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This completely revised edition reflects all of the new questions and question types that will appear on the new SAT, scheduled to be administered in Spring 2016. Students will discover: Hundreds of revised math questions with answer explanations Math strategies to help testtakers approach and correctly answer all of the question types on the SAT All questions answered and explainedHere is an intensive preparation for the SATs allimportant Math section, and a valuable learning tool for collegebound students who need extra help in math and feel the need to raise their math scores.
Dedication To Rhona: For the understanding, for the sacrifices, and for the love © Copyright 2016, 2012, 2009, 2005, 2000, and 1996 by Barron’s Educational Series, Inc All rights reserved No part of this publication may be reproduced or distributed in any form or by any means without the written permission of the copyright owner All inquiries should be addressed to: Barron’s Educational Series, Inc 250 Wireless Boulevard Hauppauge, New York 11788 www.barronseduc.com eISBN: 978-1-4380-6811-4 Publication date: June, 2016 Contents Preface LEARNING ABOUT SAT MATH Know What You’re Up Against Lesson 1-1 Lesson 1-2 Lesson 1-3 Getting Acquainted with the Redesigned SAT Multiple-Choice Questions Grid-In Questions SAT Math Strategies Lesson 2-1 Lesson 2-2 SAT Math Strategies You Need to Know Guessing and Calculators on the SAT THE FOUR MATHEMATICS CONTENT AREAS Heart of Algebra Lesson 3-1 Lesson 3-2 Lesson 3-3 Lesson 3-4 Lesson 3-5 Lesson 3-6 Lesson 3-7 Lesson 3-8 Lesson 3-9 Lesson 3-10 Lesson 3-11 Lesson 3-12 Some Beginning Math Facts Solving Linear Equations Equations with More Than One Variable Polynomials and Algebraic Fractions Factoring Quadratic Equations Systems of Equations Algebraic Inequalities Absolute Value Equations and Inequalities Graphing in the xy-Plane Graphing Linear Systems Working with Functions Problem Solving and Data Analysis Lesson 4-1 Lesson 4-2 Lesson 4-3 Lesson 4-4 Lesson 4-5 Lesson 4-6 Lesson 4-7 Lesson 4-8 Working with Percent Ratio and Variation Rate Problems Converting Units of Measurement Linear and Exponential Functions Graphs and Tables Scatterplots and Sampling Summarizing Data Using Statistics Passport To Advanced Math Lesson 5-1 Lesson 5-2 Lesson 5-3 Lesson 5-4 Lesson 5-5 Lesson 5-6 Rational Exponents More Advanced Algebraic Methods Complex Numbers Completing The Square The Parabola and Its Equations Reflecting and Translating Function Graphs Additional Topics in Math Lesson 6-1 Lesson 6-2 Lesson 6-3 Lesson 6-4 Lesson 6-5 Lesson 6-6 Reviewing Basic Geometry Facts Area of Plane Figures Circles and Their Equations Solid Figures Basic Trigonometry The Unit Circle TAKING PRACTICE TESTS Practice Test Practice Test How to Evaluate Your Performance on a Practice Test SOLUTIONS FOR TUNE-UP EXERCISES AND PRACTICE TESTS Worked out solutions for Chapters 3–6 Answer Explanations for Practice Test Answer Explanations for Practice Test T Preface his new edition of the Barron’s SAT Math Workbook is based on the redesigned 2016 SAT It is organized around a simple, easy-to-follow, and proven four-step study plan: STEP Know what to expect on test day STEP Become testwise STEP Review SAT Math topics and SAT-type questions STEP Take practice exams under test conditions STEP KNOW WHAT TO EXPECT ON TEST DAY Chapter gets you familiar with the format of the test, types of math questions, and special directions that will appear on the SAT you will take This information will save you valuable testing time when you take the SAT It will also help build your confidence and prevent errors that may arise from not understanding the directions on test day STEP BECOME TESTWISE By paying attention to the test-taking tips and SAT Math facts that are strategically placed throughout the book, you will improve your speed and accuracy, which will lead to higher test scores Chapter is a critically important chapter that discusses essential SAT Math strategies while also introducing some of the newer math topics that are tested by the redesigned SAT STEP REVIEW SAT MATH TOPICS AND SAT-TYPE QUESTIONS The SAT test redesigned for 2016 and beyond places greater emphasis on your knowing the topics that matter most from your college preparatory high school mathematics courses Chapters to serve as a math refresher of the mathematics you are expected to know and are organized around the four key SAT Math content areas: Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math (geometric and trigonometric relationships) These chapters also feature a large number and variety of SAT-type math questions organized by lesson topic The easy-to-follow topic and lesson organization makes this book ideal for either independent study or use in a formal SAT preparation class Answers and worked-out solutions are provided for all practice problems and sample tests STEP TAKE PRACTICE EXAMS UNDER TEST CONDITIONS Practice makes perfect! At the end of the book, you will find two full-length SAT Math practice tests with answer keys and detailed explanations of answers Taking these exams under test conditions will help you better manage your time when you take the actual test It will also help you identify and correct any remaining weak spots in your test preparation Lawrence S Leff Welcome to Barron’s Math Workbook for the NEW SAT e-Book version! Please note that depending on what device you are using to view this e-Book on, equations, graphs, tables, and other types of illustrations will look differently than it appears in the print book Please adjust your device accordingly This e-Book contains hundreds of hyper links that will bring you to helpful resources and allow you to click between questions and answers LEARNING ABOUT SAT MATH Know What You’re Up Against T his chapter introduces you to the test format, question types, and the mathematics topics you need to know for the redesigned 2016 SAT Compared to prior editions of the SAT, the new SAT ■ Places a greater emphasis on algebra: forming and interpreting linear and exponential models; analyzing scatterplots, and two-way tables ■ Includes two math test sections: in one section you can use a calculator and in the other section a calculator is not allowed ■ Does not deduct points for wrong answers LESSONS IN THIS CHAPTER Lesson 1-1 Getting Acquainted with the Redesigned SAT Lesson 1-2 Multiple-Choice Questions Lesson 1-3 Grid-In Questions LESSON 1-1 GETTING ACQUAINTED WITH THE REDESIGNED SAT OVERVIEW The March 2016 SAT test date marks the first administration of a redesigned SAT The mathematics content of the new version of the test will be more closely aligned to what you studied in your high school math classes The redesigned SAT is a timed exam lasting hours (or hours and 50 minutes with an optional essay) What Does the SAT Measure? The math sections of the new SAT seek to measure a student’s understanding of and ability to apply those mathematics concepts and skills that are most closely related to successfully pursuing college study and career training Why Do Colleges Require the SAT? College admissions officers know that the students who apply to their colleges come from a wide variety of high schools that may have different grading systems, curricula, and academic standards SAT scores make it possible for colleges to compare the course preparation and the performances of applicants by using a common academic yardstick Your SAT score, together with your high school grades and other information you or your high school may be asked to provide, helps college admission officers to predict your chances of success in the college courses you will take How Have the SAT Math Sections Changed? Here are five key differences between the math sections of the SAT given before 2016 and the SAT for 2016 and beyond: ■ There is no penalty for wrong answers ■ Multiple-choice questions have four (A to D) rather than five (A to E) answer choices ■ Calculators are permitted on only one of the two math sections ■ There is less emphasis on arithmetic reasoning and a greater emphasis on algebraic reasoning with more questions based on real-life scenarios and data New Math Topics Beginning with the 2016 SAT, these additional math topics will now be required: ■ Manipulating more complicated algebraic expressions including completing the square within a quadratic expression For example, the circle equation x2 + y2 + 4x − 10y = can be rewritten in the more convenient center-radius form as (x + 2)2 + (y − 5)2 = 36 by completing the square for both variables ■ Performing operations involving the imaginary unit i where i = ■ Solving more complex equations including quadratic equations with a leading coefficient greater than as well as nonfactorable quadratic equations ■ Working with trigonometric functions of general angles measured in radians as well as degrees Table 1.1 summarizes the major differences between the math sections of the previous and newly redesigned SATs TIP If you don’t know an answer to an SAT Math question, make an educated guess! There is no point penalty for a wrong answer on the redesigned SAT You get points for the questions you answer correctly but not lose points for any wrong answers (B) Consider each statement in turn I The amount of the compound present is decreasing by a fixed percent of whatever amount remains, not by a constant amount II Since the base of the exponential function is 0.8 and 0.8 = − 0.20 = − 20%, each hour the compound gets reduced by 20% of the amount present III Each hour the compound gets reduced by 20% of whatever is present rather than by the fixed amount of 80% of 500 so this statement is false (C) If x represents the number of pounds of peaches sold, then 165 − x represents the number of pounds of apples sold Thus, (C) Test the equation in each of the answer choices for x = to x = to see if it produces the correct values for y for each of the given values of x The equation in choice (C), y = 80(1.50)x, is the only equation that works For example, for x = 4, y = 80(1.50)4 = 80(5.0625) = 405 (B) Since the test score (x) must increase as the number of incorrect responses decreases (y), the linear pattern of dots must have a negative slope as in graph (2) (B) The ratio of cats to the total number of cats and dogs is proportion to solve for the total number of cats: 806 Set up a (B) Since rate × time = distance, ■ Find the distance the faster plane travels: ■ Find the distance the slower plane travels: ■ The difference in the distances traveled is 1.8 − 1.5 = 0.3 or 807 10 (A) Simplify the equations: Hence, 5y + = 3y + so 2y = −6 and y = −3 If y = −3, then x = 3(−3) + = −8 The value of 11 (B) The total amount of time swimming and running took one-half of the 12 hours and 30 minutes it took to complete the triathlon or hours and 15 minutes Let x represent the amount of time swimming and 4x represent the amount of time running the marathon Then The 26.2 mile marathon distance was run in hours or 300 minutes Since , the average rate of minutes per mile at which Dylan ran the marathon is closest to 11.5 808 12 (C) The rate of change in height when skiing from the top of the ski slope to its bottom is ■ Convert 30 miles per hour to feet per second: ■ Multiplying gives the change in height per second of the skier Thus, d(t) is the difference between the starting height of 11,000 feet and the product formed by multiplying the rate at which height is decreasing per second times the number of seconds that have elapsed in the ski run: ■ 809 13 (D) If x represents the number of years it will take for the two trees to grow to the same height, then 84 + 8x = 48 + 12x where each side of the equation repressents the height of the tree in inches Thus, 14 (B) From the table, fill in the numbers to form the ratio: 810 15 (C) If the circuit has moving parts, evaluate f(4): f(4) = 0.3(4)2 = 4.8 = t Next, evaluate g(4.8): 16 (C) For Texas, To change from millions to billions, divide by 1,000, which gives 6.3 billion 811 17 (A) Consider each statement in turn ■ I In 2015, the per capita spending for Illinois was , which was greater than the per capita spending for Texas of ■ was ■ II Per capita spending in Florida in 2015 was , where as in 2011 it so it declined in 2015 III New York rather than California had the highest per capita spending for each year 18 (D) For each point along the curve, the product of x and y is 120 Hence, an equation of the graph is xy = 120 or, equivalently, 19 (C) Find the ratio of the number of individuals earning at least $35,000 without a graduate school degree to the total number of individuals who earned at least $35,000: 20 (C) A dime is dollars, and a nickel is 05 dollars, so the total money from the dimes and nickels is × 10 + 05 × = 1.25 So, the amount that must come from quarters is 5.25 − 1.25 = 4.00 Since a quarter is 25 dollars, take 4.00 and divide it by 21 (A) Divide the number of males not taking AP classes, 23, by the total number of students in the school, 177, to get 22 (D) Find the rate of change of cruises sold: You can eliminate choices (A), (B), and (C) 23 (C) Write the equation x2 + 4x + y2 − 6y = 37 in center-radius form by completing the square for both variables: Hence, the center of the circle is at (−2, 3) Since r2 = 50, The correct combination of statements is Statements II and III 812 24 (A) Simplify the given function by factoring the numerator: 25 (C) Find the weighted average by calculating the sum of the products of the number of weeks of vacation and the number of people who receive the vacation, and then dividing the sum of the products by 25: 26 (D) If p(x) is a polynomial function and p(−1) = 3, then when x is divided by x − (−1) or, equivalently, x + 1, the remainder is 813 27 (A) A system of linear equations has no solution if the graphs of the lines are parallel Two lines are parallel when they have the same slope ■ If hy − 8x = 5, then ■ Find the value of h that makes the system of equations have no solution by setting the slopes equal: ■ 814 28 (B) Five troy ounces is equivalent to grams Since density is mass divided by volume, V: The volume is closest to cubic centimeters 29 (B) To have an increase in the confidence level for survey results, the sample size should be as large as possible and the standard deviation should be as small as possible A larger sample size will result in a data set that more completely mirrors the population, and if the standard deviation is low, the surveyor can be confident that there will not be great variation among the survey results 815 30 (A) It is given that the vertex of a parabola is (−4, k) so its equation is y = a(x + 4)2 + k Since it is also given that the parabola contains the points (0, 12) and (−3, 7), the coordinates of these points must satisfy the parabola equation ■ For (0, 12): ■ For (−3, 7): Solve the two equations simultaneously for k From the second equation, a = − k Substitute − k for a in the first equation: ■ 816 31 (28/3) Find the slope of each line ■ Since perpendicular lines have slopes that are negative reciprocals: 817 32 (405) Rate of change in depth in feet per second is Since 1.5 minutes is equivalent to 90 seconds, 1.5 minutes after his initial descent the diver will be below sea level 818 33 (3.2) Since |−3| = 3, rewrite the inequality as < 4x − < 10 Adding to each member of the inequality gives 12 < 4x < 13 Dividing each member of the inequality by makes or, equivalently, < x < 3.25 Hence, x can be any number between and 3.25, such as 3.2 34 (1,215) Assume that the ratio of the tagged animals drawn to its sample size is approximately the same If x represents the total number of fish of interest in the lake, then 35 (47) The volume of the rectangular container is 10 × 15 × 20 = 3,000 cubic inches The volume of the cylinder is πr2h = π(2)2(5) = 20π To find the maximum number of full cylindrical cups of water that can be placed into the container with no overflow, divide the volume of the rectangular container by the volume of the cylindrical cup and round the answer down: so, the maximum number of full cups of water without overflowing is 47 819 36 (42) The area of an equilateral triangle with side s is Hence, Since the pendant has sides each of which has a length of centimeters, the distance around the pendant is × = 42 centimeters 37–38 It is given that the function h(t) = −4.9t^2 + 88.2t describes the height of a projectile, in meters, after t seconds have elapsed 37 (18) The projectile will hit the ground when h(t) = 0: 38 (397) To find the maximum height, write the parabola equation in vertex form: The y-coordinate of the vertex, to the nearest meter, is 397 820 ... that represents the distance from the center of the upper base to the circumference of the lower base: 29 Since the diameter of the base is 6, the radius is ■ The height h of the cylinder is... GETTING ACQUAINTED WITH THE REDESIGNED SAT OVERVIEW The March 2016 SAT test date marks the first administration of a redesigned SAT The mathematics content of the new version of the test will be more... success in the college courses you will take How Have the SAT Math Sections Changed? Here are five key differences between the math sections of the SAT given before 2016 and the SAT for 2016 and