TOTB

THE OFFICAL WORKBOOK OF THE QUALITY ACADEMICS SAT CLASS

ABSTRACT T.O.T.B is a comprehensive collection of the questions that challenge students preparing for the S.A.T examinations. An intuitive topic driven guide to approaching questions in the Mathematics section of the paper.

By André Smith

T.O.T.B MATHEMATICS WORKBOOK 1st Edition (2013/14)

Mathematics Educator

Introduction This book addresses the Mathematics section of the S.A.T examination in a Caribbean context. Regional examinations emphasize content retention and reproduction and this examination is one which emphasizes reasoning and time management. We will address topics and strategies later in this book but below are some of the key S.A.T facts:     

The test is scored from 200 to 800 The examination is 3 hours and 45 minutes There are 54 Mathematics questions Each question must be completed on average in 1minute and 15 seconds There are 3 Math sections (20 + 18 + 16 questions), however if the experimental section (a section that E.T.S uses for analysis) is a Math section then there may be 4 Math sections.

Topics An S.A.T question may appear as a combination of the topics below. The percentages represent the prevalence of these questions on a paper and therefore also provide a guideline for where emphasis is to be placed in the syllabus. Please be aware that this is not a definitive syllabus but a suggested one, covering all topics should adequately prepare students to sit the S.A.T examination. Numbers and Operations (20–25%) ● Arithmetic word problems (including percent, ratio, and proportion) ● Properties of integers (even, odd, prime numbers, divisibility, etc.) ● Rational numbers ● Sets (union, intersection, elements) ● Counting techniques ● Sequences and series (including exponential growth) ● Elementary number theory

Algebra and Functions (35–40%) ● Substitution and simplifying algebraic expressions ● Properties of exponents ● Algebraic word problems ● Solutions of linear equations and inequalities ● Systems of equations and inequalities ● Quadratic equations ● Rational and radical equations ● Equations of lines ● Absolute value

● Direct and inverse variation ● Concepts of algebraic functions ● Newly defined symbols based on commonly used operations

Geometry and Measurement (25–30%) ● Area and perimeter of a polygon ● Area and circumference of a circle ● Volume of a box, cube, and cylinder ● Pythagorean Theorem and special properties of isosceles, equilateral, and right triangles ● Properties of parallel and perpendicular lines ● Coordinate geometry ● Geometric visualization ● Slope ● Similarity ● Transformations Data Analysis, Statistics, and Probability (10–15%) ● Data interpretation (tables and graphs) ● Descriptive statistics (mean, median, and mode) ● Probability ● Counting, Permutations & Combinations

Managing Expectations The cardinal rule of improving is that you need to work to improve. In fact the only mark of being crazy is expecting a different outcome when doing the same thing repeatedly. So to improve your score no matter, the tip or tricks, you must be practicing. As a rule of thumb, you must pick up four extra questions for every 50 points you want to raise your scores. Keeping in mind that the official statistics by ETS, the test publishers, "show" that the average combined improvement is 60 to 70 points, a 150-point improvement is quite respectable, 200 to 300 points is excellent, and 400 points is phenomenal. Improvements of 500 points are so rare that ETS often examines such answer sheets for evidence of cheating.

Strategies You will have to take these tests in a way that differs from the way you took tests in school. In a test in class you are trying to do all the questions and this way of thinking will get you into trouble. Doing an exam in this way, you will have to rush through easy questions and spend too much time on hard questions. For the S.A.T exam please follow these guidelines:

 

Make sure to get all the easy questions right Don’t get stuck on a question. If a question is taking one (1) minute, move on.

The following charts show you roughly how many questions you would need to do in order to get the given score (scoring curves will vary slightly from test to test). The chart below assumes that you do ALL QUESTIONS and get a few of the 54 Math Section questions wrong along the way. Raw Score 800 790 760 720 710 700 680 660 650 640 630 620 610 600 590 580 570 560 550 540 530

# Wrong 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Please Note: These scores are estimates based on ALL 54 QUESTIONS BEING COMPLETED.

Should I Guess? 1⁄ of a point is deducted for an incorrect answer and 1 point is 4 awarded for every correct answer. Now consider, the case where you have done all 50 questions, and have now decided to guess the remaining four (4).  



If you guess none with 50 correct your score is 720. If you get 4 wrong from guessing you are at 49 and a raw score of approximately 710 (depending on the scoring curve for that examination). If you guess 1 of the 4 correctly and get the other 3 wrong your score is 720

It may seem as though there is a penalty for guessing however if you are aiming for 700 you should be able to eliminate answer options increasing your odds of getting the correct answer. Be aware that if you are able to guess 2 of 4 correctly your raw score increases by at least 10 points.

General Math Section Instructions This section shows the general math instructions, and the instructions for "student-produced response" grids. These information boxes precede each Mathematics section in the paper and while the Resource Information box can be helpful, please be reminded that it takes time to check this information and time is at a premium in the examination. Please ensure that you familiarize yourself with these instructions

before continuing.

The Student Response Section Student-produced response questions are also known as grid-ins. That’s because instead of choosing a correct answer from a list of options, students are required to solve problems and enter their answers in the grids provided on the answer sheet. Strategies Here are some hints for answering student-produced response questions. 

Since answer choices aren't given, an approved calculator may be helpful in avoiding careless mistakes.



Write your answer in the boxes above the grid to avoid errors in gridding.



Some questions may have more than one right answer. Grid only one answer.



Keep in mind that there are no point deductions for wrong answers.



Know the gridding rules before taking the test.

How to Grid-In an Answer Here are the gridding rules: 

Mark no more than one circle in any column.



Only answers entered in the ovals in each grid area will be correct. Students do not receive credit for anything written in the boxes above the ovals.



It doesn't matter in which column students begin entering their answers; as long as the correct answer is gridded, students will receive credit.



The grid can hold only four places and can accommodate only positive numbers and zero.



Unless a problem indicates otherwise, an answer can be entered on the grid as a decimal or a fraction.



You don't have to reduce fractions like



Convert all mixed numbers to improper fractions before gridding the answer.



If the answer is a repeating decimal, you must grid the most accurate value the grid will

to their lowest terms.

accommodate. Below is a sample of the instructions you will see on the test.

Numbers and Operations (20–25%) Definitions There are a number of terms that will be used on the test, which you will want to recognize and understand. Here are some of the most common terms: Integers: Positive and negative whole numbers, and zero. NOT fractions or decimals. Prime numbers: Numbers with two factors (themselves and 1). All prime numbers are positive; the smallest prime number is 2. Two is also the only even prime number. The list of prime numbers less than 200 is given below:

It is important to note that between 0 and 200 there are 46 prime numbers. 0-50: 15 prime numbers 51-100: 10 prime numbers 101-150: 10 prime numbers 151-200: 11 prime numbers Knowing these numbers may help you to approximate the number of prime numbers within a particular range. Composite numbers: Numbers that are not prime are said to be composite. That is, these numbers have more than two (2) factors. Rational numbers: Integers, all fractions, and decimal numbers, positive and negative. Technically any number which can be expressed as a fraction of two integers – which means everything except numbers containing radicals (like √2), 𝜋 𝑜𝑟 e.

Another definition, may be numbers whose decimal form either terminates (1/4=0.25) or repeats (1/3=0.333). Irrational numbers: Numbers whose decimal form neither terminates nor repeats. Such as √2 (√16 doesn’t count because it can be simplified to 4). Also, all numbers containing 𝜋 𝑜𝑟 e. Real numbers: All numbers except the imaginary ones (i.e. natural, whole, integer, rational and irrational). Imaginary numbers: The square roots of negative numbers. That is, √−1 = 𝑖, therefore: √−4 = √4 × √−1 = 2 × 𝑖 = 2𝑖. Consecutive numbers: The members of a set of numbers in order (ascending or descending), without skipping any. Absolute value/modulus: The positive value/size of a number. You just remove the negative sign if there is one. Think of it as the distance on the number line between the number and zero. For example, the absolute value/modulus of -5, is written |−5|=5.

Factorizations The factors of a number are all the numbers by which a number can be divided. Some of the questions on the SAT exam will specifically require you to identify the factors of a given number. There are two forms of factorization:  

Simple factorization Prime factorization

The best way to compile a list of all a numbers factors is to break them into pairs beginning with 1, checking at each number to see whether the number you are factoring is divisible by that number. The other kind of factorization is prime factorization. The prime factorization of a number is the grouping of prime numbers that can be multiplied to produce that number. For example the prime factorization of 8 is 2x2x2. An easy way of finding the prime factorization is to start with the simplest prime divisor such as 2, 3 or 5 and the break the larger term into pairs. For example, if you are asked to find the prime factorization of 30. 30=2x15=2x3x5. Prime Factorization Examples What is the prime factorization of 75? (ANS. 75 = 3x5x5)

What is the prime factorization of 78? (ANS. 78 = 2x3x13) Prime Factorization Drill 1 (12 min.) Find the prime factorizations of the following: 1. 2. 3. 4. 5. 6.

64= 70= 18= 98= 68= 51=

(ANS. 64=2x2x2x2x2x2) (ANS. 70=2x5x7) (ANS. 18=2x3x3) (ANS. 98=2x7x7) (ANS. 68=2x2x17) (ANS. 51=3x17)

Prime factorizations are also useful in dealing with divisibility. For example: What is the smallest number divisible by both 14 and 12? To find the smallest number that both numbers will go into (or the smallest multiple of both numbers), look at the prime factorizations of 12 and 14. 12=2x2x3 14=2x7 Therefore the smallest multiple of both must contain the factors ‘2x7’ as well as ’2x2x3’. The smallest multiple therefore, 2x2x3x7=84. Now, try this example: What is the largest factor of 180 that is not a multiple of 15? First start by doing the prime factorization of 180. 180=2x2x3x3x5 Then all you have to do is make the biggest number possible which does not contain 15=3x5. The largest factor therefore is, 2x2x3x3=36. Prime Factorization Drill 2 (9 min.) 1. What is the smallest number divisible by both 21 and 18? (A) 42 (B) 126 (C) 189 (D) 252 (E) 378 ANS.B 2. If 𝛽𝑥 is defined as the largest prime factor of 𝑥, then for which of the following values of 𝑥 would 𝛽𝑥 have the greatest value (A) 170 (B) 117 (C) 88 (D) 62 (E) 53 ANS. E

3. If 𝑥°𝑦 is defined as the smallest integer of which both 𝑥 and 𝑦 are factors, then 10°32 is how much greater than 6°20? (A) 0 (B) 70 (C) 100 (D) 160 (E) 200 ANS. C

Divisibility Rules These rules let you test if one number is divisible by another, without having to do too much calculation! Divisible by 2 3 4 5 6 7

8 9

10 11

12

if The last digit is even (0,2,4,6,8) The sum of the digits is divisible by 3 The last two (2) digits are divisible by 4 The last digit 0 or 5 The number is divisible by both 2 and 3 If you double the last digit and subtract it from the rest of the number and the answer is:  0, or  Divisible by 7 The last three digits are divisible by 8 The sum of the digits is divisible by 9 (Note: you can apply this rule to that answer again if you want) The number ends in 0 If you sum every second digit and then subtract all other digits the answer is:  0, or  Divisible by 11 The number is divisible by both 3 and 4

Examples 128 381 (3+8+1=12, and 12 is divisible by 3) 1312 (is divisible by 4 because 12 is divisible by 4) 175 114 (it is even, and 1+1=6 and 6 ÷ 3 = 2) 𝒀𝒆𝒔 672 (Double 2 to get 4, 67-4=63, and 63 is divisible by 7

109, 816 (816 ÷ 8 = 102) Yes 1629 (1+6+2+9=18, and again, 1+8=9) yes

220 is divisible by 10 1364 ( (3+4)-(1+6) = 0 ) Yes

648 (By 3? 6+4+8=18 and 18 ÷ 3 = 6 𝑌𝑒𝑠. By 4? 48 ÷ 4 = 12 Yes) Yes

Alternatively, you could simply divide the number by any of the numbers you are testing and if the answer (quotient) is a whole number then, yes your term is divisible. Consider, the question: How many prime numbers are there between 1 and 64? Solution: Firstly, we know there are 64 terms, 32 even and 32 odd. The only even prime number is 2 so all other even numbers are not prime.

So if we list the remainder of prime possibilities we have: 2, 3, 5, 7, 9,11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63 Eliminate all multiples of 5: 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63 Eliminate all multiples of 3: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61 Eliminate all multiples of 7: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 Answer: 18 primes between 1 and 64. Prime Number Drill 1. How many integers between 2 and 20, even only, can be the sum of two different prime numbers? (A) 8 (B) 7 (C) 6 (D) 5 (E) 2 (ANS. C) 2. If 𝑝 is a prime number how many factors does 𝑝2 have? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

(ANS. D)

3. The sum of four consecutive integers is 210. Which one of these four integers is prime? (A) 47 (B) 49 (C) 51 (D) 53 (E) 57

(ANS. D)

4. If 𝑝 is a prime number greater than 5, which of the following could represent another prime number for some value of 𝑝? (A) 𝑝 + 1 (B) 𝑝 + 2 (C) 3p (D) 𝑝2 (E) 𝑝3

(ANS. B)

Note. The only even prime number is 2 and so all primes > 5 are odd and so a prime + 1 = even. Internet Resource: You can practice finding the number of prime numbers between random values and a useful tool in checking your answers can be found at: http://www.datedial.com/List_Prime_Numbers_Between_Numbers.asp

Even and Odd, Positive and Negative A number of questions on the exam deal with the result of operations on these types of numbers. Even and Odd numbers Even and Odd numbers are governed by the following Rules: Addition and Subtraction even+even=even even-even=even odd+odd=even odd-odd=even even+odd=odd even-odd=odd

Multiplication evenxeven=even evenxodd=even oddxodd=odd

Positive and Negative numbers There are fewer firm rules for positive and negative numbers. Only the rules for multiplication and division and these are listed below. Multiplication 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 × 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 ÷ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 × 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ÷ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ÷ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 ÷ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 Consecutive Numbers

The product of consecutive numbers is even. For example, If 𝑥 is a positive integer, which of the following must be an even integer? (A) (B) (C) (D) (E)

𝑥+2 2𝑥 + 1 3𝑥 + 1 𝑥2 + 𝑥 + 1 𝑥2 + 𝑥 + 2

Solution: Option (E) Note that, 𝑥 2 + 𝑥 + 2 = 𝑥(𝑥 + 1) + 2 = 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 + 2 = 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟

Order of Operations and the Calculator (Garbage In Garbage Out) What is the value of 2√𝑥 3 − 2 when x=3? (ANS. 10) The safest way to do multi-step operations like this on a calculator is one step at a time. However, that can be time-consuming, on scientific calculators, it is possible to type complex expressions into your calculator all at once. But in order for your calculator to produce the right answer the expression must be entered correctly. In order to enter the terms in an expression correctly we will need an understanding of order of operations or PEMDAS. The acronym P.E.M.D.A.S simply states the order in which an expression containing the six basic types of operations must be evaluated, that is:    

Parentheses/brackets first Exponents/powers second Multiplication/Division next Addition last (note that subtraction is the addition of a negative number)

PEMDAS Drill (10 min.) – Ask students to attempt each question in 1.25 minute intervals Using your calculator evaluate the following: 1. 2. 3. 4. 5.

4 × 4 − 3 × 3 − 16 ÷ 4 20 − (3 × 23 − 5) (5 + 2)2 − 9 × 3 + 23 (12 ÷ 3 + 4) − (42 − 6 × 2) (52 − 5) / (42 + 8 − 7 × 2)

6. (33 −

9 ) 3

(ANS. 3) (ANS. 1) (ANS. 30) (ANS. 4) (ANS. 2)

+ (4 × 3 − 32 )

(ANS. 27)

7. (7 − √9) × (42 − 3 + 1)

(ANS. 56)

8.

24 +(16−3×4) (6+32 )÷(7−4)

(ANS. 4)

Percent Change “Percent change” is a way of referring to increasing or decreasing a number. The percentage change is the amount of the increase/decrease expressed as a percentage of the original value. Percentage Change Drill 1. A 25-gallon addition to a pond containing 150 gallons constitutes an increase of what percent? (A) 14.29% (B) 16.67% (C) 17.25% (D) 20.00% (E) 25.00% (ANS. B) 2. The percent decrease from 5 to 4 is how much less than the percent increase from 4 to 5?

(A) 0% (B) 5% (C) 15% (D) 20% (E) 25%

(ANS. B)

3. Nicoletta deposits $150.00 in her savings account. If this deposit represents a 12% increase in Nicoletta’s savings, then how much does her savings account contain after the deposits? (A) $1100 (B) $1250 (C) $1400 (D) $1680 (E) $1800 (ANS. C) 4. The price of a bicycle originally sold for $250 is marked up by 30%. If this new price is subsequently discounted by 30%, then the final price of the bicycle is (A) $200.50 (B) $216.75 (C) $227.50 (D) $250 (E) $265.30 (ANS. C)

Exponents An exponent is a concise way to represent the repeated multiplication of a term. Consider 53 where 5 is the base and 3 is the exponent. There are a few important things to remember about the effects of exponents on various numbers:   

A positive number raised to any power remains positive. No exponent can make a positive number negative. A negative number raised to an odd power remains negative. A negative number raised to an even power becomes positive.

In other words, anything raised to an odd power keeps its sign. If 𝑎3 is negative, then 𝑎 is negative; if

𝑎3 is positive, then 𝑎 is positive. A term with an odd exponent only has one root (value that makes the equation true). On the other hand anything raised to an even power is positive, regardless of its original sign. One last thing to remember is that there is no term that can be squared to provide a negative number. Multiplying Exponents When Bases Are the Same Exponential terms can be multiplied when their bases are the same. Just leave the bases unchanged and add exponents: 𝑛3 × 𝑛5 = 𝑛8

3 × 34 = 35

Coefficients if they are present are multiplied normally: 2𝑏 × 3𝑏 5 = 6𝑏 6

1 3 𝑐 2

× 6𝑐 5 = 3𝑐 8

Dividing Exponents When Bases Are the Same Exponential terms can be divided when their bases are the same. Once again, the bases remain the same, and the exponents are subtracted:

𝑥8 ÷ 𝑥6 = 𝑥2

75 ÷ 7 = 74

Coefficients, if they are present, are multiplied normally: 6𝑏 5 ÷ 3𝑏 = 2𝑏 4

5

5𝑎8 ÷ 3𝑎2 = 3 𝑎6

Multiplying and Dividing Exponents When Bases Are the Same There’s one special case in which you can multiply and divide terms with different bases: when the exponents are the same. In this case you can multiply and divide under the exponents. Then the bases change and the exponents remain the same, for multiplication: 33 × 53 = 153

𝑥 8 × 𝑦 8 = (𝑥𝑦)8

And for division: 332 ÷ 32 = 112

𝑥 𝑦

𝑥 20 ÷ 𝑦 20 = ( )20

If exponential terms have different bases and different exponents, then there’s no way to combine them by adding, subtracting, dividing or multiplying. Raising Power to Powers When an exponential term is raised to another power, the exponents are multiplied: (𝑥 2 )8 = 𝑥 16

(75 )4 = 720

If there is a coefficient included in the term, then the coefficient is also raised to that power: (3𝑐 4 )3 = 27𝑐12

(5𝑔3 )2 = 25𝑔6

Using these rules, you should be able to manipulate exponents wherever you find them. Special Exponents Here are some unusual exponents you should be familiar with: Zero Any number raised to the power of zero is equal to 1 (50 = 1). Negative exponents A negative exponent is simply an inverse. 1

3−2 = 32 =

1

(3)−1 = 2

𝑎−4 = 𝑎4 𝑥 −1 = 𝑥

1

2

3

1 9

Fractional Exponents (Roots) A fractional power is a way of representing raising a number to a power and taking a root at the same time. The number on top is the power and the number at the bottom is the root. 1

3

3

273 = √271 = (√27)1 2

3

3

83 = √82 = (√8)2

5

𝑏 2 = √𝑏 5 = (√𝑏)5 4

3

3

𝑥 3 = √𝑥 4 = ( √𝑥 )4

Exponents Drill Simplifying Roots

Roots Drill

Numbers and Operations Questions 1. A survey of Town X found an average (arithmetic mean) of 3.2 persons per household and a mean of 1.2 televisions per household. If 48,000 people live in Town X, how many televisions are in Town X? (A) 15,000 (B) 16,000 (C) 18,000 (D) 40,000 (E) 57,600 2. A basketball team had a ratio of wins to losses of 3:1. After the team won six games in a row, its ratio of wins to losses became 5:1. How many games had the team won before winning six games in a row? (A) 3 (B) 6

(C) 9 (D) 15 (E) 24

3. Fifteen percent of the coins in a piggy bank are nickels and five percent are dimes. If there are 220 coins in the bank, how many are not nickels or dimes? (A) 80 (B) 176 (C) 180 (D) 187 (E) 200

4. At the beginning of 1999, the population of Rockville was 204,000 and the population of Springfield was 216,000. If the population of each city increased by exactly 20% in 1999, how many more people lived in Springfield than in Rockville at the end of 1999? (A) 2,400 (B) 10,000 (C) 12,000 (D) 14,400 (E) 43,200 5. In a list of seven integers, 13 is the lowest member, 37 is the highest member, the mean is 23, the median is 24, and the mode is 18. If the numbers 8 and 43 are then included in the list, which of the following will change? I. II. III. (A) (B) (C) (D)

The mean The median The mode I only I and II only II and III only I, II and III

6. If 𝑎2 𝑏 = 122 , and 𝑏 is an odd integer, then 𝑎 could be divisible by all of the following EXCEPT (A) 3 (B) 4 (C) 6 (D) 9 (E) 12 7. There are 250 students in 10th grade at Northgate High School. All 10th graders must take French or Spanish, but not both. If the ratio of males to females in

10th grade is 2 to 3, and 80 of the 100 French students are male, how many female students take Spanish? 8. A researcher found that the amount of sleep that she allowed her mice to get was inversely proportional to the number of errors the mice made, on average, in a maze test. If mice that got 2 hours of sleep made 3 errors in the maze test, how many errors, on average, do mice with 5 hours of sleep make? 9. For a given year, a mayor has $45,000 allotted to spend on the sanitation department, the police department, 1

and the fire department. If 5 of his money goes to the sanitation 2 3

department, and of the remaining money goes to the police department, how much does the mayor have left for the fire department? (A) $36,000 (B) $24,000 (C) $21,000 (D) $12,000 (E) $6,000 10. S is the set of all positive numbers n such that 𝑛 < 100 and √𝑛 is an integer. What is the median value of the members of set S? (A) 5 (B) 5.5 (C) 25 (D) 50 (E) 99 11. On a map, 1 centimeter represents 6 Kilometers. A square on the map with a

perimeter of 16 centimeters represents a region with what area? (A) 64 square kilometres (B) 96 square kilometers (C) 256 square kilometers (D) 576 square kilometers (E) 8,216 square kilometers 12. If 0.1% of m is equal to 10% of n, then m is what percent of 10n? (A) 1⁄1,000 % (B) 1% (C) 10% (D) 100% (E) 1,000% 13. If 𝑎 = 4𝑏 + 26, and 𝑏 is a positive integer, then 𝑎 could be divisible by all of the following EXCEPT (A) 2 (B) 4 (C) 5 (D) 6 (E) 7 14. A science class has a ratio of girls to boys of 4 to 3. If the class has a total of 35 students, how many more girls are there than boys? (A) 20 (B) 15 (C) 7 (D) 5 (E) 1 15. Which if the following is equivalent to −9 ≤ 3𝑏 + 3 ≤ 18? (A) −4 ≤ 𝑏 ≤ 5 (B) −4 ≤ 𝑏 ≤ 6 (C) −3 ≤ 𝑏 ≤ 5 (D) 3 ≤ 𝑏 ≤ 5

(E) 4 ≤ 𝑏 ≤ 6 16. If Alexandra pays $56.65 for a table, and this amount includes a tax of 3% on the price of the table, what is the amount, in dollars, that she pays in tax? 17. A researcher found that the number of bacteria in a certain sample doubles every hour. If there were 6 bacteria in the sample at the start of the experiment, how many bacteria were there after 9 hours? (A) (B) (C) (D) (E)

54 512 1536 3072 6144

18. A bakery uses a special flour mixture that contains corn, wheat, and rye in the ratio of 3:5:2. If a bag of the mixture contains 5 pounds of rye, how many pounds of wheat does it contain? (A) 2 (B) 5 (C) 7.5 (D) 10 (E) 12.5 19. If |𝑥| ≠ 0, which of the following statements must be true? (A) 𝑥 is positive (B) 2𝑥 is positive (C)

1 𝑥

is positive 2

(D) 𝑥 is positive (E) 𝑥 3 is positive 20. When 𝑛 is divided by 5, the remainder is 4. When 𝑛 is divided by 4, the

remainder is 3. If 0 < 𝑛 < 100, what is one possible value of 𝑛?

Numbers and Operations Solutions 1. 2. 3. 4. 5. 6. 7.

C C B D A D 130 6

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

1

8. 1.2 or 5 or 15 9. D 10. C

D E B D A 1.65 D E D 19

Algebra and Functions (35–40%) Solving Equations An algebraic equation contains at least one unknown (a variable). Solving as you know is the process of determining the value of that variable. Practice solving the following equations: 1.

(3𝑥 2 −7) 17

=4

2. 𝑛2 = 5𝑛 3.

2𝑎−3 3

=−

(ANS. ±5) (ANS. 0,5)

1 2

4. 𝑥 3 − 5𝑥 2 + 6𝑥 = 0 5. (11𝑥)(50) + (50𝑥)(29) = 4000

(ANS. 0.75) (ANS. 0, 3, 2) (ANS. 2)

Solving an equation must be distinguished from simplifying an expression. With simplification you are simply expected to rewrite the given terms as opposed to finding a value for an unknown (solving). Simplify the following: 1.

𝑥 5 +𝑥 4 +𝑥 3 +𝑥 2 𝑥 3 +𝑥 2 +𝑥+1

(ANS. 𝑥 2 )

2.

−3𝑏(𝑎+2)+6𝑏

(ANS. 3)

−𝑎𝑏

Algebraic Functions (Unary or Binary Operations) Algebra questions sometimes take the form of functions. A function is a set of algebraic instructions. These functions are sometimes represented using custom symbols as shown below: 𝐼𝑓 ∗ 𝑎 ∗= 𝑎2 − 5𝑎 + 4, 𝑡ℎ𝑒𝑛 ∗ 6 ∗= (A) 6 (B) 8 (C) 10 (D) 12 (E) 14 ANS. (𝑎 − 4)(𝑎 − 1) = (6 − 4)(6 − 1) = 2(5) = 10 Algebraic functions are pre-defined functions. The value of will depend on the numbers given in the question as well as the position of those numbers. That is, consider 𝐼𝑓 𝑎 ∗ 𝑏 = 2𝑎 + 3𝑏, 𝑡ℎ𝑒𝑛 7 ∗ −4 = Here we must recognize that 𝑎 = 7 𝑎𝑛𝑑 𝑏 = −4 by virtue of their position. ∴ 7 ∗ −4 = 2(7) + 3(−4) = 14 − 12 = 2 Algebraic Functions Drill 1. 𝐼𝑓 𝑦Ф = 𝑦 2 − 6, 𝑡ℎ𝑒𝑛 𝑤ℎ𝑖𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑖𝑠 (𝑦 + 6)Ф? (A) 𝑦 2 (B) 𝑦 2 − 6 (C) 2𝑦 − 36 (D) 𝑦 2 + 12𝑦 + 30 (E) 𝑦 2 + 12𝑦 + 42

(ANS. D)

2. 𝐼𝑓 [𝑥] = −|𝑥 3 |, 𝑡ℎ𝑒𝑛 [4] − [3] = (A) −91 (B) −37 (C) −1 (D) 37 (E) 91

(ANS. B)

3. 𝐼𝑓 ¥𝑐 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 5(𝑐 − 2)2 , 𝑡ℎ𝑒𝑛 ¥5 + ¥6 = (A) ¥7 (B) ¥8 (C) ¥9 (D) ¥10 (E) ¥11

(ANS. A)

Inequalities Inequalities are statements whose result provides a range of values. There are four (4) basic inequality signs, which should read like this:    

𝑎 𝑎 𝑎 𝑎

𝑏 ≤𝑏 ≥𝑏

𝑎 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑏 𝑎 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑏 𝑎 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑏 𝑎 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑏

Inequalities are treated just like equations with just one very important exception to remember:

Whenever you multiply or divide across an inequality with a negative, the inequality sign switches direction. That is, for example 4𝑛 − 20 > −3𝑛 + 15 If we multiply the equation by -1, we get −4𝑛 + 20 < 3𝑛 − 15 Inequality Drill Find the range of values which satisfy the inequalities below.

1.

6(5−𝑛) 4

≤3

(ANS.𝑛 ≥ 3)

2. 8(3𝑥 + 1) + 4 < 15

(ANS. 𝑥 < 1⁄8)

3. 4𝑛 − 25 ≤ 19 − 7𝑛

(ANS. 𝑛 ≤ 4)

4.

14𝑠−11 9

(ANS. 𝑛 ≥ 2⁄5)

≥𝑠−1

Working with Ranges Ranges use your inequality symbols to specify the boundary for acceptable solutions and often take the form, 𝑎 12 = 12 < −𝑏 < −2 −4 < 𝑎 < 5 +(12 < −𝑏 < −2) 8 5 (D) |𝑏 − 55| < 5 (E) |𝑏 − 55| > 5

14. If (𝑎 + 𝑏)2 = 49, and 𝑎𝑏 = 10, which of the following represents the value of 𝑏 in terms of 𝑎? (A)

√29 𝑎

(B) √29 − 𝑎2 (C) √39 − 𝑎 𝑎

49

(D) √10 (E) 𝑎2 √49 15. If 𝑥 + 2𝑦 = 20, 𝑦 + 2𝑧 = 9, 𝑎𝑛𝑑 2𝑥 + 𝑧 = 22, what is the value of 𝑥 + 𝑦 + 𝑧? (A) 10 (B) 12 (C) 17 (D) 22 (E) 51 16. How many solutions exist to the equation |𝑥| = |2𝑥 − 1|? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

17. Ray and Jane live 150 miles apart. Each drives toward the other’s house along a straight road connecting the two, Ray at a constant rate of 30 miles per hour and Jane at a constant rate of 50 miles per hour. If Ray and Jane leave their houses at the same time, how many miles are they from Ray’s house when they meet? (A) 40 1 2 1 56 4

(B) 51 (C)

(D) 75 1

(E) 93 4 18. If 𝑎𝑏 = 4, and 3𝑏 = 2, what is the value of 𝑎? 19. If 𝑏 is a prime number such that 3𝑏 > 5 6

10 > 𝑏, what is one possible value of 𝑏? 20. Let 𝑓(𝑥) = 𝑥 2 − 5. If 𝑓(6) − 𝑓(4) = 𝑓(𝑦), what is |𝑦|?

Algebra and Functions Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

C E D 19, 39, 59, 79 or 99 36 C A D D D

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

B C D D C C C 8 5, 7 or 11 5

Algebra Revision Test31 Time: 20 minutes for 16 questions 1. If 36 − 𝑘 = 4 + 𝑘, what is the value of 𝑘? (A) 12 (B) 16 (C) 18 (D) 20 (E) 32 2. 𝑓(𝑥) = 2𝑥 − 6 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 − 3𝑥 − 4 The functions 𝑓 and 𝑔 are defined above. What is the value of 𝑓(7) − 𝑔(3)? (A) 0 (B) 2 (C) 4 (D) 10 (E) 12

3. 3𝑦 = 2𝑥 − 6 𝑎𝑛𝑑 𝑦 = 1 − 𝑐𝑥 In the equations above, 𝑥 and 𝑦 are variables and 𝑐 is a constant. If no ordered pair of numbers (𝑥, 𝑦) satisfies both of the equations above, what is the value of 𝑐? (A) −3⁄2 (B) −2⁄3 (C) 0 (D) 2⁄3 (E) 3⁄2 4. If 𝑓 is a function defined by 𝑓(𝑥) =

(𝑥 + 4)⁄ 1 12 , for what value of 𝑥 will 𝑓(𝑥) = ⁄4 ?

(A) 7 (B) 2 (C) 1 (D) 0 (E) -1 5. If 𝑥 2 − 𝑎2 = 16 𝑎𝑛𝑑 2(𝑥 + 𝑎) = 8. What does 𝑥 − 𝑎 equal? (A) 2 (B) 4 (C) 48 (D) 64 (E) 128 6. In a class consisting of 55 boys and 45 girls, 30 students have blue eyes. What is the least possible number of girls in the class who do not have blue eyes? (A) 10 (B) 15 (C) 20 (D) 25 (E)30 7. Richard has twice as many pieces of candy as Alice. Bob has 3⁄4 as many pieces of candy as Alice. Richard has 𝑟 pieces, Alice has 𝑎 pieces and Bob has 𝑏 pieces. If 𝑟 > 0, which of the following is true? (A) 𝑎 > 𝑏 > 𝑟 (B) 𝑎 > 𝑟 > 𝑏 (C) 𝑏 < 𝑟 < 𝑎 (D) 𝑟 > 𝑎 > 𝑏 (E) 𝑟 > 𝑏 > 𝑎 8. If 𝑥 − 𝑦 + 𝑧 = 10 𝑎𝑛𝑑 − 𝑥 + 𝑦 = 3 What is the value of 𝑧? (A) 13 (B) 7 (C) 10⁄3 (D) -7 (E) -13

9. If 6𝑦 = 2𝑥 𝑎𝑛𝑑 𝑦 ≠ 0, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥⁄𝑦 ? (A) 1⁄4 (𝐵) 1⁄3 (𝐶) 3 (𝐷) 4 (𝐸) 8 𝑥 + 𝑦⁄ 10. If 𝑥⁄𝑦 = 𝑘 𝑎𝑛𝑑 𝑘 > 0, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑥 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑘? (A) 𝑘 − 1⁄𝑘 (𝐵) 𝑘 + 1⁄𝑘 (𝐶) 1⁄𝑘 (𝐷) 𝑘⁄𝑘 − 1 (𝐸) 𝑘⁄𝑘 + 1 11. 𝑚𝑥 + 𝑝𝑦 = 10 𝑎𝑛𝑑 (𝑚 + 1)𝑥 + 𝑝𝑦 = 14 Based on the equations above which of the following must be true? (A) 𝑥 = 2 (B) 𝑥 = 4 (C) 𝑦 = 6 (D) 𝑦 = 8 (E) 𝑦 − 𝑥 = 2 12. If 6 times j is 1 more than the square of k, where k is an integer, what is the smallest possible value of j? (A) -5 (B) −1⁄6 (C) 0 (D) 1⁄6(E) It cannot be determined from the information given. 13. If (𝑥 + 𝑎)(7𝑥 + 𝑏) = 7𝑥 2 + 𝑐𝑥 + 6 for all values of 𝑥, and if 𝑎 and 𝑏 are positive integers, what is one possible values of 𝑐? 14. (𝑝𝑏 )𝑎 = (𝑝2𝑐 )𝑏 In the equation above 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑝 are integers greater than 1. If 𝑐 = 4 what is the value of 𝑎? 15. For three positive prime numbers 𝑎, 𝑏 & 𝑐, 𝑎𝑏 = 33 𝑎𝑛𝑑 𝑏𝑐 = 21. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑏𝑐? 16. For all positive numbers 𝑡, let ∆𝑡 be defined by ∆𝑡 = 𝑡 − 1⁄𝑡 + 1 what is the value of ∆ 5⁄2 ? Answers 1. 2. 3. 4. 5. 6.

B E B E B B

7. 8. 9. 10. 11. 12.

D A C B B D

13. 13, 17, 23, 43 14. 8 15. 231 16. 3⁄7

Geometry and Measurement (25–30%) Plane Geometry This topic deals with questions about lines, angles, triangles and other polygons and circles.

Distance between two points Distance measures how far apart two things are. The distance between two points can be measured in any number of dimensions. However, for this exam we will focus on dimensions 1, 2 and 3. Distance is always a positive number. 1-Dimensional Distance In one dimension the distance between two points is determined simply by subtracting the coordinates of the points. For example, in this segment, the distance between -2 and 5 is calculated as: 5 − (−2) = 7

2-Dimensional Distance In two dimensions, the distance between two points can be calculated by considering the line between them to be the hypotenuse of a right angled triangle. To determine the length of this line:   

Calculate the difference in the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 Calculate the difference in the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 Use the Pythagorean Theorem

The process is illustrated below, using the variable Example: Find the distance between (-1, -1) and (2, 5). Based on the illustration tto the left:  

𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒: 2 − (−1) = 3 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒: 5 − 1 = 4

3-Dimensional Distance Consider two 3‐ dimensional points (x1, y1, z1) and (x2, y2, z2). Consider first the situation where the two z‐ coordinates are the same. Then, the distance between the points is 2‐dimensional, i.e. ,

Types of Angles

Parallel Lines and Transversals

Inequalities in Triangles

Polygons – Definitions

Pythagorean Theorem and special properties of isosceles, equilateral, and right triangles

Perimeter and Area of a Triangle

Perimeter and Area of Quadrilaterals

Maximum Area of a Triangle Max area of triangle when the triangle is a right angled triangle because the longest side must be the hypotenuse so the shorter sides must form the base and height. If these sides are not perpendicular then the height will be a slant height and the product base*height will be smaller.

The relationship between 𝜽 and 𝒍 Consider the diagram below,

𝑟 𝜃

The relationship between the variables is, 𝑙= →

𝜃 × 2𝜋𝑟 360

𝑙 𝜃 = 2𝜋𝑟 360

That is, the ratio of the length of arc to circumference equals the ratio of the angle from which the arc is subtended and 360°. Consider the following question, page 873 finish diagram

𝑦°

Note: Figure is not drawn to scale

In the figure above, AB and CD are diameters of the circle whose centre is 0. If the radius of the circle is 2 inches and the sum of the lengths of arcs AD and BC is 3𝜋 inches, then y= (A) 45 (B) 60 (C) 75 (D) 90 (E) 120

Solution: If 𝑟 = 2 → 𝐶 = 2𝜋(2) = 4𝜋 If 𝐴𝑅𝐶 𝐴𝐷 + 𝐴𝑅𝐶 𝐵𝐶 = 3𝜋 → 𝑚𝑖𝑛𝑜𝑟𝐴𝑅𝐶 𝐴𝐶 + 𝑚𝑖𝑛𝑜𝑟𝐴𝑅𝐶 𝐷𝐵 = 𝜋 → 𝑚𝑖𝑛𝑜𝑟 𝐴𝑅𝐶 𝐴𝐶 = 𝜋⁄2 𝜋

𝑦 𝑦 ∴ ⁄360° = 2⁄4𝜋 → ⁄360° = 1⁄8 → 𝑦 = 1⁄8 × 360° = 𝑦 = 45°

Solid Geometry Plane & Solid Geometry Questions 1.

In the figure above, △ 𝐴𝐵𝐷 is isosceles, and △ 𝐵𝐶𝐷 is equilateral. What is the degree measure of∠𝐴𝐷𝐶? 2.

In the rectangle above, the radius of each quarter circle is 3. What is the area of the shaded region? (A) 16 − 3𝜋 (B) 16 − 9𝜋

(C) 32 − 9𝜋 (D) 60 − 3𝜋 (E) 60 − 9𝜋 3.

In the figure above, what is the area of △ 𝐴𝐵𝐶? (A) 2√3 (B) 4 (C) 4√3 (D) 8 (E) 8√3 4.

In the figure above, segment 𝑋𝑌 joins vertices of the cube. If the length of 𝑋𝑌 is √2, what is the volume of the cube? (A) 1 (B) 2 (C) 2√2 (D) 3√2 (E) 8 5.

In the figure above, the two semicircles have radii of lengths 𝑗 and 𝑘, respectively. Which of the following expressions gives the length of the darkened curved path shown from A to C? (A) 𝑗𝑘𝜋 (B) 𝑗 2 𝑘 2 𝜋 (C) (𝑗 + 𝑘)𝜋 (D) (E) 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

𝑗+𝑘 𝜋 𝑗𝜋 𝑘𝜋 + 2 2

X X X X X X X X X X X X

18.

The circle above has center 𝑂,and ∠𝐴𝑂𝐵 measures 45°. The area of the shaded region is what fraction of the area of the circle? 19.

If a 1-inch cube of chees were cut in half in all three directions as shown above, then the total surface area of the separated smaller cubes would be how much greater than the surface area of the original 1-inch cube? (A) 2 square inches (B) 4 square inches (C) 6 square inches (D) 8 square inches (E) 12 square inches 20.

In the figure above, each of the four large circles is tangent to two of the other large circles, the small circle, and two sides of the square. If the radius of each of the large circles is 1, what is the radius of the small circle? (A) 1⁄4 (B) 1⁄2 (C)

√2−1 (approximately 2

0.207)

(D) √2 − 1 (approximately 0.414) (E)

√2 2

(approximately 0.707)

Plane & Solid Geometry Solutions 1. 2. 3. 4.

105° E C A

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

C X X X X X X X X X X X X 18. 1⁄8 19. C 20. D

Coordinate Geometry Gradient Length of Line Mid-Point

Coordinate Geometry Questions Coordinate Geometry Solutions Simultaneous Solutions (finite, none, infinite) Graphing Functions The order of a function is the highest power of 𝑥 in the equation and is always one more than the number of curves of the graph.

Function Type

Order

# of curves

Graphs

Linear

1

0

Quadratic

2

1

Cubic

3

2

Linear A linear function is one that takes the form, 𝑓(𝑥) = 𝑚𝑥 + 𝑐 Where 𝑚 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑎𝑛𝑑 𝑐 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. Quadratic A Quadratic function is one that takes the form, 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Where 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are real numbers and 𝑎 is not equal to 0. If 𝑎 is positive the curve is a minimum curve (smilie face). If 𝑎 is negative the curve is a maximum curve (frownie face). Cubic A cubic function is one that takes the form, 𝑓(𝑥) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 Where 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑑 are real numbers and 𝑎 is not equal to 0. If 𝑎 is positive the maximum is followed by a minimum. If 𝑎 is negative the minimum is followed by a maximum. Graphs of Modulo Functions Modulo functions take the form 𝑓(𝑥) = |𝑥| and must have only positive y-values. That being the case, any portion of the function that falls below the x- axis (negative values of y) must be made positive by reflection in the x-axis.

Transformations Let's start with the function notation for the basic quadratic: f(x) = x2. A function transformation takes whatever is the basic function f(x) and then "transforms" it or "translates" it, which is a fancy way of saying that you change the formula a bit and thereby move the graph around.

For instance, the graph for x2 + 3 looks like this:

This is three units higher than the basic quadratic, f(x) = x2. That is, x2 + 3 is f(x) + 3. We added a "3" outside the basic squaring function f(x) = x2 and thereby went from the basic quadratic x2 to the transformed function x2 + 3. This is always true: To move a function up, you add outside the function: f(x) + b is f(x) moved up bunits. Moving the function down works the same way; f(x) – b is f(x) moved down b units.

On the other hand, (x + 3)2 looks like this:

In this graph, f(x) has been moved over three units to the left: f(x + 3) = (x + 3)2 is f(x) shifted three units to the left. This is always true: To shift a function left, add inside the function's argument: f(x + b) givesf(x) shifted b units to the left. Shifting to the right works the same way; f(x – b) is f(x) shifted b units to the right. Warning: The common temptation is to think that f(x + 3) moves f(x) to the right by three, because "+3" is to the right. But the left-right shifting is backwards from what you might have expected. Adding moves you left; subtracting moves you right. If you lose track, think about the point on the graph where x = 0. For f(x + 3), what does x now need to be for 0 to be plugged into 𝑓? In this case, x needs to be –3, so the argument is –3 + 3 = 0, so I need to shift left by three. This process will tell you where the x-values, and

thus the graph, have shifted. At least, that's how I was able to keep track of things....

The last easy transformation is –f(x). Look at the graph of –x2:

This is just f(x) flipped upside down. Any points on the x-axis stay on the x-axis; it's the points off the axis that switch sides. This is always true: –f(x) is just f(x) flipped upside down.

For this next transformation, I'll switch to g(x) = x3:

If I put –x in for x, I get (–x)3 = –x3:

This transformation rotated the original graph around the y-axis. Any points on the y-axis stay on the yaxis; it's the points off the axis that switch sides. This is always true: g(–x) is the mirror image of g(x), as reflected in the y-axis. The transformations so far follow these rules:    

f(x) + a is f(x) shifted upward a units f(x) – a is f(x) shifted downward a units f(x + a) is f(x) shifted left a units f(x – a) is f(x) shifted right a units

 

–f(x) is f(x) flipped upside down ("reflected about the x-axis") f(–x) is the mirror of f(x) ("reflected about the y-axis")

There are two other transformations, but they're harder to "see" with any degree of accuracy. If you compare the graphs of 2x2, x2, and (1/2)x2, you'll see what I mean: 2x2

x2

_1/2 x2_

The parabola for 2x2 grows twice as fast as x2, so its graph is tall and skinny. On the other hand, the parabola for the function (1/2)x2 grows only half as fast, so its graph is short and fat. You can tell, roughly speaking, that the first graph is multiplied by something bigger than 1 and that the third graph is multiplied by something smaller than 1. But it is generally difficult to tell exactly what a graph has been multiplied by, just by looking at the picture.

Graphing Questions 1. 2.

3.

6. 4.

7.

5.

8.

9.

10.

11.

12.

13.

14.

16.

17. 15.

18.

19.

In the 𝑥𝑦 −plane above, the graph of the function 𝑓 is a line, and the graph of the function 𝑔is a parabola. The graphs of 𝑓 and 𝑔 intersect at (0,0) and (3,3). For which of the following values of 𝑎 is 𝑔(𝑎) − 𝑓(𝑎) < 0? (A) 0 (B) 2 (C) 4 (D) 5 (E) 6

20.

Graphing Solutions 1. 2. 3. 4. 5. 6. 7.

B C A C E A D

8. 9. 10. 11. 12. 13. 14.

E C D E A B D

15. A 16. E 17. 3⁄4 18. 2 19. 2 < 𝑥 < 3 20. 8.5

Data Analysis, Statistics, and Probability (10–15%) Mean, Median and Mode Mean, Median and Mode Questions 1. Which of the following CANNOT change the value of a median in a set of five numbers? (A) Adding 0 to the set (B) Multiplying each value by -1 (C) Increasing the least value only (D) Increasing the greatest value only (E) Squaring each value 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

X X X X X X X X X X X X X X X X

18. X 19. X 20. {2, 3, 9, 4, 11, 4x – 8, 3y – 4} The modes of the set above are 2 and 11. What is one possible value of 𝑥 + 𝑦?

Mean, Median and Mode Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

D X X X X X X X X X X X X X X X X X X 7.5 or 6.75

Probability Probability is the mathematical expression of the likelihood of an event. 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑥 =

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑥 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Every probability is a fraction. The biggest a probability can get is 1; a probability of 1 indicates total certainty. For example, 𝐴 𝑏𝑎𝑔 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 7 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑎𝑛𝑑 14 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑏𝑙𝑢𝑒. 𝐼𝑓 𝑜𝑛𝑒 𝑚𝑎𝑟𝑏𝑙𝑒 𝑖𝑠 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑏𝑎𝑔, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑚𝑎𝑟𝑏𝑙𝑒 𝑖𝑠 𝑏𝑙𝑢𝑒?

(A) 1⁄7 (B) 1⁄3 (C) 1⁄2 (D) 2⁄3 (E) 3⁄7

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑝(𝑑𝑟𝑎𝑤𝑖𝑛𝑔 𝐵𝑙𝑢𝑒) = 7⁄7 + 14 = 7⁄21 = 1⁄3

Probability of Multiple Events Some advanced probability questions require you to calculate the probability of more than one event. Here’s a typical example: 𝐼𝑓 𝑎 𝑓𝑎𝑖𝑟 𝑐𝑜𝑖𝑛 𝑖𝑠 𝑓𝑙𝑖𝑝𝑝𝑒𝑑 𝑡ℎ𝑟𝑒𝑒 𝑡𝑖𝑚𝑒𝑠, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑡𝑎𝑖𝑙𝑠 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 𝑡𝑤𝑖𝑐𝑒? (A) 1⁄8 (B) 1⁄5 (C) 3⁄8 (D) 5⁄8 (E) 2⁄3

When the number of possibilities involved is small enough, the easiest and safest way to do a probability question like this is to write out all of the possibilities and count the ones that give you what you want. Here are all the possible outcomes of flipping a coin three times: heads, heads, heads

tails, tails, tails

heads, heads, tails

tails, tails, heads

heads, tails, heads

tails, heads, tails

heads, tails, tails

tails, heads, heads

As you can see by counting, only three of the eight possible outcomes produce tails exactly twice. The chances of getting exactly two tails is therefore 3⁄8. The correct answer is C. Sometimes, however, you’ll be asked to calculate probabilities for multiple events when there are too many outcomes to write out easily. Consider, for example, this variation on an earlier question: 𝐴 𝑏𝑎𝑔 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 7 𝑏𝑙𝑢𝑒 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑎𝑛𝑑 14 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑏𝑙𝑢𝑒. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡ℎ𝑟𝑒𝑒 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚 𝑓𝑟𝑜𝑚 𝑡ℎ𝑖𝑠 𝑏𝑎𝑔 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑏𝑙𝑢𝑒? (A) 1⁄3 (B) 1⁄9 (C) 1⁄21 (D) 1⁄38 (E) 1⁄46 Three random drawings from a bag of 21 objects produce a huge number of possible outcomes. It’s not practical to write them all out. To calculate the likelihood of three events combined, you need to take advantage of a basic rule of probability:

The probability of multiple events occurring together is the product of the probabilities of the events occurring individually. Solution, 𝑃(𝑑𝑟𝑎𝑤𝑖𝑛𝑔 1𝑠𝑡 𝐵𝑙𝑢𝑒) = 7⁄21 𝑃(𝑑𝑟𝑎𝑤𝑖𝑛𝑔 2𝑛𝑑 𝐵𝑙𝑢𝑒) = 6⁄20 𝑃(𝑑𝑟𝑎𝑤𝑖𝑛𝑔 3𝑟𝑑 𝐵𝑙𝑢𝑒) = 5⁄19 𝑃(𝑑𝑟𝑎𝑤𝑖𝑛𝑔 𝑓𝑖𝑟𝑠𝑡 3 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑑𝑟𝑎𝑤𝑛 𝑎𝑟𝑒 𝑏𝑙𝑢𝑒) = 7⁄21 × 6⁄20 × 5⁄19 = 1⁄38 𝑐ℎ𝑜𝑖𝑐𝑒 𝐷

Probability Questions 1. A bag contains 4 red hammers, 10 blue hammers, and 6 yellow hammers. If three hammers are removed from the bag at random and no hammer is returned to the bag after removal, what is the probability that all three hammers will be blue? (A) (B) (C) (D) (E)

1 2 1 8 3 20 2 19 3 18

2. The probability that only two people in a room of three people are male is (A) (B) (C) (D) (E) 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

X X X X X X X X X X X

3 8 1 4 2 5 2 3 5 6

14. 15. 16. 17. 18. 19. 20.

X X X X X X X

Probability Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

D A X X X X X X X X X X X X X Xx X X X x

Factorials The factorial of a number, 𝑛, is a product of consecutive numbers from 𝑛 to 1 and is written 𝑛! (read ‘n factorial’). That is, 1! = 1

4! = 4 × 3!

2! = 2 × 1

3! = 3 × 2!

3! = 3 × 2 × 1

2! = 2 × 1!

4! = 4 × 3 × 2 × 1

1! = 1 × 0!

5! = 5 × 4 × 3 × 2 × 1

𝑛! = 𝑛 × (𝑛 − 1)!

What is the factorial of zero (0!)? Consider, 𝑛! = 𝑛 × (𝑛 − 1)! (𝑛 − 1)! =

𝑛! 𝑛

∴ 𝐿𝑒𝑡 𝑛 = 1 (1 − 1)! =

1! 1

∴ 0! = 1 The factorial also describes the number of ways in which a group of items can be arranged. For example, in how many ways can 3 items be arranged? The number of ways that three items can be arranged are 3! = 3 × 2 = 6.

Permutations (nPr) A permutation is an ordered arrangement of objects. So keywords/factors that may indicate a permutation question may be:       

Arrange List Order Races (order in terms of finishing position is important) Seating arrangements Elections Numbers (changing the position of a digit in a number will change its value)

A permutation effectively provides the number of ways of arranging a group of 𝑛 items 𝑟 items at a time. Consider the question: In how many ways can 3 letters (A, B, C) be arranged, a. In groups of 3 b. In groups of 2 Solution a. Let us represent these groupings ABC

ACB

BCA

BAC

CBA

CAB

From the groupings we can see that there are 6 ways in which 3 items can be arranged 3 at a time. 𝑛!

Since 𝑛𝑃𝑟 = (𝑛−𝑟)!, then 3𝑃3 =

3! 3! 3! = = = 3! = 6 𝑤𝑎𝑦𝑠 (3 − 3)! 0! 1

Now you may have noticed that when arranging the entire group of items you will get the same result as simply taking the factorial of the number of items. Solution b. Let us represent these groupings AB

BA

CA

AC

BC

CB

There are also 6 possible groupings of 3 terms arranged 2 at a time. Using the permutation calculation, 3𝑃2 =

3! 3! 6 = = =6 (3 − 2)! 1! 1

Repetition Whenever there are repeated terms we must simply divide the number of ways that all terms can be arranged by the number of ways the repeated terms can be arranged. For example, in how many ways can the following terms be arranged? i. ii. iii. iv.

ABC EYE PARALLEL JAMAICA

Solutions: i. ii. iii.

3! 𝑜𝑟 3𝑃3 = 6 3! = 3 (2! Represents the number of ways that the E’s can be arranged) 2! 8! = 3360 (2! For the numbers of ways the A’s can be arranged and 3! 2!×3!

ways the L’s can be arranged)

For the number of

7! 3!

iv.

= 840 (3! For the number of ways the A’s can be arranged)

Groups When there are terms that have to be grouped, the group must move as a unit and is counted as one term. However, we must also account for the number of ways the terms in the group can be arranged. For example, in how many ways can 4 men and 3 women be arranged, keeping the women together? Solution: 𝑤1 𝑤2 𝑤3 𝑚1 𝑚2 𝑚3 𝑚4 While this is one possible arrangement, another arrangement may look like this 𝑚1 𝑚2 𝑚3 𝑤2 𝑤1 𝑤3 𝑚4 . The important thing to note is that the entire group of women move together, but we must also remember that within the group the women can have different seating orders. Therefore, the 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 = 5! × 3! = 720 The Round Table You may be asked for the number of arrangements of 𝑛 persons about a round table. In these instances we designate on of these persons the head of the table about which the other persons are arranged and therefore the number of arrangements is (𝑛 − 1)!. For example, in how many ways could the twelve (12) Knights of King Arthur’s round table be seated about that table? 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡𝑠 = (12 − 1)! = 11! = 39,916,800

Combinations (nCr) A combination refers to the number of ways you can select a set of terms 𝑛, 𝑟 terms at a time. That is, combinations do not count arrangements of the same terms as different groupings. Let us consider the question asked earlier with respect to permutations. Consider the question: In how many ways can 3 letters (A, B, C) be 𝐬𝐞𝐥𝐞𝐜𝐭𝐞𝐝, a. In groups of 3 b. In groups of 2 Solution a. Let us represent these groupings ABC

ACB

BCA

BAC

CBA

CAB

From the groupings we can see that there is only one way to select 3 items as no matter how they are arranged it is the same 3 letters. 𝑛!

Since 𝑛𝐶𝑟 = (𝑛−𝑟)!𝑟!, then 3𝐶3 =

3! 3! 3! = = = 1 𝑤𝑎𝑦 (3 − 3)! 3! 0! 3! 1 × 3!

For combinations we divide through by the number of ways 𝑟 terms can arrange themselves which is 𝑟 factorial ways.

Solution b. Let us represent these groupings AB

CB

CA

BA

BC

AC

There are only 3 unique groupings of 3 terms selected 2 at a time. Using the combination calculation, 3𝐶2 =

3! 3! 6 = = =3 (3 − 2)! 2! 1! 2! 2

Counting Another method of approaching permutation questions is by using the counting method which basically requires you to fill the spaces given with the available option. This method is the most widely used approach to questions in which order/position is important. For example, 8 athletes run in a 100m race. Gold, Silver and Bronze medals are awarded to the first three contestants to finish (in order). How many possible lists of medal winners are there? 𝑎𝑛𝑦 𝑜𝑓 𝑡ℎ𝑒 8 𝑟𝑢𝑛𝑛𝑒𝑟𝑠 𝑐𝑎𝑛 𝑝𝑙𝑎𝑐𝑒 1𝑠𝑡 × 𝑜𝑛𝑙𝑦 7 𝑐𝑎𝑛 𝑏𝑒 2𝑛𝑑 × 𝑜𝑛𝑙𝑦 6 𝑐𝑎𝑛 𝑏𝑒 3𝑟𝑑 = 336 Another method of calculating the number of possible lists of medal winners would be to use the permutations formula 8𝑃3 = 336.

Permutations and Combinations Questions 1. Hanna is arranging tools in a tool box. She has one hammer, one wrench, one screwdriver, one tape measure, and one staple gun to place in 5 empty spots in her toolbox. If all of the tools will be placed in a spot, one tool in each spot, and the hammer and screwdriver fit only in the first 2 spots, how many different ways can she arrange the tools in the spots? 2. The Tyler Jackson Dance Company plans to perform a piece that requires 2 dancers. If there are 7 dancers in the company, how many possible pairs of dancers could perform the piece? 3. The American Ballet Repertory Company will choose 4 new corps members from its apprenticeship program. The apprentice program is made up of 6 women and 6 men. If 3 women and 1 man are to be chosen for the corps, how many different groupings are possible? 4. 1

2

3

4

5

In a certain game, a red marble and a blue marble are dropped into a box with five equally-sized sections, as shown above. If each marble lands in a different section of the bx, how many different arrangements of the two marbles are possible? (A) 5 (B) 20 (C) 25 (D) 40 (E) 100 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

X X X X X X X X X X X X

17. How many unique pairs of parallel edges does a cuboid possess? 18. If a polygon has 44 diagonals, how many sides does it have?

19. How many parallelograms can be formed from a set of 4 parallel straight lines intersecting a set of 3 parallel straight lines? 20. Everyone shakes hands with everyone else in a room. 66 handshakes result. How many persons are in the room?

Permutations and Combinations Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

12 21 120 B X X X X X X X X X X X X 18 11 18 12

Sequences and Series Series Any group of items connected by a rule is called a sequence. Mathematical sequences are those lists of terms that use a mathematical rule to produce the next term in the list. A series is the sum or partial sum of a sequence. That is, where 1, 2, 3, 4 is a sequence, 1+2+3+4 is a series with sum 10. Arithmetic Series Arithmetic sequence is the result of adding the same number to some starting value. This number that is added repeatedly to give the next term in the sequence is called the common difference, 𝑑.

Here are some sample arithmetic series: 1, 7, 13, 19, 25, 31… 3, 13, 23, 33, 43, 53…. 12, 7, 2, -3, -8, -13… Each term in the series is derived by adding a constant to the term before it. That is, consider a series with a first term 𝑎 and a common difference 𝑑, then the first n terms can be represented as shown below:

1st Term

2nd Term

𝑎

𝑎+𝑑

3rd Term

4th Term

5th Term

…𝑛th Term

(𝑎 + 𝑑) + 𝑑 = 𝑎 + 2𝑑

(𝑎 + 2𝑑) + 𝑑 = 𝑎 + 3𝑑

(𝑎 + 3𝑑) + 𝑑 = 𝑎 + 4𝑑

…𝑎 + (𝑛 − 1)𝑑

The nth term of an Arithmetic Series 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑 The Sum of an Arithmetic Series Consider, the sequence 1, 2, 3, 4. We have already stated that its sum is 10. If the first term is added to the last and the second term to the second to last and so on the individual sums are 5. The total sum is 4 times the individual sums and twice the sum of the terms. That is, 2 × 𝑆𝑢𝑚 = 4(5) → 2 × 𝑆𝑢𝑚 = #𝑜𝑓 𝑡𝑒𝑟𝑚𝑠 × (𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 + 𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚) ∴ 𝑆𝑢𝑚 =

𝑛 (𝑎 + 𝑙) 2

This formula can be used if we know the number of terms (𝑛), first term (𝑎) and last term (𝑙). If we consider the last term (𝑙) to be defined as our nth term, 𝑙 = 𝑎 + (𝑛 − 1)𝑑 𝑆𝑢𝑚 =

𝑛 (𝑎 + 𝑎 + (𝑛 − 1)𝑑) 2

∴ 𝑆𝑢𝑚 =

𝑛 (2𝑎 + (𝑛 − 1)𝑑) 2

This formula can be used if we know the number of terms (𝑛), first term (𝑎) and the common difference (𝑑). Geometric Series

1st Term

2nd Term

𝑎

𝑎+𝑑

3rd Term

4th Term

5th Term

…𝑛th Term

(𝑎 + 𝑑) + 𝑑 = 𝑎 + 2𝑑

(𝑎 + 2𝑑) + 𝑑 = 𝑎 + 3𝑑

(𝑎 + 3𝑑) + 𝑑 = 𝑎 + 4𝑑

…𝑎 + (𝑛 − 1)𝑑

Sequence and Series Questions 1. 1, -2, 3, -4, 5, -6, … The first six terms of a sequence are shown above. The odd-numbered terms are increasing consecutive positive odd integers starting with 1. The even-numbered terms are decreasing consecutive negative even integers starting with -2. What is the sum of the 50th and 51st terms of the sequence? (A) -101 (B) -1 (C) 0 (D) 1 (E) 101 2.

In the sequence above, the first term is 1210 and each term after the first is 1210 more than the preceding term. Which term in the sequence is equal to 1212? (A) The 3rd term (B) The 12th term (C) The 24th term (D) The 120th term (E) The 144th term 3. 4. 5. 6. 7. 8.

X X X X X X

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

X X X X X X X X X X X

The sequence of figures above starts with one hexagon. Each successive figure is formed by adding a ring of identical hexagons around the preceding figure. What is the total number of hexagons in the fifth figure of the sequence?

Sequence and Series Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

D E x x x x x x x x x x x x x x x x x

20. 61

Logic Some questions will require you to use your reasoning ability in order to find a solution. These are often approached by scanning the answers provided to determine which matches the criteria in the question.

Logic Questions 1. Nails were sold in 8 ounce and 20 ounce boxes. If 50 boxes of nails were sold and the total weight of the nails sold was less than 600 ounces, what is the greatest possible number of 20 ounce boxes that could have been sold? (A) 34 (B) 33 (C) 25 (D) 177 (E) 16 2. How many numbers from 1 to 200 inclusive are equal to the cube of an integer? 3. Lonnie sometimes goes to comedy movies. Greta never goes to mystery movies. If the two statements above are true, which of the following statements must also be true? I. Lonnie never goes to mystery movies. II. Greta sometimes goes to comedy movies. III. Lonnie and Greta never go to mystery movies together. (A) (B) (C) (D) (E)

I only II only III only I and III II and III

4.

A garden consists of a continuous chain of flower beds in the shape of pentagons, the beginning of which is shown in the figure above. There are 17 flower beds in the chain, and each one,

except the first and last, shares two of its sides with adjacent flower beds. If the length of each side of each bed is 1 meter, what is the perimeter of the garden? (A) 50 meters (B) 51 meters (C) 53 meters (D) 55 meters (E) 57 meters 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

X X X X X X X X X X X X X X X X

Logic Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

E E C C X Xx X X X X X X Xx X Xx X Xx

18. X 19. X 20. X