Transcripts
1. Pitch DS: data sufficiency is the most important part of Agimat conception. This part is very ambiguous, and many, many students struggle with it. In this course, I will teach you how to overcome the gym at Data Sufficiency section. I believe it is useless to solve thousands off questions without having a good strategy. That's why in discourse I will give you the best strategy to tackle any data sufficiency question. And I would show you how toe applies this strategy toe. 40 different practice questions, and once you finish, the scores will be ableto full. Any data sufficiency question that encounter do in the gym at on. Thus, both your gym outscore. I am a niece Agimat expert. I took the gym at five years ago and I scored a few 51 on that one section, and since then I have helped hundreds of students to enhance their score in this section. I will help you to do the same in this type of problems. They give you one question and to state, and you need to know if the statements are sufficient to answer the question or not and scores I will show you the best strategy to adopt in order to solve any data sufficiency problem. First, I will introduce you to data sufficient. Then I will show you the magic way. I used to prove that statement is not sufficient. Next, I would show the tactic I used to prove that statement is sufficient. And then I will show you the trap that 99% of test takers fault into. And I will show you how to get right answers. And then we'll see together 14 questions off different level off difficulty, and I will show you step by step, how to solve them in the simplest way possible. Once you master the methods I teach in this course, you will be able starting off tonight not only to solve a problem that encounters you, but also to solve them very quickly. And this will save you a lot of time during the test. This course has a 30 days money back guarantee. So if you feel that for any reason this course doesn't suit you, you can ask for a refund very easily. See you in a few minutes.
2. Introduction to GMAT Data sufficiency: So now we're gonna talk about G Matt Data Sufficiency, which is a very important part of the gym at Quantitative Best. So Jim Adidas Efficiency are about 13 to 18 questions. So it's between 1/3 of the question on half of the questions because in the quantitative part of 37 questions, So uh, it is about 40% of the questions of the Jama'at contest, and it's a very important type of questions that says it's only form of Agimat and not in other standard standardized test. So let's take a look about this type of question. So first he would ask me that question and he will give me two statements and we have five answer choices. So if the statement one alone is sufficient, but the statement two alone is not sufficient to answer the question asked. Then we answer a second statement. Two alone is sufficient, but statement one alone is not sufficient to answers. A question asked that we answer be if both statements one and two are together sufficient to answer the question asked. But neither statement alone is sufficient to answer the question asked. Then we answer. See each statement alone is sufficient to answer the question as so if we exam mine, each statement and we see that each statement alone is sufficient answers. A question then the answer is the If we examine a statement but stepped up to and they are together not sufficient to answer the question and additional that a specific are needed, then the answer is it. So we have a question, two statements and we have five answer choices. We have to know them by heart. So when we examine our statements, we can know what answer to answer directly without thinking about each off the
3. 2types of questions: So now we're gonna talk about two types of question and demand data sufficiency. The first type of question is a yes or no questions or true or false questions, for example, is B is greater than see. So here we can answer by yes or by no Uh, And then we have another example. If a minus is a minus three squares, it could then wonders were also we can answer by yes or by no. So why after uses? Because when we examine a statement, if we are sure that when using the statement, the answer to the question is always and surely yes or it's always and surely know we can know then that the statement is sufficient, we can talk about this indeed is later. So we have a yes or no question, and we have the what is the value off questions. So here you can tell me, what is the value off a or what is a value off X squared plus twos? X minus six. So I have to search for the value. The expression A or X squared with was explaining six. So if I find two values or three values or five values. It's not sufficient. If I found if I can't felt any value, it's also not sufficient. So it is sufficient on Lee when I threw one body one and unique value. So, for example, if I find that a equal to one, that's it, then it's sufficient. If I find that a is equal to two are a is equal to three or a is, you come to them that it's not sufficient if I find that a cannot be computed, that it's not sufficient.
4. Data sufficiency Basic examples: So let's take some examples. Example. One. The question is, is x greater than one? Okay, so here he asked me a question Is x greater than one? So if I think directly about this question, I cannot know the answer because I don't know X. I don't know if it's greater than one or its lesson one or it's equal to one. I don't know anything about X. So a question like that I cannot answer it. So I need some additional information's. Let's examine what hippie? What are the statements that see the state of the statements? So the statement one X is greater than two. So he tells me that X is greater than two. Is this information needed? Uh, if this information is sufficient to answer the question so I know that X is greater than two. So X is absolutely greater than one because two is greater than one. So X is absolutely greater than one. Yes, so, using the first statement, I am sure 100% that X is greater than one. So the first statement is sufficient. Let examine the second, uh, state X greater Zanzi. Okay, great extractors and zero So X could be equal to two. Is it a greater than one? Yes. X equal to greater than one. So what effects is greater than zero packs could be equal to 0.5. Is 0.5 greater than one? No. So if X is greater than zero, I cannot know if X is greater than one or if X is less than one. So the statement to is not sufficient for me to answer the question so statement to is not sufficient. So statement A is sufficient statement to is not sufficient. So our answer is a Let's move to see another example. So the question is, Is X square greater than four? Is X square greater than four? So let me examine the first statements. He tell me that X is greater than one. Okay, great. Let's stick up some examples to see. So if X is greater than one, let's say X is equal to three. That X square is equal to nine isn't greater than four. Yes, it is a great example, So it's OK for X equals three. But if I take, for example, X equal to 1.5, exit is greater than one. So X is equal to, let's say, 3/2. So let's get it. X Square Expert is equal to 9/4, and I know that 9/4 is equal to 2.25 and it's not greater than four. It's not the greater forethought. So I took two examples that are greater then one x equal three and X equal 1.5. Both are greater than one. And once he gave me X is greater. It gave me X greater than four. Then it's OK on the in the second example, it gave me X X quizzical to 2.25. Then it's it's lesson for So it's it's, uh, the answer. This question is no. So once we have the answer to this question is yes. And second time we have the answer to this question is no. So the statement one is not sufficient. Now let's examine the statement through X greater eventually. So if X is greater than three, let's take an example X equal four. Then X square. Is it to 16 greater than four? Great Yes, X equals five. It means Ext Square is equal to 25 is greater than four. And if I get bigger values off X, I will get X Square bigger and bigger and bigger. So if x greater than three, we know you can know if we are a bit familiar in algebra, that X Square is absolutely greater than nine and nine is greater than four. So I square is absolutely greater than four. So if ecstasy greater than three X squared is actually greater. Therefore, I'm 100% sure that if X is greater than three than X Square is greater than four. So statement do is sufficient so and 2nd 1 is not sufficient. So the answer is B. So let's take a look about example. Number three. What is the value off X plus y? So I need to know the value off X plus twice. Let's see first statement he gave me X Plus two y is equal to five. So can I know from this statement the value of X plus y. Let's see. Let me take an example. This is the simplest way to know. Let's take X equal to one and then I have one blessed to why is equal to five. Then two. Why is equal to five minus one? So two y is equal to four? Then why is acquitted? Two. So if X is equal to one satisfying the statement why is equal to two and then X plus y is equal to three. So let's take another example. Let's take X equals zero Mexican zero. So why is equal to fire and why is equal to 5/2? So it's 2.5 so X is equal to zero, then why is equal to 2.5, so X plus y is equal to +15 So I already took two examples and they have to values of X plus y to different values off experts y three and 2.4. And I'm sure if I think, for example, I would have different that is so I don't have a unique value off expressed why. So that's why a statement one is not sufficient. Well, it makes the mind the statement. Number two I have two x plus three y is equal to eight. So let's take ecstasy. Quint Zero than why is equal to 8/3. So X plus y is equal to 8/3. And if I take X equal to one than it would be two plus three, why is equal to eight? Three Y is equal to six. Why equal to so X equals one y equals two, then I have X Plus y is equal to three. So on the first example I have X Plus y. It's 8/3. The second example I have X Plus y is equal to three. Then I have two different values X plus flight. So I don't have a really value. And I'm sure if I think more example that will get mawr values more and more bodies, different values. So, uh, then that is a statement. Number two is not sufficient, so both statements are not sufficient now. I will exam mine both statement together. So X plus two y is equal to five and two x plus three y is equal to egg. So now I have a system system composed off two equations and to announce, So let's solve this system to see if we have a solution. So from the first equation we have X is equal to five minus two. Why? So if I take to why is this side I would have access to go to five minus two y and I will take this X here, and I will replace it in the second equation. So I'll get two times five minus two. Why plus three. Why is equal to a I will simplify. Here it's 10 to dance. Five. It's 10 minus four. Why is he plus three? Y is equal to eight to minus Why you have minus four y plus three y is equal to minus y equal to eight minus 10. So it's minus two, minus y zero to minus two. So why is equal to two? So why is he going to than X is equal to five minus two times two. So it's five minus for Bennett's secret to one. So having finished big, these calculations of the system we have X is equal to one, and why is equal to two? The exercise is equal to three. Definitely. So if I think the bulls off statements and I mix them together, I have X is equal to one, and why is he could to so exercising with three I have one and unique value off extra slice so I can compute it knowing both statements are Are you satisfied? So both statements are sufficient together to answer the question, and neither statement is sufficient alone. So this is why the answer to this question is she? So let's take the fourth example. What is the value off X minus two. Why? So what is the value of X minus two war? So let's see the first equation. The first statement. So I have two x equal. Four. Why? Plus 10. Okay, so if I take, let's say, uh, an example X equal to zero, then four y plus 10 equals zero. Then why is equal to minus 10/4? So it's minus 5/2, So it's minus 2.5. So x minus. Do I? That's do it. X minus two things minus 2.5. So the secret to zero plus two times 2.5 It's five with secret five. I tried to calculate X minus two wife and I. I got five. So let's take another example. So if X is equal to one, let's get late. Why so X is equal to one, so it's to equal. Why last 10 then four Y is equal to minus a and Y is equal to minus two. And then let's say X equals one minus two times miners to I'm calculating X minus two. Why? So let's go with the one plus four. So it's secret. Five. So we have or any to values that are equal to fight. So let me think about it twos X equal to four. Why plus 10. Let me take the four white another that the other side. So be equal to X minus four. Why sequence them? So here I see two is a multiple off to four is a multiple of two, then is a multiple of two, so I can divide everything by two. So I didn't hear by two hereby, to and hereby, to so two X over. To simplify it, I get X minus four over to its minus two y and then over to its five So x minus two. Y is five. So I can know directly from this statement that X minus two y is equal to five and this is kind off breaky question because once we see an equation was to announce, we directly tend to say OK, then I get no X. I don't know why then I can't know X minus toy. But in fact, even if I have an equation was to announce, I can still have. Or I can still compute the value of X minus two. Why? Just like we did here. So from the first statement, we can know that XT minus twe were sure 100% sure Whatever is X and y but satisfying this statement it would give me at the end X minus two Y is equal to five. So this statement is sufficient. Let me examine the seconds off the X equal six. Why, Blustein? So three X minus six y is equal to 15. So we also see that three is a multiple of 36 is a multiple of three and 15 is a multiple of three. So let's divide everything by thing to simplify this equation So three x divided by three with simplifies gonna be X minus two. Why six over phase two and equal to fight So X minus two Y is equal to five using the second statement alone so we can know the value of X minus two. Why so uh, now both statements are alone. Sufficient toe. Answer that question. So it is. Answer the So let's take a look at example. Five. So he asked me what is a value off a plus B? So let's see what he is proposing to me in the statement. One. So it's telling me that a Plus B Square is equal to nine, so a lot of people will say directly that then a plus music with three and saying that it is sufficient. But as you know from algebra, if we have excess square is equal to nine. That X is equal to three or X is equal to minus three. So here is the same. We have a plus B with this tree, and we have a plus. B is equal to minus three, so we have two possibilities of if, plus B. And we don't have a unique value off a plus B because you have two possibilities. So first statement is not sufficient to answer this question. Let's see the second. So we need to calculate a plus B, and I have a secret to be OK. So let's take a sequel before then. B is equal to a plus. Zizek with sixth Let's Take a is equal to eight than BZ, good for the A plus basic with 12. So I have two different values. We have two different a lot of different values off a plus B because statement to saying that is twice be so for 284 68 that we are plenty of examples, and each of the examples would give a different value for a plus B, so we don't have a need value. Bury a plus B. So that's why the second statement is not sufficient. Eso Let's combine both off them. We have agreed to be because the square it with mind. So let's replace a because to be in the first equation. So it will be to B plus B. It's where is it going to? Nine. Three B Square is equal to nine, so it's nine. B squares would nine, so B square is equal to one. The square is equal to one. Then be easy with one or B is equal to minus one. We have to answer to be, and what is a is a double off be so is equal to or a is equal to minus two. So let's get a plus B in the first case. So a plus music with three Let's skip in it a plus D in the second case, and we have a plus T is equal to minus three. So we have two different values off a plus B and not a league value of a plus B. So that's why, with first statement is not sufficient alone with second statement is not sufficient alone . And the Bulls statements combined together are not sufficient alone. So that's why the answer is.
5. Data Sufficiency Map: So next I'm gonna show you how to approach a gym at beta sufficiency. Question. What is the algorithm? What method we are we're gonna use to approach this type off questions. So first we what we're gonna do is exam mine statement one alone. So statement one at all we'll examine. It would see if it's sufficient. So if it's sufficient, then we are here. We like the mine is a statement to alone. So we like sometimes a statement two alone if it is sufficient that the answer is the And if statement two alone is not sufficient than the answer is a so and if that statement one alone is insufficient with examines the statement two alone, there's a statement. Be alone, the two alone's. Then the answer is B. If the statement two alone is not sufficient, then we're gonna take both off the statements and we'll examine them together and we'll see if if they are sufficient than the answer is C and if they are not sufficient than the answer is E so we can use this map to help you. Eso agimat data sufficiency Question
6. How to prove that a statement is not sufficient: so how to prove that a statement is not sufficient? So now I'm gonna teach you a very ethical, effective method to prove that a statement is not sufficient. And I'm gonna explain it using this example. So the question is is m greater than one? So we is ask me if I m is greater than one. So let's see the first statement. He tells me that and square is greater than M. So I'm squares others and I'm so let's let's take an example. So this strategy, it's about examples. So let's take the first example. Let's say AMAs equipped to and square is equal to four. If so, I'm square is equal to four and four is greater than two. So I'm square is greater than m. So the first the first statement is satisfied. So now I can see if I can answer the question or no, So is m greater than one. I am a good two. Then the answer is yes. So, first example um, equal to the answer is yes. So let me take another example. So let's say I am easy. Quick before that, Times Square is equal to 16. So 16 the greater than four. So I'm square is greater than M. So the first statement dissatisfied, built. So then after the first statement, despite eye exam, mine is the question. So I have any, What for So second example I have m equals four. And the answer if it is m is greater than one. The answer is yes. So I took 1st 2 examples and the answer is yes. But is this a guarantee that all the answers are? Yes, well, it's not so sure. So let me take another example. Let's try, for example, and equal to mine. A three and then M Square is equal to minus Terry where? So it's equal Conine. So nine is greater than minus three. So M square is greater than M. So the first statement is satisfied. But M is equal to minus three. So, um is less than one. And the answer discretion is no. So here we have different examples. Different examples. For example, I have three examples. The 1st 2 examples Give me an answer. Yes, to that question and the third example, give me an answer. That is no question. So when I use the first State and I think, examples that satisfy the first state. And I find at the end two examples. One is giving me yes. So the question on one is giving me No, the question. It means it's definitely not sufficient, because if a statement to be sufficient, it should give me always, always, always one and unique asset. That question and not different possibilities. So here we have two example. That's the chemical tour. M equals minus three for Emigrant. Through the answer, The question is yes, for em equal money threes. Answer to that question is no. So the first statement is clearly clearly not sufficient because I can find it. Two examples one answers. The question was yes. And the second answers. The question was no. So it's clearly not sufficient. So let me examine is a second, uh, statement. M plus one is greater than zero. So he's telling me that I m is greater than minus one. So if M is greater than minus one, so let me take an example. Let's take em equal to zero. And then it went to four. If I am equipped to zero and the answer to this question is clearly know if a musical before then The answer to this question is barely yes, So I have. I have taken two examples. One examples that set both example Satisfy the statement and one gives me an answer. No, The question on another on the other example gave me an answer Yes, to that question. So I can see clearly that the second statement is not sufficient to answer the question. So now I will combine both off them. So we have and square is greater than I am and I'm plus one is greater than zero. So I can say that I m is a greater than, uh, minus one. So let's let me take an example. Yes example. Let's say and is equal before and square is equal. Teoh 60. So I'm square is greater than I am satisfies the first musical for greater than my son. Satisfied second. So before we examine an example, it should satisfy the statements. So is M is greater than one. Yes, So the answer to this example is yes. I will try to take another example. For example, if I take em is equal to minus half, so I'm square is equal to one over for one over for is greater than minus half. So the first condition is satisfied. I'm equal to minus half is greater than minus one was the second condition is satisfied. Then I have to try to answer the question. His M is greater than one. A musical minus have so clearly no. So I chose. I have to. I took two examples and equal four. That gave me an answer to the question. Yes, any quote minus Have that gave me an answer to the question that is no. So I have two different answers to the question one is equal to Yes, and when is it good to know? So both statements are not sufficient to the answer is this question. So the answer is E. So as as a generalized general rule, if you want to prove that statement is not sufficient, we have to take two examples. One example that gives an answer to the question on another example that gives another answer to the question. It means one example that gives yes or and another one. It gives no or one example. Calculate the value of a ages three and another temple that give me, for example, a equal to fight, so I have to accept that gives two different answers not unique as so. When finding the exact assist examples, we can say that the answer to this question that statement is not sufficient.
7. How to prove that a statement is sufficient: So now we're gonna see how to prove that a statement is sufficient. So it's the best way to prove that statement that sufficient is to prove it mathematically . Why? Because if we choose some examples on, do you think these examples that satisfies the statements we find that, uh, the answer to this question is yes or no, or we find a value for what he is asking us for. It's not sufficient to do it this way. Why? Because if you choose two or three or four examples and the answer is also is the same, we cannot be sure that for all the examples, it's gonna be the same. So the best way is to prove it mathematically writing equations. But if the G Matt we cannot do it mathematically or it's hard to do it mathematically, we can choose two or three examples diversifying examples. Eso We can be sure, uh, that the answer is correct, but the best way is to prove it mathematically. And if we cannot do it mathematically, we can. We can choose three or four privates, happens to see how how it works and if we find it sufficient for us and we answer is that it is sufficient. Okay, so let's take a simple example. The question is, is Emma greater than one? So let's see. The first statement is telling me that M square is greater than I am. So let me take on example to see uh, that stick em is equal to two and square is equal to four. Is m squared greater than m? Yes, because for is greater than tools then the first statement is satisfied if it is emigrated than one. M is greater than one that the answer to this question is yes. So let me take another example. For example, if em is equal to minus three, M square is equal to minus three square. It's equal to nine. So M square is greater than M because lining the graters and minus C So this statement is satisfied? No, I can answer the question. Em is equal to my they so m is greater than one and not s o. M. Is less than one not greater than one. So the answer to this question is no. So here clearly is the first statement is not sufficient. That's exam line is the second statement m greater than zero. So if Emma Graters and zero M could be equal to 0.5. So the answer to this question, uh, is no and m could be equal to to advance. It is this question is yes. So also, the second statement is not sufficient. Let me eggs align both statement together. So he's telling me that M Square is greater than M and M is greater than zero. So let's see if we can do something mathematically. So let me consider the first. This is ever situations. This is the first clinic inequality, and this is the second inequality. So that's considered first. So I'm square minus m is greater than zero. Let me move the end to the to the other side and then right, it's this way Gators and zero. So am multiplied by M minus one. And the result is greater than zero. And we have the second condition that investigators and zero. So this M is greater than zero and I have that m minus one times a number that is get is and zero, and the result is greater than zero. So absolutely, we have m minus one breakers and zero So we have m is greater than one. So the answer this question is always yes, so I know that it is sufficient. So the answer to this question is she. But if I wanted to do it using examples, I have to choose examples that satisfies bulls off the conditions here. So if I choose a number between, um, equal, let's say 0.5 I'm square. It's equal Teoh 0.25. So it's less than I am, so it satisfies the second condition and not the first condition. So I'm kind off forced to choose an M that is greater than one so that EMS Square is greater than M and M is greater Sanzio. So we can I can deduce that time is good zone, but this is not the perfect way to prove it. The perfect way is to use mathematical equations, and if we don't know how it's difficult or it's time consuming to use mathematical equations. We can then move and dio empirical trying to try values and see how it performs within the inequalities or when he's ordered state
8. How to simplify a question: So now we're gonna see how to simplify a question. Sometimes the question is complicated, so we have to simply fight before beginning and sitting in. For example, if, to the power off X plus y is equal to four the bone of eight, what is the value off? Why? So here I can see this equation. It's this somehow complicated to deal about. I can see a way to simplify it. So the best a way to solve this question is to begin with, simplifying it so two to the power of X plus y is equal to +44 of eight. I know that four is two square, so I can replace four by two, square to the power A. Then I have, uh, donations saying that to the board of em and everything to the board off and is equal to the off M times. And so here it's to the part off two times eight. So go to the ball off 16. So I have to get them out of X plus. Y is equal to the heart off 16 so I can reduce that X plus y is equal. Teoh 16. So I can, uh modifies the question if X plus why is equal to 16? What is the value of white? So no, it's much, much simpler to deal with this question. So the first statement is exquisite to 81 so exquisite. Goto 81 so X is equal to nine or X is equal to minus nine. Don't forget about. It's very important. Eso. If X is equipped denying, then why is equal to 16 minus Exit 16 minus science equal to seven X equal minus nine. Then why it secretly 16 minus minus nine Zoff blast nine So that's equal to 25. So we have to values off. Uh, why eso It's not sufficient. Answer the questions. The statement. One is clearly not sufficient. Uh, let's examine statement to so I have X mine blast. Plus why Dizzy quit the 16 and X minus Y is equal to two. So let's, uh, let's some thesis equations. It's gonna be two X less why minus y is equal to 18 so two X is equal to 18. So Exit 29 when X is equal tonight and I have an X minus Y is equal to so uh, nine minus wise to do so why is clearly equal to seven? So, um, the second statement is official. So the answer to this question is clearly Bish.
9. One Thing to remember: So here is one thing to remember a very important thing that the mole that most off Jim Mattis takers forget about is that that that we have to answer this question using the statements and our aim is to answer this question and not to answer by s or answered by No . So the answer It could be no. Or could be yes. So if that there's no, it could also be sufficient. So a lot of people say that if if the answer is no, that is not sufficient. ITT's confusing. So we're gonna see this example to clarify this idea. So it is X is greater than one. Okay, let's see. First statement is X is less than two. Okay, so it's could be X is equal to 1.5 than the answer is yes, X could be equal to 0.5, and the answer is no. So the first statement is not sufficient. That examines the second statement he's standing me. Uh, the X is less than zero. So is X is less than zero than absolutely the answer to the question is no, because if X is less than zero, it could never never, never be greater than one. So is the answer. This question is no. A lot of people will answer, not sufficient. But the answer is that statement be statement to ist sufficient. Why? Because I can find an answer. And I am sure that whatever the value of X, I choose satisfying the second statement. So I can just any value that satisfies the second statement. So that X is lessons you and I can. I'm sure that the answer to this question is x greater than one is always gonna be. No, it's not gonna change so that the idea here is to answer this question Could be by s or it could be. Barb. I know, but so it's not always s. So here is the statement. Be a statement to is sufficient. So the answer is, uh, clearly be
10. Easy 1: So what is the value off X? So I need to know here, in this question, the value of X. So let me examine the first statement. So I know that X plus why is equal to minus X minus? Why? So here I have one equation. Two unknowns. So I might say that. Okay, so it's difficult to know X so I might move general mind Moved to the second statement because I have two unknowns, but not really Let us examine a bit. Dig deeper here in this statement. So let me sit if I hear a bit so it's X plus y is equal to minus X minus minus life. So it's bless why So here I have to simplify why and why? Because they are the same. Or, if you want again, take worried. That takes everything to the left side. So it's X plus y plus X minus Y is equal to zero. So here I have why minus wife zero. So I have experts exits two X is equal to zero. So if two X is equal to zero, so X is equal to zero. Definitely. So I know the value of X on Lee from the first statement. So the first statement is clearly sufficient. Execute zero. Let me examine the second statement to see if I can find the body off So I have X plus y is equal to I don't know anything about why. So let me choose. Y equals zero f y equals zero then X absolutely equal to two before I equal to one than X is equal absolutely to one because one person is able to. So I have to values off X. I have found two examples that gives me two different values off X, so I cannot find one in value of X. So second statement is clearly not sufficient. So the first statements office since this sufficient. So the answer here is clearly a
11. Easy 2: What is the value off X? So let's see the first statement. I have to factor off experts. One factor off X plus four is equal to zero. So here I have two factors that are equal to zero, so we have X Plus One is equal to zero and X plus four is equal. Zero simple algebra so X is equal to minus one, and XZ quit minus four, So I have to. Values of X X is equal to minus one on Ex addicted to minus four, so it's not possible to know the body effects because their way have to values. It's tough and it's or so Let me examine the second the statement. So I have X Square plus three x plus two with zero the eggs you where I can write it. Let's say X plus one times X plus two equals zero design and develop it. Give me X square plus X plus two X stew. So one experts to actually have sex with your it will give me the same as here. So X plus one is equal to zero or express to music of zero, so X is equal to minus one or X is equal to minus two. Where is exit with moments one or X minus. So the second statement is also not sufficient. So let me examine both statements together so its most statements together are satisfied. So X must be equal to minus one, because minus one, it satisfies the first and minus one that satisfies the second. So if I want the boats off statements to be satisfied, then it should be for X is equal to one minus one. So I know the value of X stick of when both of the statements are together. They are satisfied than X must be equal to minus one. That's months once in five East parts of statements, so I can know the value of X so X is equal to minus one. So both statements are sufficient. So the answer here is clearly definitely see
12. Easy 3: So what is the value off X So X to the border for is equal to 81 so I can know directly that X could be with three and excavated two minus three y because three to the power of four is equal to three vote of two Times Street about off to. So it's equal to nine times nine. So it's easy to 81. And if I take also minus three to the border four taking minor three square five Ministry Square So it's nine times nine equal to 81. So I have to values off. X X is equal to three or X is equal to minus e. So in G. Matt, you should know that three to the border of three years ago 27 to the border. Five adversary to this powers is simple policy. You need to know them three to the part of 46 21. It should be known by my heart eso Here I have two values off. X X is equal to three X is equal to minus three. Uh, let me see the second statement. So x Q is equal to minus 27. So here that is only one value of exit secret to minus three, because X cubed is equal to minus street board of three. And so it's minus three plans minus three times minus three. So secret to three cups leading up to his 27 minus times minus. It's plus times minus. It's minus, so it's minus 27 so X is equal to minus the and then I know X here X is equal to minus three. So second statement is clearly sufficient. So the answer here, this question is be
13. Easy 4: the perimeter off a rectangular garden is 160 feet. What is the lengths off the garden? So here we have the length of the garden, the wits off their garland. They know that perimeter is equal to 160. Perimeter is equal to times L W E could 160. So the length plus the whip's equal to 80. So lets us examine the first statement, the rent off the Godmanis street outfits with so dances with. So let's see if I replace leads by three minutes a day with us, which is a 80 forwards. 80. So what's is equal to 20 and left physical street times 27 60. So I can know from this information the length off the garden. So the statement number one is completely sufficient that May sees a statement. Number two. The difference between the lens and the wits. The garden is 40 feet, so let's minus with Says he could 40 and I have Lance plus this very last last with secret to 80. So let me see, uh, what can I do? Is this two informations So here I can say that the wit's equal to 81 of the lands like I replaced here. So let's minus 80 minded finance is equal to 40. So lengths minus 80 less than Suzie Good for E. So let's that cat sick of the two. Is it good for a plus 80? So to length is equal to one other 20. Lance is equal to 60. So also using this information I can deter mined the land off the guardians of second statement with also sufficient sof answer discretions clearly D.
14. Easy 5: What is the political effect angle are So I have a rectangle. They have lands. Wits. Uh, and he's asked me for the perimeter of this rectangle. So payment there is equal to Dunn's, uh, two times length, thus, with that's it. So the area of the rectangle is equal to fraid so areas it create. And I know that the area off the rectangle is equal to lens times with Zika 48. So how can I know the lens plus with? So let me see if I take some example. Take some example. L is equal to six, which is a good paid so two times lands. That's with six plus eight. So it's two times off. 40 18 28. This is the first example of medics. Another example. If you take a listen to music 24 so permitted, it's two times two plus 24. So it's two times 26 So 6 to 52. So we have two different answers. So, uh, clearly the 1st 1 is not sufficient. Let me see the 2nd 1 Let me. This is the 2nd 1 So the length off diagonal is equal toe 10. So here we have the lenses of diagonal so I can really see the analysis then. So if I can take this triangle here it is a rectangle triangle here. Well, I can apply the fight go to stay with him with six fingers So by the garden state omits and squared plus w square is equal Teoh 10 square So it's and square loves w square 100. So here we have the same If I think value off w I can search the value of hell Then I can replace here and chew the perimeter of group I reformed the value Then if I change W I would get another value off l and then I'll get another value the way we can. You can try it, and you will get two different values off the perimeters. So I would consider that second statement is not sufficient. So let me see those statements together. So So I know that sometimes W Z equals 48. So, um, I can multiply by two of year by two here, so it would give me two and the news equal to 96 and then I can sum up Zoe's equation. So it B l squared plus two l w plus W Square is even 196. So here what I have I have and squared plus to Hell w plus. But why square is equal to L plus W Square. So it's equal to 196. Why, this is very well known algebra if l plus w squares and square plus w squared plus two l w so N s W square is equal toe 196. So plus w z 14 off course, we neglect the minus 14 because possibly be negative. So helpless Musical 14. So maybe there is a two times 40 and 6 to 28. Hey, I know the value of the primitive finally, so I can answer the question using both off statements, so it's clearly answer C.
15. Easy 6: his ex between zero and two. So here is a question. Zero lesson X. Listen to Is this true? All right, so let's see the first statement. He's telling me access between minus 0.2 and 2.2. So mine is your point to his lesson. X is listen to point to. So if I know that X is between minus 0.2 and 2.2 am I sure that, uh, access between zero to So let's take a couple of examples. So if I take First X equal to one so his ex is between zero point on 02 yes, zeroes, that's an honest listen to. So the answer is, the question is yes, but also X could be with 2.1, so it satisfies first statement. But 2.1 is greater than two. So it's not between zero and two. So the answer to the first question is no. So I have two examples, one that gives me that the answer is yes. And the fact second gives me that the answer is no. So the first statement is clearly not sufficient. Second statement that seven force is 3/4 more than X. So 7/4 is equal before more than X. So here I can compute value off X. I will multiply everything by four So four times 744 times four times x So why I'm multiplying by four just so I can simplify So here for it for four and four are considered . If I so I can have seven is equal to three plus for X Then I can say for access he could seven minor, very good four. So xz quit one. So using the second statement, I know that xz quit one So X is surely between zero and two. So now I can answer that question That X is between zero into the answer is yes. So stick statement number two A sufficient and answer This question is clearly being
16. Easy 7: so his ex less than 15. So the question is X. Is it less some 15? So let's see the first statement, the some off X and Y's lessons. 15. So X plus why is less than 15 So as usually, let's take some examples. So let's take, for example, X is equal to then why is equal to two so X plus y equals 12 It this letter 15. So it satisfies the first statements so I can use it to test the question. So it's X here. Let's some 15 Yes, X is equal to 10 so X is less than 15. So the answer is this question is yes. No, I will try to find an example that gives me an answer to the question. No, so I can have two different answers. Yes, and also I can say that it's not sufficient. But if I cannot find the next example, then I will try to demonstrate mathematically that it is correct. So let's see if I can find another example. Let's choose, for example, X is equal to 20 and why is equal to minus Stan? So 20 minus 10 is equal to then this this less and 15 so I can use this example since it satisfies the first statement so I can use this example here. X is greater than 15 because X is equal to 20. So the answer this question is no. So I have two examples, one that gives me that. The answer to the question is yes. The second is giving me the advances. This question is no so clearly the first statement is not sufficient. Second statement. Why is less than 16? OK, but I don't know anything about extracts. Could be going to 10. X could be equal to 20. So once is less than 15 and here it's more than 15. Great is not 50 so I can never know anything about excusing on. Lee is the second condition. So clearly first conditions second condition are not sufficient. Let's examine both of them. Both of our X plus y is equal. This is nets and 15 and why is less than 16? I can use examples from that I already used in the first statement. Let's say X e quitted then why is equal to two then? Here we satisfy bows conditions and the answer is a question then, is less than 15. So the answer to this question is yes. Let's be choose the same example I choose. So X is equal to 20. Why is equal to minus Stan? Why is here less than 16? It's my understanding, Honest analyst is less than 16. So here I have 20 is greater than 15. So the answer there is no So I have different answers. So both off statements used together are not sufficient to answer the question. So here it's clearly his answer each.
17. Easy 8: What is the remainder? One b is divided by three if B is positive and bigger. So I know that B is positive. Antigua and I want to know what is the remainder when being divided by three. So B is equal to three Q plus remainder. So I need to know this remainder. So let's see the first statement when these divided by 18 the remainder is four. So let's say B is equal to 18 SK plus four so I can write it this way, since 18 is a multiple off three. So I can't this wait three times six. Okay, since three times six equals 18 plus four is equal to three plus one. I want to show the three here to see how many stories we have so is equal to B. B is equal to tree. Let's take it as a factor. Six. K plus one plus one. So here I have for three times constant plus one If I take this constant equal to six K plus one, so here clearly the remainder is equal to one and one is less than three, so the remainder should be less than three. When I divide by three. So here the remainder is clearly one here. So what I did is 18 k 18 3 times six So erected this way. Three times six Gay and I took these Here I I took three as a factor and then three. Factor off six Cape lost one plus one. So here I have three times constant +11 should be less than three. So that's why I did. The manipulation was four because four is greater than three. So I need to I need to manipulate. I mean, I need to manipulate with it to make it less than three. So that's why I did this. So the answer to this question is, the remainder is one, so it's sufficient so clearly. Statement. One alone is sufficient. Let's see Statement number two When B is divided by 12 13. Understand so B is equal to 12 k plus. Then I would do the same technique music with +23 times four K plus nine plus one. So B is equal to three fact it off four K plus three plus one. So B is equal to three Hugh plus one. Because I took Humi Could four K plus three. So, uh, here also I can say that the remainder is one, so I can know the remainder using only the statement to SOS answered. This question is clearly
18. Easy 9: does s contain any even numbers. So let's say s is a set. Ah, contain any even numbers. So let's see, There are no prime numbers in s, so no problem numbers and s okay, so as to be that it could be nine. I could be present in s because it's not that prime number and mess. Uh, and also, let's say, uh, eight could be present and said s so yes, it can contain even numbers and it can not to contain even numbers. So it can be nine and 33 for example. Here we have we don't have even numbers on another example. We have nine and eight. We have even numbers. So clearly I have took two examples, one that contains even numbers. And when that doesn't contain any even number. So the first statement is not sufficient. Let's examine the 2nd 1 There are no multiple off five and s Okay, here I uses it. Bulls off examples. Um, no multiple fights. OK, but I still can have even numbers and not even numbers. The 2nd 1 is not sufficient. Using both together, I can also use the same example. See, we don't have multiple of five, and we don't have prime numbers on one example contained even number and another example doesn't put in any even numbers who really both are not sufficient. So here of answer is
19. Easy 10: if X is greater than zero. Is X over y greater than X? Where is it? Let's exam mines The first statement. So as you know, this lesson, why is less than one and we have to compare? X is divided by why and X So I know if I think the number X and I divided by a number that between zero and on and excess posted course so X would be, ah, the results would be greater than X. For example, X is equal. Then why is equal to 0.2 so X, divided by why is equal to then divided by 0.2. So it's gonna be a good 10 divided by two divided by 10. So it's sequel to 10 times 10 divided by 2 to 50. So we began was then we ended up with 50. So when I divide X always whenever excess positive when I divided with the number between zero and one, the result is gonna be bigger. So bigger than X so x develop Always Great is the next statement. Number one is very sufficient. So statement number two x is greater than two. OK, but I don't know anything about why. So let's take exit with then why is equal to two, then? X over y is equal to five and five is less than then and let's take another example in. So if X is equipped to then and why is equal to 0.1 so X, divided by why is equal to 10 divided by zero. What once was 100 So it's greater than then. So here's answer is no. Here's answer is yes. I found two examples of give me a No. One Noor one s so obscurely second statement Not sufficient. So the 1st 1 as the obsolete before is sufficient. So the answer is a
20. Easy 11: so 8 13 nine 15. And what is the value off end in the list of? Also, I have a list a 39 15 and then I want to know the value of ends first statement and is greater than then. Okay, it's a beautiful so and could be 11 and could be 12 and could be 124 by, so I cannot know the value off n clearly not suspicions. The median off the numbers in the list is 11. So the median here is 11 as I know that, uh, if I have and not a number off terms, uh, the media, the media is it is the one in the middle, so let us rearrange them. So 89 13 15. So, uh, we have n and I know that the media is equal to 11. So what I have what could end be so n it is forced to be 11 here because if n is equal to 11 then the median is 11. Because I have two values bego 11 to values greater 11. So if the median is 11 here in this case, so N is it is gonna be 11 because, let's say, for example, that N is equal to seven. Then here the media could be nine. And if N is equal to 18 then the 1,000,000 is searching. So the only way that the media is 11 is that N is equal to 11 here in the middle. So second statement is clearly sufficient. So that's it. This question is the
21. Medium1: So let's examine this question. What is the ratio off? Why two x? So let me see is the first statement he's telling me that why is equal to them x. So I can say directly that why over X is equal to then? Then I can know that Asia y over X is he could have done them the statement one is sufficient, but it is a huge, huge YouTube huge mistake because in algebra, when I want to put it this way. So here I'm dividing by X. So I have Why is equal to 10 X? I'm dividing bulls by eggs. So that's why it gives me why over X is it would then I said, if I hereby x so there is a rule that I cannot divide by X. The X could be equal to zero. So let's examine Here is X could be zero. So if X is equal to zero, then 10 times zeros equals zero, then why is equal to zero? So X equals zero y zero, and this is satisfying this condition. So zero a couple 00 is satisfying this statement. So in this case, if why's he could do both is equal to 10 x extra bacon to zero and like a big zero and then the racial 0/0, it is nuts known. It's not did their mind. We cannot know What is it? Mathematically speaking, it's not defined 0/0. So I actually went to zero. And why is it zeal that we cannot know that Asia if X is different than zero? Then the nation wise over X is equal to 10 So we have two different answers. This question first time is not known. And second on the stand, so clearly is the first statement. It's official. It's not sufficient here. It's a very, very, very, very tricky question. So if he if he asked me the question, why is it that the 10 X and X is different than zero? Then, yes, it is gonna be sufficient because answer is always equal to 10. So let's examine the second statement. X is equal to five. We don't know anything anything about why, so we cannot know the ratio. Why open X? Because let's take y is equal to 15. So why over X eyes equal to three? And if we take, why is equal to 10 that why, over excessive good to do so. Two different answers. So I cannot know the answer of this question. But if we combine both west both statements together so X is equal to five and why is equal to 10 X? So it's equal to 10 times five secret to 50. So then why over xz quinto 50/5. So it's equal to then s o we can know the ratio of whatever x. So the answer is both together are sufficient. Uh, so the answer is C.
22. Medium2: so we're next. If X and why are integers is why is greater than zero so X? Why are diggers? Is is why greater than zio his life represent you. So let's take the mind examining the first statement. The first statement is stunning. Is that X plus 0.5 gators on zero? I don't know anything Anything about why? So why could be going to 10? Why would be to minus tan? So once is answer is yes. Then that's it is no. I have found two examples that one examples give me dance of yes is a question of the 2nd 1 gives me No, So it's clearly not sufficient. So the first statement is really not sufficient. I looked. The man is the 2nd 1 He's standing that next time. Why his graters? Zero so x Times y is gators and zero. So I have two possibilities. X raters and zero. And why is the greatest theory or X? Is lesson zero unwise lessons here who, since I don't know anything about X, except that it is an integer so X could be positive or negative. So why also could be positive or negative? So here's the answer to that question is, yes, Here that's special is no eso. So, uh, it's not sufficient. So statement Number two is not sufficient. So let me examine both statements, so the 1st 1 is X is greater Zahn minus cedar 0.5 so X could be equal to 01 to the excess. The second statement is X is next times lies, graters and you. So let's look about X So here. I can't say that X is definitely different on zero because X is equal to zero, then x times y Z zero and not greater than zero. So X cannot be equal to zero in this case if because the X Times Y is Gildas and zero than X is absolutely positive because it's greater than minus 0.5. Still, it's equal to its none negative. But here I can be sure that it's not people to zero, because if excessive capacity on X Times Y is equal to zero of this example, it's not then greater than zero. So actually enough who could not be equal to zero? So X is with 123 So, uh, this way, if X is equal to 1234 It's positive and bigger than absolutely why is a positive? Bigger? So why is positive? Because it positive? Because why times that positive number gets posted number. It's absolutely posted. Why cannot be negative. But if y is negative than explore x times, what is negative and here I have to satisfy both conditions. So the answer this is the question is C Both question along are about both statements alone are not sufficient, and two statements together are sufficient.
23. Medium3: if x times y difference and zero here. It's very good. Very good expression. X we can. I can conclude that X is different than zero and why is different than zero and is one over X plus one over Y is equal to eight who is one over X plus one over. Why is he ate? Isn't so the first statement. Staying a X is equal to why. So let's take an example. If X is equal to two. Why is equal to and 1/2 plus 1/2 is equal to one, so it's different than eight. So the answer to suspicion is no. Let me choose another example. I will try to choose an example so that it gives me a good band. So if I have an answer, no. And if I can find the answer, yes, then I can say that it's not sufficiently. Let let me. I want to try to find finance. It has AIDS, so I find it. Have to find that this is equal before, and this is it would before the X equal to y. So if X, let's say is equal to one over for and why is he won over for so one over X plus one over why is equal to 1/1 over four, plus 1/1 over four and 1/1, of course of secret before plus four. So it's equal to eight. So then the answer to this question is, yes. I have home to example one that gives me an answer. No one doesn't answer. Yes, so the first statement is clearly not sufficient for those who do not know how to do the 1/1 over four. So it's equal to 1/1. I'm divided by 1/4, so it's sequence 1/1 times four times one I I there turns that divide the division here into multiplication, and I turned the one over foreign to floor over one. So it gives me for over once what's he could before. So, uh, first statement is not sufficiently smooth to the second state man. So that's that question is one over X plus one over wise, equal to eight z question, and he's telling me here is X plus. Why is equal off a eggs less X time for? So can I use the statement. Tour it to It's I can I can see that it's similar here. Where the age here It's the same between bulls off the equations. So here we have eight alone and here we have eight with X and y So why we don't isolate X. So we see if we can get something similar. Toe this Soto Eisen 88 I should divided by X times lie so I can simplify by X times life so to divide in order to divide by X times Why I should divide hereby extend Roy and divide here by explains why so now I can sit If I hear extends why extent Loy I can sit If I hear the why and the why and here is the X on the axle then I would have won over why less one over X is equal to a so using the second statement I can and there is a question by yes proven mathematically. So it's clearly is answer be
24. Medium4: does X plus y is equal to seven. Uh, so let me examine the first statement. Five x plus y z could 31. So I will take an example she that satisfies this thing. This equation. So let me take exit with six of them. Five times six is 30 plus. Let me take why is he going to one? So it's gonna make with 31 So x equals six. Why is equal to one X plus y is equal to seven? Yes, So experts wise it would be seven banter is yes. But what if I take another example that satisfies this equation? Let me take let's say, as X equal to four. So five times four is equal to 2020 plus 11. So why's it with 11 equals 31? So here we have X plus y z equals 15 that that there is no so using the first statement I have two answers. X plus y is equal to seven and exercise. You could do 15 So once the answer is yes and another time second times answer is no. So clearly not sufficient. Here we have the same X plus fire. Why is equal to. So I can say that X is equal to one and why is equal to two? Because one plus five times two swords, one plus 10 to 11. So here express wise, he would do the Let me take another example. So if X is equal, do let's say X is equal to six and Y is equal to one, then six plus five times once or six past 5 11 words. So experts wise equal to seven. So he advances No chances. Yes, so the same argument off the statement. One. They are not sufficient. So let's combine both of them. Five x plus. Why Here we have X plus five. Why is a very 1 11? So I want X plus. Why? So I can solve this system off immigration so enough by by simple methods, substitution methods, let's say But here I have something very interesting. I have five x plus y zing 31 x plus five. Why, I have five year five year one here, one here. So if I some both of the vehicle Asians, I would have five x plus x that you give me six packs. Why last five? Why it would give me plus six. Why so 31 plus 11 and before two? So it's gonna be six. Factor off X plus wise it would avoid too. So X plus y is equal to 40 to over 6 42/60 with seven so x plus y z could seven and both west both statements together are sufficient on The answer to this question is C. This is very tricky stuff to some equations or sub Strack equations. I can use it to find easily what I want to find, because if I want to do the substitution, it's gonna consume time. And Diamond Diamond Jama'at is a test off time. Time is very important, so I have to gain every second.
25. Medium5: So if X Times Y is different than zero so explain. Why didn't zero so X is different? Zero. And why is different than zero? Oh, this is what is the value of X Times supplies? What is the very next? Let's see. The 1st 1 is sending me that y is equal to X plus two. So let's take an example. Why is equipped with seven X is good five. Seven times times equal to X Times y z to certify. Let's take as an example why equal experts to wait with the three and X is equal one so x times y z three times one to see. So we have two different values off X times y. So once I have excellent bicycle to 35 here I have experts will times why is equal to three eso Uh, it's not secure enough sufficient. Let's examine the 2nd 1 Why is he going to X squared? Plus two x is equal to zero x can a big zero So X equals one. So one square it's one plus two secret, so x times. Why the answer off Extent voices Could you see? Let me take another example, so X is equal to two. Then why is equal square plus toe for close to its six? So expands wise Â£6 to a sick with 12. So here I found two different examples one that gives me that X Times y I think the three and the second is giving me Eckstein Isaac with so clearly it's not sufficient. Let me examine both statements. So why is he good X plus two and why is it X squared? Plus two. So here I can say that X plus two is equal squared plus two and then I have actually could X square, So x squared minus x zero. So let me take X here. Eyes right. Axel X is equal to X factor off X minus 180 So here I have two possibilities. Have two possibilities. X is equal to zero Rx minus one is equal to zero. Actually sick of those you know it's impossible because expanse wise different. You also exit different, you know. So here we have X minus one is equal to zero. So excessive with one. And if X is equal to one. But I can calculate why so why is equal to three. So X is equal to one. And why is it extends? Why is it could do three and I have the answer. So here, each sticking down is not sufficient on both together are sufficient. So the answer is clearly see.
26. Medium6: So what is a value off? Nine. The bar off. Why Time Street? The power off X. So let's exam mine first statement. But before let us see. Maybe I can simplify that question. So nine is equal to treat the power of to the power off. Why times read the part off X. So three to the point of to to the butterfly is equal to treat the part off to why it's sterile. Three. The poor Off em, the war off N is equal to treat the poor off em pines n. So I have here treats the poor off to time times why times treated apart off X. So it's equal to three to the power off to y plus X. Because if I have treated the power off expanse, it's part of why it's a good three. Part of experts. Why it's a rule. Basically, we're off, uh, exponents. So what is the value off? Three to the power off to Y plus X. This is a question. I simply fight the question. So let me see the first statement. X is equal to minus two. Why? So if excessive wants us so it's gonna be three to the power off to why minus two I. So it's like with three to borrow 06 foot one. So simplifying by question made it very easier. So if X is equal to minus two, why then it's gonna be treated for residents because one so so I can know the value off mine bought off. Why time status for excellent vision is sufficient. 2nd 1 wise equipped to do so. Why is it going to so it's gonna be trade to the park off four plus X. But I don't know the value of X X could be to do 12 people do. I'm gonna be It's going to give a different values off expression when taking different values of X. So the second statement is absolutely not sufficient. So the 1st 1 is sufficient was Answer is clearly a
27. Medium7: is X cubed equal. Teoh 27 Excuse equal to 27. So with excuses 27 so X is absolutely equal to three because there is no other value of X, so that X cubed is equal to 27. Let's take execute toe Minor three. It's gonna excuse is gonna give me minus 27 So it's not seven, so it's only three. So the question here is X is equal to three. So the X is equal to city. I simply fight that question. I made it easier if X is equal to city so that my ex is less than four. Okay, so extra big with two extra because one extra people to three. So I cannot know the answer here. It's impossible to know because X could be equipment three it could be not. The equal position vis alone is not sufficient. Let me excellent. The second The statement X is greater than two. So x could be could three. So the answer's yes, extra people four. So the answer is no. So also, the 2nd 1 is not something. Let me examine booze So we're studying me. X's lesson for on access gators on two. So excuse frequently. Three. Uh, it's just in four and its benefits greater than two was. Answer here is yes, but I can't find another value of X. So here's the question. Here's the tricky question because it didn't tell me. And the problem that X is an integer, so X could be equal to three and extra be equal to 2.5 or extradicted to 3.5. So if he told me that X is an integer that okay, X is equal cities, this is the only answer. But here I have excessive photo 2.5 x physical people can find out, for example, So the answer is no. So once the answer is yes, and I have different examples that gives me that the answer is no. So it's clearly both of them are not sufficient. So the answer is clearly, uh is
28. Medium8: What is the value off digger? A. So he wants to know what is a lot of anti Garay? There is an anti go for him. So let's take the first statement X squared plus three a minus Stanic with zero. So it is an equation off second degree. So I want to get relate A So I have to search for Delta that is going to be square minus for a C. So it's sequence to here. So here we have a is one B three and SES minus 10. So B square. So it's nine minus four times one times minus 10. So secret Remind Plus four e shows 49 Death has 49 So let me get a one is equal minus D minus three miles squared off 49 divided by two times one and a two to minus three minus school 49 Divided by 21 It's a formula. It's general formula is very well known, so it's minus three minus seven, divided by two. Here I minus plus my sleep last seven divided by two So a one is equal to minus stand over to its minus 582 is equal to four over to its equal to, uh, do so I have to values off a So a one is equal. A one here is equal to minus five on a Tuesday Desire to varies off a So it's not. Ah, sufficient Answer the question. It's not sufficient. Answers a question I'm gonna see. Second, second statement. So it's to the boat off to A is equal to 16. Okay, so 16 I can write that I can remove to see that right? To do that to the boat off four. So it's gonna be to a is equal to four. So is equal to two. So statement number two is sufficient and the answer is bish.
29. Medium9: So the next question is be greater than one. So he's asked me if he is greater than one. So let's exam on the first statement. He's telling me that B Plus two is graters and zero so I can move the to to the other side . So it's d greater than minus two. So be is greater than minus two, so we could be equal to zero, which is less than one. So the answer to that question is no. And he would be good to 10 which is greater than one. So the answer to this question is yes. So I have two examples. Is that satisfied? The first statement and one is giving me no as an answer to the question and the other is giving me. Yes, as an answer to the question so clearly, Statement One is not sufficient. Let's examine statements to so a statement to be square is greater than two. So what can I, uh, know about? Be so let's say let's take an example, be equal to five. This square is equal to 25 greater than two, so it's OK and use it five. So it's greater than one. So it's answer to that question is yes. But let me take. B is equal to minus four. The square is equal to 16 greater than two. So statement dissatisfied but be is great. Is that is lesson one. So the answer to that question is no. So also, statement B is not sufficient because I have found two examples when gives me yes, as an answer and one giving me no as an answer. So it's not sufficient statement. Let me examine both together so the 1st 1 is is greater than minus two on a B square is greater than two. So I have a system off to in equations. So I know from math rules that if the square is greater than two, then be is greater, um, squared off to or B is less than minus squared off to see if I can draw it. So here is zero. Here is where the two here is minus where the two years minus two because my skirt was minus one point for So using the 1st 2nd inequality I have, there is solution, so be is greater than square root of two, and B is less than minus square the to. But it has to verify also the second condition, so be is greater than minus two. So here is the solution. So let me take for anything. So here it could be a solution here and here. It could be another solution. So if I take a number in this area, it's Biggers and square root of two. So it's bigger than one. But if I take a number and assist area, it's satisfies those conditions and is it is less than one. So I have two examples that say, here I I'm gonna take one for minus 1.5. Answer is no. And there I will take. Ford's answer is yes. We have two examples satisfying those conditions but giving me different answers to the questions. So both they are not so together. They are not sufficient. So the answer to this question is he
30. Medium10: So what is the value off integer? A. Let's say someone. The first statement, a minus two is less than zero. So ese less than two. So a could be good zero a could be one A B minus two. So clearly is the first statement is not sufficient because I can form and other values of a And so I cannot know the value off A from this statement. So statement one is clearly not suspicion. Let me examine statements Teoh A at the bottom, it is less or equal to zero. So I have to find an example that satisfies this condition. So if I take for example, A is equal to to aid the whole eight support off eighth, it's gonna give me a positive number. So it's not satisfying. So any positives, the number I choose here I will give me a positive number and return So it's not working. Let's take a is equal to minus two any other number. Negative ember minus two to the power of eight. So since the power is an even number, so minus two, a power and even power the minus two and even power is always positive because that state once, two times minus 25 minus two by minus two minus times minus. It's plus Afghanistan once it's plush and so on. So if we have a and even exponents, it will give a positive number at them. So, uh, if is negative, that would give me also a positive number. So it cannot be negative with the honest is, the only solution that works is a good zero, because if a good zero a the part off Physical co zero and zero is less or equal, but it's equal zero. So the only possibility is it's very tricky questions very, really very tricky. So the only possibilities that satisfies this condition is a equals zero. So I can know from this statement it is sufficient for me to know the value of a with zero . So the second statement is sufficient. So my answer is B
31. Medium11: if X Times why is different than zero? So X difference and zero and why is different than zero is X over? Why is less than zero? So he's asking me Is X over? Y is less than zero. Let me see the first state so two over X less than two over right so I can divide by two bulls off park, both offsides. So because two is posted, I will not change the sign off the inequality. So it's gonna be one over X less than one over why? But is X over Y negative? Let's take on example. So, um, that's a X is equal to one. Uh oh are excellent before And why is equal Teoh, too? So, uh, so 1/4 is less than 1/2 and x over wife examine X Over y is positive because for over two is equal to its positive. So the answer to this question the first example I took is no, it's not negative, but let me take another example. Let's say X's equal to minus two y is equal to two, so minus half is less than half, so it satisfies the condition, the statement on minus 2/2 is equal to minus one, so it's negative. So the answer to that question is yes. So I found two examples. Satisfying is the first statement, and one is giving me yes to the an answer to the question, and the other is giving me no. So clearly, First statement is not sufficient. Second statement is telling me that excess gators and zero Okay, it's nice, but I don't know anything about why Why could be equal to do. And it's gonna give them an answer. No to the question or right could be good minus day. And that could be a yes to that question, because excess positive. So it's not sufficient. So let me take both conditions. So one over X is less than one over why and X is supposed so. I can see that X is positive. And if excess positive one over X here it's also positive, and one over why is greater than one over X? Because one of Rex's Lesson one over Why so one over? Why is greater than one over excellent over Wise also also so positive. So if one over wise also wasn't there, then absolutely. Why is positive and then x over y is positive. And the answer is the question is always no. So we have an answer. We are sure that the answer is no. So both statement together are sufficient. Let me repeat it. So excess positive. So one over X is positive and one over why is greater than a one over X. So one over why is positive? And since one over why it's posted, then why is positive so exposed? Why is positive X over y is greater than zero? So the answer is always no to the question. So we have one answer to the question. So both statements together a sufficient so the answer is question to see.
32. Medium12: This is a value offic off a closer to 30 than 45. So is telling me that he wants to know if a is closer toe 30 or 45. So let's examine that the statement. I have 45 minus a greater than a minus three. So let us take the ages. Other side and the minor circuit has decided to become like this bigger than to a So I can say that to a is less than 30 plus 45. So this answer five because 75 here it's five is greater than two a day. So two weighs less than 75. So a is less than 75/2. So a is less and seven vote with 37 when five. So let's stay here. We have three here we have 25. What is the method point Between them? The midpoint is 37.5 and I know that a is less than 37.5. So a is here, so definitely a is closer to 30. Then Teoh 45. So first statement is sufficient. Second statement is greater than 37. So here we have 37 and your EPS 37.5. The middle point between both and I can choose in a here that is equal to 37.2, or I can choose in a here that's equal to 40. So here I have two values of a. The 1st 1 is lows after 30 because it's less sense than the 37 point point five and the other one is closer to 45 so it could not be sufficient to know the answer of the question because I have already two answers to different answers. The 2nd 1 is definitely not sufficient, So the answer to this question is clearly a
33. Medium13: This isn't eager ex divisible by four. So he's asked me to to see if X is divisible by four. So let me exam mind First statement. He's telling me that eight X is divisible by 16 so eight X is equal to 16 times gay. So let's say they gave the fact of So when I say eight exit visible I 16. So a thanks is equipped to 16 k so I can divide by eight. So x is equal two. Okay, so X is a multiple off to so X could be two extra four x six x eight So here is not divisible here is divisible that it was inevitable. So once is visible on the other times not visible. So I cannot know if X is divisible by four from the first statement. So first statement is clearly not sufficient. Let's examine the second sequence or nine eggs. Zeke with 12. Yeah, so I can divide by three bowls of them. So the X is equal before. Okay, so So K is equal to the X divided by four on. I know that Kay is an digital gays and a teacher. And here I have three Times X divided by four I know that three is not divided by four. So in order to get a K as an integer X must be divisible by four. For example, if X is equal to three is not divisible by four on two is not. There is life or so three times to my wife for so X must be divisible by four. So, for example, X is equipped with well, then three times 12. It's 36 rebel. What's nine? Because off gays and is anything so K is an integer so three X over four must be an integer . So since three is not divisible by four than X must be divisible by, uh for so the second statement is sufficient. And the answer this question is be
34. Medium14: ISTEA divisible by 11. So let's me see. The first statement t is equal to X minus y where X and Y are integers So I don't know anything about X. Anything about why so I don't know anything about T, so I cannot know if he's with this. Is this visible by 11 or not? So give is if a statement is not sufficient. That means he's a 2nd 1 x is divisible by 11 and why is not divisible by 11? So here I have. If I want to divide de by 11 I will must example Divide explain 11 on why by 11 So if he is divisible by 11 it must give me an A digger s o X is divisible by 11. So here it's gonna give me and Tigger. But because he's standing is that excess divisible by Antigo What you'll ever so X is divisible by 11. So the result is the went by 11 going man and figure and what is not divisible by 11. So why divided by 11 is gonna give me This is my number So an antigen minus that this is my number is gonna give me Let's see my number. So the is not the divisible by 11. So that's it is this question is no. So the 2nd 1 is sufficient sues Answer is B because when using the second statements, the answer to the question is no. Why? Because we're divided by 11. I don't get an integer. I get a decimal, so he's not visited by
35. Medium15: So when X is divided by 33 men dinners one so X is equal to three q plus one when X is divided by force, Remember, zero So X is equal for Okay, so let's be just see what X can be. So let me take the 1st 1 second one. So the 1st 1 is equal Teoh. So if you equals zero, that X is equal to one. If you requisite one or seventh and then the 13 every time I had sees or 19 it's too. So on the 2nd 1 it's gonna be a month of four. So for a 12 16 uh, 2024 uh, when a city to so let me see which are common to both of them. So four extra big before extra people will 16. So let me see. Let's take four divided by it. So four is equal to zero times less force of the remainder is for And if I take 16 it's equal to two times eight plus zero. So the remainder zeal. So I cannot know that remainder when X is divided by eight. Because once I have thirst time. The first example that they took The remainder is four, and the 2nd 1 is. The remainder is zero. So I cannot know those answers is the first statement is clearly not sufficient that this is the 2nd 11 X is divided by 16 remainders for so X is equal to 16 Q plus four, so I can write it in another way. So X is equal to do eight times too few plus four so exited through eight times K plus four . Considering that Casey to you, which is an anti eager also so excited to eight k plus four. So the when x divided by eight the remainder is for So the statement do is sufficient to know that Amanda, when X is divided by eight. So statement Teoh alone a sufficient So the answer is B.
36. Medium16: What is the value off the racial? X two y square so X divided by why Square So let's take some of the first state man the issue off X squared y z Good tonight. So X square to why, uh, Why is equal to nine? So I need to know what is the value of X divided by X by Y scripts. Well, let's see the statement So X squared is equal to nine. Why? So let's take an example. Off X is equal to X is equal to nine. Then why is equal to nine also so X divided by y score. So it's 9/81. So its secrets who won over nine. So it's equal to 1/9. So let me take another example. Let's say X is equal to six. So X squared is equal to 36. So then nine times wise, it could 36 wise it before. So why is he could before so X over y square. So 6/16 is gonna give me the over eight. So first example gave me 1/9. On second example, it gave me the over eight. So it's not the same. So the first statement is not sufficient. Answers a question. The 2nd 1 the ratio off X 2 to 3. So x divided by two with three So x is equal to six. Exit with six. But I don't know what is why. So I cannot search the value of extra wife square. So yes, it's not sufficient. 2nd 1 So lets us combine both of them. So x square. So why is equal to nine on, uh, X is equal to six. First one and 2nd 1 X square. Oh, why is nine and execute six? So, uh, X squares with nine. Why? So 36 is equal to nine white. So why is it good for so X over y square? Uh, that's gonna be 6/16. So it's going to feel about AIDS. So I have the answer using both the questions combined. So the answer is clearly. See
37. Medium17: if any of the great is on 150 what is a unit? Digit off. And so and is the greatest honor 50. He wants to know the inhibition off him. So let's see the first statement that any digits off end is the same as the prodigious and square. So they digit off em is the same off to India and square. So, in order, if if we have the number and, for example is equal to 2186 if you want to know the inner digit off and square, we should only look to the in addition, off end and squared. So six square 36. So the in addition off and square is gonna be six also because 36 ends by six and so on. So, uh, tradition are the same. So let's pull that and ends was one toe end square will end with also one two. It's gonna be four three. It's gonna be nine. So here doesn't work, or it's gonna be 16. So, six, it's not gonna work. Five. It's gonna be 25 5 So it's gonna work one also in our six so it's 36. I think the six has been a word. Seven. It's nine because it's 49. Gonna work. Eight. It's Square City, so it's four. I'm gonna work nine 81. So it's one. It's not gonna work. So only have three choices. Three choices. 156 So it's one and five and six. So I cannot know the in addition to them because it could be one. It could be. Five. It could be six. So it's not sufficient. Let's see is a second statement. Then it's digit off end is the same as any businessman and Cubes. So let's see one. So it's gonna be one, because one time I love that 112 cubits. Eight. So it's not gonna work today. You it's gonna be 27 7 is not gonna work. So for four times for 16 6 times four is 24. So it works for so already we have Teoh board. Four times four is 16 6 We s let's four. So it works five. It's absolutely It's working because every time, like I haven't been by 55 times five is 25 times cited 125 66 times, six times six is also works because six is always, uh, this weight and always by 67 that seven. It's 29 9.76 sixties, so it's not working eight times. Date is 64 4 times at 32. So on by to its doesn't work nine times 9 81 1 times nine It's gonna end by 729. That's it. So it works. So we have different disabilities. Are the one 56 on, uh, nine Also, uh, So we have different possibilities. Let's combine them together. So we still have ah three possibilities in Ghana 15 and six. So it's not sufficient. Cannot know the units digit off n because it could be 151 1 other than 55 156 and still have instant satisfy balls off those off the state so that this question is clearly he
38. Medium18: What is what is the value off X when X is multiplied by nine? The result is between 55 on day 65. So, uh, let's say it's nine times seven. It's gonna be 60 city, so X could be equal to seven. But they didn't tell me anywhere. That X is an integer, uh so X could be equal to become seven times 6.99 It's gonna give me also 62.9 or something like that. So X could take different values because nobody tell pulled me. That X is an integer so extra not could be non interior eso. The first statement is not sufficient because extra beak of the seven and extra big 6.99 or 7.1 maybe 6.7, I don't know. So we have different possibilities off X. So first statements not sufficient second statement when X is double, the result is between 11 and 17. So when X is doubled, the result is between X incident in the same here. So if X is equal to X, doubles was gonna give 16. Um, extra could be Quindio seven, also on Devon's to be 14 and 15 11 17. So it's not sufficient that different possibilities off X and now so is the second is not sufficient and no one combining both of them. Uh, we went, combining both off the statements X is also there could be 76.9 lines. It certifies it satisfies both of statements, and it's not. It's clearly not sufficient, So the the answer to this question is.
39. Medium19: the price off a certain property decreased by 10% in the first year, increased by 20% in the second year and decreased by 30%. Serve year. What was amount off dollar increase in the property during the second year? So I know that the property increased by 10% then increased by when he put decreased by 10 person because by 10% increased by 20% and then decreased by 3%. What was amount of dollar increase in the property amount of better please during the second. Yeah, so the price off the property at the end of the second year Waas City toe suited to 7 400 So after increasing 20% it's became 32 400. So I have the price X. I increase it by 20%. So times 1.2 I will get 32,400 so I can search for X the price before the increase of 20% when I can no x so I can know the difference. And I know the amount of dollar in off the during these the property plays dealing with second here. So 1st 1 is clearly sufficient. Let's examine the second statement the increase in property price over the 1st 2 years. Waas seven 1003 102 lessons a decrease in the property price during the search year. So So here's first year, but first at the beginning, I have X. After the first year, I have 0.9 x and after second year, I have 1.2 times 0.9 x and then at the end, I have 0.7 times 1.2 times zero mine X. This is a press. The increases the property price over the 1st 2 years. Waas okay, over the 1st 2 years. So the prices is in the second year is this is a price at the beginning. This is a price increase. It is the difference between balls. So 1.2 times 0.9 uh, ex, uh, minus X. So this is Shankly's waas 7320 less than the dick reasons. A property price eso was equal to that decrees the property prices a 30 year. So the decreases one point two times 0.9 x minus zio 0.27 times 1.2. That was your point. My X minus seven, uh, 1000. The other than 20. So here I have, uh, an equation with X. So one sided their mind X. I can undermine everything I want. So here. I'm not gonna come next. As I can see from this equation, I can, so, thanks. I can search it. Search for it easily. For example, Here I have the same value here and here, so I can remove it. I can't take this value to here, So it's gonna be, I think think it's gonna be seven out that sometimes it's the other than 20 is equal to ex fires minus 07 times, two times. Bless one. So exes, I can copulate Ecstasies initial. I can no X. I can know the value here, the value here and then I can say abstract and get the increase of the amount of the property price tuning in second a year. So the second is also sufficient on The answer to this question is clearly the
40. Medium20: in 1990 there were 400 female includes that company A So in 1990 400 female please. If number off few military that company increased by 70% from 1990 to 2000 sold themselves , so we have less 70%. So why we can calculate it So 400 bless 70% off 400. So the fire zero so four times simply started and a so it's 100 personality and a key will take up to 600 off and 80 so 680 employees in the 2000. So by what percentage? The number of human agrees that company A is from 2000 to 2000 and then. So I want to know the percentage of increase females from 2000 to 2000 and then So let's take some lines for a statement from one from 1990 to 2000 and 10 the number of humility and Louise increased by 100% opening day, so it's doubled, increasing the number by 100% it meanings who are doubling its So in 2010 you have 800 employees, so we have the number of him. Joey's off even a reason to sell 2000. We have the number of employees in 2000 and 10 so we can correct calculate the percentage of increase So we can say that it's 800 minus 680 divided by 680 times 100 we can get percentage off trees. So the 1st 1 is leaving so efficient. Let's see, the 2nd 1 In 1990 there were 600 made enquiries at Company A. So we have We're leaving here with Malin. Please. You don't have any information about team and please. So clearly is a second statement is not sufficient. So here's answer is clearly a
41. Medium21: at what Speed wasa car traveling on a trip when it had completed half off the total distance off the trip. So I have a car moving from point A to point B. So I want to know the speed of the car when it had completed half of the total distance off that, uh, true. So let me examine the first statement. The trip WAAS 160 kilometers long and took two hours to complete. So I have the total distance. It's a total time. I can use it to calculate the average speed, its secret, the distance over time it secret 1 60 divided by two. So it's 80 kilometers per hour. So I know the average speed. But I don't know what is what waas the speed of the car when it had completed half of the total distance of that. I don't have any information about this so clearly. The first statement is not sure. The car traveled at an average rate of 80 kilometers per hour on the trip, so it's the same information here, so he's standing that the average IQ is 80 kilometers per hour, so it it is very valuable so it can be 100 at the beginning and then 60 and then that can change in function off time. I don't know any information about the speed of the car when it had completed the half the distance off the tip, so I only have the average, uh, speed. So clearly the 2nd 1 is not sufficient. When I combine both statements, I only have the average speed of the whole distance. So it's clearly not sufficient bulls, so it's answer is clearly he.
42. Medium22: the average weight off the woman in the room is 58 kilogram, and the average rate off the men in the room is 83 kilogram. What is the average weight off the people in the room? So average off the man? It's equal. Teoh Hotel Wait off men. You find it by the number off men. Let me say that it's M so I can deduce that the total weight off men is equal to average. Bine stems a lexical to 83 and the total weight off woman is Sequent to the same. It's 58 times. Woman W is the number of from, So I want the average weight off people in the room so average it's equal to total weight off man blushed to the local for one. So it's 83 men plus 68 woman divided by employers that the number of men but some of former is equal to so assists the question here. This is what I want to deter mine. So there are total off 99 people in the So I know that I m plus W Z quits in 99. So here I can replace it by 99 but I don't know em. And I know I don't know, w So if I take em is equal to 88 and w zika to on, it's gonna give me a result here. And if I take em is equal to one W's equal to 90 oh eight. So it would give me another reason. Surely so it's not sufficient because I don't know that, uh, I don't know the number of men and the number off woman, so I would take the mind. The second statement There are twice as many women as men in the room, so I have w was equal to em, so I can replace it to see what give it would give me. So 83 m lost 58 times two is equal to M plus two and Mexico to 3 a.m. So where it's equal to 83 plus two times if they to concept eight is 116 116 plus, So 160 plus 83. So 110 plus 8100 and 90 nine. So it's 199 you wanted by 30. Uh, money on em divided by three AM so I can signify by him so I can know the average weight. Using only is the second information I used. W is equipped. Teoh. Am I replaced here in the formula and I got the answer on De. So that's it's a second statement is sufficient, so the answer is fish.
43. Medium23: this is a median offset s even. So let me examine First state Set s is composed of consecutive even integer So set s consecutive and fingers Let me see. So, uh, let me take an example for 68 These are consecutive even into use The median here is six. The main is equal to six also. So here the median is equal toe. So here, I mean, is equal to the median. So, um, the media is even so it disaster it is a question is yes. Let's take another example for six. Eight, then. Okay, so here are consecutive. Even in diggers, the median is that immediate value between six and eight. So the median is able to sever and the mean is equal Teoh seven also. So we need to know here just, uh, an important information that if I have consecutive, even integers the median is equal to the mean. So if the meanest six median six years a median, its seventh and demean is also seven, So here's the mean is ah, the mean is uh oh, so that's it is a question is no. So I have to his example. One example that gives me announce it is. That question is yes. It was an example that gives me an answer to the question is no eso The 1st 1 is not sufficient. That makes the mind the second statement. Does that mean offset s is even so let me see is I mean off surfaces. Even so, let me take up 468 it is I mean is even the meeting is even so that this question is yes. Let me take another example. Let's say, for example, 11 for here the mean is one plus one plus four divided by 30 to 60. But reason to who was even so the example set despise the second statement and let me see the media and the media is clearly here one so the median is owed. So the answer to this question is no. But I have two examples. One, it's telling me Yes, The second is staying No. So clearly the 2nd 1 is not sufficient. Let me examine both off the statements. So I have consecutive even integers consecutive even and figures It means I told you before that mean is equipped with the media. Next is telling me the media of services evens with the median is even on the mean is equal to the mean. The median is equipped with the means, so that mean eso the media is even because the mean off the sicknesses even and they're both sequel using good statements. So I know that the media is even using both statements, so it's clear his answer, see?
44. Medium24: So what is the value off X? So the 1st 1 absolute value of X is equal to minus X. So I know from zebras that if absolute value off X is equal to minus X, that does need that X is negative. So X could begin to minus two. Exit convicted, minus Stan extradicted to any negative numbers, so it's clearly not sufficient with me. Let me take an example. Absolute Very Afghanistan is equal to 10 and minus minus. 10 is equal to 10 so it's a value. Afghanistan is equal to minus minus stand. So everybody, it's a despise. It's a condition on the value of X. Could be, uh, minus. Then the same for minus two of the self reliance wanted for any and they get no birds, so it's clearly her statement is not sufficient. Second statement. X squared is equipped with 16 so X could be equal. Four or X could be equal to minus four. We have two values. Pay attention to this when you have exquisite with the 16 then X is equal to or minus four . We have big balls values, so I cannot choose a value for extra guy over half to values to different values for and minus four since clearly the 2nd 1 is not sufficient now remaining both off statements from the first statement. I know that X is negative For the second statement I know x equal to four or exit with minus four. So since X is negative that I have to choose here. Exactly. The Highness four. Let before. So when I combine both statements Xnegative so X must be minus four. So I know the value of X. So both off statements are sufficient. Those answer is C.