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GMAT: Data sufficiency 24

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

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  • blobby green style avatar for user Ayush Agrawal
    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)
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    • winston default style avatar for user Tetsuya
      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)
  • blobby green style avatar for user jay sharpe
    question 105 around in; 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)
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    • mr pants teal style avatar for user Ben
      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.