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## GMAT

### Course: GMAT > Unit 1

Lesson 2: Data sufficiency- GMAT: Data sufficiency 1
- GMAT: Data sufficiency 2
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- GMAT: Data sufficiency 21 (correction)
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# GMAT: Data sufficiency 24

103-106, pg. 287. Created by Sal Khan.

## Want to join the conversation?

- I did not get the explanation of question 105.. It was to be proved that |x| = y-z ... From statement 1 we get |x| = -x & statement 2 we get x<0. How does these two statements together solve the question?(4 votes)
- Sal's explanation was like this:

In order to prove that |x| = y-z from statement 1, |x| has to be -x, which means that x has to be 0 or a negative number, but we weren't told so, until the second statement said that x is less than 0. So we need both statements to prove the given equation.

Let me explain it this way:

Statement 1)

x+y = z --> x = z-y = -(y-z) --> -x = y-z

If x is 0 or a positive number, |x| = x = -(y-z)

If x is 0 or a negative number, |x| = -x = y-z

But we don't know what x is.

Statement 2)

x is a negative number.

From both statements, |x| = y-z(6 votes)

- question 105 around6:59in; i saw the explanation below but im still unclear on how it works. if |x| = -x doesnt this tell us that the value of x is negative? since |x| = positive, the only way -x is positive is if x itself is negative (a (-)(-#) situation, giving a +#. whereas if x is positive than -x is negatvie and cannot be = |x|.(1 vote)
- I had to think about this one for a minute.

As Tetsuya said, statement 1 effectively tells us that -x = y-z. We can plug this back into |x| = y-z to get |x| = -x, but it's important to note that this formula is the*question*, not a statement in itself.

We wanted to know whether |x| = y-z, and given what we know from statement 1, we can instead ask*whether*|x| = -x. The answer to that is: "when x < 0", and that's all we can get from statement 1.

Then of course statement 2 tells us exactly what we need to know.

It's interesting to note that if statement 2 were something like "x >= 0" or "x = 2", that would be sufficient as well. In that case |x| would not be equal to y-z, but the GMAT answer would still be "both statements together are sufficient to determine an answer".(1 vote)

## Video transcript

We're on problem 103. And it says, if n and k
are greater than 0, is n over k an integer? So essentially they're asking,
is k divisible into n. So let's see, statement
1 says, n and k are both integers. And that's good that they told
me that, because I was assuming it incorrectly. So they're saying n and
k are both integers. Well that still doesn't
help me much. n could be 2 and
k could be 100. And 2 over 100, is 1/50. That's not an integer. Although n could be 4
and k could be 2. And that would be an integer. 4 over 2 is 2. So statement 1 by itself
doesn't help me. Statement 2 says n squared and
k squared are both integers. Well once again, I don't see
this as helping me much. If n and k are both integers, of
course when you square them they're going to integers. You take any integer and you
square it you get an integer. You take any other integer
and you square it you get an integer. So this actually gives me no new
information relative to 1. So, I'm just trying to see
if I'm missing something. But as far as I can
tell it's E. Both statements together
are not sufficient to answer this question. I mean, I can give you a case. n could be 4, k could be 2. And then, n over k would
be an integer. And both of these would
satisfy both of these conditions. Or it could be, 2 and 4. In which case, n over
k aren't integers. They don't actually
differentiate between n and k at all. So it's E. Next problem. 104. If the average arithmetic mean
of six numbers is 75, how many of the numbers are
equal to 75? OK, so six numbers
are equal to 75. How many of them are
equal to 75? Let me go scroll down
a little bit. The first statement says,
none of the six numbers is less than 75. So none less than 75. So this is interesting. If I have six numbers. And their average is 75 and none
of them are less than 75, they essentially all
have to be 75. Let's say the first
number is 75. Now let's just say that I had
some number above 75 in the list. Let's just say for
argument that there was a 76 in the list. It doesn't
have to be an integer. They're not assuming that. But let's say I had 76 in the
list. In order for the average to be 75, I would have to offset
this number that's one more than 75 with another
number that's one less than 75. But they told us that none of
the numbers are less than 75. So if I have a number greater
than 75, I have to have a corresponding number less than
75, or several of them to balance off that number. Or you could have several
balancing several. It can get complicated. But the general idea is, is that
if no numbers are less than 75, you can't
have any numbers greater than 75 either. So this actually tells you
that all six are 75. So statement 1 alone
is sufficient. Statement 2 says, none
of the six numbers is greater than 75. Well it's the same argument. If I have a number,
let's call it a. Let's say it's one of the
numbers that's in my list. And it's greater than 75. In order to average down to
75, I'm going to have to have-- well let's say
a is less than 75. Because we're using statement
number 2. If a is less than 75--
let's say 75 is here in the number line. I'm just doing ad hoc
diagrams. So a is less than 75. In order for the whole group
to average to 75, I have to have some number that's bigger
than 75 to balance off the a. Let's call that b. But they're saying that
we have no numbers greater than 75. So we have no numbers
greater than 75. We can't have any numbers less
than 75 either and all the numbers have to be
equal to 75. All six of them. So once again 2 is
also sufficient. So each of these statements,
independently, are sufficient to answer this question. Actually they both answer the
question that all six are equal to 75. That was interesting. Problem 105. Is the absolute value of
x equal to y minus z? OK. Absolute value of x equal
to y minus z. All right. So in statement number 1,
they tell us that x plus y is equal to z. So let's see if we can do
some substitution. z is equal to x plus y. So if we substitute z into this
top equation we get the absolute value of x is equal
to y minus x plus y. So that means that the absolute
value of x is equal to y minus x minus y. And that says that the absolute
value of x is equal to minus x. Because the y's cancel out. Well I don't know if
this is true still. This would apply to any
negative number. If I have a negative
1, negative 1 absolute value is 1. And negative negative
1 is 1 as well. So this would apply if x is
0 or some negative number. But they don't tell us that. So we don't know if
this is true. Statement 2 tells us-- well
there you go-- statement 2 tells us that x is
less than 0. So this tells us x is definitely
a negative number. And if you put any negative
number into this, which we got from this condition and this
condition, if you put this into that, you know this is true
for any negative number. The absolute value of any
negative number is the opposite of that number. So you need both of
these statements. And obviously statement
2 alone is useless. Because if you just say x is
less than 0, you don't know anything about y or z without
this condition right here. So both statements together
are sufficient. But individually, they are
reasonably useless. 106. We're getting up there. What was the total amount of
revenue that a theater received from the sale of 400
tickets, some of which were sold at x percent of full price
and the rest of which were sold at full price? So let's say we have the reduced
price tickets and the full price. So R is the number of reduced
price tickets and F is the number of full price tickets. So R plus F. So the the total number
of tickets was 400. What was the total amount
of revenue? And then the total amount of
revenue generated is going to be equal to the price
of full price times the full price tickets. So P times F where P is the
price of full priced. Plus the number of reduced
tickets and that's sold at x percent of the full
price tickets. Whatever x is. x percent. we'll convert it to a decimal
at the appropriate time. So this is what the initial
problem description tells us. That the total number of
tickets is 400 and then whatever-- of its x percentage
of P-- so whatever the full price is times the full price
tickets plus-- this would be the reduced price times the
reduced price tickets-- equals the total revenue. Now let's see if we can
figure this one out. x is equal to 50. Well this by itself just
tells us that this is p over 2 right here. This x is 0.5. It's 50%. Remember they said x percent. So this just makes this
equation, it turns it into, 0.5 times P times R
plus P times F is equal to the revenue. And we still have
one, two, three unknowns and two equations. So that by itself
isn't enough. We actually have
four unknowns. R, P, F and revenue. Four unknowns, two equations. Not good enough. 2, full price tickets
sold for $20 each. So the price of a full
price ticket is $20. So now, this is interesting. Now we could take this
and substitute it into this equation. Because we now know what the
price is, the P variable. And so if we use both of these
conditions, we have 50% of a full price-- well, that's $10--
times the number of reduced price tickets plus $20,
times the number of full price tickets, is equal to
the revenue generated. And then we know that a total
of 400 tickets are sold. R plus F is equal to 400. Well even now, we still just
don't have enough information. They would have to tell
us either R or F. Or they would have to tell us
the total amount of revenue generated in order for us--
well, actually that's what they want to figure out. They want to know the total
amount of revenue. So in order to figure that out,
we would have to have at least one more condition
to solve for R. And then be able
to solve for F. We have to know R or F to
figure out the revenue. Let me just make sure I'm not
missing something here. Yeah, there's no other way to
just-- you can't just separate out an R plus F. So I'm going to say that E, both
statements together are not sufficient. I'm pretty sure I'm right. I'm don't think I'm missing
anything right here. Some of which were sold at x
percent of full price and the rest were sold at full
price, which is $20. So R plus F is 40. And then we have 10R is
equal to revenue. Yeah. We have three unknowns and two equations, two linear equations. Not good enough. Next problem. Oh, I'm over time. See you in the next video.