GRE® MATH WORKBOOK
ELEVENTH EDITION


Table of Contents
GRE Math Workbook
Cover
Title Page
Copyright
Acknowledgments
How to Use This Book
Related Titles
Part One: Getting Started
Chapter 1: Introduction to GRE Math
Understanding the GRE Quantitative Reasoning Sections
MST Mechanics
Question Types
Quantitative Comparison
Problem Solving
Data Interpretation
Part Two: Quantitative Reasoning
Chapter 2: Quantitative Comparison
Quantitative Comparison
Quantitative Comparison Questions
Quantitative Comparison Practice Set
Quantitative Comparison Practice Set: Answer Key
Quantitative Comparison Practice Set: Answers and
Explanations
Chapter 3: Problem Solving

Problem Solving Questions (Single Answer)
Problem Solving (All-That-Apply)
Problem Solving Questions (Numeric Entry)
Problem Solving Practice Set
Problem Solving Practice Set: Answer Key
Problem Solving Practice Set: Answers and Explanations
Chapter 4: Data Interpretation
Data Interpretation Questions (One Answer)
Data Interpretation Questions (All-That-Apply)


Data Interpretation Questions (Numeric Entry)
Data Interpretation Practice Set
Data Interpretation Practice Set: Answer Key
Data Interpretation Practice Set: Answers and Explanations
Chapter 5: Quantitative Reasoning Practice
Quantitative Reasoning Practice Set 1
Quantitative Reasoning Practice Set 1 Answer Key
Quantitative Reasoning Practice Set 1 Answers and
Explanations
Quantitative Reasoning Practice Set 1 Diagnostic Tool
Quantitative Reasoning Practice Set 2
Quantitative Reasoning Practice Set 2 Answer Key
Quantitative Reasoning Practice Set 2 Answers and
Explanations
Quantitative Reasoning Practice Set 2 Diagnostic Tool
Quantitative Reasoning Practice Set 3
Quantitative Reasoning Practice Set 3 Answer Key
Quantitative Reasoning Practice Set 3 Answers and
Explanations

Quantitative Reasoning Practice Set 3 Diagnostic Tool
Quantitative Reasoning Practice Set 4
Quantitative Reasoning Practice Set 4 Answer Key
Quantitative Reasoning Practice Set 4 Answers and
Explanations
Quantitative Reasoning Practice Set 4 Diagnostic Tool
Quantitative Reasoning Practice Set 5
Quantitative Reasoning Practice Set 5 Answer Key
Quantitative Reasoning Practice Set 5 Answers and
Explanations
Quantitative Reasoning Practice Set 5 Diagnostic Tool
Quantitative Reasoning Practice Set 6
Quantitative Reasoning Practice Set 6 Answer
KeyQuantitative Reasoning Practice Set 6 Answer Key
Quantitative Reasoning Practice Set 6 Answers and
Explanations
Quantitative Reasoning Practice Set 6 Diagnostic Tool
Part Three: Math Content Review


Chapter 6: Arithmetic
Arithmetic
Real Numbers
Number Operations
Number Operations Exercises
Number Operations Exercises Answer Key
Number Properties
Number Properties Exercises
Number Properties Exercises Answer Key
Averages (Arithmetic Means)

Averages (Arithmetic Means) Exercises
Averages (Arithmetic Means) Exercises Answer Key
Ratios
Ratios Exercises
Ratios Exercises Answer Key
Percents
Percents Exercises
Percents Exercises Answer Key
Powers and Roots
Powers and Roots Exercises
Powers and Roots Exercises Answer Key
Chapter 7: Algebra
Understanding Algebra
Operations with Algebraic Expressions
Operations with Algebraic Expressions Exercises
Operations with Algebraic Expressions Exercises Answer Key
Rules of Exponents
Rules of Exponents Exercises
Rules of Exponents Exercises Answer Key
Solving Linear Equations
Solving Linear Equations Exercises
Solving Linear Equations Exercises Answer Key
Quadratic Equations
Quadratic Equations Exercises
Quadratic Equations Exercises Answer Key
Solving Inequalities
Solving Inequalities Exercises


Solving Inequalities Exercises Answer Key

Translation into Algebra
Translation into Algebra Exercises
Translation into Algebra Exercises Answer Key
Functions
Functions Exercises
Functions Exercises Answer Key
Coordinate Geometry
Coordinate Geometry Exercises
Coordinate Geometry Exercises Answer Key
Graphs of Functions
Graphs of Functions Exercises
Graphs of Functions Exercises Answer Key
Chapter 8: Geometry
Understanding Geometry
Lines and Angles
Lines and Angles Exercises
Lines and Angles Exercises Answer Key
Triangles and Pythagorean Theorem
Triangles and Pythagorean Theorem Exercises
Triangles and Pythagorean Theorem Exercises Answer Key
Polygons
Polygons Exercises
Polygons Exercises Answer Key
Circles
Circles Exercises
Circles Exercises Answer Key
Multiple Figures
Multiple Figures Exercises
Multiple Figures Exercises Answer Key
Three-Dimensional Figures (Uniform Solids)

Three-Dimensional Figures (Uniform Solids) Exercises
Three-Dimensional Figures (Uniform Solids) Exercises
Answer Key
Chapter 9: Data Interpretation
Understanding Data Analysis
Counting Methods


Counting Methods Exercises
Counting Methods Exercises Answer Key
Descriptive Statistics
Descriptive Statistics Exercises
Descriptive Statistics Exercises Answer Key
Data Graphs and Tables
Data Graphs and Tables Exercises
Data Graphs and Tables Exercises Answer Key
Probability
Probability Exercises
Probability Exercises Answer Key
Part Four: Advanced Math Practice
Chapter 10: High-Difficulty Question Sets
Introduction to High-Difficulty Question Sets
Advanced Math Questions Practice Set 1
Advanced Math Questions Practice Set 1 Answer Key
Advanced Math Questions Practice Set 1 Answers and
Explanations
Advanced Math Questions Practice Set 2
Advanced Math Questions Practice Set 2 Answer Key
Advanced Math Questions Practice Set 2 Answers and
Explanations

Advanced Math Questions Practice Set 3
Advanced Math Questions Practice Set 3 Answer Key
Advanced Math Questions Practice Set 3 Answers and
Explanations
Advanced Math Questions Practice Set 4
Advanced Math Questions Practice Set 4 Answer Key
Advanced Math Questions Practice Set 4 Answers and
Explanations
Part Five: GRE Resources
Appendix: Math Reference


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© 2019 Kaplan, Inc.
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Acknowledgments
Special thanks to the team that made this book possible:
Arthur Ahn, Matthew Belinkie, Shannon Berning, Lauren T. Bernstein, Kim
Bowers, Gerard Cortinez, Elisa Davis, Lola Disparte, Boris Dvorkin, John
Evans, Paula Fleming, Darcy Galane, Joanna Graham, Adam Grey, Allison
Harm, Jack Hayes, Adam Hinz, Gar Hong, Sunny Hwang, Cinzia Iacono, Avi
Lidgi, Kate Lopaze, Keith Lubeley, TJ Mancini, Jennifer Moore, Jason Moss,
Walt Niedner, Robert Reiss, S. Ross, Derek Rusnak, Emily Sachar, Stephanie
Schrauth, Sheryl Stebbins, Glen Stohr, Sascha Strelka, Gene Suhir, Martha
Torres, Liza Weale, Lee A. Weiss, and many others who have contributed
materials and advice over the years.


How to Use This Book
Kaplan has prepared students to take standardized tests for more than 75
years. Our team of teachers and researchers knows more about
preparation for the GRE than anyone else, and you’ll find Kaplan’s
accumulated knowledge and experience throughout this book. The GRE is
a standardized test, so every test covers the same content in roughly the
same way. This is good news for you; it means that the best way to prepare
is to focus on the sort of questions you are likely to see on Test Day. The
main focus of this book is on strategic reviews, exercises, and practice sets
with explanations that will help you brush up on any math skills you may
have forgotten. There is a lot of math to absorb, and it’s hard to do it all at
once. Instead, plan out a study schedule and work through this book over
the course of several weeks.

GETTING STARTED

In Part 1 of this book, “Getting Started,” we’ll provide you with background
information on the Quantitative Reasoning section of the test, what it
covers, and how it’s organized.

QUANTITATIVE REASONING SECTION
The Quantitative Reasoning section of the GRE contains three main
question types: Quantitative Comparison, Problem Solving, and Data


Interpretation. Part 2 of this book covers these types with strategies and
sample questions. Your focus here should be to familiarize yourself with
the question types so you won’t be trying to figure out how to approach
them on Test Day. And practice, practice, practice!

MATH CONTENT REVIEW
Once you have the big picture, focus on the content. Part 3 of this book,
“Math Content Review,” gives you a complete tour of the math that you
will see on Test Day. The material in the math content review is divided
into particular subjects. Each subject begins with a review, followed by
practice exercises organized by level of difficulty. This structure makes it
easy for you to pinpoint the math concepts you need to review and quickly
get your skills up to speed.
We suggest that you read the content review that introduces a section and
then try some of the exercises. After checking your answers to those
exercises, decide whether you need to spend more time with the content
review before completing the sets of exercises. Keep track of your progress
as you proceed with the content review. Not only will this build your skills,
it will increase your confidence leading up to Test Day.
If you find that you would like access to more of Kaplan’s practice tests
and quizzes, as well as in-depth instruction on the question types and

strategies, look into the variety of course options available at
kaptest.com/GRE .


If you have questions about what you’re studying, ask our expert GRE
faculty on our Facebook page: www.facebook.com/KaplanGradPrep

.

Thanks for choosing Kaplan. We wish you the best of luck on your journey
to graduate school.


For more GRE® prep, Kaplan offers a range of print and digital products,
available in stores and online, including:
Kaplan GRE® Prep Plus with 3 Practice Tests and 500-Question Qbank
Kaplan GRE® Prep with 2 Practice Tests
GRE® Verbal Workbook
Kaplan GRE® Vocabulary Flashcards
To learn more about Kaplan’s comprehensive prep courses for the GRE®,
please visit kaptest.com/GRE .


Part One

GETTING STARTED


Chapter 1


Introduction to GRE Math
Understanding the GRE Quantitative
Reasoning Sections
The Quantitative Reasoning sections of the GRE test your ability to reason
quantitatively—to read a math problem, understand what it is asking, and
solve it. The basic math disciplines covered on the GRE are arithmetic,
algebra, geometry, and data interpretation. In this way, the GRE has topics
similar to those covered on the SAT. There is no trigonometry or calculus
on the GRE. The aim of the test is to provide an accurate indication of your
ability to use your knowledge of the fundamental topics and apply
reasoning skills to the various question types. The goal is to make the test
a true indicator of your ability to apply given information, think logically,
and draw conclusions. These are skills you will need at the graduate level
of study.
The GRE contains two math sections, each having 20 questions. The
allotted time for each section is 35 minutes. To perform well on the
Quantitative Reasoning sections, you want to perform the fundamental


math skills efficiently and apply your reasoning skills at the same time to
answer as many questions correctly as possible. The various question
types will be explained in detail in part 2 of this book. The foundations of
arithmetic, algebra, geometry, and data interpretation will be reviewed in
part 3 of this book. In every chapter, there will be plenty of opportunities to
practice and check your answers.


MST Mechanics
The GRE is a multi-stage test (MST). While working within a section of the
test during the time allotted, you are allowed to skip questions and return

to them as long as time remains for the section. The test is computerbased with built-in capabilities such as the “Mark” button to indicate a
question you want to examine later (within the time allowed for that
section) and the “Review” button to see your progress on the entire set of
questions in a section, an optional time display, and an on-screen
calculator. As you prepare for Test Day, always consider how you plan to
manage your time for each section and how these computer capabilities
can help you.
On the GRE, every question has the same value. So you do not want to
waste time dwelling on one question while you may be able to answer
several others correctly. This is one way the Mark and Review buttons can
assist you in managing your time. You can turn the time display on or off as
you wish. Some test takers will find it helpful; others will find it annoying.
The availability of an on-screen calculator is something you will want to
consider as you practice for the test. Questions in the Quantitative
Reasoning sections are not designed to be calculator-intensive. If you find
yourself doing a long calculation to answer a question, you may be missing
the reasoning aspect completely. That means most questions on the GRE
are written to be answered without a calculator. You will want to reserve


the calculator for an isolated calculation or a quick check of an answer, but
not use it as your main resource for the questions. To help you plan for the
best use of the on-screen calculator, an image of it is shown below. Notice
that it is a simple, four-function calculator with a square root key and
change of sign key.

As you review mathematical foundations and practice answering the
various question types in this workbook, use your own calculator
strategically. This will lead you to rely on your critical thinking skills.



Question Types
A Quantitative Reasoning section with 20 questions will be composed of
three question types:
Quantitative Comparison
Problem Solving
Data Interpretation
Each question type will require you to draw upon your ability to combine
your knowledge of mathematical concepts with your reasoning skills in a
particular way. However, the question types are not distributed evenly
within a section. The chart below shows how many questions you can
expect of each type, as well as the average amount of time you should
spend on each question type.

Number of

Quantitative

Problem

Data

Comparison

Solving

Interpretation

Approx. 7–8


Approx. 9–10

Approx. 3

1.5 minutes

1.5–2 minutes

2 minutes

Questions
Time per Question


Quantitative Comparison
A Quantitative Comparison question will not ask you to identify a
particular value as the answer, but it will ask you to make a comparison
between two quantities and identify a relationship between them. The four
answer choices for a Quantitative Comparison are always the same—same
wording, same order. Because the answer choices are consistent, you will
become familiar with them quickly and be able to concentrate on the
comparison at hand.
The Kaplan Method for answering Quantitative Comparison questions and
strategies to help solve them efficiently can be found in chapter 2 .


Problem Solving
A Problem Solving question may appear as a math problem, or it may
appear as a word problem dealing with a real-world situation. These
questions deal with percents, ratios,

and other proportions; linear
and quadratic equations; basic probability and statistics; and two- and
three-dimensional geometry.
The Kaplan Method for answering Problem Solving questions and
strategies to help solve them efficiently can be found in chapter 3

.


Data Interpretation
Questions that require you to interpret data will be based on information
located in graphs or tables. You will see a set of three or four questions
based on each data presentation. Frequently, you will have to use
information from more than one data source to answer a question. Data
Interpretation questions need to be answered thoughtfully, only after you
have taken time to analyze the contents of each graph or table of data.
The Kaplan Method for answering Data Interpretation questions and
strategies to help solve them efficiently can be found in chapter 4 .