GRADUATE RECORD EXAMINATIONS®
Official GRE Quantitative Reasoning
Practice Questions, Volume 1
Chapter 4 – Algebra
Questions
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Chapter 4
Algebra
This chapter contains GRE Quantitative Reasoning practice questions that involve algebra.
Algebra topics include operations with exponents; factoring and simplifying algebraic expressions; relations, functions, equations, and inequalities; solving linear and quadratic equations and inequalities; solving simultaneous equations and inequalities; setting up equations to solve word problems; and coordinate geometry, including graphs of functions, equations, and inequalities, intercepts, and slopes of lines.
The questions are arranged by question type: Quantitative Comparison questions, followed by both types of Multiple-choice questions, and then Numeric Entry questions.
The answer key and the explanations for the answers for the Algebra Questions are in the separate document titled “GRE Chapter 4 Algebra Answers and Explanations.doc.”
Review the answers and explanations carefully, paying particular attention to explanations for questions that you answered incorrectly.
Before answering the practice questions, read the Quantitative Reasoning section directions that begin on the next page. Also, review the directions that precede each question type to make sure you understand how to answer the questions.
Quantitative Reasoning Section Directions
Directions: For each question, indicate the best answer, using the directions given.
Notes: All numbers used are real numbers.
All figures are assumed to lie in a plane unless otherwise indicated.
Geometric figures, such as lines, circles, triangles, and quadrilaterals, are not necessarily drawn to scale. That is, you should not assume that quantities such as lengths and angle measures are as they appear in a figure. You should assume, however, that lines shown as straight are actually straight, points on a line are in the order shown, and more generally, all geometric objects are in the relative positions shown. For questions with geometric figures, you should base your answers on geometric reasoning, not on estimating or comparing quantities from how they are drawn in the geometric figure.
Coordinate systems, such as x y-planes and number lines, are drawn to scale; therefore, you can read, estimate, or compare quantities in such figures from how they are drawn in the coordinate system.
Graphical data presentations, such as bar graphs, circle graphs, and line graphs, are drawn to scale; therefore, you can read, estimate, or compare data values from how they are drawn in the graphical data presentation.
Quantitative Comparison Questions
For Questions 1 to 7, compare Quantity A and Quantity B, using the additional information given, if any. Select one of the following four answer choices.
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
A symbol that appears more than once in a question has the same meaning throughout the question.
Example 1:
Quantity A: 2 times 6
Quantity B: 2+6
Solution: Quantity A is equal to 12 and Quantity B is equal to 8, so the correct answer for Example 1 is answer choice A, Quantity A is greater.
Example 2:
Refer to the figure.
Figure for Example 2
Begin skippable part of figure description.
The figure for example 2 shows triangle PQR, where P is the leftmost vertex of the horizontal base PR and vertex Q is above PR. Point S lies on horizontal base PR and appears to be the midpoint of PR. Line segment QS is drawn from vertex Q to point S. The lengths of PS and SR appear to be equal.
End skippable part of figure description.
Quantity A: The length of PS
Quantity B: The length of SR
Solution: The correct answer for Example 2 is answer choice D. The relationship between the lengths of PS and SR cannot be determined from the information given since equal measures cannot be assumed, even though the lengths of PS and SR appear to be equal in the figure.
Question 1.
It is given that the fraction with numerator x, times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x+3, close parenthesis, times open parenthesis, x minus 4, close parenthesis squared, end fraction, = 0.
Quantity A: x
Quantity B: negative 2
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 2.
It is given that x is greater than 0.
Quantity A: the fraction 1 over x.
Quantity B: The fraction with numerator x+1, and denominator x squared.
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 3.
Quantity A: the absolute value of m + 25, end absolute value
Quantity B: 25 minus m
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 4.
During an experiment, the pressure of a fixed mass of gas increased from 40 pounds per square inch (p.s.i) to 50 p.s.i. Throughout the experiment, the pressure, P p.s.i, and the volume, V cubic inches, of the gas varied in such a way that the value of the product PV was constant.
Quantity A: The volume of the gas when the pressure was 40 p.s.i
Quantity B: 1.2 times the volume of the gas when the pressure was 50 p.s.i
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 5.
It is given that x is greater than or equal to 0.
Quantity A: x percent of 100x
Quantity B: x squared
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 6.
It is given that open parenthesis 4 x minus 2 y close parenthesis, times, open parenthesis, 6 x + 3 y, close parenthesis, = 18.
Quantity A: 4, x squared, minus, y squared
Quantity B: 6
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Question 7.
Figure for Algebra Question 7
Begin skippable part of figure description.
The figure shows a line and three points in the first quadrant of the xy-plane. One of the points is on the line, one of the points is above the line, and one of the points is below the line.
The line starts at the origin, slants upward and to the right. The point on the line has coordinates 6 comma 6. The point above the line has coordinates c comma d, and the point below the line has coordinates w comma z.
End skippable part of figure description.
Quantity A: w+d
Quantity B: c+z
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.
Multiple-choice Questions—Select One Answer Choice
Question 8.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
If x, y squared = 12 and x y = 4, then x =
A. 1
B. 2
C. the positive square root of 3
D. 2 thirds
E. 4 thirds
Select and indicate the best one of the answer choices given.
Question 9.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
The total cost of 8 bagels at a bakery is x dollars. At this cost per bagel, which of the following represents the total cost, in dollars, of y bagels?
A. the fraction 8 over x y
B. the fraction 8x over y
C. the fraction 8y over x
D. the fraction x y over 8
E. the fraction x over 8y
Select and indicate the best one of the answer choices given.
Question 10.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
Which of the following is equal to the fraction with numerator 2 raised to the x minus y power, and denominator 2 raised to the x + y power for all integers x and y?
A. 4 raised to the negative x power
B. 4 raised to the negative y power
C. 2 raised to the x y power
D. 4 raised to the x power
E. 4 raised to the y power
Select and indicate the best one of the answer choices given.
Question 11.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
How many integers are in the solution set of the inequality x squared minus 10 is less than 0?
A. Two
B. Five
C. Six
D. Seven
E. Ten
Select and indicate the best one of the answer choices given.
Question 12.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
A group of 5,000 investors responded to a survey asking whether they owned stocks and whether they owned bonds. Of the group, 20 percent responded that they owned only one of the two types of investments. If r is the number of investors in the group who owned stocks but not bonds, which of the following represents the number of investors in the group who owned bonds but not stocks, in terms of r?
A. 5,000 minus r
B. 1,000 minus r
C. r minus 1,000
D. 1,000r
E. 0.2, times open parenthesis, 5,000 minus r, close parenthesis
Select and indicate the best one of the answer choices given.
Question 13.
This question has five answer choices, labeled A through E. Select the best one of the answer choices given.
If the fraction with numerator r + s and denominator 4 + 5 is equal to the fraction r over 4, + the fraction s over 5, which of the following statements must be true?
A. r=s
B. 5r=4s
C. 5r equals negative 4s
D. 25r=16s
E. 25r equals negative 16s
Select and indicate the best one of the answer choices given.
Multiple-choice Questions—Select One or More Answer Choices
Question 14.
This question has five answer choices, labeled A through E. Select all the answer choices that apply.
In the xy-plane, triangular region R is bounded by the lines x=0, y=0, and 4x+3y=60. Which of the following points lie inside region R ?
Indicate all such points.
A. the point with coordinates 2 comma 18
B. the point with coordinates 5 comma 12
C. the point with coordinates 10 comma 7
D. the point with coordinates 12 comma 3
E. the point with coordinates 15 comma 2
Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one or as many as all five of the answer choices.
Question 15.
This question has five answer choices, labeled A through E. Select all the answer choices that apply.
At the beginning of a trip, the tank of Diana’s car was filled with gasoline to half of its capacity. During the trip, Diana used 30 percent of the gasoline in the tank. At the end of the trip, Diana added 8 gallons of gasoline to the tank. The capacity of the tank of Diana’s car was x gallons. Which of the following expressions represent the number of gallons of gasoline in the tank after Diana added gasoline to the tank at the end of the trip?
Indicate all such expressions.
A. the fraction x over 2, end fraction, minus, the fraction 3x over 20, end fraction,+ 8
B. the fraction 7x over 20, end fraction + 8
C. the fraction 3x over 20, end fraction + 8
D. the fraction x over 2, end fraction, + the fraction 3x over 20, end fraction, minus 8
E. the fraction 7x over 2, end fraction, minus 8
Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one or as many as all five of the answer choices.