GED Math Test Tutor, For the 2024-2025 GED Test: Certified GED Aligned Prep

2nd Edition

Ged® Math

Test Tutor

Research & Education Association

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Garden City, NY 11530

Email: [email protected]

GED® Math Test Tutor, 2nd Edition

Published 2024

Copyright © 2016 by LSC Communications Book LLC d/b/a Research & Education Association, a division of Lakeside Book Company. Prior edition copyright © 2014 by Research & Education Association. All rights reserved. No part of this book may be reproduced in any form without permission of the publisher.

Library of Congress Control Number 2015959982

ISBN-13: 978-0-7386-1292-8

ISBN-10: 0-7386-1292-8

The content of this eBook is also contained in the print version of this title, GED Math Test Tutor (Second Edition), cataloged under ISBN-13: 978-0-7386-1210-2 and ISBN-10: 0-7386-1210-3.

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Contents

About the Author

Author Acknowledgments

Author Dedication

About REA

REA Acknowledgments

What Our Award-Winning Content Means to You

Introduction: About the GED® Mathematical Reasoning Test

How This Book Can Help

What to Expect on the GED® Mathematical Reasoning Test

GED® Performance Levels

Types of Questions

On-Screen and Other Test-Day Resources

Supplementary Prep Resources

At-Home Testing Option

International Testing

Calculator Instructions

GED® Mathematical Reasoning Diagnostic Pretest

Answer Sheet

Answer Key

Performance Check

Detailed Solutions

Chapter 1: I Was Born Knowing That

The Number Line

Addition and Subtraction

Multiplication

Division

Exercises

Solutions

Chapter 2: The Parts of the Whole

Decimals

Fractions

Ratios and Proportions

Percentages

The Relations Among Decimals, Fractions, and Percentages

Real-World Situations

Exercises

Solutions

Chapter 3: Power Play

Powers

Roots

Order of Operations

Scientific Notation

Exercises

Solutions

Chapter 4: Algebra—Not a Four-letter Word

Setting up Equations

Working with Algebraic Expressions

Evaluating Algebraic Expressions

Solving Linear Equations

Simultaneous Equations

Inequalities

Exercises

Solutions

Chapter 5: The X (Squared) Factor

Polynomials

Quadratic Equations

Functions

Exercises

Solutions

Chapter 6: The Shape of Things

Lines and Angles

Triangles

Quadrilaterals

Polygons

Circles

Composite Two-Dimensional Figures

Three-Dimensional Figures

Composite Three-Dimensional Figures

Exercises

Solutions

Chapter 7: What Are the Chances?

Counting

Permutations and Combinations

Probability

Data

Central Tendency

Spread of Data

Statistics

Exercises

Solutions

GED® Mathematical Reasoning Practice Test 1

Answer Sheet

Answer Key

Detailed Solutions

GED® Mathematical Reasoning Practice Test 2

Answer Sheet

Answer Key

Detailed Solutions

Glossary

Appendix

Practice Test Performance Checks

GED® Mathematics Formula Sheet

Index

About the Author

Sandra Rush earned a Bachelor of Arts degree in mathematics from Temple University in Philadelphia. While an undergraduate student, she served as a math tutor for members of the basketball team as well as for local public school students. In these early years of her own higher education, she realized that teaching mathematics had a special appeal.

When awarded a research assistantship and later a Ford Foundation Fellowship for an advanced degree in the Department of Atmospheric Sciences at the University of California, Los Angeles, Sandra pursued graduate school and earned a Master of Arts degree in ionospheric physics. Upon graduation, however, she turned down offers to become a physicist in favor of a career as a mathematics instructor at the secondary school level. She quickly became chair of the mathematics department at Notre Dame Academy in Massachusetts.

At about that same time, she tutored the first of several young adults pursuing a high school equivalency certificate. That student passed the math portion of the test on her first attempt, and with the certificate she was able to advance in her career. Sandra was hooked on teaching. She has taught mathematics and physics from the junior high school to university levels in three states. Sandra has also continued her side career of tutoring and coaching students at all levels. Her one-on-one approach with young adults has yielded excellent results.

Sandra’s interest in education has extended beyond the classroom to the field of publishing, including writing and editing manuals on preparation for standardized tests at all levels. This book is the latest of her efforts to make mathematics both understandable and enjoyable for generations of students.

Author Acknowledgments

I think that people read acknowledgments just to see whether their names are mentioned (and spelled correctly). Authors write acknowledgments to give thanks to people who have been significant in encouraging their efforts. My list begins with my children, Mike and Kara, who have always supported me in my endeavors, no matter how bizarre they sometimes seemed. My twin sister Sue and friends Joann, Carol, Dave, and Herb have provided much-appreciated day-to-day support and love. My longtime friends from Humanities have influenced me and given me confidence in my abilities. Thank you all, even those who touched my life but whose names aren’t specifically mentioned here.

Author Dedication

This book is dedicated in memory of two of my high school teachers, Erasmo S. Ciccolella, who taught me how to write, and Frederick W. Drewes, who encouraged my love of math. What a profound and lasting influence great teachers have on their students.

About REA

Founded in 1959, Research & Education Association (REA) is dedicated to publishing the finest and most effective educational materials—including study guides and test preps—for students of all ages.

Today, REA’s wide-ranging catalog is a leading resource for students, teachers, and other professionals. Visit www.rea.com to see our complete catalog.

REA Acknowledgments

We would like to thank Leanne Fucci, M.S., for her technical edit. Ms. Fucci has taught Adult Basic Education and been a GED® instructor for two decades. We also extend gratitude to veteran math instructors Steve Reiss and Stu Schwartz for their review and feedback.

In addition, we would like to acknowledge the team that brought this book to market: Pam Weston, Publisher, for setting the quality standards for production integrity and managing the publication to completion; Larry B. Kling, Editorial Director, for coordinating development and updates of this edition; Diane Goldschmidt, Managing Editor, for proofreading and continuity check; Caragraphics for typesetting this edition; Jennifer Calhoun for file prep; and Amnet ContentSource for accessible ePUB3 conversion.

REA gratefully acknowledges Texas Instruments’ permission to use an image of the TI-30XS MultiView™ calculator.

What Our Award-Winning Content Means to You

REA has made every effort to ensure this test-prep product is fully aligned to the GED® test itself. REA is an Official GED® Publisher Partner and we have been honored with the ProCert Excellence in Education Award for Best-in-Class GED® Prep. The award comes from Utah-based ProCert Labs, which develops and applies quality assurance standards to print and digital educational and training products.

This book, the GED® Math Test Tutor, and our companion book, GED® Total Solution, were both prepared under the supervision of GED Testing Service®, and are guaranteed to cover 100% of the test objectives—fully and accurately.

Drop us a line at [email protected] to let us know how you do on your test!

Introduction

About the GED® Mathematical Reasoning Test

This introductory chapter is as important as any of the subject review chapters that follow. The subject review chapters will show you what you need to know about mathematics for the GED® test, but if you don’t know how to take the test and what to expect, you may not be successful, even if you understand the math. To be successful with the GED® Mathematical Reasoning test, it is essential that you know three skill sets: math skills, calculator skills, and some computer skills.

How This Book Can Help

For all three skill sets, practice, practice, practice will pay off. This book helps with the mathematical reasoning skills, so read the book carefully and be sure you understand each section before you advance to the next. Numerous examples will give you the practice and confidence you need to do well. To paraphrase GED Testing Service®, it’s not just about knowing how, but rather why!

Remembering isn’t the same as memorizing. This book is designed to help you remember facts that you need without having to memorize. To help you get off to a running start, the book features a diagnostic pretest up front. Test-takers often overestimate how prepared they are for a test, so we suggest taking the diagnostic pretest before you delve into the review. Your performance will help you pinpoint what you know and what you don’t so you can spend time on the areas that need more attention. Then move on to the subject review, where you’ll find everything you need to excel on the GED® test covered in strict alignment with GED® standards. In the review, each mathematical topic is introduced, accompanied by several example questions with answers and explanations that are designed as further teaching aids to help you remember what has been presented. Exercises at the end of each chapter allow you to assess how well you understood the concepts introduced in the chapter. These exercises are presented in GED® test formats so you can get used to how questions will be presented on the actual test.

You probably have heard the saying, We learn from our mistakes. If you do make a mistake on any of the exercises, use that as a further learning experience and don’t be discouraged. If you ever locked the key in your car, that was a learning experience (a frustrating one, for sure) and, at least for a while after that, you always made sure you had the key in your hand when you locked the door. So if you choose the wrong answer to an exercise in the book, go to the explanation of the correct answer and learn from it so you won’t make that type of mistake again.

The final part of the book contains two full-length practice GED® Mathematical Reasoning tests followed by the correct answer choices with explanations. Time yourself (115 minutes) to get a sense of how quickly you must come up with answers. You can go back to any question at any time and you can change your answers if you want to, but you need to do it all within the 115-minute time frame. Admittedly, taking the test on paper is different from taking the test on a computer, but the goal here is to learn the math and strategy to answer the problems correctly.

What to Expect on the GED® Math Test

Today’s GED® Mathematical Reasoning test emphasizes understanding and logical thinking over rote memorization, and it is based on real-world problem-solving skills. This next-generation test expects you to understand what a problem is asking for and proceed to answer it based on the information given. Use the following four-part plan for reasoning skills to answer the questions:

Figure out what is known, what is necessary to solve the problem, what information is missing (usually your solution), and what is unnecessary (just because something is mentioned doesn’t mean it has anything to do with your solution).

Devise a strategy to solve the problem. This may involve making a sketch or a table (use the erasable note board you will be given at the testing center) or looking for a pattern, for example.

Solve the problem according to your strategy and choose or enter the correct answer.

Make sure the answer makes sense.

As an example of the reasoning aspect of today’s GED® test, rather than asking 10 − (5 + 2.21) = ? as might have appeared on previous GED® math tests, the current version asks, If Liz buys two items at the store that cost $5.00 and $2.21, how much change will she receive from a $10 bill? It is the same arithmetic, but the test-taker has to know how to set up that arithmetic. The GED® Mathematical Reasoning test measures how well you can figure out what is asked for and then apply mathematical skills, whether from your own memory bank or by using the tools that are available on the test, to come up with answers. This book shows you how to do this.

The problems on the GED® test mimic the real world, which is why it is computer-based. Almost everyone is carrying around a calculator—it’s in the form of a cellphone. Almost no one multiplies 63 × 24 on paper, much less in their heads—they all reach for the calculator. Many store clerks don’t count out change anymore, they just give the customer what the register (also a calculator) tells them to. In addition, with so much knowledge available via the Internet, memorization of many formulas is no longer necessary because people can look them up and avoid misremembering.

The total test time is 115 minutes, but that includes 2 minutes for an introduction and wrap-up, so the actual testing time is only 113 minutes.

The math test has a maximum of 49 raw score points. The value of each question varies with the difficulty of the question. Therefore, the number of questions on the test is not 49. In fact, it is 46, as seen in the practice tests at the end of this book. GED Testing Service® converts total raw score points into a final score.

Questions are presented one at a time on the computer screen. One chart, table, or problem scenario may have several questions associated with it. The computer uses a split screen for this type of question. On a split screen, the graphic or scenario stays on one half of the screen, and the questions appear one by one on the other half. Fortunately, there is a Previous button if any of these questions depend at all on a response to a prior question.

Fifty-five percent of the questions involve algebra, but that doesn’t mean you are just given an equation and asked to solve it. Questions on any of the math subjects on the test (each of which is covered in a chapter of this book) may involve algebraic reasoning. The other 45 percent of the questions are computational.

Your score is based strictly on the number of correct answers, with no penalty deducted for wrong answers or bad guesses. Therefore, you should answer every question. Once you have selected an answer to a question, take an extra second or two to make sure it makes sense. As an obvious example, the change you receive from a purchase shouldn’t be more than the money you gave the clerk, or the discounted cost of an item shouldn’t be more than the original price.

GED® Performance Levels

Your final score is reported on a scale that ranges from 100 to 200 on each of the four content-area tests. The passing standard for each of the four GED® test modules is 145. At the end of 2015, GED Testing Service® revised its Performance Levels on the GED® test. These levels were set following extensive analysis of candidate performance in the first year of the latest version of the test. The current Performance Levels are as follows:

Performance Level 1: Below Passing (100–144 scaled score points)

Performance Level 2: Pass / High School Equivalency (145–164 scaled score points)

Performance Level 3: GED® College Ready (165–174 scaled score points)

Performance Level 4: GED® College Ready + Credit (175–200 scaled score points)

Test-takers scoring in Performance Levels 3 or 4, in any content area, can obtain an ACE Credit® transcript.

For scores in Performance Level 3 (GED® College Ready; 165–174 scaled score points), the ACE Credit® transcript documents your readiness for placement in credit-bearing courses in postsecondary education as well as recommended exemption from placement testing.

For Mathematical Reasoning test scores in Performance Level 4 (GED® College Ready + Credit; 175–200 scaled score points), the ACE Credit® transcript provides a recommendation that the institution you plan on attending grant 3 undergraduate semester hours in College Algebra.

Types of Questions

The GED® Mathematical Reasoning test has five types of questions: multiple-choice, drop-down, fill-in-the-blank, hot-spot (also called select-an-area), and drag-and-drop.

Multiple-Choice Items

Traditional multiple-choice questions are a major part of the test. These questions have four answers from which to choose, designated A, B, C, and D. You don’t necessarily have to come up with the correct answer from scratch—you only have to be able to identify the correct answer among the four choices. You should be able to eliminate one or more of the answer choices without any calculation if they are obviously wrong (for example, the problem asks for a whole number and one answer choice is a fraction or decimal), or inappropriate (for example, the problem indicates that the answer will be in the set {1, 2, 3, 4, 5} and the answer choice is 6). To choose an answer, click next to the letter for that choice.

Drop-Down Items

Drop-down items are a variation of multiple-choice questions. These are responses that are embedded directly in the text, and you click on one of three to five given choices to make the statement true. Here’s a simplified example: The number 3 is {  less than  equal to  greater than  } the value of 2(5) – 7.

Fill-in-the-Blank Items

Fill-in-the-blank questions involve problems for which you type your answer in a box. It can be the value you got from doing a specific calculation, a one-word or one-phrase answer to a question, or an equation you would use to solve a problem. Fill-in-the-blank questions may have more than one blank to fill in. These questions are similar to multiple-choice questions except that you aren’t given any choices for your answer. You must come up with the answer on your own and type it in a box that is part of the question. Don’t be concerned about entering extra spaces as you type your answer because the extra spaces are automatically deleted when the test is scored.

A symbol selector sheet is provided on-screen so that you can use symbols in your answer. Each question of this type will state which symbols may be used in your answer. See the section on the symbol selector tool later in this introduction.

Hot-Spot (aka Select-an-Area) Items

Hot-spot items require you to move the cursor exactly where you want it. Such questions are also called select-an-area items. That’s because they have you use your mouse to insert a point or select an area on a coordinate grid, number line, dot plot, or an edge on a geometric figure. On the test, a hot spot will appear as a red ‘X’ or a blue dot (•), whereas a hot area will appear as a highlighted section of the response area. If you change your mind about the placement of a point, clicking that location again will delete it and let you insert a new point. Occasionally, you will be asked to plot, say, three points. If you try to insert a fourth one, you will be alerted that you have already inserted the requested number of points. You can then delete one of the other points to replace it with the new point.

Drag-and-Drop Items

Likewise, drag-and-drop items require computer dexterity in moving the cursor to drag small images on the screen, such as words or parts of numerical expressions (e.g., numbers, operators, or variables), and drop them into boxes where they make sense. Typically, you are given some choices to drag into boxes. For example, you may write an inequality by dragging 5 items from a choice of 8 items: x, +, 7y, >, and 9, in that order, to write x + 7y > 9. The test format allows you to replace any choice you have made by simply dragging a new item onto where the old item was. Then the old item goes back into the area of choices and the new item you selected goes into the proper box in the question.

On-Screen and Other Test-Day Resources

The test provides many helpful resources (virtual calculator, calculator reference sheet, a formula sheet, and erasable note boards that are to be used as scratchpads—even a time clock so you can pace yourself). It is essential that you be familiar with each of these so you can easily use them during the test. They will cut down on your test-taking time as well as help with the accuracy of your answers.

The two choices at the lower right corner of each screen are important to bear in mind. One is Next, which you have to click to go to the next question. The other is Previous, which comes in handy when all of a sudden you remember how to do a former question, or when you need the results from the last question to answer the present question. This may happen in a split-screen situation, where a graphic or table, for example, is displayed on the left side of the screen, and two or more questions about the same visual are asked on the right side of the screen.

Calculator

The GED® test is completely computer-based. Therefore, just as important as sharpening your mathematical reasoning skills is sharpening your computer and calculator skills. Calculator pages following this introduction give step-by-step instructions for using the virtual TI-30XS calculator that is supplied on the test as a drop-down item on 90% of the questions. Note that these calculator instructions (on pages 11–25) are also provided in the chapters for which a certain calculation is relevant.

The virtual calculator will help you do the computations, so you won’t have to worry about making mathematical errors—the calculator does most of the computations for you. Your main focus should be on understanding what is being asked. For that reason, this book presents the skills needed to do a problem and then lots of word problems so you can improve your mathematical reasoning skills. The actual computations are the easier parts of the problems.

Also available in your drop-down menu during the test is a calculator reference sheet; this is a chart that summarizes how to use the virtual calculator. Even though you will have this at hand, you should become familiar with a calculator before the test to avoid spending time during the test learning what you should already know. The step-by-step instructions in this book are more detailed than what’s on the calculator reference sheet.

Formula Sheet

A formula sheet (see appendix at the back of this book) is a drop-down item on the GED® test. It provides essential information necessary for answering problems. This formula sheet provides:

The areas of a parallelogram and trapezoid

The surface area and volume of various three-dimensional figures

The slope of a line and the slope-intercept form and point-slope form of the equation of a line

The standard form of a quadratic equation as well as the quadratic formula

The Pythagorean Theorem

The formula for simple interest

Throughout this book, whenever one of these formulas is used, the text tells whether it is on the formula sheet. You are expected to know certain formulas that are not on the sheet. These are also identified in this book when they are used. The other foundational formulas that you are expected to know (and therefore are not included on the formula sheet) are the following:

The total cost formulas that take into account markups and markdowns (see Chapter 2)

The distance formula, d = rt (see Chapter 4)

Certain geometric formulas for perimeters, circumference, and areas of two-dimensional figures (see the table in Chapter 6)

The measures of central tendency, such as the mean, median, and mode (see Chapter 7)

Symbol Selector Tool

For fill-in-the-blank questions, you may have to use symbols that are not readily available on the keyboard. To facilitate using symbols, the GED® test provides a symbol selector tool as a drop-down item for those questions that require a symbol. When you choose a symbol, be sure to click on Choose and Insert from the bottom of the symbol selector tool so that the symbol is included in your answer.

The symbol tool contains symbols, but it does not explain what they mean. The table below includes the symbols as well as their meanings.

What the Symbols Mean

Follow for extended description

Erasable Note Board

You will be given an erasable note board on which to do your calculations or make notes (see gedtestingservice.com and search for note boards). You are not allowed to use the note board before the test begins. If you need an additional note board during the test, you can request a fresh one but you must turn in the one that is used. You may have only one note board at a time, and you must turn in the note board at the end of the test. The note board takes the place of scratch paper, so use it for that purpose. It is erasable.

The best advice for using the note board is to practice writing small so you don’t have to get another note board. Erase when you are sure you won’t need that information anymore. Before you even start with question 1 (but certainly after the test has begun), write (small) at the top of the note board anything you may need to jog your memory during the test. As examples, write PEMDAS to remind you of the order of operations, or the formulas for the circumference or area of a circle—whatever you think you might need and perhaps forget in the stress of taking the test. Leave these reminders on the note board for the duration of the test. That’s why it’s a good idea to write small—you’ll still have space on the board to do calculations or sketch a figure.

Timer

A test-timer clock is located on the upper right-hand corner of the test display. You can minimize the clock or maintain it in view. However, during the final few minutes of the test, in order to make certain that you are aware that time is running out, the clock will appear and cannot be minimized.

It is a good idea to keep the clock in view to help you to pace yourself. The test allows a little less than two minutes per question, so if any multiple-choice or drop-down question seems to be taking more than a minute of your time, answer it to the best of your ability (eliminating obvious wrong choices), flag it for review (see next section), and go on to the next question. It might be a good idea to mark on your note board the test number and the answers that you should look at again, so when you go back you aren’t rereading answers that you have already eliminated. Mark for review any other type of question that baffles you. Then, when you finish all of the questions, you should have enough time to go back to these questions.

Flag for Review

If you think you want to go back to any question, answer it to the best of your ability and then press the Flag for Review button to return to it later. If you can eliminate one or two choices, you can make an informed guess, but rather than taking too much time right then, flag that question, mark the number of the question and the remaining choices on your note board, go on to the next question, and come back later. Often, your brain is working in the background while you continue the test, still mulling over the ones you flagged, and when you return to them, the answers may come to you right away. You definitely do not want any unread questions at the end of the test just because you ran out of time by spending too much time on a question that momentarily baffled you. By the way, the question number (written in the form Question 1 of 46, for example) is in the upper right corner above the Flag for Review option.

Flagging a question for review is an excellent idea. If it is taking more than a minute to figure out an answer, especially on a multiple-choice or drop-down question, mark your best guess (eliminating all of the obviously wrong options) and click