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GRADUATE RECORD EXAMINATIONS Official GRE Quantitative ReasoningPractice Questions, Volume 1Large Print (18 point) EditionChapter 4—AlgebraAnswer Key with Answers and ExplanationsCopyright 2014 by Educational Testing Service. Allrights reserved. ETS, the ETS logo, GRADUATE RECORDEXAMINATIONS, and GRE are registered trademarks ofEducational Testing Service (ETS) in the United States andother countries.LP18 GRE Prep Materials[This footer should NOT be printed.] 7/17/2014Chapter 4 Algebra Ans/Explan - rev 00
AlgebraThis document begins with the answer key for questions foundin the Chapter 4 Algebra Practice Questions document. Followingthe answer key are the complete explanations for each question.If you wish to work through the questions before consulting theanswers and explanations, please use the Chapter 4 AlgebraPractice Questions document.Answer Key1. A: Quantity A is greater.2. B: Quantity B is greater.3. D: The relationship cannot be determined from theinformation given.4. A: Quantity A is greater.5. C: The two quantities are equal.6. B: Quantity B is greater.-2LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Algebra Ans/Explan rev03
7. A: Quantity A is greater.8. E:439. D:xy810. B: 4 - y11. D: Seven12. B: 1,000 - r13. E: 25r -16s14. The answer to question 14 consists of two of the answerchoices.B: (5, 12)D: (12, 3)-3LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
15. The answer to question 15 consists of two of the answerchoices.x 3x 82 207x 8B:20A:16. In question 16 you were asked to enter an integer or a decimal.The answer to question 16 is 7.2.17. In question 17 you were asked to enter an integer or a decimal.The answer to question 17 is 17.-4LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
Answers and ExplanationsThis is a quantitative comparison question.x ( x - 2)( x 3)( x - 4)2 0Quantity AQuantity Bx-21.Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-5LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationTo compare x with -2, you should first solve the equationx ( x - 2) 0 for x. To solve the equation, recall that a2( x 3)( x - 4)afraction is equal to 0 only if a 0 and b π 0. So the fractionbx ( x - 2)( x 3)( x - 4)2is equal to 0 only if x ( x - 2) is equal to 0 and( x 3)( x - 4)2 is not equal to 0. Note that the only values of x forwhich x ( x - 2) 0 are x 0 and x 2, and for these two values,( x 3)( x - 4)2 is not equal to 0. Therefore the only values of x forwhich the fractionx ( x - 2)( x 3)( x - 4)2is equal to 0 are x 0 andx 2. Since both 0 and 2 are greater than Quantity B, -2, thecorrect answer is Choice A.This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-6LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
This is a quantitative comparison question.x 0Quantity AQuantity B1xx 1x22.Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-7LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
Explanation1x 1x , can be expressed as, which222xxx111can be simplified to . Note that Quantity A is , and for allxx x2Note that Quantity B,nonzero values of x,1111 0. It follows that ; that is,x x2xx2Quantity B is greater than Quantity A. Thus the correct answer isChoice B.This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-8LP18 GRE Prep Materials[This footer should NOT be printed.] 10/22/2014Chapter 4 Algebra Ans/Explan - rev 02
This is a quantitative comparison question.3.Quantity AQuantity Bm 2525 - mQuantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-9LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationTo compare Quantity A m 25 with Quantity B 25 - m, first notethat because the absolute value of any quantity is greater than orequal to 0, it must be true that m 25 0 for all values of m. Butyou know that 25 - m is less than 0 when m is greater than 25. SoQuantity A is greater than Quantity B when m 25.On the other hand, both m 25 and 25 - m are equal to 25 whenm 0. So in one case Quantity A is greater than Quantity B, and inthe other case the two quantities are equal. Thus the correct answeris Choice D.This explanation uses the following strategies.Strategy 11: Divide into CasesStrategy 13: Determine whether a Conclusion Follows from theInformation Given-10LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
This is a quantitative comparison question.During an experiment, the pressure of a fixed mass of gasincreased from 40 pounds per square inch (psi) to 50 psi.Throughout the experiment, the pressure, P psi, and thevolume, V cubic inches, of the gas varied in such a waythat the value of the product PV was constant.Quantity BQuantity A4.The volume of the gaswhen the pressure was40 psi1.2 times the volume of thegas when the pressure was50 psiQuantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-11LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationYou are given that the relationship between the pressure andvolume of the gas throughout the experiment was PV C ,where C is a positive constant.Quantity A is the volume of the gas when the pressure was 40 psi.At this pressure, the equation PV C becomes 40V C. Solvingthis equation for V, you get V C 0.025C; that is, the volume40of the gas was 0.025C cubic inches.Quantity B is 1.2 times the volume of the gas when thepressure was 50 psi. At this pressure, the equation PV Cbecomes 50V C , or V C 0.02C. Therefore50(1.2)V (1.2)( 0.02C ) 0.024C ; that is, Quantity Bis 0.024C cubic inches. Thus the correct answer is Choice A.This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or AlgebraicRepresentationStrategy 8: Search for a Mathematical Relationship-12LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a quantitative comparison question.x 05.Quantity AQuantity Bx percent of 100xx2Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-13LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
ExplanationNote that Quantity A, x percent of 100 x, can be expressedalgebraically as( )x(100 x ). Simplifying this algebraic expression100gives x 2 , which is Quantity B. The correct answer is thereforeChoice C.This explanation uses the following strategy.Strategy 1: Translate from Words to an Arithmetic or AlgebraicRepresentation.-14LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a quantitative comparison question.(4 x - 2 y )(6 x 3 y ) 186.Quantity AQuantity B4x 2 - y 26Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-15LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
ExplanationThe given equation ( 4 x - 2 y )(6 x 3 y ) 18 can be simplified asfollows.Step 1: Note that(4 x - 2 y ) 2 (2 x - y ) and (6 x 3 y ) 3 (2 x y ) , so the givenequation can be rewritten as ( 2)( 2 x - y )(3)( 2 x y ) 18.Step 2: Dividing both sides of the rewritten equation by 6 gives(2 x - y )(2 x y ) 3.Step 3: Multiplying out the left side of the equation in Step 2 gives4 x 2 - y 2 3.Since Quantity A is 4 x 2 - y 2 , it follows that Quantity A is equalto 3. Since Quantity B is 6, the correct answer is Choice B.This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-16LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
This is a quantitative comparison question.Quantity AQuantity Bw dc z7.Quantity A is greater.Quantity B is greater.The two quantities are equal.The relationship cannot be determined from theinformation given.-17LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationThe line in the xy-plane passes through the origin and the point withcoordinates (6, 6) , so the equation of the line is y x. Since thepoint with coordinates (c, d ) is above the line, it follows thatd c, and since the point with coordinates ( w, z ) is below the line,it follows that w z. From the fact that d c and w z , you getw d c z ; that is, Quantity A is greater than Quantity B. Thusthe correct answer is Choice A.This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-18LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
This is a multiple-choice question. To answer this question, selectone answer choice.8. If xy 2 12 and xy 4, then x 1232343-19LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationFrom the given equations xy 2 12 and xy 4 , it follows that12 xy 2 ( xy ) y 4 y, and so y 3. Substituting y 3 in theequation xy 4 gives 3x 4 , or x is Choice E,4. Thus the correct answer34.3This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-20LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a multiple-choice question. To answer this question, selectone answer choice.9. The total cost of 8 bagels at a bakery is x dollars. At this costper bagel, which of the following represents the total cost, indollars, of y bagels?8xy8xy8yxxy8x8y-21LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationThe total cost of 8 bagels is x dollars, so one bagel costsTherefore the total cost of y bagels is ythe correct answer is Choice D,()xdollars.8xyx, ordollars. Thus88xy.8This explanation uses the following strategy.Strategy 1: Translate from words to an Arithmetic or AlgebraicRepresentation-22LP18 GRE Prep Materials[This footer should NOT be printed.] 12/17/2014Chapter 4 Algebra Ans/Explan rev04
This is a multiple-choice question. To answer this question, selectone answer choice.2x- y10. Which of the following is equal tofor all integers xx y2and y ?4- x4- y2 xy4x4y-23LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
Explanation2x- ySimplifying the fractionyieldsx y22x- y 2 x - y - ( x y ) 2 x - y - x - y 2 -2 y .2x yThe only answer choice with 2 as the base is Choice C, 2 xy , whichis clearly not the correct answer. Since all the other answer choiceshave 4 as the base, it is a good idea to rewrite 2 -2 y as anexpression with 4 as the base, as follows.2 -2 y 1 2y21( )y22 1 4- y4yThus the correct answer is Choice B, 4 - y.This explanation uses the following strategies.Strategy 5: Simplify an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship-24LP18 GRE Prep Materials[This footer should NOT be printed.] 12/17/2014Chapter 4 Algebra Ans/Explan rev04
This is a multiple-choice question. To answer this question, selectone answer choice.11. How many integers are in the solution set of the inequalityx 2 - 10 0 ?TwoFiveSixSevenTen-25LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationThe inequality x 2 - 10 0 is equivalent to x 2 10.By inspection, the positive integers that satisfy this inequalityare 1, 2, and 3. Note that 0 and the negative integers-1, -2, and -3 also satisfy the inequality, and there are no otherinteger solutions. So there are seven integers in the solution set:-3, -2, -1, 0, 1, 2, and 3. Thus the correct answer is Choice D,seven.This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship-26LP18 GRE Prep Materials[This footer should NOT be printed.] 01/15/2015Chapter 4 Algebra Ans/Explan - rev 05
This is a multiple-choice question. To answer this question, selectone answer choice.12. A group of 5,000 investors responded to a survey askingwhether they owned stocks and whether they owned bonds.Of the group, 20 percent responded that they owned only oneof the two types of investments. If r is the number of investorsin the group who owned stocks but not bonds, which of thefollowing represents the number of investors in the group whoowned bonds but not stocks, in terms of r ?5,000 - r1,000 - rr - 1,0001,000r(0.2)(5,000 - r )-27LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationTwenty percent of the 5,000 investors that responded to the surveysaid they owned either stocks or bonds, but not both. So the numberof investors in that group is (0.20)(5,000) , or 1,000. Given thatr members of that group owned stocks but not bonds, the numberof investors in that group who owned bonds but not stocks is1,000 - r. Thus the correct answer is Choice B, 1, 000 - r.This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or AlgebraicRepresentationStrategy 8: Search for a Mathematical Relationship-28LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a multiple-choice question. To answer this question, selectone answer choice.r sr s , which of the following statements must be4 5 4 5true?13. Ifr s5r 4 s5r -4 s25r 16 s25r -16 s-29LP18 GRE Prep Materials[This footer should NOT be printed.] 12/17/2014Chapter 4 Algebra Ans/Explan rev04
ExplanationYou can simplify each side of the equationas follows.Left-hand Side of Equation:r sr s ,4 5 4 5r sr s 4 59Right-hand Side of Equation:r s5r 4 s 4 520r sr s can be rewritten as4 5 4 5r s5r 4 s .920So the equationCross multiplying in the rewritten equation gives20 (r s ) 9 (5r 4 s ) , which simplifies to20r 20 s 45r 36 s, or -16 s 25r. Thus the correct answer isChoice E, 25r -16 s.This explanation uses the following strategy.Strategy 5: Simplify an Arithmetic or Algebraic Representation-30LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a multiple-choice question. To answer this question, selectall the answer choices that apply.14. In the xy-plane, triangular region R is bounded by thelines x 0, y 0, and 4x 3y 60. Which of the followingpoints lie inside region R ?Indicate all such points.(2, 18)(5, 12)(10, 7 )(12, 3)(15, 2)-31LP18 GRE Prep Materials[This footer should NOT be printed.] 12/17/2014Chapter 4 Algebra Ans/Explan rev04
ExplanationConsider the three lines that bound triangular region R. Theline x 0 is the y-axis, and the line y 0 is the x-axis. Theline 4 x 3 y 60 intersects the x-axis at (15, 0) and intersectsthe y-axis at (0, 20). The figure below shows region R.From the figure, you can see that all points inside region R havepositive coordinates and lie below the line 4x 3y 60. Notethat the equation 4x 3y 60 can be rewritten in the form4x. In this form, you can see that points inside region R34satisfy the inequality y 20 - x. Since all of the answer3y 20 --32LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
choices have positive coordinates, you need only to checkwhether the coordinates in each answer choice satisfy theinequality y 20 -4x, or equivalently 4 x 3 y 60.3Choice A, ( 2, 18) : 4 x 3 y 4 (2) 3 (18) 62 60.So Choice A is not in region R.Choice B, (5, 12) : 4 x 3 y 4 (5) 3 (12) 56 60.So Choice B is in region R.Choice C, (10, 7 ) : 4 x 3 y 4 (10) 3 (7 ) 61 60.So Choice C is not in region R.Choice D, (12, 3) : 4 x 3 y 4 (12) 3 (3) 57 60.So Choice D is in region R.Choice E, (15, 2) : 4 x 3 y 4 (15) 3 (2) 66 60.So Choice E is not in region R.-33LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
Thus the correct answer consists of choices B and D; that is,( 5, 12) and (12, 3) .This explanation uses the following strategies.Strategy 3: Translate from an Algebraic to a GraphicalRepresentationStrategy 8: Search for a Mathematical Relationship-34LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
This is a multiple-choice question. To answer this question, selectall the answer choices that apply.15. At the beginning of a trip, the tank of Diana’s car was filledwith gasoline to half of its capacity. During the trip, Dianaused 30 percent of the gasoline in the tank. At the end of thetrip, Diana added 8 gallons of gasoline to the tank. Thecapacity of the tank of Diana’s car was x gallons. Which ofthe following expressions represent the number of gallons ofgasoline in the tank after Diana added gasoline to the tank atthe end of the trip?-35LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
Indicate all such expressions.x 3x 82 207x 8203x 820x 3x -82 207x-820-36LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
ExplanationThe capacity of the car’s tank was x gallons of gasoline. Beforethe trip, the tank was half full and therefore containedxgallons of2gasoline. During the trip, 30 percent of the gasoline in the tankwas used, so the number of gallons left was 70 percent ofx, or2(107 )( 2x ) 720x . After Diana added 8 gallons of gasoline to the tank,the total number of gallons in the tank wascorrect choice is Choice B,7x 8. Thus one207x 8.20However, the question asks you to find all of the answer choicesthat represent the number of gallons of gasoline in the tank at theend of the trip. So you need to determine whether any of the other7xchoices are equivalent to 8. Of the answer choices, only20Choices A and C have the same constant term as Choice B: 8.So these are the only choices that need to be checked. Choice A,x 3x 8, can be simplified as follows.2 20x 3x10 x 3 x7x 8 8 82 20202020-37LP18 GRE Prep Materials[This footer should NOT be printed.] 12/08/2014Chapter 4 Alegbra Ans/Explan rev03
So Choice A is equivalent to7x3x 8. Choice C, 8, is clearly20207x 8. Thus the correct answer consists of20x 3x7x 8 and 8.choices A and B; that is, 2 2020not equivalent toThis explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or AlgebraicRepresentationStrategy 5: Simplify an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship-38LP18 GRE Prep Materials[This footer should NOT be printed.] 12/17/2014Chapter 4 Algebra Ans/Explan rev04
This is a numeric entry question. In this question you must enteryour answer as an integer or as a decimal.16. Machine A, working alone at its constant rate, producesx pounds of peanut butter in 12 minutes. Machine B, workingalone at its constant rate, produces x pounds of peanut butterin 18 minutes. How many minutes will it take machines A andB, working simultaneously at their respective constant rates, toproduce x pounds of peanut 8 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationTranslating the given information into an algebraic expression,xpounds of peanut butter in12xpounds of peanut butter in1 minute, and machine B produces18you see that machine A produces1 minute. Therefore, working simultaneously, machine A andmachine B producexx pounds of peanut butter in 1 minute.12 18-40LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
Letting t be the number of minutes it takes machines A and B,working simultaneously, to produce x pounds of peanut butter,you can set up the equation()xx t x.12 18Solving for t, you get()11 t 112 185t 136t 36 7.25The correct answer is 7.2 minutes.This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or AlgebraicRepresentationStrategy 8: Search for a Mathematical Relationship-41LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
This is a numeric entry question. In this question you must enteryour answer as an integer or as a decimal.17. The function f has the property that f ( x ) f ( x 1) for allnumbers x. If f ( 4) 17, what is the value of f (8) 123456789.0123456789.0123456789-42LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
ExplanationThe property that f ( x ) f ( x 1) for all numbers x implies thatf (4) f (5) , f (5) f (6) , f (6) f (7 ) , and f (7 ) f (8) .Therefore, since f ( 4) 17, it follows that f (8) 17. Thus thecorrect answer is 17.This explanation uses the following strategy.Strategy 7: Find a PatternThis is the end of Chapter 4 – Algebra Answer Key with Answersand Explanations.-43LP18 GRE Prep Materials[This footer should NOT be printed.] 09/08/2014Chapter 4 Algebra Ans/Explan - rev 01
in the Chapter 4 Algebra Practice Questions document. Following the answer key are the complete explanations for each question. If you wish to work through the questions before consulting the answers and explanations, please use the Chapter 4 Algebra Practice Questions document. Answer Key 1
