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    MATH1

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    MATH1 SAT

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    Welcome to the SAT Teaching SystemsWeve developed our educational package to integrate you, your students, the video component,

    and the supplemental materials into an effective learning system.

    The program delivers information in a clear, concise, example-lled manner that teaches with the perspectiveof the learner in mind. The supplemental material allows students structured opportunities to practiceand enhance their knowledge of basic and advanced concepts.

    Each module contains the following items: a lesson plan, worksheets, and various testing components,and a practice exam.

    The Lesson Plan has three parts:

    Pre-viewing reviews the basic elements of the SAT test.

    Viewing the program offers a fun fast-paced way to teach important concepts.

    Post-viewing provides worksheets to reinforce the concepts taught in the video.

    Testing components consist o:

    Worksheets that have your students practice the material to reinforce the concepts and topicsintroduced.

    PracticeTestwhichcoversallthelearningobjectivesandcanbeusedeitherasahomework

    assignmentorasapracticetestinclass.

    We hope that you and your students nd Teaching Systems benecial and enjoyable. Be sure to check outCerebellum.com for special offers, new subjects, and other great resources!

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    MATH1 SAT

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    Lesson PlanVideo: 150 minutes Lesson: 3 days

    Pre-Viewing

    :00 Warm Up:The Math section covers arithmetic, percentages, decimals, order of operations, fractions, averages,

    ratios, statistics, probability, geometry, functions, and algebra (including higher level Algebra II).

    :00 Test-Prep:In each math section of the SAT, the questions are arranged in order of difculty. To help you allocate yourtime on the SAT, we like to label the questions with three degrees of difculty: the Good, the Bad andthe Ugly. The student-produced response questions are the only part of the SAT Math Section that are

    not multiple choice. These questions require you to ll in your own answer by marking the ovals on youranswer grid.

    Viewing

    :04 Playing Video:Since this workbook and the SAT Math module cover the same material, you can watch one wholemodule then do the relevant workbook part, watch part of the module and work on that part of theworkbook, try the workbook rst and then watch the DVDits all up to you! The great thing about theDVD is that you can always go back and review any sections or subjects that are giving you trouble. Theworkbook and the video are an unbeatable tagteam combo.

    :04 Wrap Up:When youre ready, you can have students take the Practice Tests provided on the CD-ROM. The idea isthat if you take these tests in a setting similar to the real tests your students will be better prepared cometest day.

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    MATH1 SAT

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    SAT Math Section Contents

    Drill 1: The Good, the Bad, and the Ugly 7

    Drill 2: Student-Produced Response Questions 8

    Drill 3: Denitions 9

    Drill 4: Percentages 9

    Drill 5: Percent Increase & Decrease 10

    Drill 6: Decimals 11

    Drill 7: Fractions 11

    Drill 8: Average Questions 13Drill 9: Median & Mode 14

    Drill 10: Square Roots 15

    Drill 11: Exponents 16

    Drill 12: Ratios 17

    Drill 13: Proportions 19

    Drill 14: Algebraic Manipulation 20

    Drill 15: Inequalities 22

    Drill 16: Simultaneous Equations 23

    Drill 17: Absolute Value, Direct & Inverse Variation 24

    Drill 18: Quadratic Equations 26

    Drill 19: Functions 26

    Drill 20: Domain and Range 28

    Drill 21: Functions as Models 30

    Drill 22: Algebra: Experiments 33

    Drill 23: Algebra: Using Actual Numbers 34

    Drill 24: Algebra: Working Backwards With the Answers 36

    Drill 25: Probability 37

    Drill 26: Geometry: Angles 38

    Drill 27: Geometry: Triangles 40

    Drill 28: Geometry: Perimeter, Area, Parallel Lines 43

    Drill 29: Geometry: Circles and Volume 45

    Drill 30: Coordinate Plane and Slope 47

    The math portion of this workbook needs to be completed with a calculator, the same one that you will use on the actual test. It is alsoimportant that you practice your SAT work under quiet, test-like conditions to create the same kind of environment that you will experienceon the day of the test.

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    MATH1 SAT

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    Drill 1: The Good, the Bad, and the UglyIn each math section of the SAT, the questions are arranged in order of difculty. To help you allocate your timeon the SAT, we like to label the questions with three degrees of difculty the Good, the Bad and the Ugly.Every student needs to do all the questions listed as Good and Bad. Picking up points on these questions iscrucial. Do not rush through the Good and Bad questions to get the Ugly ones.

    On the 25-minute, 20-question multiple-choice math section:

    The GoodQuestions 1 to 8.

    The BadQuestions 9 to 17.

    The UglyQuestions 18 to 20.

    Everyone needs to do at least 117.

    On the 20-minute, 16-question multiple-choice math section:

    The GoodQuestions 1 to 5.

    The BadQuestions 6 to 12.

    The UglyQuestions 13 to 16.

    Everyone needs to do at least 111.

    As you attempt each question, you need to know if it is Good, Bad, or Ugly. The expectation of how difcult the

    question is will help you avoid traps.If you are shooting for a 500 you need to do all the Good and Bad questions. Do not worry about the Uglyquestions.

    If you are shooting for a 600 you need to do 18 questions on both of the 20-question sections and 14 on the16-question section.

    If you are shooting for a 700 you need to do all the questions.

    Everyone wants to score as high as possible on the SAT. However, you cant realistically shoot for 700 untilyou can get to 600. Likewise, you cant shoot for 600 until you can get 500. Improvements come in steps.Increasing the number of questions you attempt in a section leaves less time for the Good questions. Thus,doing more questions before you are ready can actually lower your score.

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    MATH1 SAT

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    Drill 2: Student-Produced Response QuestionsThe student-produced response questions are the only part of the SAT Math Section thatare not multiple choice. These questions require you to ll in your own answer by marking the ovals on youranswer grid.

    Your answer will be graded as correct whether it is entered as a fraction or a decimal, as long as the answer tsinto the grid. If your answer to a question is 100 over 200, this would need to be reduced because it does not tinto the four slots available to grid-in your numbers.

    Practice entering in your answers.Hint: Ifittsintheanswergrid,youhaveyouranswer.Dontreduceorroundoff

    ifyouareabletotinyourresponse.Savethetime.

    1. 5/10 2. 2.5 3. 15/35 4. 15 1/2 5. .5767

    (Answersareonpage48-50)

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    Drill 3: DefnitionsIntegers: All numbers except fractions and decimals. For example: -7, 0, 2 are all integers.

    Even: Divisible by 2. -4 is even. 8 is even. And dont forget that 0 (zero) is even, too.

    Odd: Not divisible by 2.

    Positive: Greater than 0. 1/2 is positive. So is 0.4 and 100.

    Negative: Less than 0.

    Prime numbers: A prime number is only divisible by itself and 1.

    Whole numbers: Any number except for fractions and negatives.

    Digits: The numbers 0 through 9.

    Consecutive numbers: Numbers that are in order. 2, 3, 4, 5, etc.

    Distinct: Numbers that are different. 4 and 3 are distinct. 4 and 4 are not distinct.

    Order of Operations: PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

    Divisibility: Dividing so that there is nothing left over. For example, 8 is divisible by 4 since 4 divides perfectly into8. 9 is not divisible by 4 since 4 does not go perfectly into 9.

    Remainder: The part left over when you divide. When 9 is divided by 4 the remainder is 1.

    Multiples: Numbers that your original number divides into perfectly. Multiples of 4 would be 4, 8, 12, 16, 20, 24,etc. Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, etc.

    Factors:All the numbers that divide perfectly into your original number. The factors of 16 are 1, 2, 4, 8, 16. Thefactors of 50 are 1, 2, 5, 10, 25, 50.

    Zero: Zero is even and an integer. However, 0 (zero) is neither positive nor negative.

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    Drill 4: PercentagesREMEMBER: You can convert a percent to a decimal by moving the decimal point two places to the left. (35% =.35) Or you can convert a percent to a fraction by placing it over 100. (35% = 35/100) Remember, a percent issimply a part over a whole, times 100.

    Problems: (Answers are on pages 40-43.)

    1. What is 40% of 70?

    2. 80% of 25 =

    3. Which is greater, 30% of 45, or 45% of 30?

    4. 60 percent of 40 percent of 300 is equal to which of the following?

    (A) 12 percent of 300

    (B) 18 percent of 300

    (C) 20 percent of 300

    (D) 24 percent of 300

    (E) 30 percent of 300

    5. In a class of 24 students, 9 students scored between 80% and 90% on a test, 3 scored over 90%, and 4scored between 70% and 80%. What percentage of students scored below 70% on the test?

    (A) 66%

    (B) 50%

    (C) 33%

    (D) 24%

    (E) 13%

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    Drill 5: Percent Increase & DecreaseHere is the formula:

    Percent increase or decrease = number increase or decrease / original whole

    Problems: (Answers are on pages 40-43.)

    1. A student was able to read 30 pages in an hour. After taking a speed reading course, the student was able toread 45 pages an hour. By what percent did the students reading ability increase?

    (A) 15%

    (B) 30%

    (C) 45%

    (D) 50%

    (E) 75%

    2. During the rst semester at law school, there were 350 students enrolled. At the start of the second semester,there were 270 students. By approximately what percent did the rst-year student body decrease?

    (A) 15%

    (B) 23%

    (C) 31%

    (D) 37%

    (E) 45%

    3. After a stern memo was circulated at the ofce, monthly production levels of new computers went up 25%. If232 computers a month were being produced before the memo, how many were being produced a month afterthe memo?

    (A) 240

    (B) 258

    (C) 290

    (D) 312

    (E) 348

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    Drill 6: DecimalsDont be fooled into wasting time on decimal problems like these below. Pull out your calculator and startpushing buttons. When practicing on these problems, be sure to use the same calculator you intend to use whenyou take the SAT.

    Problems: (Answers are on pages 40-43.)

    1. 4.02 + 6.679 =

    2. 5.31 + 7.006 =

    3. 4.9 - 6.23 =

    4. 7.67 x 3.1 =

    5. 9.24 3.67 =

    Drill 7: Fractions

    Here are the basics: To add or subtract fractions, nd a common denominator. To multiply fractions, just multiplythe numerators by the numerators, and the denominators with the denominators. And to divide fractions, ip thesecond fraction over, and then multiply them.

    Problems: (Answers are on pages 40-43.)

    1. 2/3 + 4/5 =

    2. 3/4 + 7/12 =

    3. 8/9 7/3 =

    4. 3/7 7/8 =

    5. 6/7 x 19/21 =

    6. 12/13 x (-5/8)

    7. 2/5 3/7 =

    8. 9/25 5/3 =

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    Drill 8: Average QuestionsAverage Questions (also known as arithmetic mean) will always involve three pieces of information:the average, the number of items being averaged, and the total sum of all the things being averaged.

    A typical SAT question will give you two of these three pieces of information. Your job is to gure outthe missing piece.

    There are 3 scenarios:

    1) To solve for an average, divide the total sum by the number of items.

    2) To nd the total sum of all the items being averaged, multiply the average bythe number of items being averaged.

    3) To nd the number of items being averaged, divide the total sum by the average.Problems: (Answers are on pages 40-43.)

    1. Bobby took 3 tests and scored an 87, 93, and 99.What was the average (arithmetic mean) of his threetest scores?

    2. If Dougs average phone bill for the year came outto 40 dollars per month, how much money did Doug

    spend on his phone bill for the entire year?

    3. If 25 is the average of 14, x, and 40, what is thevalue of x?

    4. The average (arithmetic mean) of three numbers is

    29. If two of the numbers are 21and 24, what is the third number?

    (A) 13

    (B) 29

    (C) 42

    (D) 45

    (E) 87

    5. What is the average (arithmetic mean) of all evenintegers from 1 to 20 inclusive?

    (A) 8

    (B) 10

    (C) 11

    (D) 12

    (E) 20

    6. The top three students at Tony Clifton High Schoolaveraged a 96 test score on the Spanish nal. If theaverage of two of the students was 94, what did thethird student score on the test to bring their collectiveaverage up to 96?

    (A) 90

    (B) 94

    (C) 96

    (D) 98

    (E) 100

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    7. An airline sold 60 coach tickets, each at a price of$200. This same airline also sold 20 rst-class tickets,each for $600. What was the average cost of a ticketon this ight?

    (A) 200

    (B) 300

    (C) 400

    (D) 510

    (E) 710

    8. If a basketball player averaged 34 points a gameduring a 6-game series, and scored 54 during the sixth

    and nal game of this series, how many points did theplayer average over the rst 5 games?

    (A) 28

    (B) 30

    (C) 34

    (D) 38

    (E) 54

    9. For the 5 months from January 1st until the endof May, a bus service that operates between LosAngeles and Las Vegas sold an average of 600 roundtrip tickets per month. If the company sold 900 ticketsin January and 600 tickets in February, what is theaverage number of tickets that were sold in March,

    April and May?

    (A) 500

    (B) 600

    (C) 900

    (D) 1500

    (E) 3000

    Drill 9: Median & Mode

    The median is dened as the middle number in a group of numbers. In order to determine what the median is,it is important to put your numbers in ascending order. If your set has an even amount of numbers, take theaverage of the middle two numbers to nd your median.

    The mode is the number that appears most often in a set of numbers. One good way to remember what modemeans is to think the most since mode and most sound alike. Also important to note is that there can be morethan one mode in a set of numbers.

    Problems: (Answers are on pages 40-43.)

    Find the median in each of the following sets ofnumbers:

    1. (6, 9, 10, 2, 5)

    2. (2, 3, 4, 5, 1)

    3. (8, 2, 4, 1)

    Find the mode in each of the following sets ofnumbers:

    4. (3, 2, 4, 5, 2, 4, 4)

    5. (1, 1, 2, 3, 4, 5, 6, 6, 7)

    6. (12, 15, 15, 15, 16, 17, 17)

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    7. What is the median of the rst 3 positive multiplesof 7?

    (A) 3

    (B) 7

    (C) 14

    (D) 21

    (E) 28

    Set A = (12, 13, 15, 19, 23, 23, 24, 30)

    Set B = (12, 8, 27, 25, 31)

    8. What is the average (arithmetic mean) of the modeof set A and the median of set B?

    (A) 12

    (B) 23

    (C) 24

    (D) 25

    (E) 31

    Drill 10: Square Roots

    The square root of a number is the number that needs to be multiplied by itself, or squared, to get to thatnumber.

    Rules:

    1) You CAN multiply or divide square roots.

    For example: 2 5 = 10

    2) You CANNOT add or subtract square roots.For example: 2 + 5 7

    Hints:

    1) The square root of zero is still equal to zero. 0 = 0

    2) The square root of one equals one. 1 = 1

    3) The square root of any fraction between zero and one gets LARGER. 1/4 > 1/4

    Thus, when taking the quantitative comparison section of the SAT, dont automatically assume that a square rootmakes a number smaller. It can be equal or even larger.

    Problems: (Answers are on pages 48-50.)

    1. 316 39 =

    (A) 3

    (B) 4

    (C) 5

    (D) 37

    (E) 7

    2. 10 5 =

    (A) 5

    (B) 52

    (C) 225

    (D) 252

    (E) 505

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    3. 1/4 + 1/25 =(A) 2/29

    (B) 1/5

    (C) 1/4

    (D) 1/2

    (E) 7/10

    Drill 11: Exponents

    An exponent is the number of times a base is raised to a power.

    Rules:

    1) When you multiply exponents together, you actually add them. When you divide exponents, you subtract. x2multiplied by x3 equals x5.

    2) As was the case with square roots, you CANNOT ADD or SUBTRACT exponents. X2 + X3 does not equal X5.

    Hints:

    1) Zero squared equals zero and one squared still equals one. A number squared can still equal itself.

    Also squaring a fraction or decimal actually makes the number smaller.

    1/4 = 1/16

    These are both important concepts when attacking quantitative comparison questions.

    2) Please note that -102 = -100 since a negative times a negatives gives a positive. So when you are solving anexponent question the answer can be the positive or negative version of itself. Thus if x2 = 100, x can equal 10or -10. This concept is very important for quantitative comparison questions.

    Problems: (Answers are on pages 40-43.)

    1. (7) (104) + (2) (103) + 4 =

    (A) 7,204

    (B) 70,204

    (C) 72,004

    (D) 72,040

    (E) 72,404

    2. x2 = 8, then x4 =

    (A) 64

    (B) 32

    (C) 16

    (D) 8

    (E) 4

    3. If y4 = 81, then 2y =

    (A) 2

    (B) 3

    (C) 4

    (D) 8

    (E) 16

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    Drill 12: Ratios

    Aratio compares how many parts you have of two or more things.

    If a 1st grade class trip has 1 parent for every 5 children, then the ratio of parents to children is 1 to 5. This canbe written as 1:5 or 1/5; they both mean the same thing. The important thing to note is that these expressionsare comparing the ratio of parents to children. These numbers are not necessarily the actual number of parentsand children on the trip, but for every 1 parent on this trip, there will be 5 children as well. This relationship willremain constant.

    Most SAT ratio questions are designed to have you add up the parts in your ratio. 1 parent for every 5 childrenmeans that 6 is your total number of parts, or in this case, people. This simple addition step is usually the rststep for solving any ratio question.

    Now were in a position to answer some basic ratio questions. What fraction of the trips participants are

    parents? 1/6 (1 out of every 6 people). What fraction of the trips participants are children? 5/6 (5 out of every 6people). In order to answer either of these questions, we rst needed to determine that 6 is the total number ofpeople from which these fractions would be judged. So the rst thing we do when we come to a ratio questionis to add up the parts to nd the total.

    1 parent for every 5 children dealt with 6 people at a time. This is what we call the Before. If we are told thatthere are 30 people on the trip, this is what we call the After. In orderto get your numbers to the After, you must gure out something that is known as the ratio jumper.

    In order to solve this problem you must ask yourself, how do I get from the Before to the After or specically,What number do I multiply 6 by to get to 30? Or you can go in reverse and gure out what you need to divide30 by to get to 6. Either way, this work needs to be done to determine the ratio jumper.

    In this case the ratio jumper is 5. Once you establish this number, the rest of this question becomes calculatorwork because you will multiply all of the Before numbers by the same ratio jumper number.

    Parents Children Totalpeople

    Before 1 5 6

    RatioJumper 5 5 5

    After 5 25 30

    Problems: (Answers are on pages 48-50.)

    1. The ratio of attendance at a college basketballgame was recorded as 14 students for every 1professor. If there were 3000 people at the game, howmany of them were professors?

    (A) 1

    (B) 14

    (C) 200

    (D) 256

    (E) 2800

    2. If the ratio of dogs to cats at an animal shelteris 7 to 5, and dogs and cats are the only animals atthe shelter, what fractional part of the animals at thisshelter are cats?

    (A) 7/5

    (B) 5/7

    (C) 7/12

    (D) 5/12

    (E) 12/35

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    3. If x:y:z = 1:3:9 and z = 27, then x + y =(A) 4

    (B) 12

    (C) 13

    (D) 18

    (E) 39

    4. In a room containing only children, the ratio of boysto girls is 2:3. Boys are what fractional part of the totalchildren in the room?

    (A) 3/2(B) 2/3

    (C) 3/5

    (D) 2/5

    (E) 2/9

    5. The instructions for sewing a sweater suggest using6 feet of red yarn for every 3 feet of white yarn and1 foot of blue yarn. If the total sweater uses 15 feetof white yarn, how much yarn is used on the entiresweater?

    (A) 15

    (B) 50

    (C) 90

    (D) 150

    (E) 300

    6. A college basketball team has a win-to-loss ratio of4 to 3. If the team has played a total of 35 games, howmany more games has the team won than lost?

    (A) 1

    (B) 4

    (C) 5

    (D) 7

    (E) 20

    Drill 13: Proportions

    A typical proportion question gives you two sets of fractions with one of the four numbers missing. Your job isto cross-multiply to solve for the missing variable. These are GREAT calculator questions. The key is to keepyour numbers consistent.

    Problems: (Answersareonpages40-43.)

    1. A weight of 3 pounds is equal to 48 ounces. Aweight of 1/2 pound is equal to how many ounces?

    (A) 48

    (B) 32

    (C) 16

    (D) 8

    (E) 4

    2. A wheel turns 60 times every 3 minutes. At thisrate, how many times will the wheel turn in 4 minutes?

    (A) 20

    (B) 40

    (C) 60

    (D) 80

    (E) 100

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    3. If a recipe that feeds 4 people uses 6 ounces ofavoring, how many ounces of avoring are needed tofeed 6 people?

    (A) 6

    (B) 8

    (C) 9

    (D) 12

    (E) 24

    4. At a kindergarten lunch, each child will eat one slice

    of pizza. If each pizza contains 8 slices, and there are256 children at the kindergarten, how many pizzas areneeded to ensure that each child has one slice?

    (A) 256

    (B) 128

    (C) 32

    (D 16

    (E) 8

    5. Jeff requires 7 hours of sleep per night during theve-day school week, and 9 hours of sleep per nightover his two-day weekends. How many hours ofsleep does Jeff get during the course of a 16-weeksemester?

    (A) 35

    (B) 53

    (C) 256

    (D) 716

    (E) 848

    Student-produced Response Question:

    6. There are 22 students for every 1 teacher at anelementary school. If 14 teachers work at the school,how many students go to the school?

    Drill 14: Algebraic ManipulationMost basic Algebra problems are designed to make you solve for a missing variable. In order to do these

    questions your job will be to isolate that missing variable on one side of the equation. You can add, subtract,multiply or divide, but remember to do it to both sides of the equation. That is the key step in algebrayou cando whatever you want, as long as you do the same thing to both sides of the equation.

    One good way to attack these questions is to think in terms of opposites. If you are solving for x, and one side ofthe equation contains 2x (which is 2 times x), you will want to divide both sides of the equation by 2. If one sideof the equation contains x - 3, you will want to add 3 to both sides of the equation. Doing the opposite will oftenhelp get rid of the unwanted parts and isolate the variable on one side of the equation.

    Problems: (Answers are on pages 40-43.)

    1. Solve for x. 3x + 10 = 34

    2. Solve for y. 2y 5 = 19

    3. If a = 4 then (2 a)/2 =

    (A) -2

    (B) -1

    (C) 0

    (D) 1

    (E) 2

    4. If 14 - y = 3y 2, then y =

    (A) 0

    (B) 2(C) 4

    (D) 6

    (E) 8

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    5. If (t + 3) (4 + 22) = 40, then t =(A) 1

    (B) 2

    (C) 3

    (D) 4

    (E) 8

    6. If 4x + 20y = 88, then x + 5y =

    (A) 4

    (B) 8

    (C) 11

    (D) 22

    (E) 44

    7. If 14/a = 42/9, then a =(A) 2

    (B) 3

    (C) 4

    (D) 6

    (E) 9

    8. If 5x-7 = 28, then 3x =

    (A) 7

    (B) 14

    (C) 21

    (D) 28

    (E) 35

    Drill 15: Inequalities

    < Means less than

    > Means greater than

    The rule to memorize

    WHEN YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER, THE LESS THAN OR GREATER THAN SYMBOL FLIPS.

    If you forget to switch the sign, you will get this question wrong.

    Problems: (Answers are on pages 40-43.)

    1. If -2x + 10 < 20, then

    (A) x < -5

    (B) x > -5

    (C) x < 5

    (D) x > 5

    (E) x < 10

    2. If 4x 6 < 18 + 6x, then

    (A) x < 12

    (B) x > 12

    (C) x < -12

    (D) x > -12

    (E) x < 24

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    3. If 2 < x < 5 and 3 < y < 8, which of the following

    must be true for x + y?(A) 1 < x + y < 8

    (B) 2 < x + y < 8

    (C) 3 < x + y < 8

    (D) 3 < x + y < 13

    (E) 5 < x + y < 13

    4. If 3 < A < 7 and 4 < B < 10, which of the following

    must be true for b a?(A) -3 < b a < 7

    (B) 1 < b a < 3

    (C) 4 < b a < 6

    (D) 7 < b a < 17

    (E) 10 < b a < 14

    Drill 16: Simultaneous Equations

    When dealing with simultaneous equations, rst line them up and then combine the two equations by either

    adding or subtracting. The most common mistake is to try to deal with the two equations separately. When youcombine the equations by adding or subtracting, one of the variables will drop out and you will be able to solvefor the other variable. Once you have one variable, simply substitute it back into one of the original equations tosolve for the other variable.

    Problems: (Answers are on pages 40-43.)

    1. If 2a + 5b = 20 and 3a 5b = 30, then a = ?

    2. If 2x + 4y = 10 and 3x + 5y = 20, then 5x + 9y = ?

    3. If 3x + 4y = 24 and 4x + 3y = 25, then x + y = ?

    4. If a = 4 + b and 3a = 12 2b, what is the value of a?

    (A) 24

    (B) 12

    (C) 8

    (D) 4

    (E) 3

    5. If a + b = 16, b + c = 20, and a + c = 40, then a + b + c = ?

    (A) 30

    (B) 32

    (C) 34

    (D) 36

    (E) 38

    6. If 3b + 4c = 30, then 12b + 16c = ?

    (A) 15

    (B) 30

    (C) 60

    (D) 90

    (E) 120

    7. Three roses and two tulips cost $10.00 and fourroses and ve tulips cost $18.00. How much do onerose and one tulip cost?

    (A) 2

    (B) 4

    (C) 7

    (D) 14

    (E) 28

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    Drill 17: Absolute Value, Direct & Inverse VariationAbsolute Value

    Simply put, the absolute value of a number is the distance between that number and 0 (zero) on the number line.

    Very important point! Absolute value is a distance, so its ALWAYS POSITIVE. Again, because the absolute valueof a number is a distanceits distance from 0 on the number lineit is always positive.

    What is the absolute value of 8? 8.

    What is the absolute value of negative 8? 8 again.

    Absolute value = always a positive number.

    Heres how to write an absolute value:

    l8l = 8. This says that the absolute value of 8 is 8.

    l-8l = 8. This says that the absolute value of -8 is 8.

    (Notice that two lines are on either side of the number were trying to nd the absolute value of.)

    Heres another way to think of absolute value that will help you out. Take the number8 again. 8 is the absolute value of what 2 numbers?

    Thats right, 8 and -8. Every positive number is the absolute value of two numbers.That number itself and its negative.

    Direct Variation

    The equation for direct variation is:

    x = ky where k is a constant, and x and y are variables.

    This seems like a tricky equation, but all its saying is that y changes directly as x does.

    That means, when x changes, y changes in the same way. If x doubles, y doubles. Ifx triples, y triples. And so on.

    Just remember the equation x = ky.

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    Inverse VariationThe equation for Indirect Variation is:

    xy = k where k is a constant, and x and y are variables.

    All this is saying is that y changes inversely as x changes. When x changes, y changes in an opposite way. If xdoubles, y gets cut by half. For any inverse variation question, just remember the equation xy = k.

    Then just plug in the numbers they give you, and use your algebra knowledge to solve forx or y, whichever the question asks for.

    Problems (Answers are on pages 40-43.)

    1. Give the absolute value of the followingexpressions:

    A) -4B) -1/2

    C) 64

    D) 3/4

    E) -12

    Give the formulas for:

    2. Direct variation

    3. Inverse variation

    Drill 18: Quadratic Equations

    FOIL... stands for First, Outside, Inside, Last. This is still the best way to convertan unfactored equation into a factored one.

    The most common quadratic equations used on the SAT are:(x + y) 2 = x 2 + 2x y + y 2

    (x y) 2 = x 2 2x y + y 2

    (x + y) (x y) = x 2 y 2

    Memorizing the above 3 equations will save you time on any SAT question thatinvolves factoring.

    Problems: (Answers are on pages 40-43.)

    1. If x y =5 and x2 y 2 = 15, then x + y =? 2. If a/b + b/a = 8

    what is the value of (a + b) (1/a + 1/b)?(A) 6

    (B) 10

    (C) 16

    (D) 32

    (E) 64

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    Drill 19: FunctionsFunctions are one of the most intimidating-looking types of problems on the math sections of the SAT. However,these questions are usually more bark than bite. Function questions will give you a strange looking symbolalong with a formula next to it. Your job is to run some numbers through this formula to come up with an answer.Remember, the symbols in these problems have no mathematical value other than what the problem assigns tothem. Dont worry if you dont recognize the symbolsno one else will either.

    Problems: (Answers are on pages 40-43.)

    Questions 1, 2, and 3 refer to the following function:

    a and b are distinct integers.

    a b equals the larger of the numbers a and b.

    a b equals the smaller of the two numbers a and b.

    1. What is the value of (32)?

    (A) 1

    (B) 2

    (C) 3

    (D) 4

    (E) 5

    2. What is the value of (2

    3)

    (5

    4)?(A) 1

    (B) 2

    (C) 3

    (D) 4

    (E) 5

    3. What is the value of (46) (21)?

    (A) 1

    (B) 2

    (C) 4

    (D) 5

    (E) 6

    Questions 4, 5, and 6 refer to the following function:

    For all positive integers x greater than 1, let x be the product of all positive integers less than x.For example, 4 = 3 2 1 = 6

    4. What is the value of 3 3?(A) 3

    (B) 4

    (C) 6

    (D) 9

    (E) 81

    5. What is the value of 5 - 4?(A) 1

    (B) 2

    (C) 6

    (D) 18

    (E) 24

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    6. What is the value of 4

    ?(A) 6

    (B) 16

    (C) 24

    (D) 64

    (E) 120

    7. a and b are non-zero integers and a $ b = 5 a / b. What is the value of 4$10?

    (A) 10$4

    (B) 7$4

    (C) 6$5

    (D) 6$15

    (E) 2$$10

    Function Notation

    On test day, youll see ofcial function notation. Dont worry, its not difcult!

    F(x) This is read as F of x.

    Heres a sample function problem.

    F(x) = 10x + 2 This is read as F of x equals 10 times x plus 2.

    F(3)

    Your task is to solve this function when x = 3. [We get that from F(3).]

    All we do is plug in 3 for x. 10 times 3 equals 30. 30 plus 2 equals 32. Thats it.

    Just be aware that they might throw in another letter other than F, but its all the same.

    G(x) This is read as G of x. No real difference!

    H(x) This is read as H of x. Again, no real difference. Just be prepared!

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    Drill 20: Domain and RangeOther concepts about functions that youll need to know are domain and range.The domain of a function is the set of values for which the function is dened.

    The range of a function is the set of the results of the function.

    What does all that mean? For domain, its just the set of all input values for x. That is, domain describes whatmakes sense to plug in for x. For range, thats just the results or the output of the function.

    Lets look at a function and determine its domain.

    F(x) = X + 2X 3

    Now, heres something the SAT will check to see if you know; you cant have a 0 (zero) in the denominator of afraction. So, for this function, what numbers for x will result in a zero in the denominator?

    3. If we plug in 3 for x, in the denominator we have:

    3 3 = 0.

    We cant have that! -3 is ne for x, because -3 3 = -6.

    The domain of our function is the set of all number except 3. We write that as

    F(x) = X + 2X 3

    x 3.

    Heres another thing to remember that will help you out with the domains of functions: you cant get the squareroot of a negative numberit just doesnt exist.

    The -1? Doesnt exist.

    -2? Doesnt exist.

    So, if in a function problem you see x (your domain) under a square root, you have to make sure the numbersyou plug in for x wont result in a negative number.

    Example:

    F(x) = x - 2

    If we plug in anything less than 2 for x, we get a negative number under the square root.

    For instance, lets plug in 1 for x.

    1 2 = -1We cant take the square root of that.

    So the domain of this function is all numbers greater than or equal to 2. We write this as:

    F(x) = x - 2

    Domain = x 2.

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    Lets gure out the range of function. Remember, the range of a function is the set of all the possible values thatare the result of applying the function. What is the range of this function?

    F(x) = x2

    Well, look at x2. We know that any number squared wont end up negative, so x wont end up a negativenumber. So, x has to be a number greater than or equal to zero. The range is all POSSIBLE values, so

    The range of F(x) = x2 > 0

    Problems: (Answers are on pages 48-50.)

    1. If the function F is dened by f(x) = x2 6, then f(a b) is equivalent to

    (A) a2 2ab + b2 6

    (B) a2 2ab + b2 + 6

    (C) a2 + b2 36

    (D) a2 + b2 + 36

    (E) 2ab + b2 + 6

    2. Let the function K be dened by k(x) = 2 4x. If the domain of the function k is -2 < x < 4, what is thesmallest value in the range of the function?

    (A) -20

    (B) -14

    (C) 7

    (D) 14(E) 20

    3. Let the function F be dened by f(x) = 12 x2. If the domain of function f is -12 < x < 1, what is the largestvalue in the range of the function?

    (A) -12

    (B) -6

    (C) 0

    (D) 6

    (E) 12

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    Drill 21: Functions as ModelsThe SAT will ask you to consider some real-life situations involving functions. These require a little bit of thought,but are not particularly difcult.

    The graph above tells us about Suzys lemonade stand. It shows usthe number of cups of lemonade she sold at different prices.

    On the horizontal axis, you can see she sold cups of lemonade for 10cents, 20 cents, 30 cents, and 40 cents.

    The vertical axis shows us how many cups sold100, 200, 300, 400.

    The graph shows us the function of howthe price affects the number of cups sold.

    The question might ask something like this:

    If Suzy wants to sell the maximum number of cups of lemonade, what price should she set for a cup?

    Looking at the graph, you see that the line peaks at around 300 cups. Looking down, we see the price for thosecups was 20 cents. So 20 cents is our answer.

    Linear Functions

    Lets look at linear functions. A linear function is just an equation whosegraph is a straight line. Like this.

    Youll need to know this formula:

    y = mx + b

    The values of x and y can vary.

    m is the slope of a line.

    b is the y-intercept. This is where the line intercepts the y axis.

    400

    300

    200

    100

    $0.10 $0.20 $0.30 $0.40 $0.50

    0

    Suzys Lemonade Stand

    Price

    CupsSold

    Y

    X

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    Take a look at this graph of a linear function below:

    For this line, m (the slope) = -2.

    b (the y-intercept) = 4.

    Lets look at a problem with a graph.

    If the line above, line g, has a slope of -3, what is the y-intercept ofline g?

    Lets look at our formula: y = mx + b

    First off, we know that the y-intercept is b, so well be solving for b.Also, they give us the slope, -3.

    y = -3x + b

    Now we need to nd x and y. We can tell from the graph above thatthe point (2, 4) is on line g. Lets plug in these numbers.

    4 = -3(2) + b

    4 = -6 + b

    Lets get b by itself by adding 6 to each side.

    4 + 6 = b10 = b

    Our y-intercept is 10. Now we know that line g hits the y axis at 10.

    1

    2

    3

    4

    -4

    -3

    -2

    -1

    -4 -3 -2 -1 1 2 3 4

    0

    -1

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    -2

    -3

    -4

    -5

    -6

    -7

    -8

    -9

    -10

    1 2 3 4 5 6 7 8 9 10- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2

    0

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    Problem: (Answer is on page 40-43.)1. If an Alaskan dog sled team starts a race and travels 800 miles to the nish, as shown in the graph above,between what two points did the team reach its greatest average speed?

    (A) A to B

    (B) B to C

    (C) C to D

    (D) D to E

    (E) E to F

    0

    1

    A

    B

    C

    D

    E

    F

    200 300 400 500 600 700 800 900 1000

    2

    3

    4

    5

    6

    7

    8

    9

    10

    Drill 22: Algebra: Experiments

    Anytime you see a problem with variables, like x or y, you can avoid doing the algebra by experimenting withyour own numbers. Put in your own trial numbers for each of the variables to set up your experiment. After you

    assign numbers for all the variables, answer the question with those numbers. Whatever answer you come upwith will be YOUR answer to the question. Now go down to the answer choices and substitute in the numbersthat you have invented. The answer choice that gives you YOUR answer will be the correct answer to thequestion.

    Do you need to try it again with different numbers? No. Unlike the Quantitative Comparisons, there is nothing tomess up here. Run a set of numbers through the problem, come up with your answer, and nd it in the answerchoices. This will help you answer even the toughest-looking algebra questions.

    It is important to know that YOU WILL STILL GET THE ANSWER TO THE PROBLEM NO MATTER WHAT NUMBERS YOU USE ASLONG AS YOU ARE CONSISTENT. Since you can use any numbers you want, our recommendation is to use numbersthat make the math easy to do. So 10 is probably a better number to pick than 167.

    Problems: (Answers are on pages 40-43.)

    1. Chip can do x pushups every minute. How many pushups can Chip do in one hour?(A) x

    (B) 3x

    (C) 6x

    (D) 30x

    (E) 60x

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    2. Charles is 4 years older than Alex and 2 years olderthan Bob. If Alex is a years old, then in terms of Alex,the sum of their ages =

    (A) 3 a 4

    (B) 3 a 2

    (C) 3 a + 2

    (D) 3 a + 4

    (E) 3 a + 6

    3. Howard is now 5 years older than John was 2 years

    ago. John is now j years old. In terms of j, how manyyears old is Howard now?

    (A) j 5

    (B) j 3

    (C) j 2

    (D) j + 2

    (E) j + 3

    4 If x/4, x /5, and x /6 are integers, which of thefollowing is NOT necessarily an integer?

    (A) x /60(B) x /30

    (C) x /20

    (D) x /12

    (E) x /8

    5. The sum of three positive consecutive even integersis x. What is the value of the smallest of the threeintegers?

    (A) (x 6)/3

    (B) (x + 6)/3

    (C) x /3 6

    (D) x /3 + 6

    (E) 3 x 6

    6. If x is an odd integer, which of the following must

    also be an odd integer?(A) x 1

    (B) x + 1

    (C) 2 x

    (D) 2 x + 1

    (E) 2 x + 2

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    Drill 23: Algebra: Using Actual NumbersIfaquestionasksyoutondwhatfractionorpercentsomethingis,youcanuseyourownnumberstospeedup

    theprocess.ThisworksforanytimetheSATtriestogethypotheticalwithyou.

    Once again, it is better to think in terms of actual numbers as opposed to variables. In everyday life you dont doalgebra, you use actual numbers. Whenever you go shopping. Whenever you go to eat. This is what you do. So,it makes sense to use actual numbers wherever possible on the SAT.

    Your job is to pick numbers that meet the requirements of the question.

    Problems: (Answers are on pages 40-43.)

    1. If a and b are two consecutive odd integers thenb a =

    (A) 0

    (B) 1

    (C) 2

    (D) 3

    (E) 5

    2. If a (b + c) is a positive number, which of thefollowing must be positive?

    I. a II. b + c III. a + b + c

    (A) None

    (B) I only

    (C) II only

    (D) III only

    (E) I, II, and III

    3. As part of a Christmas sale, an electronics storereduces its stereo prices by a 20% discount. Thenlooking to spark even more business, this same storereduces its discounted price by another 25% on New

    Years Day. By what overall percent has the stereo beenreduced in price?

    (A) 50%

    (B) 45%

    (C) 40%

    (D) 25%

    (E) 20%

    4. Charlie does 1/3 of his homework during his lunchbreak and 1/2 of what remains on his ride home on

    the school bus. What fractional part of his homeworkremains?

    (A) 1/6

    (B) 1/3

    (C) 1/2

    (D) 2/3

    (E) 5/6

    5. A clothing designer discounts last yearsmerchandise by 50% of the original price. After nding

    no increase in sales, the designer discounts the newsales price by an additional 20%. By what overallpercent has the merchandise been reduced in price?

    (A) 20%

    (B) 30%

    (C) 60%

    (D) 70%

    (E) 75%

    6. On Monday, Joey read 1/4 of a novel for his English

    class. On Tuesday he read 1/3 of what was leftof the book. What fraction of the book did Joey readon Monday and Tuesday?

    (A) 1/4

    (B) 2/7

    (C) 1/3

    (D) 1/2

    (E) 2/3

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    Drill 24: Algebra: Working Backwards With the AnswersWhen you are attacking a word problem and are having a hard time setting up an equation, all is not lost. Veryoften you can work your way out of this predicament by going down to the answer choices. The great thingabout a multiple-choice test is that one of the ve answers MUST work. It may take a few tries, but you areguaranteed to nd the answer.

    Answer choices on the SAT always go in increasing or decreasing order. This means that when experimentingwith your answer choices, you want to start in the middle with answer choice C. This way, even if its notthe correct answer, it can still help you determine if you need a bigger or smaller number.

    Thus, if C gives you an answer that is too big, you certainly dont need to try the two answer choices that willbe larger (probably D and E). The answer has to be A or B. Try either one. If it works, youve got an answer.

    If it doesnt, you still have your answer as it will have to be the one you didnt try. This elimination techniqueis a good way to speed up and not have to try all ve answer choices.

    This technique will come in handy on Algebra problems as well as any time you nd yourself stuck on the test.Take advantage of the test being multiple-choice.

    Problems: (Answers are on pages 40-43.)

    1. When x is divided by 9, the remainder is 6, andwhen x is divided by 6, the remainder is 0. Which ofthe following numbers could be x?

    (A) 36

    (B) 100

    (C) 106(D) 108

    (E) 114

    2. If t + 3 is an even positive integer then t could bewhich of the following?

    (A) -3

    (B) -2

    (C) -1

    (D) 0

    (E) 2

    3. If 3a + 2 = 94 a, what is the value of a?

    (A) 0

    (B) 1

    (C) 2

    (D) 3(E) 4

    4. To celebrate his SAT Math score, Jesse orderedhimself a set of personalized pencils. If Jesse lost 1/4of the pencils the rst week he used them and 1/2 ofthe pencils that were left the second week, and Jessenow has 3 pencils remaining, how many pencils didJesse order originally?

    (A) 20

    (B) 16

    (C) 12(D) 8

    (E) 6

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    5. Frank weighs twice as much as George but 30 pounds less than Harry. If Harry weighs 110 pounds more thanGeorge, how much does Harry weigh?

    (A) 110

    (B) 160

    (C) 190

    (D) 200

    (E) 220

    Drill 25: Probability

    Probability might only be the topic of one or two questions on the entire test. To solve a probabilityquestion,you need to gure out the total number of occurrences and divide it by the number of times that the eventspecied in the question can happen. For example:

    What is the probability of rolling a die and getting a 4?

    If you roll a die, there are 6 different numbers you could get, 1 through 6. This means that 6 is the total numberof occurrences. Getting a 4 (which is the requirement of the question) would be 1 of 6 possible things that canhappen. Thus there is a 1/6 probability of rolling a die and getting a 4.

    In probability situations involving more than one event, gure out the individual probability of each occurrenceand multiply the results together.

    For example:If there is a 1/6 chance of rolling a 4, the probability of rolling two 4s would be

    1/6 1/6, which equals 1/36.

    Problems: (Answers are on pages 40-43.)

    1. A two-sided coin is ipped twice. What is theprobability that the coin will come up tails on bothips?

    2. What is the probability that it will come up heads onboth ips?

    3. Someone rolls two dice with faces ranging from 1to 6. What is the probability that the dice will add upto 7?

    4. A bubble gum machine contains gum balls coloredred, white and blue. If blue gum balls are 1/6 of thetotal and there are 21 red gum balls and 14 white gumballs, what are the odds of choosing at random a whitegum ball?

    (A) 1/6

    (B) 1/3(C) 1/2

    (D) 2/3

    (E) 5/6

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    Drill 26: Geometry: AnglesThe geometry on the SAT is pretty straightforward. Basically, you need to know a little bit about circles, triangles,rectangles, and parallel lines. You wont have to memorize any complex formulas just some basic ones. And getthiseven if you forget one of the formulas, dont worry about it. All the formulas you need are provided for youat the beginning of each math section.

    ALL DIAGRAMS, UNLESS OTHERWISE STATED, ARE DRAWN TO SCALE!

    Angles Review:

    There are 360 in a circle.

    This means that there are 180 in a half circle.

    There are also 180 in a straight l ine. The sum of the angles of anytriangle equals 180.

    The sum of the angles of any four-sided gure equals 360.

    When two straight lines cross each other, they form vertical angles. Vertical angles arealways equal.

    A right angle is an angle that equals ninety degrees.All the angles of a rectangle are right angles.

    Bisect means to divide something into two equal parts. You can bisect a line or an angle.

    360

    180

    a

    b c

    a bc

    d

    a

    b

    c

    d

    Problems: (Answers are on pages 40-43.)

    1. In the gure below, line l is an angle bisectorforming the smaller angles y, z, w, and v. What is thevalue of v + x + y?

    (A) 45(B) 90

    (C) 135

    (D) 180

    (E) 270

    2. In the gure below, a + c =

    (A) 90

    (B) 130

    (C) 140

    (D) 170

    (E) 180

    xw

    vl

    yz

    a bc

    d

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    3. In the gure below, line l bisects an angle to formthe smaller angles b and c.What is the value of a + b?

    (A) 45

    (B) 80

    (C) 90

    (D) 135

    (E) 180

    4. In the gure below, angle a is equal to angle h, andangle g = 35.a + d + f =

    (A) 145

    (B) 180

    (C) 290

    (D) 325

    (E) 435

    5. In the gure below, angle f = 25.a + b + c + d + e + f + g + h =

    (A) 90

    (B) 135

    (C) 180

    (D) 210

    (E) It cannot be determined

    xw

    v

    yz

    a bc d

    e fg h

    a

    b

    cd e

    fg

    hl

    Drill 27: Geometry: TrianglesTriangle Review:

    An isosceles triangle is any triangle with two equal sides.

    In an isosceles triangle, the angles opposite the equal sides are also equal.

    If a triangle has three equal sides, it is called an equilateral triangle.

    In an equilateral triangle, all angles equal 60.

    Isosceles Triangle

    60

    Equilateral Triangle

    60 60

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    If a triangle has three sides of varying lengths, then the longest side is acrossfrom the longest angle, and the smallest side is across from the smallest angle.

    This is known as a scalene triangle.

    Aright triangle has an angle of ninety degrees. If you know two sidesof a right triangle, you can nd the length of the third side by using thePythagorean theorem.

    Pythagorean theorem:

    a2 + b2 = c2

    Pythagorean Triplets:

    3, 4, 5 right triangle

    6, 8, 10 right triangle

    5, 12, 13 right triangle

    7, 24, 25 right triangle

    Special Right Triangles:

    There are two special right triangles that show up on the SAT. These are the 30-60-90 right triangle, and the 45-45-90 right triangle. These triangles can have sides ofany length, but the cool thing is those lengths will always be in the same relationship toeach other. On a 30-60-90 right triangle if the shortest side has a length of x, then thehypotenuse will have a length of 2x, and the third side will have alength of x3

    For the 45-45-90 right triangle, if the length of a side is s, then the other side will have

    a length of s as well, and the hypotenuse will have a length of s2.Both of these special right triangles are included with the formulas at the beginning ofeach Math Section on the SAT. Remember this: if you see a right triangle on the SAT,chances are its either a special right triangle or a Pythagorean triplet. By recognizingthese on the test, you can save yourself a lot of time and work.

    Largest Agnle

    Largest Side

    a

    b c

    3

    4 5

    x

    x3 2x

    30

    60

    s2s

    s

    45

    45

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    Problems: (Answers are on pages 40-43.)1. Triangle ABC is an isosceles triangle. If AngleB = 40, what is the value of angle C?

    (A) 40

    (B) 45

    (C) 70

    (D) 90

    (E) It cannot be determined

    2. In the gure below, what is the measure of angle B?

    (A) 30

    (B) 45

    (C) 60

    (D) 75

    (E) It cannot be determined

    3. If, in the gure below, ABCD is a rectangle withdiagonals BD and AC that bisect each other, then AD =

    (A) 5

    (B) 6(C) 52

    (D) 8

    (E) 10

    4. In the gure below, square ABEF shares a side withrectangle BCDE. If AC = 17, than diagonal BD =

    (A) 7

    (B) 10

    (C) 13

    (D) 25

    (E) 26

    5. In the gure below, square ABCD has two diagonals,AC and BD, that bisect each other. If AB = 4, than DE=

    (A) 2

    (B) 2

    (C) 22

    (D) 4

    (E) 42

    6. What is the value of x in the gure below?(A) 50

    (B) 60

    (C) 110

    (D) 145

    (E) 155

    7. In the gure below, what is the value of x?

    (A) 90

    (B) 60(C) 45

    (D) 40

    (E) 30

    b

    a c

    b c

    d

    e

    a

    6

    5

    b c

    def

    a

    5

    b

    cd

    e

    a

    4

    35

    2595

    x

    30

    80

    x

    b

    a c

    843

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    Drill 28: Geometry: Perimeter, Area, Parallel LinesTo nd the perimeter of any gure, simply add the lengths of the sides together.

    Area is a little trickier than perimeter. Here are the formulas:

    The formula for area of a rectangle is length times width, or l w.

    The formula for the area of a square is simply side squared, or s2.

    The formula for the area of a triangle is one half times base times height, or 1/2bh.

    If you have an odd shaped quadrilateral, like this, just drop a line in and divide it

    into two recognizable shapes.

    Lets check out parallel lines. When two parallel lines are cut by a third line,eight angles are created. If you know one of those angles, you can ndthe measure of the other seven. For example, if angle a equals 120, then angleb equals 60, because together they form a straight angle. Angle d equals 120,because a and d are vertical angles. And angle c equals 60, because angleb and c are vertical. On the bottom, angle e equals 120 because anglee and angle c are opposite interior angles. Angle f equals 60, because e and fform a straight angle, angle g equals 60, because f and g are vertical angles,and angle h equals 120, because e and h are vertical angles.

    Problems: (Answers are on pages 40-43.)

    1. In the gures below, triangle ABC and quadrilateralDEFG have the same perimeter.What is the value of x?

    (A) 4

    (B) 5

    (C) 6

    (D) 7

    (E) 8

    2. The gure below is made up of twelve identicalsquares, each with side of length 2.What is the perimeter of the gure?

    (A) 96

    (B) 48

    (C) 36

    (D) 32

    (E) 24

    b e

    d g

    f

    a c

    8

    5

    6

    3

    4

    x7

    a bc d

    e fg h

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    3. In the gure below, triangle ABD is inscribed in rectangle ABCE. What is the areaof triangle ABD?

    (A) 49

    (B) 28

    (C) 16

    (D) 12

    (E) 6

    4. A triangle with base 4 and height 7 has an area that is one-third the area of a rectangle with a width 6. Whatis the length of the rectangle?

    (A) 4(B) 7

    (C) 12

    (D) 14

    (E) 28

    5. In the gure below, rectangle ABCD is cut by two parallel lines, each of length 5. What is the area of theshaded region?

    (A) 8

    (B) 12

    (C) 14

    (D) 20

    (E) 32

    6. In the gure below, lines l and m are parallel, and they are intersected by two other lines. What is the value of x + y?

    (A) 70

    (B) 110

    (C) 140

    (D) 180

    (E) 210

    7. If l1 is parallel to l2 in the gure above, than a + b =

    (A) 300

    (B) 270

    (C) 245

    (D) 135

    (E) 110

    b

    cde

    a 8

    4

    b

    cd

    a 8

    45

    70

    110

    70x

    y

    l

    m

    l

    l45 a

    b

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    Drill 29: Circles and VolumeWhen dealing with circles and volume on the SAT, its important to know when to use which formula. The goodthing is that all the formulas youll need are printed at the beginning of the math section on the test, so refer tothem whenever you need to refresh your memory. If all else fails, circle problems are great for estimating. Andremember, whenever you can, eliminate answer choices.

    Here are the formulas:

    The radius, r, is the distance from the center of the circle to any point on the edge of the

    circle. Once you know the radius, you can gure out everything else.

    The diameter, d, is the straight line that runs from one side of the circle to the other, passing

    through the center. The diameter equals 2 times the radius.

    The circumference, C, is the distance around the outside of the circle, kind of like a perimeter.The formula for circumference is 2r or d.

    The area, A, of a circle is the amount of space inside the circle. The formula for the area of a circle is r2.

    The formula for calculating the volume of a rectangle is L x W x h.

    The formula for calculating the volume of a cylinder is r2h.

    d

    rc

    Problems: (Answers are on pages 40-43.)

    1. A circle has an area of 16. What is its diameter?

    (A) 16

    (B) 8

    (C) 6(D) 4

    (E) 2

    2. In the gure to the below, a right triangle is inscribedin a circle, with the hypotenuse of the triangle passingthrough the center of the circle. What is the area of the

    shaded region?

    (A) 100

    (B) 100 48

    (C) 25 48(D) 25 24

    (E) 16 24

    3. In the gure below, two identical circles areinscribed in a rectangle. If the area of the rectangle is72, then what is the area of one of the circles?

    (A) 6

    (B) 9

    (C) 12

    (D) 18

    (E) 36

    4. Kevin rolls a tire with a diameter of one foot downthe street. If he rolls the tire 41 feet, approximatelyhow many revolutions has the tire made?

    (A) 25

    (B) 20

    (C) 16(D) 13

    (E) 8

    6

    8

    6

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    5. Jennifer purchases a box at a garage sale. If thebox measures 4 feet by 8 feet by 3 feet, what is thevolume of the box?

    (A) 24

    (B) 32

    (C) 56

    (D) 96

    (E) 120

    6. The base of a tin can has a radius of 4 and the canhas a height of 7. What is the volume of the can?

    (A) 28

    (B) 49

    (C) 112

    (D) 128

    (E) 142

    Drill 30: Coordinate Plane and SlopeProblems: (Answers are on pages 40-43.)

    1. In the gure below, a line is to be drawn throughpoint P so that it never crosses the y-axis. Throughwhich of the following points will the line never pass?

    (A) (3, 0)

    (B) (3, -1)

    (C) (-3, -2)

    (D) (3, -2)

    (E) (3, 4)

    2. In the gure below, what is the distance from pointP to the origin?

    (A) 3

    (B) 4

    (C) 4.5

    (D) 5

    (E) 7

    3. If l1 contains points A (4, -2) and B (-7, -2), what isthe slope of the line?

    (A) 3/4

    (B) 4/7

    (C) 0

    (D) 2/3

    (E) 3/2

    4. If l1 has a slope of 3/5 and contains points (3, 4) and(a, 7), what is the value of a?

    (A) 8

    (B) 4

    (C) 3/4

    (D) 3(E) 8

    x

    P(4,3)

    y

    x P(3,2)

    y

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    MATH DRILL ANSWERSDrill 2: Student-Produced Response

    Questions

    1. 5/10 okay as is

    2. 2.5 okay as is

    3. 15/35 = 3/7 need to reduce

    4. 15.5 = 31/2 need to change toimproper fraction

    5. .5767 = .576 or .577

    need to round off or drop last number(either is acceptable)

    Drill 4: Percentages

    1. 28

    2. 20

    3. They are equal; both equal 13.5%

    4. D.24 percent of 300 (take 60 percent of 40to get 24)

    5. C.33 percent (9 + 3 + 4 = 16, so 8 scoredbelow 70%. 8 is 33% of 24.)

    Drill 5: Percent Increase and Decrease

    1. D. 50% (15/30 = 50%)

    2. B. 23%(350 270 = 80, 80/350 = 22.857%and question asks forapproximate answer.)

    3. C. 290(x/232 = 25%, x = 58, 58 + 232 = 290)

    Drill 6: Decimals

    1. 10.699

    2. 12.316

    3. -1.33

    4. 23.777

    5. 2.5177

    Drill 7: Fractions

    1. 22/152. 4/3

    3. -13/19

    4. -25/56

    5. 114/147

    6. -15/26

    7. 14/15

    8. 27/125

    Drill 8: Average Questions

    1. 93

    2. 480

    3. x = 21

    4. C. 42

    5. C. 11

    6. E. 100

    7. B. 300

    8. B. 30

    9. A. 500

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    Drill 9: Median and Mode

    1. 6

    2. 3

    3. 3

    4. 4

    5. 1, 6 (two modes)

    6. 15

    7. C. 14

    8. C. 24 (mode of set A = 23, medianof set B = 25)

    Drill 10: Square Roots

    1. A. 3

    2. B. 52

    3. E. 7/10

    Drill 11: Exponents1. C. 72,004

    2. A. 64

    3. D. 8

    4. B. 9/1021

    Drill 12: Ratios

    1. C. 200 (ratio jumper = 200)

    2. D. 5/12

    3. B. 12 (ratio jumper = 3)

    4. D. 2/5

    5. B. 50 (ratio jumper = 5)

    6. C. 5 (ratio jumper = 5)

    Drill 13: Proportions

    1. D. 8

    2. D. 80

    3. C. 9

    4. C. 32

    5. E. 848

    6. 308

    Drill 14: Algebraic Manipulation

    1. x = 8

    2. y = 12

    3. B. -1

    4. C. 4

    5. B. 2

    6. D. 22 (divide the rst equation by 4)

    7. B. 3

    8. C. 3x = 21

    Drill 15: Inequalities

    1. B. x > -5

    2. D. x > -12

    3. E. 5 < x + y < 13

    4. A. -3 b a 7

    (test the range by making b a as big and as smallas possible)

    MATH DRILL ANSWERS

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    MATH DRILL ANSWERSDrill 16: Simultaneous Equations

    1. a = 10

    2. 5x + 9y = 30 (add the equationstogether and you have the answer)

    3. x + y = 7 (add the equations together,divide by 7, and you have the answer)

    4. D. 4

    5. E. 38 (add all 3 equations togetherand divide by 2)

    6. E. 120 (multiply the rst equation by 4to get the answer)

    7. B. 4

    Drill 17: Absolute Value, Direct & Inverse

    Variation

    1. A) 4

    B) 1/2

    C) 64 or 8

    D) 3/4

    E) 122. Direct variation: x = ky

    3. Inverse variation: xy = k

    Drill 18: Quadratic Equations

    1. x + y = 3

    2. B. 10

    Drill 19: Functions

    1. C. 3

    2. E. 53. B. 2

    4. B. 4

    5. D. 18

    6. E. 120

    7. D. 615

    Drill 20: Domain and Range

    1. A

    2. B

    3. E

    Drill 21: Functions as Models

    1. E

    Drill 22: Algebra: Experiments

    1. E. 60x

    2. E. 3a + 6

    3. E. j + 3

    4. E. x/8(do an experiment where x = 60)

    5. A. (x 6)/3

    6. D. 2x + 1

    Drill 23: Algebra: Using

    Actual Numbers

    1. C. 2

    2. A. None3. C. 40%

    4. B. 1/3

    5. C. 60%

    6. D. 1/2

    Drill 24: Algebra: Working Backwards with

    the Answers

    1. E. 114

    2. C. 1

    3. C. 24. D. 8

    5. C. 190

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    Drill 25: Probability

    1. 1/4

    2 1/4

    3. 1/6

    4. B. 1/3(total gum balls = 42. Oddsof selecting white = 14/42 or 1/3)

    Drill 26: Geometry: Angles

    1. D. 180

    2. E. 180

    3. D. 135

    4. D. 325

    5. C. 180

    Drill 27: Geometry: Triangles

    1. E. It cannot be determined.(Cannot determine which 2 sidesare equal.)

    2. A. 30(Its a 30 - 60 - 90 right triangle withsides 4, 43, 8. The shortest angle isacross from the shortest side)

    3. D. 8 (Pythagorean triplet: 6, 8, 10)

    4. C. 13 (Pythagorean triplet: 5, 12, 13)

    5. C. 22

    6. E. 155

    7. D. 40

    Drill 28: Geometry: Perimeter, Area, Parallel

    Lines

    1. D. 7

    2. D. 32

    3. C. 16

    4. B. 7

    5. D. 20 (Find area of the rectangleand subtract out the areas of the

    2 smaller triangles.)6. D. 180

    7. B. 270

    Drill 29: Geometry: Circles and Volume

    1. B. 8

    2. D. 25 24 (nd area of circle andsubtract out area of triangle)

    3. B. 9 (radius of circle is 3, soA = r2 or 9)

    4. D. 13 (the radius equals 1/2, soC = . Use = 3.14 and divide into 41to get an approximate answer of 13)

    5. D. 96

    6. C. 112

    Drill 30: Coordinate Plane and Slope

    1. C. (-3, -2)

    2. D. 5 (It forms a 3, 4, 5 right triangle.)

    3. C. 0 (A horizontal line has azero slope.)

    4. E. 8

    MATH DRILL ANSWERS