REA's GED® Math Test Tutor - Focus on the most challenging part of the GED® test! Revised Second Edition - Recognized with ProCert's 2018 Excellence in Education Award for Best-in-Class GED® PrepIt's like having your own tutor… Test-takers report that the Mathematical Reasoning test is the toughest of all the GED® tests. That's why REA offers the most complete guide to success on the GED® Math test.Written by Sandra Rush, a GED® Math tutor and test expert - This revised second edition of our best-selling test prep is completely aligned with the content and format of the current GED® test. A comprehensive review covers all the mathematics topics tested on the exam, including fractions, decimals, percentages, algebra, polynomials, scientific notation, and more.Know your way around the TI-30XS MultiView™ calculator - A step-by-step tutorial gets you up and running in no time.Helpful hints you need to know - Hundreds of hints, shortcuts, and exercises reinforce key math points, including secrets of rounding, calculating with fractions, knowing the difference between domain and range, a neat way to remember negative and absolute values, and much more!Diagnostic pretest and two full length practice tests - A diagnostic pretest with comprehensive feedback helps you pinpoint your strengths and weaknesses. Two full-length practice tests feature every type of question (including drag-and-drop, hot-spot, and drop-down test items), subject area, and skill tested on the GED® Mathematical Reasoning test. Each practice test comes with detailed answer explanations to help you identify where you need extra review.If you're concerned about your math score on the GED®, you need REA's GED® Math Test Tutor!
About the Author
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.
Read an Excerpt
"I Was Born Knowing That"
Have you ever thought when asked "How do you know that?" that you were born knowing it. You cannot remember how you learned it, but as far back as you can remember, you always knew it. An example is the simple "one plus one equals two." You actually have a lot of math knowledge that you haven't even thought of. You know it, but you just don't know that you know it.
This book builds on your current knowledge to develop a mindset that promises to help you on many levels, such as everyday situations (checking to make sure you got the correct change when you buy something) and workplace problems (figuring what the deductions from your paycheck should be), as well as passing the Math Reasoning portion of the GED® test. All it involves is some insight and calling on the knowledge you already have.
Just as important as knowing math basics is knowing what to expect on the GED® test, so be sure you understand the information in the Introduction to this book. It is important to know how the test will be structured as well as to develop familiarity with the virtual calculator and other aids provided on the test (formula sheet, calculator reference, and symbol sheet).
USE LOGIC! Calculations are secondary in importance to finding a logical solution pathway that focuses on problem solving, modeling, thinking, and reasoning. The calculations, if done correctly, will always give you the correct solution, but if you don't know what the problem is asking for, or if you cannot model it, or if you cannot reason through a solution pathway, how will you know what calculations to make? This book will help you with the calculations and give you ways to improve your logical skills — skills that you were "born knowing."
Sometimes it is possible to get mathematically correct results even if you use incorrect lines of reasoning. This can be because two "wrongs" made a "right" or due to several other scenarios. And sometimes you can use what seems to be a correct line of reasoning but come up with an incorrect answer. Be sure to check whether your answer makes sense.
The following popular riddle shows the importance of checking your answers. If you bought a cupcake and a cookie at a bake sale for $1.10, and the cupcake costs a dollar more than the cookie, how much did the cookie cost? A surprising number of people answer this right away as 10 cents. But if you checked that answer, you would see right away that it was wrong because then that total is $1.20, not $1.10. (The correct answer is 5 cents.)
When starting a problem, ask yourself first: What is the problem asking? What information is it giving? Is any of this information unnecessary or missing? As an example of a problem with missing information, suppose Tim has twelve coins. He tells you that five of them are dimes and seven of the coins are worth five cents or more. He will give you all of the coins if you can tell him how many quarters he has. The problem is asking how many quarters there are. You know that five are dimes and seven are nickels, dimes, and quarters (and perhaps half-dollars). You can figure out that two of them must be coins that are nickels, quarters, or half-dollars. But you don't have enough information to say how many of each, so you can't take Tim's challenge. You have to have another piece of information, such as that he has two nickels, in which case you know there aren't any quarters. If Tim told you the total value of the coins, you could figure out how many are dimes, even if it is by trial and error. But without the last piece of information, this is an impossible question to answer correctly, except by a guess. Can you see that the answer must be 0, 1, or 2?
Suppose Tim asked instead how many pennies he had. Could you answer that? Of course you can because you know that seven of the coins are nickels or greater, so there must be (12 – 7 =) five pennies.
Let's try another example: Three people — Adele, Benjamin, and Contessa — are playing a card game that ends when the totals of all points equals 50. Can you tell who is the winner? No, of course not — you don't have enough information. What more do you have to know? If you know that Contessa has 15 points, will that help? No, not yet, because all that tells you is that Adele and Benjamin have a total of (50 - 15 =) 35 points, but you don't know how that is distributed. Will knowing just two scores, but not all three, help? Yes, because you can then add them and subtract the total from 50 to get the third score, and then you can tell who the winner is. And, of course, if you are given all three scores, you can tell right away who won.
In fact, if you are given all three scores and the total to end the game, what would be the extra, unnecessary information? To solve the problem, if you knew two scores and the fact that 50 ends the game, you can figure out the third score to compare to the other two. Conversely, if you knew all three scores, there is hardly any challenge in finding which is higher, and you don't have to know that a total of 50 ends the game.
A final example has to do with horse racing. The Kentucky Derby is run on a 2kilometer track. How fast on average was the winning horse going? You cannot tell if you only know the length of the course. What other information do you need? You need the time it took for the horse to finish the race. Without that, you cannot tell how fast the horse was running.
The point is that rather than jumping in to solve a problem or choose a multiple-choice answer, take a moment to see what is given and what is asked for and what you have to figure out yourself. Then, when you have answered the problem, take another moment to make sure the answer makes sense.
This chapter starts with some basic math that you probably already know — again, you just know it — and builds on this information. For this chapter, try the examples near the beginning of each section first — if you think you already know the topic and can do all of the examples correctly, skip to the next section. Throughout the book, concentrate on learning the topics you aren't sure about. You can always go back if you forget something (use the index to find the page numbers).
The GED® virtual TI-30XS calculator will be a time-saver on the test, so it is essential that you also study how to use it. If you don't know what the many keys mean, it won't be as useful to you and will take up your time instead of saving time. Be sure you are familiar with the calculator, even though you probably can do most of the problems in this chapter without a calculator. Some problems on the test (about 10%) will not allow you to use the calculator, so be prepared for that. In fact, it's a good idea to try to do the arithmetic in this chapter both ways: with and without a calculator.
Although the chapters in this book are divided into the topics on the GED® test, each chapter includes all of the information learned in the previous chapters. In addition, "calculator boxes" in the chapters, such as the one at the beginning of this chapter, present how to use the virtual calculator for a particular subject area or type of problem. At the end of the instructional chapters, you will have all of the information you need to go on to do the practice tests confidently.
The Number Line
A brief history of how numbers evolved is interesting. About three millennia ago, the only numbers that were used were the positive numbers, 1, 2, 3, ... These were counting numbers, also called natural numbers. There was no number to indicate "nothing," such as if a farmer had three cows and sold them all, how many he would have? There was a concept of zero, but no symbol for it. The symbol "0" came about as a placeholder, so "3" no longer meant 3 or 30 or 300. These numbers are called whole numbers, 0, 1, 2, 3, ... (natural numbers plus 0). Eventually, the idea of negatives came to be; integers are the numbers shown on the number line below, ... -3, -2, -1, 0, 1, 2, 3,... All of the numbers (including fractions, decimals, etc.) between any two integers plus all of the integers together are called real numbers.
The above line is just a portion of the number line, which really goes on forever (to infinity, or ∞) in both the positive and negative directions. Integers are marked on the line, but there are infinitely many numbers between each two integers — for example, fractions and decimals, which we discuss in the next chapter. Numbers have various classifications, such as odd or even, and positive or negative.
The numbers get larger as we go to the right on the number line. That is true whether we consider positive numbers or negative numbers. Using the number line above and the symbols < for "less than" and > for "greater than," we see that it is true, although not always obvious, that
10 > 3 3 > -1 -1 > -4 -10 < -5 -3 < 0 5 < 9.
Combining an equal sign with an inequality sign yields the symbols ≥ and ≤=, which include the number. So all the integers less than 8 (< 8) include ... 3, 4, 5, 6, and 7; whereas all the integers less than or equal to 8 include ... 3, 4, 5, 6, 7, 8. When graphed on the number line,
Notice that the circle at 8 is an open circle for < 8.
Notice that the circle at 8 is filled in for ≤ 8.
To remember which of the symbols, < and >, is which, look at the left-hand sides. The "less than" sign (<) has the smaller (lesser) side on the left, and the "greater than" sign (>) has the bigger (greater) side on the left.
Rearrange the following numbers by size, starting with the smallest:
8 9 -10 2 16 -3 -5 0 -7 10
-10 < -7 < -5 < -3 < 0 < 2 < 8 < 9 < 10 < 16
The number -29 belongs to which number sets? Choose all that apply.
A. Natural numbers
C. Whole numbers
D. Real numbers
(B) and (D), integers and real numbers
One way to remember negative numbers is to think of temperature: -10 degrees is colder (less warm) than -5 degrees, so -10 < -5.
Addition and Subtraction
Our number system is based on 10 (called the decimal system). It is believed that this is because the ancient mathematicians who "invented" the method of counting used their ten fingers (okay, eight fingers and two thumbs) to keep track when counting. The fact that we call the numbers 1 to 9 "digits" lends credibility to this theory because digit is another word for finger.
Numbers greater than 9 have two digits; the digit on the far right represents ones (also called units), and the next place to the left represents tens, then to the left of that is hundreds, and so on, as shown in the diagram. Note that commas are added to make it easier to read a large number.
So the number 83,724 has 8 ten-thousands, 3 thousands, 7 hundreds, 2 tens, and 4 ones, and it is read as "eighty-three thousand, seven hundred twenty-four." Note that we say "eighty-three thousand" instead of "eight ten thousand, three thousand." We usually say the hundreds, tens, and ones together in each group of three numbers that are separated by the commas.
Note above that we filled in zeros to keep the columns straight. The zeros simply mean there are no units digits for the 2 tens, and no tens or units digits for the 7 hundreds. If you can't add larger numbers in your head, write them in column form. For any addition or subtraction problems, it is important to keep the digits in their correct columns. Of course, on the GED® test, you may be able to use the calculator.
When you hear the word arithmetic, what usually comes to mind right away is 1 + 1 = 2, 2 + 2 = 4, and so on. Or perhaps you also thought 3 - 1=2. This is information you were "born knowing." For example, try the following problems — the correct answers follow.
Perform the following operations:
a. 6 + 7 =
b. 23 + 15 =
c. 34 + 43 =
d. 8 - 3 =
e. 36 - 12
f. 67 - 51
If you got these correct, go to the next section. Otherwise, look again at the number line to get an idea of how we got these answers.
For example, for 6 + 7, start at 6 and move seven places to the right (when adding, move to the right). The answer is 13. If you have 6 pencils and you get 7 more, you now have 13 pencils.
For 8 - 3, which is the same as 8 + (-3), start at 8, but this time move three spaces to the left (when subtracting, or adding a negative, move to the left). Or think of "more" for addition and "less" or "fewer" for subtraction. If you start with 8 pens and you have 3 fewer, you now have 5 pens.
For problems involving more than one digit, write them in column form, and add each column separately:
Now we'll step it up a bit. Regrouping involves carrying and borrowing digits. Do the problems in the following example — if you get all of it right, skip to the next section.
Perform the following operations:
a. 13 + 29 =
b. 57 + 55 =
c. 75 + 36 =
d. 34 - 16 =
e. 73 - 59 =
f. 82 - 79 =
These problems involve carrying and borrowing. "Carrying" happens when the two numbers add to more than 9. Example 1.4(a) written in column form looks like this:
Start at the units column and work to the left. The units column has the numbers 3 and 9, for a total of 12, which has a 1 in the tens column and a 2 in the units column, so let's put the 1 where it belongs, in the tens column.
Similarly, in subtraction, if you have to subtract a larger number from a smaller number in any column, you can "borrow" 1 from the column to the left, keeping in mind that the 1 really is 1 of whatever its original value was. Example 1.4(d), if written in column form, looks like this:
The units column has the numbers 4 and 6, but you have to subtract 6 from 4. If you borrow 1 from the tens column, that makes the 4 into a 14. Of course, it also changes the 3 in the tens column into a 2.
You can check your answer in a subtraction problem (appropriately called the "difference") by adding the answer to the number being subtracted. If you get the other number in the problem, it checks out; otherwise, find out where you made a mistake. For the example above, 16 + 18 indeed equals 34.
The word absolute in absolute value may make you think of a positive number. Actually absolute values are unsigned values. Try the following examples — if you get them all correct, skip this section and go on to the next.
Evaluate the following absolute values:
a. |-3| =
b. |475| =
c. |2 - 2| =
d. |3 - 1| =
e.|1 - 3| =
f.|3 + 4 - 2 - 8| =
Positive numbers don't have to have a sign in front of them. The number 8 means +8. Negative numbers always have to have a negative sign in front of them; otherwise, how would we know they aren't positive? Treat negative numbers the same as positive numbers in addition and subtraction with the following rule: keep track of the signs, specifically that minus a negative (two negative signs in a row) is a positive.
It is often helpful to put parentheses around a negative number until it is combined with the operation in front of it. For example, 8 - (-5) is the same as 8 + 5 = 13.
As we have seen, numbers are really a measure on the number line of how many units each number is from 0. So 8 is 8 units from 0, and -5 is 5 units from 0 (on the other side of 0, but still 5 units). The difference, which measures the total distance from 8 to -5, is 13 units because there are 13 units between 8 and -5. The difference of the two numbers that are on the same side of zero is simply that — the difference in their distances. So -8 - (-5) = -3, which indicates 3 units on the negative side of 0. Note again how minus a negative becomes the addition of a positive: -8 - (-5) = -8 + 5 = -3, or 3 units on the negative side.
Think of "difference" as "distance on the number line." Sketch or think of where the two values are on the number line — the distance between them on the number line is the difference.
The distance of a number from 0 on the number line is actually its absolute value. It is the number of units, and it doesn't matter whether the number is positive or negative. In other words, the numbers +8 and -8 are each 8 units from 0 on the number line. Therefore, the distance from the point -8 to the point 8 on the number line is 16 units.
Excerpted from "GED Math Test Tutor"
Copyright © 2016 Research & Education Association, Inc..
Excerpted by permission of Research & Education Association.
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Table of Contents
About the Author
What “Content Aligned” Means to You
Introduction: About the GED® Mathematical Reasoning Test
How This Book Can Help
What to Expect on the GED® Mathematical Reasoning Test
New GED® Performance Levels
Types of Questions
On-Screen and Other Resources
GED® Mathematical Reasoning Diagnostic Pretest
Chapter 1: “I Was Born Knowing That”
The Number Line
Addition and Subtraction
Chapter 2: The Parts of the Whole
Ratios and Proportions
The Relations Among Decimals, Fractions, and Percentages
Chapter 3: Power Play
Order of Operations
Chapter 4: Algebra—Not a Four-letter Word
Setting up Equations
Working with Algebraic Expressions
Evaluating Algebraic Expressions
Solving Linear Equations
Chapter 5: The X (Squared) Factor
Chapter 6: The Shape of Things 195
Lines and Angles
Composite Two-Dimensional Figures
Composite Three-Dimensional Figures
Chapter 7: What Are the Chances?
Permutations and Combinations
Spread of Data
GED® Mathematical Reasoning Practice Test 1
GED® Mathematical Reasoning Practice Test 2