GED Math Preparation - Chapter 1-7 PDF

CHAPTER 1 Whole Numbers and Operations After years of having a calculator available for things like addition, subtraction, division, and multiplication, it is easy to forget some of the steps needed to do these types of problems by hand. For most of the GED® test math section, you won’t have...

CHAPTER 1 Whole Numbers and Operations After years of having a calculator available for things like addition, subtraction, division, and multiplication, it is easy to forget some of the steps needed to do these types of problems by hand. For most of the GED® test math section, you won’t have to worry about this—there is an onscreen calculator. However, there are a handful of questions for which the calculator is not available. These questions are designed to make sure that you remember how to do things like add large numbers or perform long division. To make sure you’re ready, let’s go through and review these operations. Addition To add two or more numbers, line up the numbers in columns by place value. This means that the ones place of the first number is lined up with the ones place of the next number. After this, add each column from right to left. In addition, the solution is called the sum. EXAMPLE 1 112 + 26 EXAMPLE 2 Add: 56 + 11 + 2. When the total in one column is larger than 10, “carry” the tens digit over to the next column to the left. EXAMPLE 3 Add: 242 + 18 + 195. Although the examples simply go in the same order as in the question, addition is commutative. This means that the order in which numbers are added does not matter. Subtraction To subtract, line up numbers in columns the same way as with addition and then subtract each column from right to left. In subtraction, the solution is called the difference. EXAMPLE 4 Subtract: 2055 - 13. If there are not enough ones to subtract, regroup 1 ten as 10 ones. In other words, subtract 1 from the number in the tens place and place a 1 in front of the number in the ones place. Then subtract the ones. If there are not enough numbers in other columns, use regrouping in the same way. EXAMPLE 5 Subtract: 1758 - 909. Your final answer in any subtraction question can always be double- checked by comparing the top term with the sum of the difference and the bottom term. For instance, in the example above, 849 + 909 = 1758, which is the original number on the top line. If the numbers added up to something else, you would know that there was a mistake in the subtraction. Multiplication Multiplication is a bit more complicated than addition and subtraction. Remember that the solution to a multiplication problem is called the product. Let’s look at the process of multiplication step-by-step. EXAMPLE 6 Multiply: 185 × 6. In this example, multiply each of the digits in 185 by 6, from right to left. If the result in any case is larger than 10, then the tens digit will be carried over to the next column. When the second number has more than one digit, the steps stay the same but you need to indent and then add to find the final answer. EXAMPLE 7 Multiply: 289 × 24. As in addition, the order in which numbers are multiplied does not matter. Division Division is calculated in a process that is commonly called “long division.” Remember that the solution of a division problem is called the quotient. Let’s look at the division process step-by-step. EXAMPLE 8 Divide: 1638 ÷ 7.  If at any point in the process you reach a number that is too small to be divided, write a zero and then bring down the next number. Look at the next example. EXAMPLE 9 Divide: 2018÷4. Since 2 is not divisible by 4, place a decimal point and a zero after the dividend to make 2018 into 2018.0. Then add a decimal point after the last digit in the quotient. Bring that zero down to the remainder of 2 to make 20. Four goes into 20 five times, so put a 5 after the decimal in the quotient. Calculators and the GED® Mathematical Reasoning Test The GED® Mathematical Reasoning test includes an onscreen version of the TI-30XS calculator. You may also bring a TI-30XS Multiview Scientific Calculator if you wish. No other type of calculator will be allowed. Even though you may not need it for every question, the calculator is available for a majority of the math questions. Make sure you are completely familiar with this calculator before the test by either using one to practice or reviewing as much about the calculator as you can. Even though we will review all of the steps for using the calculator, you may find it useful to check the website of the GED® Testing Service at http://gedtestingservice.com/testers/calculator. There you will find demonstration videos and guides.  To get a final answer for any operation, you will hit the ENTER key. There is no need to clear your screen in between problems, but if you wish to do so, you may use the CLEAR key at any time. If you do not clear your screen, the up and down arrows allow you to view your previous work.  For addition, subtraction, multiplication, and division you will type in the first number, the operation key, and then the second number, and then press the ENTER key. Important: When performing subtraction, make sure to press the subtraction key on the right of the calculator and not the negative key on the bottom of the calculator. Otherwise, you will get an error. EXERCISE 1 Operations with Whole Numbers Directions: Perform the indicated operation and check your final answer on your calculator. 1. 76 + 1088 2. 2133 – 1849 3. 65 + 14 + 930 4. 852 ÷ 7 5. 168 × 19 6. 4322 – 24 7. 85 – 72 8. 2018 × 89  9. 1084 + 9417 10. 1528 ÷ 12 11. 890 – 104 12. 72 + 215 13. 1999 + 631 14. 810 ÷ 2 15. 9380 ÷ 20 16. 75 × 15 17. 49 × 82 18. 4003 – 209 19. 4429 ÷ 6 20. 1600 × 28 Answers are on page 513. Word Problems Involving Basic Operations When working on word problems involving the basic operations, there are some key phrases or ideas that can help you keep track of whether you should multiply, add, divide, or subtract. Addition: Find a total, find a sum, determine how many “altogether” Subtraction: Find the difference, determine how many are left over Multiplication: Find a total when given groups, find the product, find how many “altogether” (when working with groups)  Division: Find an amount or rate “per,” split into equal parts, share, find how many in each group given a total The list does not show every possibility, and there are many other ways that these operations are indicated in word problems. The best way to get comfortable with the wording is to work plenty of problems. With that in mind, let’s go through a few examples together. EXAMPLE 10 Nathan runs errands for a small fee. In the past four days, he has earned $13, $45, $20, and $32 running errands. In total, how much money has Nathan made during this time period? Because this problem is asking for a total, add the given numbers: 13 + 45 + 20 + 32 = 110 Nathan has earned a total of $110 during this time period. EXAMPLE 11 An office begins the month with 20 reams of paper. If the office used 3 reams the first week and 8 the second, how many reams of paper remain? In this question, you are asked how many reams of paper remain after some are used. Every time a ream of paper is used, there is one less remaining. In other words, this problem requires subtraction. There are 20 – 3 = 17 – 8 = 9 reams of paper remaining. EXAMPLE 12 A storage area can hold 110 large storage bins. If each storage bin can hold 30 hardcover books, how many hardcover books can be stored in the storage area? Like example 10, this problem is also asking for a total. What makes it different is that the total involves groups: groups of books in storage bins. Anytime you are finding a total involving groups, you will multiply.  Each bin can hold 30 books, and there are 110 bins in the storage area. 110 × 30 = 3300 books can be stored in the storage area. EXAMPLE 13 In the past 30 days, a website has received a total of 37,500 visitors. If the site received the same number of visitors each day, how many did it receive each day? Here, you are asked to divide the total number of visitors evenly, over the 30 days. Dividing up groups is represented by division. Therefore, you can say that there were 37,500÷30=1250 visitors to the website each day. EXAMPLE 14 Kate has 30 scented candles. She plans to give 6 of the candles to her friend Lisa. She then plans to sell the remaining candles for $8 each in order to raise money for charity. If she sells all of the remaining candles, how much money will she raise? There is a lot more going on in this problem than in the ones we have seen so far. First, some of the candles are given away (subtraction), and then you are asked to find a total based on the remaining group of candles (multiplication). The order in which these things happen matters because selling 24 candles would be different from selling 30. After Kate gives Lisa 6 candles, she has 30 – 6 = 24 candles remaining. She will sell each of these for $8, so the total will be $8 × 24 = $192 raised. EXERCISE 2 Word Problems Directions: Solve each of the following word problems. Pay close attention to the language used in the problem to determine which operation makes the most sense. 1. Rich has five paychecks left before his summer job ends. If he saves $35 from each of these remaining paychecks, how much will he have saved when his job ends? 2. Two brothers and a sister share the cost of renting a 3-bedroom apartment. Because the sister is still in college, the brothers let her pay only $300 a month toward the rent. The two brothers split the remainder evenly. If the total rent for the apartment is $1400 a month, how much does each brother pay? 3. Since the start of the year, a small company has made $45,690 in sales. If the company makes $39,115 in sales over the remainder of the year, what will be its total sales for the year? 4. On Monday, the balance in a checking account was $1,250. On Tuesday, two bills were paid from this account: an electric bill of $142 and a cable bill of $95. Assuming there were no other transactions, what is the balance of the checking account at the end of the day on Tuesday? 5. A towing company charges drivers $200 to tow a car to the company’s lot and $40 a day to store it. If a car was towed to the lot 26 days ago, how much has its driver accumulated in fees? Answers are on page 513.  CHAPTER 2 Exponents, Roots, and Number Properties Exponents Exponents are a common shorthand used in mathematics. You have probably seen terms like 42 or 34. Both of these are written with exponents. These are read as “four to the second power” and “three to the fourth power,” respectively. This is why you sometimes hear people call exponents “powers.” In a term like the one shown, the exponent tells you how many times to multiply the base by itself. For example, 42 = 4 × 4 = 16 and 34 = 3 × 3 × 3 × 3 = 81. The following are a few special rules and some terminology you should be familiar with in regard to exponents. Any number with an exponent of 0 is equal to 1. This means that , and 12150 = 1. (The only exception is 00, which is considered an indeterminate form. This will not be on the GED® test, however!) Any number to the power of 1 is just that number. For example,. Any number to the power of 2 is said to be “squared.” Any number to the power of 3 is said to be “cubed.”  For example, 52 can be read as “five squared” and 53 can be read as “five cubed.” Negative Exponents Negative exponents have a special meaning. They can be thought of as a way to rewrite a fraction. As a general rule, for any number A that is not zero: In other words, any number with a negative exponent can be rewritten as a fraction with 1 above the fraction bar and the same number with a positive exponent below the fraction bar. Here are some examples. EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 EXERCISE 1 Exponents Directions: Evaluate each of the following expressions containing exponents. 1. 32 2. 15 3. 121 4. 73 5. 22 6. 180 7. 8-2 8. 6-1 9. 4-3 10. 1-4 Answers are on pages 513-514. The Rules of Exponents Expressions with exponents have rules that allow you to simplify them. These are commonly called the rules or laws of exponents. These rules apply whether the exponents are negative or positive, or even if they are not whole numbers. Rule 1: When Multiplying Two Terms with the Same Base, Add the Exponents  An × Am = An+m Remember that this rule applies only if the base numbers are the same. For example, 23 × 2-1 can be simplified using rule 1 because in both terms the base is 2. 32 × 410 cannot be simplified in that way because the base is 3 in one term but 4 in the other. EXAMPLE 4 Using the laws of exponents, write a numerical expression that is equivalent to 44 × 42. Because the two terms with exponents have the same base, you can apply rule 1: 44 × 42 = 44+2 = 46. EXAMPLE 5 Using the laws of exponents, write a numerical expression that is equivalent to. Again, the two terms have the same base, so. Rule 2: When Dividing Two Terms with the Same Base, Subtract the Exponents On the GED® test, you may see division written with the usual division symbol, or you may see it written as a fraction. Both of these are shown in this rule, and both mean the same thing. EXAMPLE 6  Using the laws of exponents, write a numerical expression that is equivalent to. The two terms have the same base and are being divided, so apply rule 2:. EXAMPLE 7 Using the laws of exponents, write a numerical expression that is equivalent to 45 ÷ 4. Both terms have the same base, but the second 4 does not appear to have an exponent. However, recall that any number to the power –of 1 is that same number. Therefore, 4 can be thought of as 41. Thus, 45 ÷ 4 = 45-1 = 44. Rule 3: To Take a Term with an Exponent to a Power, Multiply the Powers (Am)n = Am×n For this rule, you do not need to worry about the base. Just remember that when a power is taken to a power, the exponents are multiplied. For example,. When simplifying some expressions, you may need to use more than one of the rules provided above. In these cases, the order in which you apply the rules does not matter so long as anything within parentheses is simplified first. EXAMPLE 8 Simplify the expression (43 × 42)2 using the laws of exponents.  Working inside the parentheses first, (43 × 42)2 = (45)2 = 410. EXERCISE 2 Rules of Exponents Directions: For each of the following, select the expression that is equivalent to the one given. 1. 23 × 25 A. B. 2 C. 22 D. 28 2. A. B. C. 54 D. 55 3. A. B.  C. D. 4. (84)4 A. B. 1 C. 88 D. 816 5. A. 1 B. C. 0 D. 34 6. (32 × 3)3 A. 33 B. 36 C. 37 D. 39 7. A. B. 710  C. D. 77 8. A. 45 B. 48 C. 415 D. 421 9. 18 × 18 × 182 A. 1 B. 18 C. 182 D. 184 10. (93)4 × 9 A. 92 B. 94 C. 98 D. 913 Answers are on page 514. Square Roots and Cube Roots Square roots and cube roots are two ways of “undoing” exponents. For example, because 62 = 36, the square root of 36 is 6. Also, because 23 = 8, the cube root of 8 is 2. The symbol used for any type of root is called a radical. For a square root, the symbol is. For the cube root, the symbol is. For the first two examples, you could have written and. Numbers for which the square root or the cube root is a whole number are referred to as “perfect squares” or “perfect cubes.” It is a good idea to know some of the common perfect squares and perfect cubes. The most common of these are listed here: Expressions involving square roots and cube roots can be simplified by using your knowledge of the perfect squares and cubes. The next two examples show how this works. EXAMPLE 9 Simplify:. Anytime you need to simplify a square root term, try to rewrite the number under the radical as an expression containing a perfect square. For example, 72 = 2 ×36 and 36 is a perfect square. Therefore,. Because the square root of 36 is 6, the 36 under the radical can be rewritten as 6 in front of the radical. The final answer is read as “6 times the square root of 2.”  EXAMPLE 10 Simplify:. Because 12 can be rewritten as the product of 3 and 4, you can simplify this term as. The same process can be applied to perfect cubes and cube roots. For example,. Note that not all expressions with square roots or cube roots can be simplified. If the number cannot be rewritten as an expression containing a perfect square or a perfect cube, then the radical cannot be simplified. EXERCISE 3 Square Roots and Cube Roots Directions: Simplify each of the following using your knowledge of perfect squares and perfect cubes. 1. 2. 3. 4. 5. 6. 7. 8.  9. 10. Answers are on page 514. Exponents and Roots on the TI-30XS Here are the basics of how to calculate exponents and roots on the TI- 30XS calculator. Exponents To calculate the value of any exponent, type in the base first followed by the caret key located in the left-hand column of the calculator. For example, to find 34: This works for any power, but there is a shortcut key that can be used for squaring numbers. Roots The radical symbols on the calculator are both above the regular buttons in green writing. This means that you must utilize the 2nd key to access them. For cube roots, the calculator will give a decimal approximation, but for square roots, the calculator will give the simplified version by default. In order to get the decimal approximation, you will need to press (the toggle key). (finds the simplified answer) (converts to a decimal approximation) Cube roots are found using a different key, as you can see in the following example. Notice that the 3 from the root symbol is typed in first. Go back and try the practice problems with the calculator. For square roots, make sure to practice getting the decimal approximation as well as the simplified versions of the answers. Order of Operations Which of the following is equal to 3 + 4(1 + 2)? Is it 15, or 27, or is it 9? Which answer you get depends on the order in which you did the operations, but only one answer is correct. Getting the correct answer depends on following a set of rules called the “order of operations.” You may have heard this referred to as PEMDAS or even by a common saying used to remember it: “Please Excuse My Dear Aunt Sally.” The following is the process you should follow anytime you are simplifying a numerical expression. Parentheses. Complete any operations inside of parentheses first. If there is more than one set, start with the innermost parentheses. Exponents. Find the value of any exponents. Multiplication and division. Perform any multiplication or division from left to right. Addition and subtraction. Perform any addition or subtraction from left to right. The example was the expression 3 + 4(1 + 2). Following the order of operations: 3 + 4(1 + 2) Addition inside of the parentheses is performed first. = 3 + 4(3) Multiplication is performed next because there are no terms with exponents. (Remember that multiplication can be written as 4(3) or 4 × 3.) 3 + 4(3) = 3 + 12 Addition is performed last. = 15 Let’s go through a couple more examples to make sure it all makes sense.  EXAMPLE 11 Find the value of. Just as there are different ways to write multiplication, there are different ways to write division. The term. However, before the division can be completed, the term with the exponent must be calculated (this is the first step because there are no parentheses): EXAMPLE 12 Find the value of 2(2 + 5)2 - 6. 2(2 + 5)2 - 6 = 2(7)2 - 6 = 2(49) - 6 = 98 - 6 = 92 EXERCISE 4 Order of Operations Directions: Find the value of each of the following numerical expressions. 1. 8 ÷ 22 + 1 2. 3. 4 + 2(8 + 6) - 1 4. 3(6 - 5)3 5. 4 × 3 - 2  6. 3 -1 + 22 × 5 7. (8 + 1 - 5)3 8. (2 + 4) ÷ 2 - 1 9. 10. 16 + 4(3 + 2) Answers are on page 514. The Distributive Property Using the order of operations, you know that 2(1 + 5) = 2(6) = 12, but a test question may ask you simply to rewrite this expression instead of calculate its final value. This can be done using the distributive property. If A, B, and C are any numbers, then: A(B + C) = A(B) + A(C) and A(B - C) = A(B) - A(C). This property says that with an expression such as 2(1 + 5), you can first add the terms inside the parentheses as previously shown, or you can multiply each term inside the parentheses by the 2 to get 2(1 + 5) = 2(1) + 2(5). If you calculate the value, you will see that the result is 12, just as before. The distributive property will be very useful later on when you work with algebraic expressions. Some important things to note: The distributive property only works if the operation inside the parentheses is addition or subtraction. It will not work with any other operations. However, the property will work no matter how many terms are added or subtracted within the parentheses. The expression 4(5 + 6 - 1 + 2) is equivalent to 4(5) + 4(6) - 4(1) + 4(2). Notice that the 4 is multiplied by (“distributed to”) each term inside the parentheses and that the terms keep the same operation (addition or subtraction).  CHAPTER 3 Decimal Numbers and Operations Decimals appear in a wide variety of questions on the GED® test either in basic calculations or in word problems, especially those that involve money. In this chapter, you will review how to work with decimals. Decimals and Place Value The following chart shows how to refer to each place value in a decimal. Using this table, a number such as 5.38 is read “5 and thirty-eight hundredths,” while 0.237 is read as “two hundred thirty-seven thousandths.” Notice that the place value of the last number determines how it is read. Rounding Decimals Often, you will need to be able to round a decimal to a specific place. For example, a GED® test question may say “to the nearest hundredth meter, what is the radius of the circle?” This indicates that the final answer must be rounded to the nearest hundredth.  When rounding decimals, look at the number immediately to the right of the place you are rounding to. If that number is 5 or larger, add 1 to the place you are rounding. If that number is smaller than 5, keep the place the same. Remove all numbers to the right of the place you are rounding to. EXAMPLE 1 Round 1.09256 to the nearest thousandth. The number in the thousandths place is 2. The number immediately to the right of this number is 5. Therefore, the 2 is rounded up to 3 to get 1.093. EXAMPLE 2 Round 8.63 to the nearest tenth. Because the number to the right of the 6 is 3, the 6 stays the same and the final result is 8.6. Comparing Decimals Some questions on the GED® test will ask you to determine which decimal in a group or pair is largest or smallest. You may also have to order the decimals from largest to smallest or the other way around. A common mistake is to think that a decimal with more numbers is always the largest. For example, 0.12 is smaller than 0.3. In order to compare two decimals, it is helpful to rewrite the decimals so that they have the same number of digits. Any zero written on the end of a decimal number does not change its value. 0.3 is the same as 0.30, and that is the same as 0.300. By rewriting 0.3 as 0.30 and comparing it to 0.12, you can see that 0.12 is smaller because 12 is smaller than 30. This works anytime you are comparing two decimal numbers. To indicate that one number is larger or smaller than another, mathematics uses the symbols > and or < to indicate which number is larger. 0.098 _______ 0.22 Rewriting the decimals so that they each have three digits, you are comparing 0.098 and 0.220. Because 220 is larger than 98, 0.22 must be larger. Using one of the symbols: 0.098 < 0.22 This is read as “Ninety-eight thousandths is less than twenty-two hundredths.” EXAMPLE 4 Use > or < to indicate which number is larger. 0.001 _______ 0.0009 Again rewriting, the decimals are 0.0010 and 0.0009, and because 10 is larger than 9: 0.001 > 0.0009 When the open part of the symbol faces left, it is read as “greater than.” Therefore, this statement is read as “one thousandth is greater than nine ten thousandths.” EXERCISE 1 Comparing Decimals Directions: Use >, , ,