The entire GMAT® quantitative part of the exam takes place in one section of the test. This section contains 37 questions and must be completed in 75 minutes. Therefore, the test taker can spend about two minutes per question, on average. The questions in this section consist of two different multiple-choice formats: prob- lem solving and data sufficiency. Each type of question has five possible choices for answers. These questions test a person’s knowledge of mathematical concepts and their applications, along with thinking and reason- ing skills. Examinees will be asked to recall the mathematics that they learned in middle school and high school and apply these skills in an advanced manner for the questions on the test. Although scrap paper is allowed, the use of calculators is prohibited on the GMAT exam. Since the Quantitative section is only administered as a CAT, this section will be taken on a computer. As each person takes the exam, the computer randomly generates the sequence of questions administered based on the participant’s ability. The test begins with a question of average difficulty. If the question is answered correctly, points are added to the score and a more difficult question follows. If the question is answered incorrectly, there is no penalty, but an easier question follows. Keep in mind that harder questions carry more weight and will result in a higher score. Because of the CAT format, each question must be answered and confirmed before proceeding to the next question. Since randomly guessing an incorrect answer CHAPTER Quantitative Pretest 18 307 will lower your score, making an educated guess by eliminating one or more of the answer choices should result in a better score. Your score is based on both the number of questions you answer and the level of dif- ficulty of the questions; the more difficult questions you answer, the better. Even though the Quantitative section is administered on a computer, minimal computing skills are nec- essary. Free GMAT tutorials can be downloaded from various Internet sites and taken ahead of time. The test- ing site also offers a tutorial that can be completed immediately before commencing the test. These skills, such as using the mouse and the HELP feature, should be practiced before beginning the test. Once you start a par- ticular section of the exam, the clock cannot be stopped. Time spent asking for help will be counted in the total time for that particular section. The quantitative portion will not test how well you recall a lot of facts and figures; instead, it will test how well you use your existing knowledge of math and how well you apply it to various situations. In addi- tion, this section of the test will not evaluate your personality, work ethic, or ability to work with others. Although the problems may seem difficult at times, they will not be assessing the undergraduate work you may have completed in college or any particular course you may have taken; the math will be high school level. Even though the test is used as a precursor for business school, the questions will not require knowledge of business-related skills. This section of the test contains a number of trial questions that are being field-tested for future use. These particular questions will not be counted toward your total score; however, the actual questions are not distinguished from the trial questions. Do your best on all of the questions and treat them as if they all count.  Problem Solving Questions – QUANTITATIVE PRETEST – 308 1.abcde 2.abcde 3.abcde 4.abcde 5.abcde 6.abcde 7.abcde 8.abcde 9.abcde 10. a b c d e 11. a b c d e 12. a b c d e 13. a b c d e 14. a b c d e 15. a b c d e 16. a b c d e 17. a b c d e 18. a b c d e 19. a b c d e 20. a b c d e ANSWER SHEET 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. 1. If both the length and the width of a rectangle are tripled, then the area of the rectangle is a. two times larger. b. three times larger. c. five times larger. d. six times larger. e. nine times larger. 2. If a set of numbers consists of ᎏ 1 4 ᎏ and ᎏ 1 6 ᎏ , what number can be added to the set to make the average (arithmetic mean) also equal to ᎏ 1 4 ᎏ ? a. ᎏ 1 6 ᎏ b. ᎏ 1 5 ᎏ c. ᎏ 1 4 ᎏ d. ᎏ 1 3 ᎏ e. ᎏ 1 2 ᎏ 3. Given integers as the measurements of the sides of a triangle, what is the maximum perimeter of a tri- angle where two of the sides measure 10 and 14? a. 34 b. 38 c. 44 d. 47 e. 48 4. In 40 minutes, Diane walks 2.5 miles and Sue walks 1.5 miles. In miles per hour, how much faster is Diane walking? a. 1 b. 1.5 c. 2 d. 2.5 e. 3 5. If x Ϫ2, then a. x Ϫ 2 b. x Ϫ 10 c. 5x + 2 d. x + 2 e. 5x Ϫ 2 6. If five less than y is six more than x + 1, then by how much is x less than y? a. 6 b. 7 c. 10 d. 11 e. 12 5x 2 Ϫ 20 5x ϩ 10 ϭ – QUANTITATIVE PRETEST – 309 7. If x dozen eggs cost y dollars, what is the cost, C,ofz dozen eggs? a. C ϭ xyz b. c. d. C ϭ xy + z e. C ϭ x + y + z 8. At a certain high school, 638 students are taking biology this year. Last year 580 students took biology. Which of the following statements is NOT true? a. There was a 10% increase in students taking biology. b. There were 90% more students taking biology last year. c. There were 10% fewer students taking biology last year. d. The number of students taking biology this year is 110% of the number from last year. e. The number of students taking biology last year was about 91% of the students taking biology this year. 9. Two positive integers differ by 7. The sum of their squares is 169. Find the larger integer. a. 4 b. 5 c. 9 d. 12 e. 14 10. Quadrilateral WXYZ has diagonals that bisect each other. Which of the following could describe this quadrilateral? I. parallelogram II. rhombus III. isosceles trapezoid a. I only b. I and II only c. I and III only d. II and III only e. I, II, and III  Data Sufficiency Questions Directions: Each of the following problems contains a question that is followed by two statements. Select your answer using the data in statement (1) and statement (2) and determine whether they provide enough infor- C ϭ yz x C ϭ xy z – QUANTITATIVE PRETEST – 310 mation to answer the initial question. If you are asked for the value of a quantity, the information is suffi- cient when it is possible to determine only one value for the quantity. 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. 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). 11. Is k even? (1) k + 1 is odd. (2) k + 2 is even. 12. Is quadrilateral ABCD a rectangle? (1) m ∠ ABC ϭ 90 ° (2) AB ϭ CD 13. Sam has a total of 33 nickels and dimes in his pocket. How many dimes does he have? (1) There are more than 30 nickels. (2) He has a total of $1.75 in his pocket. 14. If x is a nonzero integer, is x positive? (1) x 2 is positive. (2) x 3 is positive. 15. The area of a triangle is 36 square units. What is the height? (1) The area of a similar triangle is 48 square units. (2) The base of the triangle is half the height. 16. What is the value of x? (1) x 2 ϭϪ6x Ϫ 9 (2) 2y Ϫ x ϭ 10 – QUANTITATIVE PRETEST – 311 17. What is the slope of line m? (1) It is parallel to the line 2y ϭ 3 + x. (2) The line intersects the y-axis at the point (0, 5). 18. If two triangles are similar, what is the perimeter of the smaller triangle? (1) The sum of the perimeters of the triangles is 30. (2) The ratio of the measures of two corresponding sides is 2 to 3. 19. While shopping, Steve spent three times as much money as Judy, and Judy spent five times as much as Nancy. How much did Nancy spend? (1) The average amount of money spent by the three people was $49. (2) Judy spent $35. 20. A cube has an edge of e units and a rectangular prism has a base area of 25 and a height of h. Is the volume of the cube equal to the volume of the rectangular prism? (1) The value of h is equal to the value of e. (2) The sum of the volumes is 250 cubic units.  Answer Explanations to the Pretest 1. e. Suppose that the length of the rectangle is 10 and the width is 5. The area of this rectangle would be A ϭ lw ϭ 10 × 5 ϭ 50. If both the length and width are tripled, then the new length is 10 × 3 ϭ 30 and the new width is 5 × 3 ϭ 15. The new area would be A ϭ lw ϭ 30 × 15 ϭ 450; 450 is nine times larger than 50. Therefore, the answer is e. 2. d. Let x equal the number to be added to the set. Then is equal to ᎏ 1 4 ᎏ . Use the LCD of 12 in the numerator so the equation becomes . Cross-multiply to get , which simplifies to ᎏ 5 3 ᎏ +4x ϭ 3. Subtract ᎏ 5 3 ᎏ from each side of the equation to get 4x ϭ ᎏ 4 3 ᎏ . Divide each side by 4. . Another way to look at this problem is to see that and . Since you want the average to be , then the third number would have to be to make this average. 3. 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 ϭ 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 ϭ 47. 4 12 ϭ 1 3 1 4 ϭ 3 12 1 6 ϭ 3 12 1 4 ϭ 3 12 x ϭ 4 3 Ϭ 4 ϭ 4 3 × 1 4 ϭ 1 3 . 4x 4 ϭ 4 3 4 41 5 12 2ϩ 4x ϭ 3 3 12 ϩ 2 12 ϩ x 3 Ϫ 5 12 ϩ x 3 ϭ 1 4 1 4 ϩ 1 3 ϩ x 3 – QUANTITATIVE PRETEST – 312 4. b. Since the distance given is out of 40 minutes instead of 60, convert each distance to hours by using a proportion. For Diane, use . Cross-multiply to get 40x ϭ 150. Divide each side by 40. Diane walks 3.75 miles in one hour. For Sue, repeat the same process using . Cross-multiply to get 40x ϭ 90 and divide each side by 40. So Sue walks 2.25 miles in one hour. 3.75 Ϫ 2.25 ϭ 1.5. Diane walks 1.5 miles per hour faster than Sue. 5. a. Factor the expression and cancel out common factors. The expression reduces to x Ϫ 2. 6. e. Translate the sentence into mathematical symbols and use an equation. Five less than y becomes y Ϫ 5, and six more than x + 1 becomes x + 1 + 6. Putting both statements together results in the equation y Ϫ 5 ϭ x + 1 + 6. This simplifies to y Ϫ 5 ϭ x + 7. Since you need to find how much is x less than y, solve the equation for x by subtracting 7 from both sides. Since x ϭ y Ϫ 12, x is 12 less than y, which is choice e. 7. c. Substitution can make this type of problem easier. Assume that you are buying 10 dozen eggs. If this 10 dozen eggs cost $20, then 1 dozen eggs cost $2. This is the result of dividing $20 by 10, which in this problem is . If is the cost of 1 dozen eggs, then if you buy z dozen eggs, the cost is , which is the same as choice c,. 8. b. Use the proportion for the percent of change. 638 Ϫ 580 ϭ 58 students is the increase in the num- ber of students. . Cross-multiply to get 580x ϭ 5,800 and divide each side by 580. x ϭ 10. Therefore, the percent of increase is 10%. The only statement that does not support this is b because it implies that fewer students are taking biology this year. 9. d. Let x ϭ the smaller integer and let y ϭ the larger integer. The first sentence translates to y Ϫ x ϭ 7 and the second sentence translates to x 2 + y 2 ϭ 169. Solve this equation by solving for y in the first equation (y ϭ x + 7) and substituting into the second equation. x 2 + y 2 ϭ 169 x 2 + (x + 7) 2 ϭ 169 Use FOIL to multiply out (x + 7) 2 : x 2 + x 2 + 7x + 7x + 49 ϭ 169 Combine like terms: 2x 2 + 14x + 49 ϭ 169 Subtract 169 from both sides: 2x 2 + 14x + 49 Ϫ 169 ϭ 169 Ϫ 169 2x 2 + 14x Ϫ 120 ϭ 0 Factor the left side: 2 (x 2 + 7x Ϫ 60) ϭ 0 2 (x + 12)(x Ϫ 5) ϭ 0 Set each factor equal to zero and solve 2  0 x + 12 ϭ 0. x Ϫ 5 ϭ 0 x ϭϪ12 or x ϭ 5 Reject the solution of Ϫ12 because the integers are positive. Therefore, the larger integer is 5 + 7 ϭ 12. A much easier way to solve this problem would be to look at the answer choices and find the solution through trial and error. 58 580 ϭ x 100 C ϭ yz x y x × z y x y x 5x 2 Ϫ 20 5x ϩ 10 ϭ 51x 2 Ϫ 4 2 51x ϩ 2 2 ϭ 51x ϩ 2 21x Ϫ 2 2 51x ϩ 2 2 ϭ 1x Ϫ 22. 1.5 40 ϭ x 60 2.5 40 ϭ x 60 – QUANTITATIVE PRETEST – 313 10. b. The diagonals of both parallelograms and rhombuses bisect each other. Isosceles trapezoids have diagonals that are congruent, but do not bisect each other. 11. d. Either statement is sufficient. If k + 1 is odd, then one less than this, or k, must be an even number. If k + 2 is even and consecutive even numbers are two apart, then k must also be even. 12. e. Neither statement is sufficient. Statement (1) states that one of the angles is 90 degrees, but this alone does not prove that all four are right angles. Statement (2) states that one pair of nonadjacent sides are the same length; this also is not enough information to prove that both pairs of opposite sides are the same measure. 13. b. Since statement (1) says there are more than 30 nickels, assume there are 31 nickels, which would total $1.55. You would then need two dimes to have the total equal $1.75 from statement (2). Both statements together are sufficient. 14. b. Substitute possible numbers for x.Ifx ϭ 2, then (2) 2 ϭ 4. If x ϭϪ2, then (Ϫ2) 2 ϭ 4, so statement (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 × 2 × 2 ϭ 8; the value is positive. Therefore, from statement (2), x is positive. 15. b. Using statement (2), the formula for the area of the triangle, can be used to find the height. Let b ϭ the base and 2b ϭ the height. Therefore, the base is 6 and the height is 12. The information in statement (1) is not necessary and insufficient. 16. a. Statement (1) only has one variable. This quadratic equation can be put in standard form (x 2 + 6x + 9 ϭ 0) and then solved for x by either factoring or using the quadratic formula. Since statement (2) has variables of both x and y, it is not enough information to solve for x. 17. a. Parallel lines have equal slopes. Using statement (1), the slope of the line can be found by changing the equation 2y ϭ 3 + x to slope-intercept form, y = ᎏ 1 2 ᎏ + 3. The slope is ᎏ 1 2 ᎏ . Statement (2) gives the y- intercept of the line, but this is not enough information to calculate the slope of the line. 18. c. Statement (1) is insufficient because the information does not tell you anything about the individ- ual triangles. Statement (2) gives information about each triangle, but no values for the perimeters. Use both statements and the fact that the ratio of the perimeters of similar triangles is the same as the ratio of their corresponding sides. Therefore, 2x + 3x ϭ 30. Since this can be solved for x, the perime- ters can be found. Both statements together are sufficient. 19. d. Either statement is sufficient. If the average dollar amount of the three people is $49, then the total amount spent is 49 × 3 ϭ $147. If you let x ϭ the amount that Nancy spent, then 5x is the amount Judy spent and 3(5x) ϭ 15x is the amount that Steve spent. x + 5x + 15x + 21x. ϭ $7. Using state- ment (2), if Judy spent $35, then Nancy spent $7 (35 Ϭ 5). 147 21 36 ϭ 1 2 12b 21b 2ϭ b 2 . A ϭ 1 2 bh, – QUANTITATIVE PRETEST – 314 20. c. Statement (1) alone will not suffice. For instance, if an edge ϭ 3 cm, then Recall that volume is length times width times height. However, if you assume the volumes are equal, the two vol- ume formulas can be set equal to one another. Let x ϭ the length of the cube and also the height of the rectangular prism. Since volume is basically length times width times height, then x 3 ϭ 25x. x 3 Ϫ 25x ϭ 0. Factor to get x (x Ϫ 5)(x + 5) ϭ 0. Solve for x to get x ϭ 0, Ϫ5, or 5. Five is the length of an edge and the height. Statement (2) is also needed to solve this problem; with the information found from statement (1), statement (2) can be used to verify that the edge is 5; therefore, it follows that the two volumes are equal. 3 3  25 ϫ 3. – QUANTITATIVE PRETEST – 315 . the next question. Since randomly guessing an incorrect answer CHAPTER Quantitative Pretest 18 307 will lower your score, making an educated guess by eliminating. questions and treat them as if they all count.  Problem Solving Questions – QUANTITATIVE PRETEST – 308 1.abcde 2.abcde 3.abcde 4.abcde 5.abcde 6.abcde 7.abcde