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盘点GMAT逻辑推理中常见的数学考点

编辑: 长安来自: 环球教育

  从逻辑学的角度来说数学是一种更加系统和严谨的逻辑,在GMAT逻辑的考查中当然是少不了它的出现。在这里,广州环球教育侯宇飞老师把GMAT逻辑中涉及的几种常见的数学考点给同学们罗列一下,希望对参加GMAT考试的同学有所帮助。


  一、关于一些基本数学概念的考查。


  接下来看一下这些概念在GMAT题目中是怎么考查的?先来一道简单的题目!

  例:If it is true that the streets and the sidewalks are wet whenever it is raining, which of the following must also be true?

  Ⅰ.If the streets and sidewalks are wet, it is raining.

  Ⅱ.If the streets and sidewalks are not wet, it is not raining.

  Ⅲ.If it is not raining, the streets and sidewalks are not wet.

  A.Ⅰonly

  B.Ⅱonly

  C.Ⅲonly

  D.Ⅰand Ⅱonly

  E.Ⅱand Ⅲonly

  解析:命题Ⅰ是原命题的逆命题,所以错误。

  命题Ⅱ是原题目的逆否命题,逆否命题等于原命题,所以该命题是正确的。

  命题Ⅲ是原命题的否命题,否命题不等同于原命题,所以命题Ⅲ错误。所以上面题目的答案选B。

  One more question!!!

  例.Increase in the level of high-density lipoprotein (HDL) in the human bloodstream lower bloodstream-cholesterol levels by increasing the body's capacity to rid itself of excess cholesterol .levels of HDL in the bloodstream of some individuals are significantly increased by a program of regular exercise and weight reduction.

  Which of the following can be correctly inferred from the statements above?

  A.Individuals who are underweight do not run any risk of developing high levels of cholesterol in the bloodstream.

  B.Individuals who do not exercise regularly have a high risk developing high levels of cholesterol in the bloodstream late in life.

  C.Exercise and weight reduction are the most-effective methods of lowering bloodstream cholesterol levels in humans.

  D.A program of regular exercise and weight reduction lowers cholesterol levels in the bloodstream of some individuals.

  E.Only regular exercise is necessary to decrease cholesterol levels in the bloodstream of individuals of average weight.

  解析:这道题目的核心信息在最后一句,这也是逻辑题目普遍的特点,结构属于机制功能型题目,题目的核心意思是,通过规律的锻炼和减肥可以提高cholesterol(胆固醇)的量,接下来选项我们逐一排除选项。

  A属于无中生有的信息。B可以看作是原题目信息的否命题,依然错误。C无中生有,而且还有极端词most。D原文最后一句的相同内容的转述,没问题。E无中生有,而且伴有极端词only。

  二、否命题的考查:

  P and q  否命题:not p or not q

  P or q 否命题 not p and not q

  例:Black Americans are, on the whole, about twice as likely as White Americans to develop high blood pressure, This likelihood also holds for westernized Black Africans when compared to White Africans, Researchers have hypothesized that this predisposition in westernized Blacks may reflect an interaction between western high-salt diets and genes that adapted to an environmental scarcity of salt.

  Which of the following statements about present-day, westernized Black Africans, if true, would most tend to confirm the researchers' hypothesis?

  (A)The blood pressures of those descended from peoples situated throughout their history in Senegal and Gambia where salt was always available, are low.

  (B)The unusually high salt consumption in certain areas of Africa represents a serious health problem.

  (C)Because of their blood pressure levels, most White Africans have markedly decreased their salt consumption.

  (D) Blood pressures are low among the Yoruba, who throughout their history, have been situated far inland from sources of sea and far south of Saharan salt mines.

  (E)No significant differences in salt metabolism have been found between people who have had salt available throughout their history and those who have not.

  解析:我们可以从题干中的重要hypothesized判断出来,该题目是一道因果关系的题目,对于因果关系削弱题目我们常用的手段就是非因非果即否定原因进而否定其结果。接下来我们看一下该题目的主干逻辑:

  选项A就是一个非因非果的思路的题目:世代生活在塞内加尔和赞比亚salt available的人(当然是黑人了)血压低。首先血压低否定了原题干中的黑人血压高,结果被否定掉。

  另外,“世代生活在salt available的这两个国家的人”该信息否定了原题干两个原因中的“genes that adapted to an environmental scarcity of salt.”这里涉及了上面讲过的P and q 否命题:not p or not q的数学概念的应用。

  这道很多同学会错选D,接下来分析一下D选项的问题:该选项和A选项很相似,结果同样是被否定了,但是我们不能把远离盐矿错误地理解为就是缺盐,偷换概念也是逻辑错误选项常用的设置的一种手段,同学们要引起足够的重视,而且即使我们把远离盐矿错误地等同于缺盐,对题目也是一种削弱的作用。

  三、利润=收入-支出

  该关系也是逻辑题目经常涉及的一类考题,请看题!

  例.In 1984 Exco, which sells it's products only through mail-order catalogs, began distributing its catalog to people who had never purchased Exco's products, while it continued to distribute the catalog to previous customers. Total dollar sales increased in 1984, but Exco's profits that year were smaller than in 1983.

  Which of the following, if true, contributes most to an explanation of Exco's smaller profits in 1984as compared to 1983?

  (A)There was a two percent increase in 1984in the sales tax that consumers had to pay on all purchase from the catalog.

  (B)A greater number of catalogs were sent to previous customers than to people who never purchased products from Exco.

  (C)In 1984 Exco's product-manufacturing costs increased by a smaller amount than the products' selling prices Increased.

  (D)Customers who never previously purchased products from Exco purchased, on the average, fewer products in 1984 than did previous customers.

  (E)The increase between 1983 and 1984 in the cost of mailing the catalogs was greater than the increase in sales from l983 to 1984.

  解析:最后一句话说总的销量增加了,但是例如减少了,说明成本上升了,答案选E。

  四、控制变量型题目表现为:A=B*C;A=B/C;A=B+C.

  1.A=B*C型

  例. Sparva, unlike Treland's other provinces, requires automobile insurers to pay for any medical treatments sought by someone who has been involved in an accident; in the other provinces, insurers pay for nonemergency treatment only if they preapprove the treatment. Clearly, Sparva's less restrictive policy must be the explanation for the fact that altogether insurers there pay for far more treatments after accidents than insurers in other provinces, even though Sparva does not have the largest population.

  Which of the following, if true, most strengthen the argument?

  A.Car insurance costs more in Sparva than in any other province.

  B.The cost of medical care in Sparva is higher than the national average.

  C.Difference insurance companies have different standards for determining what constitutes emergency treatment.

  D.Fewer insurance companies operate in Sparva than in any other province.

  E.There are fewer traffic accidents annually in Sparva than in any of the provinces of comparable or great population.

  解析:文中要加强是“less restrictive policy”,对该信息可以表现为两种方式。

  一种是题目一开始提到的,别的地方不赔的,Sparva赔,另外一种形式就是:别的地方赔少的, Sparva赔多。E选项就是第二种情况:total payments=the number of accident*the payment per accident。total payments多,而the number of accident少,说明the payment per accident多,从而加强了文中的less restrictive policy。

  2.A=B/C型

  例:One state adds a 7 percent sales tax to the price of most products purchased within its jurisdiction. This tax, therefore, if viewed as tax on income, has the reverse effect of the federal income tax: the lower the income, the higher the annual percentage rate at which the income is taxed.

  The conclusion above would be properly drawn if which of the following were assumed as a premise?

  A.The amount of money citizens spend on products subject to the state tax tends to be equal across income levels.

  B.The federal income tax favors citizens with high incomes, whereas the state sales tax favors citizens with low incomes.

  C.Citizens with low annual incomes can afford to pay a relatively higher percentage of their incomes in state sales tax, since their federal income tax is relatively low.

  D.The lower a state's sales tax, the more it will tend to redistribute income from the more affluent citizens to the rest of society.

  E.Citizens who fall to earn federally taxable income are also exempt from the state sales tax.

  解析:题目大意“一个州计划向买东西的人征收7%其消费额的作为税,因此,如果把这个税制看作是收入所得税的话,要对联邦的税制产生相反的效果——收入越低,收入所得税占收入的比例越高”。

  Income tax percentage=income tax/income。而题目中又给了income tax= sales tax,那么题目最后已经限定了income↓,income tax percentage↑,那么这里就要control 关系中的唯一变量income tax= sales tax,选项A满足该要求,A选项说市民按照税率花比收入水平的钱都一样,那么分子不变。

  3.A=B+C型

  例. In tests for paranoia, a serious disease, a false positive result indicates that people have paranoia when, in fact, they do not; a false negative result indicates that people do not have paranoia when, in fact, they do, To detect paranoia most accurately, physicians should use the laboratory test that has the lowest proportion of false positive results.

  Which of the following, if true, gives the most support to the recommendation above?

  A.The accepted treatment for paranoia does not have damaging side effects.

  B.The laboratory test that has the lowest proportion of false positive results causes the same minor side effects as do the other laboratory tests used to detect paranoia.

  C.In treating paranoia patients, it is essential to begin treatment as early as possible, since even a week of delay can result in loss of life.

  D.The proportion of inconclusive test results is equal for all laboratory tests used to detect paranoia.

  E. All laboratory tests to detect paranoia have the same proportion of false negative.

  解析:false result=false positive result +false negative result.

  题干说在诊断paranoia病的时候会出现两种错误结果,现在实验的方法可以把错误的positive result降到最低,文中所提及的方案最终是要把错误率降到最低,而现在false positive result降到最低,那么现在唯一需要的就是实验室情况下把false negative result 降低或者相等,那么最后总的错误结果就会降低或者实验的方法有效,答案选E。

  五、百分比、人均、总量等相关概念的考查。

  In the United States, of the people who moved from one state to another when they retired, the percentage who retired to Florida has decreased by three percentage points over the past ten years. Since many local businesses in Florida cater to retirees, this decline is likely to have a noticeably negative economic effect on these businesses.

  Which of the following, if true, most seriously weakens the argument?

  A.Florida attracts more people who move from one state to another when they retire than does any other state.

  B.The number of people who move out of Florida to accept employment in other states has increased over the past ten years.

  C.there are far more local businesses in Florida that cater to tourists than there are local businesses that cater to retirees.

  D.The total number of people who retired and moved to another state for their retirement has increased significantly over the past ten years.

  E.The number of people who left Florida when they retired to live in another state was greater last year than if was ten years ago.

  解析:影响Florida经济的实质是移入当地的退休的实际人数,而不是百分比,所以在该题干论证忽略了这个问题,而选项D的情况出现的话,移入到Florida的退休人数有可能比之前增多,有助于当地的经济,所以答案选D。

  小结:这里罗列了GMAT逻辑里面常涉及地五种数学概念的考查,数学的知识点比较简单,对于GMAT考生只要能够敏锐地识别出其中的考点,正确答案的方向是很容易确定的。

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