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盤點GMAT邏輯推理中常見的數學考點

編輯: 長安來自: 環球教育

  從邏輯學的角度來說數學是一種更加系統和嚴謹的邏輯,在GMAT邏輯的考查中當然是少不了它的出現。在這里,廣州環球教育侯宇飛老師把GMAT邏輯中涉及的幾種常見的數學考點給同學們羅列一下,希望對參加GMAT考試的同學有所幫助。


  一、關于一些基本數學概念的考查。


  接下來看一下這些概念在GMAT題目中是怎么考查的?先來一道簡單的題目!

  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:命題Ⅰ是原命題的逆命題,所以錯誤。

  命題Ⅱ是原題目的逆否命題,逆否命題等于原命題,所以該命題是正確的。

  命題Ⅲ是原命題的否命題,否命題不等同于原命題,所以命題Ⅲ錯誤。所以上面題目的答案選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:這道題目的核心信息在最后一句,這也是邏輯題目普遍的特點,結構屬于機制功能型題目,題目的核心意思是,通過規律的鍛煉和減肥可以提高cholesterol(膽固醇)的量,接下來選項我們逐一排除選項。

  A屬于無中生有的信息。B可以看作是原題目信息的否命題,依然錯誤。C無中生有,而且還有極端詞most。D原文最后一句的相同內容的轉述,沒問題。E無中生有,而且伴有極端詞only。

  二、否命題的考查:

  P and q  否命題:not p or not q

  P or q 否命題 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:我們可以從題干中的重要hypothesized判斷出來,該題目是一道因果關系的題目,對于因果關系削弱題目我們常用的手段就是非因非果即否定原因進而否定其結果。接下來我們看一下該題目的主干邏輯:

  選項A就是一個非因非果的思路的題目:世代生活在塞內加爾和贊比亞salt available的人(當然是黑人了)血壓低。首先血壓低否定了原題干中的黑人血壓高,結果被否定掉。

  另外,“世代生活在salt available的這兩個國家的人”該信息否定了原題干兩個原因中的“genes that adapted to an environmental scarcity of salt.”這里涉及了上面講過的P and q 否命題:not p or not q的數學概念的應用。

  這道很多同學會錯選D,接下來分析一下D選項的問題:該選項和A選項很相似,結果同樣是被否定了,但是我們不能把遠離鹽礦錯誤地理解為就是缺鹽,偷換概念也是邏輯錯誤選項常用的設置的一種手段,同學們要引起足夠的重視,而且即使我們把遠離鹽礦錯誤地等同于缺鹽,對題目也是一種削弱的作用。

  三、利潤=收入-支出

  該關系也是邏輯題目經常涉及的一類考題,請看題!

  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:最后一句話說總的銷量增加了,但是例如減少了,說明成本上升了,答案選E。

  四、控制變量型題目表現為: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:文中要加強是“less restrictive policy”,對該信息可以表現為兩種方式。

  一種是題目一開始提到的,別的地方不賠的,Sparva賠,另外一種形式就是:別的地方賠少的, Sparva賠多。E選項就是第二種情況:total payments=the number of accident*the payment per accident。total payments多,而the number of accident少,說明the payment per accident多,從而加強了文中的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:題目大意“一個州計劃向買東西的人征收7%其消費額的作為稅,因此,如果把這個稅制看作是收入所得稅的話,要對聯邦的稅制產生相反的效果——收入越低,收入所得稅占收入的比例越高”。

  Income tax percentage=income tax/income。而題目中又給了income tax= sales tax,那么題目最后已經限定了income↓,income tax percentage↑,那么這里就要control 關系中的唯一變量income tax= sales tax,選項A滿足該要求,A選項說市民按照稅率花比收入水平的錢都一樣,那么分子不變。

  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.

  題干說在診斷paranoia病的時候會出現兩種錯誤結果,現在實驗的方法可以把錯誤的positive result降到最低,文中所提及的方案最終是要把錯誤率降到最低,而現在false positive result降到最低,那么現在唯一需要的就是實驗室情況下把false negative result 降低或者相等,那么最后總的錯誤結果就會降低或者實驗的方法有效,答案選E。

  五、百分比、人均、總量等相關概念的考查。

  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:影響Florida經濟的實質是移入當地的退休的實際人數,而不是百分比,所以在該題干論證忽略了這個問題,而選項D的情況出現的話,移入到Florida的退休人數有可能比之前增多,有助于當地的經濟,所以答案選D。

  小結:這里羅列了GMAT邏輯里面常涉及地五種數學概念的考查,數學的知識點比較簡單,對于GMAT考生只要能夠敏銳地識別出其中的考點,正確答案的方向是很容易確定的。

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