Awa practise 7 doc

The Quantitative section tests your overall understanding of basic math concepts. The math presented in this section will be comparable to what you encountered in middle school and high school, and the ques- tion level may seem similar to that on the SAT ® exam or ACT Assessment ® . Even though the questions are presented in different formats, reviewing some fundamental topics will be very helpful. This section tests your ability to use critical thinking and reasoning skills to solve quantitative problems.You will want to review how to solve equations, how to simplify radicals, and how to calculate the volume of a cube. However, the major- ity of the questions will also ask you to take the problem one step further to assess how well you apply and reason through the material. The two types of questions in the Quantitative section are problem solving and data sufficiency. You have already seen both types of questions in the pretest. Each type will be explained in more detail in the next section.  About the Types of Questions The two types of questions — problem solving and data sufficiency — each contains five answer choices. Both types of questions will be scattered throughout the section. Problem solving questions test your basic knowl- edge of math concepts — what you should have learned in middle school and high school. Most of these ques- tions will ask you to take this existing knowledge and apply it to various situations. You will need to use reasoning skills to analyze the questions and determine the correct solutions. The majority of the questions will contain a multistep procedure. When answering problem-solving questions, try to eliminate improba- ble answers first to increase your chances of selecting the correct solution. A Sample Problem Solving Question Directions: Solve the problem and choose the letter indicating the best answer choice. The numbers used in this section are real numbers. The figures used are drawn to scale and lie in a plane unless otherwise noted. Given integers as the lengths of the sides of a triangle, what is the maximum perimeter of a triangle where two of the sides measure 10 and 14? a. 27 b. 28 c. 48 d. 47 e. 52 Answer: d. Use the triangle inequality, which states that the sum of the two smaller sides of a triangle must be greater than the measure of the third side. By adding the two known sides of 10 + 14 = 24, this gives a maximum value of 23 for the third side because the side must be an integer. Since the perimeter of a polygon is the sum of its sides, the maximum perimeter must be 10 + 14 + 23 = 47. – ABOUT THE QUANTITATIVE SECTION– 318 The other type of question in this section is data sufficiency. Data sufficiency questions give an initial question or statement followed by two statements labeled (1) and (2). Given the initial information, you must determine whether the statements offer enough data to solve the problem. The five possible answer choices are as follows: a. Statement (1), BY ITSELF, will suffice to solve the problem, but NOT statement (2) by itself. b. Statement (2), BY ITSELF, will suffice to solve the problem, but NOT statement (1) by itself. c. The problem can be solved using statement (1) and statement (2) TOGETHER, but not ONLY statement (1) or statement (2). d. The problem can be solved using EITHER statement (1) only or statement (2) only. e. The problem CANNOT be solved using statement (1) and statement (2) TOGETHER. This type of question measures the test taker’s ability to examine and interpret a quantitative problem and distinguish between pertinent and irrelevant information. To solve this particular type of problem, the test taker will have to be able to determine at what point there is enough data to solve a problem. Since these questions are seldom used outside of the GMAT exam, it is important to familiarize yourself with the for- mat and strategies used with this type of question as much as possible before taking the exam. Strategies can be used when answering data sufficiency questions. For example, start off by trying to solve the question solely by using statement (1). If statement (1) contains enough information to do so, then your only choice is between a (statement [1] only) or d (each statement alone contains enough information). If statement (1) is not enough information to answer the question, your choices boil down to b (statement [2] only), c (the statements need to be used together), or e (the problem cannot be solved using the infor- mation from both statements, and more information is needed). A Sample Data Sufficiency Question Directions: The following problem contains a question followed by two statements. Select your answer using the data in statement (1) and statement (2) and determine whether they provide enough information to answer the initial question. If you are asked for the value of a quantity, the information is sufficient when it is possible to determine only one value for the quantity. a. Statement (1), BY ITSELF, will suffice to solve the problem, but NOT statement (2) by itself. b. Statement (2), BY ITSELF, will suffice to solve the problem, but NOT statement (1) by itself. c. The problem can be solved using statement (1) and statement (2) TOGETHER, but not ONLY statement (1) or statement (2). d. The problem can be solved using EITHER statement (1) only or statement (2) only. e. The problem CANNOT be solved using statement (1) and statement (2) TOGETHER. The numbers used are real numbers. If a figure accompanies a question, the figure will be drawn to scale according to the original question or information, but it will not necessarily be consistent with the infor- mation given in statements (1) and (2). – ABOUT THE QUANTITATIVE SECTION– 319 If x is a nonzero integer, is x positive? (1) x 2 is positive. (2) x 3 is positive. Answer: b. Substitute possible numbers for x.Ifx ϭ 2, then (2) 2 ϭ 4. If x ϭϪ2, then (Ϫ2) 2 ϭ 4, so state- ment (1) is not sufficient. Substituting into statement (2), if x ϭϪ2, then (Ϫ2) 3 ϭ (Ϫ2)(Ϫ2)(Ϫ2) ϭϪ8; the value is negative. If x ϭ 2, then 2 3 ϭ 2 ϫ 2 ϫ 2 ϭ 8; the value is positive. Therefore, from statement (2), x is positive. – ABOUT THE QUANTITATIVE SECTION– 320 The following lessons are designed to review the basic mathematical concepts that you will encounter on the GMAT® Quantitative section and are divided into three major sections: arithmetic, algebra, and geometry. The lessons and corresponding questions will help you remember a lot of the primary content of middle school and high school math. Please remember that the difficulty of many of the questions is based on the manner in which the question is asked, not the mathematical concepts. These questions will focus on criti- cal thinking and reasoning skills. Do not be intimidated by the math; you have seen most of it, if not all of it, before.  Types of Numbers You will encounter several types of numbers on the exam: ■ Real numbers. The set of all rational and irrational numbers. ■ Rational numbers. Any number that can be expressed as , where b  0. This really means “any num- ber that can be written as a fraction” and includes any repeating or terminating decimals, integers, and whole numbers. a b CHAPTER Arithmetic 20 321 ■ Irrational numbers. Any nonrepeating, nonterminating decimal (i.e., ͙2 ෆ, ␲, 0.343443444 ). ■ Integers. The set of whole numbers and their opposites { ,–2,–1,0,1,2,3, }. ■ Whole numbers. {0,1,2,3,4,5,6, }. ■ Natural numbers also known as the counting numbers. {1,2,3,4,5,6,7, }.  Properties of Numbers Although you will not be tested on the actual names of the properties, you should be familiar with the ways each one helps to simplify problems. You will also notice that most properties work for addition and multi- plication, but not subtraction and division. If the operation is not mentioned, assume the property will not work under that operation. Commutative Property This property states that even though the order of the numbers changes, the answer is the same. This prop- erty works for addition and multiplication. Examples a + b = b + a ab = ba 3 + 4 = 4 + 3 3 × 4 = 4 × 3 7 = 7 12 = 12 Associative Property This property states that even though the grouping of the numbers changes, the result or answer is the same. This property also works for addition and multiplication. a + (b + c) = (a + b) + ca(bc) = (ab)c 2 + (3 + 5) = (2 + 3) + 5 2 × (3 × 5) = (2 × 3) × 5 2 + 8 = 5 + 5 2 × 15 = 6 × 5 10 = 10 30 = 30 Identity Property Two identity properties exist: the Identity Property of Addition and the Identity Property of Multiplication. ADDITION Any number plus zero is itself. Zero is the additive identity element. a + 0 = a 5 + 0 = 5 – ARITHMETIC– 322 MULTIPLICATION Any number times one is itself. One is the multiplicative identity element. a × 1 = a 5 × 1 = 5 Inverse Property This property is often used when you want a number to cancel out in an equation. ADDITION The additive inverse of any number is its opposite. a + (–a ) = 0 3 + (–3) = 0 MULTIPLICATION The multiplicative inverse of any number is its reciprocal. a × = 1 6 × = 1 Distributive Property This property is used when two different operations appear: multiplication and addition or multiplication and subtraction. It basically states that the number being multiplied must be multiplied, or distributed, to each term within the parentheses. a (b + c) = ab + ac or a (b – c) = ab – ac 5(a + 2) = 5 × a + 5 × 2, which simplifies to 5a + 10 2(3x – 4) = 2 × 3x – 2 × 4, which simplifies to 6x – 8  Order of Operations The operations in a multistep expression must be completed in a specific order. This particular order can be remembered as PEMDAS. In any expression, evaluate in this order: PParentheses/grouping symbols first E then Exponents MD Multiplication/Division in order from right to left AS Addition/Subtraction in order from left to right Keep in mind that division may be done before multiplication and subtraction may be done before addi- tion, depending on which operation is first when working from left to right. 1 6 1 a – ARITHMETIC– 323 . what is the maximum perimeter of a triangle where two of the sides measure 10 and 14? a. 27 b. 28 c. 48 d. 47 e. 52 Answer: d. Use the triangle inequality, which states that the sum of the two smaller. works for addition and multiplication. Examples a + b = b + a ab = ba 3 + 4 = 4 + 3 3 × 4 = 4 × 3 7 = 7 12 = 12 Associative Property This property states that even though the grouping of the numbers. numbers. {0,1,2,3,4,5,6, }. ■ Natural numbers also known as the counting numbers. {1,2,3,4,5,6 ,7, }.  Properties of Numbers Although you will not be tested on the actual names of the properties,