
on the real GMAT but I just wanted to show you how
easy it was to solve. Statement number two,
let's see if this is independently useful. The selling price was
250% of the original purchase price of $36,000. We chose p as the selling
price, right? The selling price minus-- right,
everything we talked about before was the
selling price. And now they introduce
this thing called the purchase price. So the price selling is equal
to-- and that's what we're going to figure out--
is equal to 250%. So that's 2.5 times the purchase
price of $36,000. So times $36,000. Actually we didn't have
to write this. So the price is 2.5 times 36,000
which is, I'm guessing, let's see, 72, yep, it's
$90,000 again. We didn't have to do that. But once again, this is just a
very-- this is actually not even algebra. You just have to multiply
2.5 times 36,000 and your get the answer. So each of these, independently,
are enough to solve this problem. So that's D. Next problem. 60. This yellow's a little
bit over the top. Let me do more muted color. So they write, if the square
root of x/y is equal to-- what does that say-- d? Right, is that what
they wrote? Is equal to d? No, that's equal to n I think. Alright, is equal to n, what
is the value of x? So statement number one,
they tell us that yn is equal to 10. Well, this is pretty useful
because this first equation they give us, we would just
multiply both sides by y, we get square root of x is
equal to yn, right? y times both sides
gets us this. And if yn is equal to 10 then
we know that the square root of x is equal to 10. So x is equal to 100. So statement one alone
is enough. Now what does statement
two do for us? They tell us y is equal to
40 and n is equal to 1/4. So once again, they substituted
everything else, so we just have to
solve for x. So both of these individually
are enough so the answer is D. But we could solve for
it just for fun. We get square root of x/40 is
equal to 1/4 and then you get, essentially solving this,
you get square root of x is equal to 10. And you get the same thing, x
is equal to 100 and you are all done, right? And you might say, oh, but isn't
there a plus or minus? No, because we're saying the
square root of something is equal to 10. We're not saying the square
root of 100 is what? If you said what is the square
root of a 100 you'd say it's plus or minus 10. But if we said the square root
of something is 10 that something has to be 100. Anyway, next problem. Don't want to dilly-dally. 61. I want to imbue you with a sense
of urgency for the GMAT. How many integers are there
between, but not including, r and s. Fair enough. So essentially we have to figure
out what r and s are or how far part they are. So statement number one-- so
integers between r and s, but not including, remember,
not including. So statement number one
tells us that r minus s is equal to 10. Now this is an interesting
question because if we knew that r and s were integers then
we could just pick. r minus s is, maybe r is equal
to 11 and s is equal to 1. And then if you actually just
want me to write it out you could say, well, 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11. How many integers are between
the two and not equaling the two? 1, 2, 3, 4, 5, 6,
7, 8, 9, right? But that's if we assume that r
and s are integers and this would work for any two sets of
integers that are 10 apart. But what happens if we do it
a little bit differently? What happens if we say that
this is 1.1, 2.1, 3.1-- oh no, sorry. What if it's between
1.1 and 11.1? So if it's 1.1 is r-- oh no,
let me write it this way. What if r was 11.1 and
s was equal to 1.1? Then what are the integers
between them but not equal? So if you start at 1.1 you would
have 2, 3, 4, 5, 6, 7, 8, 9, 10-- and now 11 would be
included because 11 would still be less than r-- 10, 11. So now you have 1, 2, 3,
4, 5, 6, 7, 8, 9, 10. So statement number one is
actually not enough because they didn't tell us that
r and s are integers. If r and s were integers
then the answer would be 9 in this case. But if r and s are not
integers then the answer could be 10. So statement number one alone
is not enough, the answer could be 9 or 10. OK, what does statement
number two do for us. Let me do it in a
different color. Statement number two. This is an interesting
problem. There are 9 integers between,
but not including, r plus 1 and s plus 1. OK. So this is interesting. So they're telling us that there
are 9 integers between r plus 1 and s plus 1. So, I mean, actually just
looking at it, inspection, you could say, oh, well then
there are 9 integers between r and s. And they say but
not including. So this is enough. This alone is sufficient and
you don't need statement number one. Statement number one actually
doesn't get you anywhere. So let me prove to you that if
there are 9 integers between r plus 1 and s plus 1-- Well,
think of it this way: if there are 9 integers between r plus 1
an s plus 1, if you subtract 1-- so s is one of the one's
that are between-- now I want to make sure I do this right--
Let me think of a good way for me to prove it to you. Well, I'll just do it
with an example. So let's say that the example
is 11 and 1, right? This isn't a proof but I just
want to give you the intuition, right? So if r is 11 and s is 1, so
then you would have 12-- well, I don't want to do
it that way. The easiest way to think about
it is, it doesn't matter how much you are adding-- if you
add the same amount to both the bottom of the range and the
top of the range it should not change the total
number of integers that are between them. So there would also be 9
integers between r minus 1 and s minus 1. You're just shifting the range
along the number line but you're not going to actually
change the number of integers in between them. So if they're 9 integers between
r plus 1 and s plus 1, there's going to be 9 integers
between r and s. So two is all you need. Hopefully that's a satisfactory
explanation. I didn't want to go into
something rigorous when we're trying to imbue you with
a sense of urgency. See if I have time for
the next problem. OK, 62. What is the number of members of
club x who are at least 35 years of age. So number who's age is at least
35, so it's greater than or equal to 35. The number who's age is true. OK. They tell us statement number
one, exactly 3/4 of the members of club x are under
35 years of age. So 3/4 are less than 35. That is fair enough but that
doesn't tell us the number that are at least 35. This tells us the percentage. This tells us that 1/4 above or
we say 1/4 greater than or equal to 35. 1/4 of total members greater
than or equal to 35. So it tells us proportion
but it doesn't tell us the total amount. And I just got the 1/4 from 3/4
are less than 35, 1/4 are going to be greater than or
equal to 35 years of age. So one by itself doesn't
help us. Let's see what two
does for us. The 64 women-- am I reading
the same problem? OK, yeah-- the 64 women in club
x constitute 40% of the club's membership. OK. So 64 is equal to 40% of
the total membership. Now we're set, right? Because we can use this equation
to figure out the total membership. Call it t for total. t is going to be equal
to 64 divided by 0.4. They just told us that
essentially 64 is 40% percent of the total. They said it's the women
and all that. But they could've just told us
that 64 is 40% of the total. So then you can solve
for the total. And then you substitute that
here and you say 1/4 of the total are 35 five years or older
and you get what we were trying to solve for. And so the answer is 1/4 times
64/0.4, and whatever that is, you can put it in
your calculator. But all we had to care is that
when you use both of these equations combined, you're able
to solve for the number that are greater than or
equal to 35 years old. And each statement alone isn't
sufficient, so you need both statements. And that is C, both statements
together are sufficient. See you in the next video.