GMAT考试：Testprep数学精解2

　　If I eat nuts， then I break out in hives. This in turn can be symbolized as NH.

　　Next， we interpret the clause there is a blemish on my hand to mean hives， which we symbolize as H. Substituting these symbolssintosthe argument yields the following diagram：

　　NH

　　H

　　Therefore， N

　　The diagram clearly shows that this argument has the same structure as the given argument. The answer， therefore， is 。

　　Denying the Premise Fallacy

　　AB

　　~A

　　Therefore， ~B

　　The fallacy of denying the premise occurs when an if-then statement is presented， its premise denied， and then its conclusion wrongly negated.

　　Example：

　　The senator will be reelected only if he opposes the new tax bill. But he was defeated. So he must have supported the new tax bill.

　　The sentence The senator will be reelected only if he opposes the new tax bill contains an embedded if-then statement： If the senator is reelected， then he opposes the new tax bill. This in turn can be symbolized as R~T. The sentence But　the senator was defeated can be reworded as He was not reelected， which　in turn can be symbolized as ~R. Finally， the sentence He must have supported the new tax bill can be symbolized as T. Using these symbols the argumen

　　2006年9月16日 　　Therefore， T

　　 This diagram clearly shows that the argument is committing the fallacy　of denying the premise. An if-then statement is made; its premise is negated; then its conclusion is negated.

　　Transitive Property

　　AB

　　BC

　　Therefore， AC

　　These arguments are rarely difficult， provided you step back and take a birds-eye view. It may be helpful to view this structure as an inequality in mathematics. For example， 5 4 and 4 3， so 5 3.

　　Notice that the conclusion in the transitive property is also an if-then statement. So we dont know that C is true unless we know that A is true. However， if we add the premise A is true to the diagram， then we can conclude that C is true：

　　AB

　　BC

　　A

　　Therefore， C

　　As you may have anticipated， the contrapositive can be generalized to the transitive property：

　　AB

　　BC

　　~C

　　Therefore， ~A

　　Example：

　　If you work hard， you will be successful in America. If you are successful in America， you can lead a life of leisure. So if you work hard in America.

　　Let W stand for you work hard， S stand for you will be successful in America， and L stand for you can lead a life of leisure. Now the first sentence translates as WS， the second sentence as SL， and the conclusion asWL. Combining these symbol statements yields the following diagram：

　　WS

　　SL

　　Therefore， WL

　　The diagram clearly displays the transitive property.

　　DeMorgans Laws

　　~ = ~A or ~B

　　~ = ~A ~B

　　If you have taken a course in logic， you are probably familiar with these formulas. Their validity is intuitively clear： The conjunction AB is false when either， or both， of its parts are false. This is precisely what ~A or ~B says. And the disjunction A or B is false only when both A and B are false，which is precisely what ~A and ~B says.

　　You will rarely get an argument whose main structure is based on these rulesthey are too mechanical. Nevertheless， DeMorgans laws often help simplify， clarify， or transform parts of an argument. They are also useful with games.

　　Example：

　　It is not the case that either Bill or Jane is going to the party.

　　This argument can be diagrammed as ~， which by the second of DeMorgans laws simplifies to 。 This diagram tells us that neither of th

　　A unless B

　　~BA

　　A unless B is a rather complex structure. Though surprisingly we use it with little thought or confusion in our day-to-day speech.

　　To see that A unless B is equivalent to ~BA， consider the following situation：

　　Biff is at the beach unless it is raining.

　　Given this statement， we know that if it is not raining， then Biff is at the beach. Now if we symbolize Biff is at the beach as B， and it is rainingas R， then the statement can be diagrammed as ~RB.

　　CLASSIFICATION

　　In Logic II， we studied deductive arguments. However， the bulk of arguments on the GMAT are inductive. In this section we will classify and study the major types of inductive arguments.

　　An argument is deductive if its conclusion necessarily follows from its premisesotherwise it is inductive. In an inductive argument， the author presents the premises as evidence or reasons for the conclusion. The validity of the conclusion depends on how compelling the premises are. Unlike deductive arguments， the conclusion of an inductive argument is never certain. The truth of the conclusion can range from highly likely to highly unlikely. In reasonable arguments， the conclusion is likely. In fallacious arguments， it is improbable. We will study both reasonable and fallacious arguments.

　　We will classify the three major types of inductive reasoninggeneralization， analogy， and causaland their associated fallacies.

　　Generalization

　　Generalization and analogy， which we consider in the next section， are the main tools by which we accumulate knowledge and analyze our world. Many people define generalization as inductive reasoning. In colloquial speech， the　phrase to generalize carries a negative connotation. To argue by generalization， however， is neither inherently good nor bad. The relative validity of　a generalization depends on both the context of the argument and the likelihood that its conclusion is true. Polling organizations make predictions by　generalizing information from a small sample of the population， which hopefully represents the general population. The soundness of their predictions depends on how representative the sample is and on its size. Clearly， the less comprehensive a conclusion is the more likely it is to be true.　

　　During the late seventies when Japan was rapidly expanding its share of the　American auto market， GM surveyed owners of GM cars and asked them whether they would be more willing to buy a large， powerful car or a small， economical car. Seventy percent of those who responded said that they would prefer a　large car. On the basis of this survey， GM decided to continue building large cars. Yet during the80s， GM lost even more of the market to the Japanese

　　

　　Which one of the following， if it were determined to be true， would best explain this discrepancy.

　　 Only 10 percent of those who were polled replied.

　　 Ford which conducted a similar survey with similar results continued to　build large cars and also lost more of their market to the Japanese.

　　 The surveyed owners who preferred big cars also preferred big homes.

　　 GM determined that it would be more profitable to make big cars.

　　 Eighty percent of the owners who wanted big cars and only 40 percent of　the owners who wanted small cars replied to the survey.

　　The argument generalizes from the survey to the general car-buying population， so the reliability of the projection depends on how representative the sample is. At first glance， choice seems rather good， because 10 percent does not seem large enough. However， political opinion polls are typically based on only .001 percent of the population. More importantly， we dont know　what percentage of GM car owners received the survey. Choice simply states that Ford made the same mistake that GM did. Choice is irrelevant. Choice ， rather than explaining the discrepancy， gives even more reason for　GM to continue making large cars. Finally， choice points out that partof the survey did not represent the entire public， so is the answer.

　　Analogy

　　To argue by analogy is to claim that because two things are similar in some　respects， they will be similar in others. Medical experimentation on animals　is predicated on such reasoning. The argument goes like this： the metabolism of pigs， for example， is similar to that of humans， and high doses of saccharine cause cancer in pigs. Therefore， high doses of saccharine probably stronger the argument will be. Also the less ambitious the conclusion the　stronger the argument will be. The argument above would be strengthened by changing probably to may. It can be weakened by pointing out the dissimil arities between pigs and people.

　　Example：

　　Just as the fishing line becomes too taut， so too the trials and tribulations of life in the city can become so stressful that ones mind can snap.

　　Which one of the following most closely parallels the reasoning used in the　argument above?

　　 Just as the bow may be drawn too taut， so too may ones life be wasted pursuing self-gratification.

　　 Just as a gamblers fortunes change unpredictably， so too do ones career opportunities come unexpectedly.

　　 Just as a plant can be killed by over watering it， so too can drinking too much water lead to lethargy.

　　 Just as the engine may race too quickly， so too may life in the fast lane lead to an early death.

　　 Just as an actor may become stressed before a performance， so too may dwelling on the negative cause depression.

　　The argument compares the tautness in a fishing line to the stress of city life; it then concludes that the mind can snap just as the fishing line can.

　　So we are looking for an answer-choice that compares two things and draws a　conclusion based on their similarity. Notice that we are looking for an argument that uses similar reasoning， but not necessarily similar concepts. In fact， an answer-choice that mentions either tautness or stress will probably　be a same-language trap.

　　Choice uses the same-language trapnotice too taut. The analogy between a taut bow and self-gratification is weak， if existent. Choice offers　a good analogy but no conclusion. Choice offers both a good analogy and　a conclusion; however， the conclusion， leads to lethargy， understates the　scope of what the analogy implies. Choice offers a strong analogy and a　conclusion with the same scope found in the original： the engine blows， th　e person dies the line snaps， the mind snaps. This is probably the best　answer， but still we should check every choice. The last choice， ， uses l　anguage from the original， stressful， to make its weak analogy more tempt　

　　Causal Reasoning

　　Of the three types of inductive reasoning we will discuss， causal reasoning　is both the weakest and the most prone to fallacy. Nevertheless， it is a useful and common method of thought.

　　To argue by causation is to claim that one thing causes another. A causal argument can be either weak or strong depending on the context. For example， to claim that you won the lottery because you saw a shooting star the night before is clearly fallacious. However， most people believe that smoking causes cancer because cancer often strikes those with a history of cigarette use.

　　Although the connection between smoking and cancer is virtually certain， as　with all inductive arguments it can never be 100 percent certain. Cigarette　companies have claimed that there may be a genetic predisposition in some people to both develop cancer and crave nicotine. Although this claim is highly improbable， it is conceivable.

　　There are two common fallacies associated with causal reasoning：

　　1. Confusing Correlation with Causation.

　　To claim that A caused B merely because A occurred immediately before B is clearly questionable. It may be only coincidental that they occurred together， or something else may have caused them to occur together. For example， the　fact that insomnia and lack of appetite often occur together does not mean　that one necessarily causes the other. They may both be symptoms of an underlying condition.

　　2. Confusing Necessary Conditions with Sufficient Conditions.

　　A is necessary for B means B cannot occur without A. A is sufficient for B　means A causes B to occur， but B can still occur without A. For example，　a small tax base is sufficient to cause a budget deficit， but excessive spen　ding can cause a deficit even with a large tax base. A common fallacy is to　assume that a necessary condition is sufficient to cause a situation. For example， to win a modern war it is necessary to have modern， high-tech equipment， but it is not sufficient， as Iraq discovered in the Persian Gulf War.　

　　Contradiction

　　A Contradiction is committed when two opposing statements are simultaneously　asserted. For example， saying it is raining and it is not raining is a co　ntradiction. Typically， however， the arguer obscures the contradiction to the point that the argument can be quite compelling. Take， for instance， the following argument：

　　We cannot know anything， because we intuitively realize that our thoughts are unreliable.

　　This argument has an air of reasonableness to it. But intuitively realize　means to know. Thus the arguer is in essence saying that we know that we dont know anything. This is self-contradictory.

　　Equivocation

　　Equivocation is the use of a word in more than one sense during an argument.

　　This technique is often used by politicians to leave themselves an out. If someone objects to a particular statement， the politician can simply claim　the other meaning.

　　Example：

　　Individual rights must be championed by the government. It is right for one　to believe in God. So government should promote the belief in God.

　　In this argument， right is used ambiguously. In the phrase individual rights it is used in the sense of a privilege， whereas in the second sentence right is used to mean proper or moral. The questionable conclusion is possible　only if the arguer is allowed to play with the meaning of the critical word　right.

　　Circular Reasoning

　　Circular reasoning involves assuming as a premise that which you are trying　to prove. Intuitively， it may seem that no one would fall for such an argument. However， the conclusion may appear to state something additional， or the　argument may be so long that the reader may forget that the conclusion was　

　　Example：

　　The death penalty is appropriate for traitors because it is right to execute

　　those who betray their own country and thereby risk the lives of millions.

　　This argument is circular because right means essentially the same thing as appropriate. In effect， the writer is saying that the death penalty is a　ppropriate because it is appropriate.

　　Shifting The Burden Of Proof

　　It is incumbent on the writer to provide evidence or support for her position. To imply that a position is true merely because no one has disproved it is to shift the burden of proof to others.

　　Example：

　　Since no one has been able to prove Gods existence， there must not be a God

　　There are two major weaknesses in this argument. First， the fact that Gods　existence has yet to be proven does not preclude any future proof of existence. Second， if there is a God， one would expect that his existence is independent of any proof by man.

　　Unwarranted Assumptions

　　The fallacy of unwarranted assumption is committed when the conclusion of an　argument is based on a premise that is false or unwarranted. An assumption is unwarranted when it is falsethese premises are usually suppressed or vaguely written. An assumption is also unwarranted when　it is true but does not apply in the given contextthese premises are usually explicit.

　　Example：

　　Either restrictions must be placed on freedom of speech or certain subversive elements in society will use it to destroy this country. Since to allow the latter to occur is unconscionable， we must restrict freedom of speech.

　　The conclusion above is unsound because　 subversives do not in fact want to destroy the country the author places too much importance on the freedom of speech the author fails to consider an accommodation between the two alternatives the meaning of freedom of speech has not been defined　

　　The arguer offers two options： either restrict freedom of speech， or lose the country. He hopes the reader will assume that these are the only options a　vailable. This is unwarranted. He does not state how the so-called subversive elements would destroy the country， nor for that matter， why they would　want to destroy it. There may be a third option that the author did not mention; namely， that society may be able to tolerate the subversives and it may even be improved by the diversity of opinion they offer. The answer is 。

　　Appeal To Authority

　　To appeal to authority is to cite an experts opinion as support for ones own opinion. This method of thought is not necessarily fallacious. Clearly， the reasonableness of the argument depends on the expertise of the person being cited and whether she is an expert in a field relevant to the argument.

　　Appealing to a doctors authority on a medical issue， for example， would be　reasonable; but if the issue is about dermatology and the doctor is an orthopedist， then the argument would be questionable.

　　Personal Attack

　　In a personal attack ， a persons character is challenged instead of her opinions.

　　Example：

　　Politician： How can we trust my opponent to be true to the voters? He isnt　true to his wife!

　　This argument is weak because it attacks the opponents character， not his p　ositions. Some people may consider fidelity a prerequisite for public office History， however， shows no correlation between fidelity and great political leadership.

　　I would fly you to the moon and back

　　If youll be if youll be my baby

　　Got a ticket for a worldswhereswe belong　

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　　Data Sufficiency　　

　　INTRODUCTION DATA SUFFICIENCY

　　Most people have much more difficulty with the Data Sufficiency problems than with the Standard Math problems. However， the mathematical knowledge and s　kill required to solve Data Sufficiency problems is no greater than that required to solve standard math problems. What makes Data Sufficiency problems　appear harder at first is the complicated directions. But once you become familiar with the directions， youll find these problems no harder than standard math problems. In fact， people usually become proficient more quickly on　Data Sufficiency problems.

　　THE DIRECTIONS

　　The directions for Data Sufficiency questions are rather complicated. Before　reading any further， take some time to learn the directions cold. Some of the wording in the directions below has been changed from the GMAT to make it　clearer. You should never have to look at the instructions during the test.

　　Directions： Each of the following Data Sufficiency problems contains a question followed by two statements， numbered and 。 You need not solve the　problem; rather you must decide whether the information given is sufficient　to solve the problem.

　　The correct answer to a question is　A if statement ALONE is sufficient to answer the question but statement　alone is not sufficient;　B if statement ALONE is sufficient to answer the question but statement alone is not sufficient;

C if the two statements TAKEN TOGETHER are sufficient to answer the question， but NEITHER statement ALONE is sufficient;D if EACH statement ALONE is sufficient to answer the question;E if the two statements TAKEN TOGETHER are still NOT sufficient to answer the question.

　　Numbers： Only real numbers are used. That is， there are no complex numbers.

　　Drawings： The drawings are drawn to scale according to the information given　in the question， but may conflict with the information given in statements

　　You can assume that a line that appears straight is straight and that angle　measures cannot be zero.

　　You can assume that the relative positions of points， angles， and objects are as shown.

　　All drawings lie in a plane unless stated otherwise.

　　Example：

　　In triangle ABC to the right， what is the value of y?

　　 AB = AC

　　 x = 30

　　Explanation： By statement ， triangle ABC is isosceles. Hence， its base angles are equal： y = z. Since the angle sum of a triangle is 180 degrees， we　get x + y + z = 180. Replacing z with y in this equation and then simplifying yields x + 2y = 180. Since statement does not give a value for x， we cannot determine the value of y from statement alone. By statement ， x= 30. Hence， x + y + z = 180 becomes 30 + y + z = 180， or y + z = 150. Since statement does not give a value for z， we cannot determine the value of y from statement alone. However， using both statements in combination，we can find both x and z and therefore y. Hence， the answer is C.

　　Notice in the above example that the triangle appears to be a right triangle

　　 However， that cannot be assumed： angle A may be 89 degrees or 91 degrees，we cant tell from the drawing. You must be very careful not to assume any more than what is explicitly given in a Data Sufficiency problem.

　　ELIMINATION

　　Data Sufficiency questions provide fertile ground for elimination. In fact，it is rare that you wont be able to eliminate some answer-choices. Remember， if you can eliminate at least one answer choice， the odds of gaining points by guessing are in your favor.

　　The following table summarizes how elimination functions with Data Sufficie　

　　Statement Choices Eliminated is sufficient B， C， E　is not sufficient A， D is sufficient A， C， E is not sufficient B， D is not sufficient and is not sufficient A， B， D

　　Example 1： What is the 1st term in sequence S?

　　 The 3rd term of S is 4.

　　 The 2nd term of S is three times the 1st， and the 3rd term is four times　the 2nd.is no help in finding the first term of S. For example， the following sequences each have 4 as their third term， yet they have different first terms：

　　0， 2， 4

　　-4， 0， 4

　　This eliminates choices A and D. Now， even if we are unable to solve this problem， we have significantly increased our chances of guessing correctlyfrom 1 in 5 to 1 in 3.

　　Turning to ， we completely ignore the information in 。 Although contains a lot of information， it also is not sufficient. For example， the following sequences each satisfy ， yet they have different first terms：

　　1， 3， 12

　　3， 9， 36

　　This eliminates B， and our chances of guessing correctly have increased to 1　in 2.

　　Next， we consider and together. From ， we know the 3rd term of S　is 4. From ， we know the 3rd term is four times the 2nd. This is equivalent to saying the 2nd term is 1/4 the 3rd term： 4 = 1. Further， from， we know the 2nd term is three times the 1st. This is equivalent to saying the 1st term is 1/3 the 2nd term： 1 = 1/3. Hence， the first term　of the sequence is fully determined： 1/3， 1， 4. The answer is C.

　　Example 2： In the figure to the right， what is the area of the triangle?　

　　Recall that a triangle is a right triangle if and only if the square of the　longest side is equal to the sum of the squares of the shorter sides 。 Hence， implies that the triangle is a right triangle. So　the area of the triangle is /2. Note， there is no need to calculate the areawe just need to know that the area can be calculated. Hence， the answer is either A or D.

　　Turning to ， we see immediately that we have a right triangle. Hence， again the area can be calculated. The answer is D.

　　Example 3： Is p

　　 p/3

　　 -p + x -q + x

　　Multiplying both sides of p/3

　　Hence， is sufficient. As to ， subtract x from both sides of -p + x

　　-q + x， which yields -p -q.

　　Multiplying both sides of this inequality by -1， and recalling that multiplying both sides of an inequality by a negative number reverses the inequality， yields p Hence， is also sufficient. The answer is D.

　　Example 4： If x is both the cube of an integer and between 2 and 200， what is the value of x?

　　 x is odd.

　　 x is the square of an integer.

　　Since x is both a cube and between 2 and 200， we are looking at the integers：

　　which reduce to

　　8， 27， 64， 125

　　Since there are two odd integers in this set， is not sufficient to uniquely determine the value of x. This eliminates choices A and D.

　　Next， there is only one perfect square， 64， in the set. Hence， is suffi　

　　 ABC is the base word.

　　 If C immediately follows B， then C can be moved to the front of the codeword to generate another word.

　　From ， we cannot determine whether CAB is a code word since gives no　rule for generating another word from the base word. This eliminates A and D

　　

　　Turning to ， we still cannot determine whether CAB is a code word since now we have no word to apply this rule to. This eliminates B.

　　However， if we consider and together， then we can determine whether

　　CAB is a code word：

　　From ， ABC is a code word.

　　From ， the C in the code word ABC can be moved to the front of the word：

　　CAB.

　　Hence， CAB is a code word and the answer is C.

　　UNWARRANTED ASSUMPTIONS

　　Be extra careful not to read any moresintosa statement than what is given.

　　The main purpose of some difficult problems is to lure yousintosmaking an unwarranted assumption.

　　If you avoid the temptation， these problems can become routine.

　　Example 6： Did Incumbent I get over 50% of the vote?

　　 Challenger C got 49% of the vote.

　　 Incumbent I got 25，000 of the 100，000 votes cast.

　　If you did not make any unwarranted assumptions， you probably did not find t　his to be a hard problem. What makes a problem difficult is not necessarily　its underlying complexity; rather a problem is classified as difficult if many people miss it. A problem may be simple yet contain a psychological trap that causes people to answer it incorrectly.

　　The above problem is difficult because many people subconsciously assume that there are only two candidates. They then figure that since the challenger　received 49% of the vote the incumbent received 51% of the vote. This would　be a valid deduction if C were the only challenger 。

　　But we cannot assume that. There may be two or more challengers. Hence，　　

　　Now， consider alone. Since Incumbent I received 25，000 of the 100，000 votes cast， I necessarily received 25% of the vote. Hence， the answer to the question is No， the incumbent did not receive over 50% of the vote. Therefore， is sufficient to answer the question. The answer is B.

　　Note， some people have trouble with because they feel that the question　asks for a yes answer. But on Data Sufficiency questions， a no answer is　just as valid as a yes answer. What were looking for is a definite answer.

　　CHECKING EXTREME CASES

　　When drawing a geometric figure or checking a given one， be sure to includedrawings of extreme cases as well as ordinary ones.

　　Example 1： In the figure to the right， AC is a chord and B is a point on the　circle. What is the measure of angle x?

　　Although in the drawing AC looks to be a diameter， that cannot be assumed. All we know is that AC is a chord. Hence， numerous cases are possible， three　of which are illustrated below：

　　In Case I， x is greater than 45 degrees; in Case II， x equals 45 degrees; in　Case III， x is less than 45 degrees. Hence， the given information is not sufficient to answer the question.

　　Example 2： Three rays emanate from a common point and form three angles with　measures p， q， and r. What is the measure of q + r ?

　　It is natural to make the drawing symmetric as follows：

　　In this case， p = q = r = 120， so q + r = 240. However， there are other drawings possible. For example：

　　In this case， q + r = 180. Hence， the given information is not sufficient to　answer the question.

　　Problems：

　　1. Suppose 3p + 4q = 11. Then what is the value of q?

　　 p is prime.

　　 q = -2p

　　 is insufficient. For example， if p = 3 and q = 1/2， then 3p + 4q = 3

　　+ 4 = 11. However， if p = 5 and q = -1， then 3p + 4q = 3 + 4 = 1

　　1. Since the value of q is not unique， is insufficient.

　　Turning to ， we now have a system of two equations in two unknowns. Hence， the system can be solved to determine the value of q. Thus， is suffici　　

　　2. What is the perimeter of triangle ABC above?

　　 The ratio of DE to BF is 1： 3.

　　 D and E are midpoints of sides AB and CB， respectively.

　　Since we do not even know whether BF is an altitude， nothing can be determined from 。 More importantly， there is no information telling us the absolute size of the triangle.

　　As to ， although from geometry we know that DE = AC/2， this relationship　holds for any size triangle. Hence， is also insufficient.

　　Together， and are also insufficient since we still dont have information about the size of the triangle， so we cant determine the perimeter. The answer is E.

　　3. A dress was initially listed at a price that would have given the store a　profit of 20 percent of the wholesale cost. What was the wholesale cost of　the dress?

　　 After reducing the asking price by 10 percent， the dress sold for a net　profit of 10 dollars.

　　 The dress sold for 50 dollars.

　　Consider just the question setup. Since the store would have made a profit of 20 percent on the wholesale cost， the original price P of the dress was 120 percent of the cost： P = 1.2C. Now， translating sintosan equation yields：

　　P - .1P = C + 10

　　Simplifying gives

　　9P = C + 10

　　Solving for P yields

　　P = /.9

　　Plugging this expression for PsintosP = 1.2C gives

　　/.9 = 1.2C

　　Since we now have only one equation involving the cost， we can determine the　cost by solving for C. Hence， the answer is A or D. is insufficient since it does not relate the selling price to any other　information. Note， the phrase initially listed implies that there was more　than one asking price. If it wasnt for that phrase， would be sufficien　

　　4. What is the value of the two-digit number x?

　　 The sum of its digits is 4.

　　 The difference of its digits is 4.

　　Considering only， x must be 13， 22， 31， or 40. Hence， is not sufficient to determine the value of x.

　　Considering only， x must be 40， 51， 15， 62， 26， 73， 37， 84， 48， 95， or 59. Hence， is not sufficient to determine the value of x.

　　Considering and together， we see that 40 and only 40 is common to the two sets of choices for x. Hence， x must be 40. Thus， together and　are sufficient to uniquely determine the value of x. The answer is C.

　　5. If x and y do not equal 0， is x/y an integer?

　　 x is prime.

　　 y is even.

　　 is not sufficient since we dont know the value of y. Similarly， is　not sufficient. Furthermore， and together are still insufficient since there is an even prime number2. For example， let x be the prime number

　　2， and let y be the even number 2 。 Then x/y = 2/2 = 1， which is an integer. For all other values of x and y， x/y is not an integer. The answer is E.

　　6. Is 500 the average score on the GMAT?

　　 Half of the people who take the GMAT score above 500 and half of the people score below 500.

　　 The highest GMAT score is 800 and the lowest score is 200.

　　Many students mistakenly think that implies the average is 500. Suppose　just 2 people take the test and one scores 700 and the other scores 400 。 Clearly， the average score for the two test-takers is not 500. is less tempting. Knowing the highest and lowest scores tells us　nothing about the other scores. Finally， and together do not determine the average since together they still dont tell us the distribution of　

　　I) If x is in S， then 1/x is in S.

　　II) If both x and y are in S， then so is x + y.

　　Is 3 in S?

　　 1/3 is in S.

　　 1 is in S.

　　Consider alone. Since 1/3 is in S， we know from Property I that 1/　= 3 is in S. Hence， is sufficient.

　　Consider alone. Since 1 is in S， we know from Property II that 1 + 1 = 2 is in S. Applying Property II again s

　　hows that 1 + 2 = 3 is in S. Hence， is also sufficient. The answer is D.

　　8. What is the area of the triangle above?

　　 a = x， b = 2x， and c = 3x.

　　 The side opposite a is 4 and the side opposite b is 3.

　　From we can determine the measures of the angles： a + b + c = x + 2x + 3

　　x = 6x = 180

　　Dividing the last equation by 6 gives： x = 30

　　Hence， a = 30， b = 60， and c = 90. However， different size triangles can have these angle measures， as the diagram below illus