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The real Number system What is the real number system? All the numbers we use for counting, measuring and calculating are called real numbers. All the numbers you will come across at this stage are real numbers. They are made up of a collection of different types of numbers. Types of Numbers Natural numbers (counting numbers) These are the numbers 1, 2, 3, 4, etc. The smallest is 1. We also call these numbers COUNTING NUMBERS. There is no largest natural number. This set of numbers is denoted by the symbol N. We may therefore write this set as follows: N 1; 2; 3; 4;... Whole numbers (W) Whole numbers include all natural numbers and zero. The numbers 0, 1, 2, 3,… are called whole numbers. Remember that every natural number is a whole number. Zero is not a natural number therefore not every whole number is a natural number. Integers (Z) All the positive and negative whole numbers including zero are integers. This set is denoted by the symbol Z. We may write this set as follows: Z ..., 4, 3, 2, 1, 0, 1, 2, 3, 4,... You can represent these numbers on a number line as shown below: Rational numbers (Q) These are numbers of the form a where a and b are integers and b 0 . This means that b the numbers we can write down as one integer divided by another integer so long as the bottom integer is not zero. These integers can be positive or negative. The list of rational numbers is also 1 , 3 , 11, 201, 3 3 is also a rational number, 2 5 6 49 a as 3 . because we can express 3 in terms of b 1 endless. Some rational numbers are This collection is denoted by the symbol Q. Thus we can state that: Q a such that a and b areintegersand b 0 b Irrational numbers ( Q ) Any number which is not rational is referred to as an IRRATIONAL NUMBER. Examples: (which has the value 3.141592654...), 2, 5, 2.7182818284590452358 and so on. These numbers cannot be written as a quotient of two integers. More generally, the square roots of all prime numbers are irrational numbers. We shall denote the collection of all irrational numbers by Q. Primes and Composites x 1 is said to be prime if and only if it is divisible by 1 and itself. A natural number Meaning, a prime number has only two natural factors. Otherwise, a number which is not a prime is called a composite - i.e. it is composed of more than two natural factors. EXAMPLES Primes: Composites: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, etc. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, etc. NOTE: Zero and number 1 are not prime numbers. Factors The factors of a number are those numbers which divide exactly into a given number. E.G. The factors of 24 are 1, 2, 3, 4, 6, 12, 24 The pairs of factors of 30 are: 1 x 30 2 x 15 3 x 10 and 5x6 Therefore, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30 Multiples A multiple of a number n is k n where k is a counting number. Examples: Some multiples of 5 are 5,10,15, … Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, … Factors and Multiples (open slides) Prime Factors Remember that the factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30 The prime factors of 30 are: 2, 3, and 5 We can find the prime factors by expressing a number as a product of a primes. How to express a number as a product of primes or product of prime factors: E.G. Express 44 as a product of primes (prime factors) 1. Write down the 1st few prime numbers e.g. 2, 3, 5, 7, 11, 13 Divide 44 by the 1st prime number (2) as many times as possible until it can no longer divide exactly into that number. 2. 3. 4. Divide 44 by the next prime number (3) as many times as possible and so on until you get 1. Write down the product of all the prime numbers you divided in. 44 : 22 : 11 : 1 2 2 11 e.g. or 2 44 2 22 11 11 1 Therefore 5. 44 2 2 11 Write any repeated prime number as powers (using index form) e.g. 44 22 11 Exercises (a) (b) Or Express 6960 as a product of primes Express each of the following numbers as a product of prime numbers. Decompose the following numbers into prime factors 1200 4464 8 000 2464 Highest Common Factors ( HCF) or Highest Common Divisor(HCD) To find the HCF we can apply the following: Method 1: List all factors of the given numbers and Find the highest common factor of 8 and 12 We know that factors of 12 are: 1, 2, 3, 4, 6, 12 identify the HCF. and factors of 8 are: 1, 2, 4, 8 The common factors of 8 and 12 are 1, 2, and 4. But the highest common factor is 4, hence HCF is 4. Method 2: Factorise the given numbers into their prime factors respectively. Select those common factors (with lowest power) and multiply them together. e.g. for 12: 12 : 6 : 3 :1 2 2 3 12 22 3 and For 8: 8 : 4 : 2 :1 2 2 2 3 8 2 2 2 2 Hence, the HCF of 12 and 8 is 22 4 Exercises: Find the HCF of (a) (b) (c) (d) (e) 24, 72, 96 and 300 25, 455, 1050 9 and 45 12, 26 and 36 255 and 75 Multiples A multiple of a number n is k x n where k is a counting number. Examples: Multiples of 5 are 5,10,15, … Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, … Lowest Common Multiple (LCM) Method 1: List the multiples of the given numbers and LCM. identify the Find the LCM of 12 and 8 8 = 8, 16, 24, 32 12 = 12, 24, 36 24 is the LCM of 12 and 8 Method 2: Factorise the given numbers into their prime factors respectively. Select every number (prime factors) with highest power which occur in any of the decompositions (prime factors) of each of the given numbers and multiply them together. The LCM of 12 and 8 For 12: 12 : 6 : 3 :1 2 2 3 12 22 3 And For 8: 8 : 4 : 2 :1 2 2 2 8 2 2 2 23 Hence, the LCM of 8 and 12 is = Exercises: Find the LCM of 23 3 24 (a) 24, 72, 96 and 300 (b) 25, 455, 1050 (c) 9 and 45 (d) (e) 12, 26 and 36 255 and 75 Problem sums on HCF and LCM can be sometimes tricky as they are not easy to identify. The main focus here is how to determine when to find the HCF and when to find the LCM of the numbers involved in the problem sums. First let’s take a look at a problem involving the HCF. 3 strings of different lengths, 240 cm, 318 cm and 426 cm are to be cut into equal lengths. What is the greatest possible length of each piece?6 LCM problem: Two lighthouses flash their lights every 20s and 30s respectively. Given that they flashed together at 7pm, when will they next flash together? One method to finding the next time the lighthouses flash together is: 20, 40, 60 30, 60, 90 60 is a multiple common to 20 and 30, and thus the lighthouses will flash together in 60s’ time, i.e. at 7:01pm. This is the same as finding the lowest common multiple, or LCM: More Examples: 1. As a humanitarian effort, food ration is distributed to each refugee in a refugee camp. If a day’s ration is 284 packets of biscuits, 426 packets of instant noodles and 710 bottles of water, what is the greatest possible number of refugees are there in the camp? [142 refugees] 2. 294 blue balls, 252 pink balls and 210 yellow balls are distributed equally among some students with none left over. What is the biggest possible number of students? [42 students] 3. A group of girls bought 72 rainbow hairbands, 144 brown and black hairbands, and 216 bright-coloured hairbands. What is the largest possible number of girls in the group? [72 girls] 4. A man has a garden measuring 84 m by 56 m. He wants to divide them equally into the minimum number of square plots. What is the length of each square plot? [28 m] 5. Leonard wants to cut identical square as big as he can from a piece of paper 168 mm by 196 mm. What is the length of each square? [] 6. Candice, Gerald and Johnny were jumping up a flight of stairs. Candice did 2 steps at a time, Gerald 3 steps at time while Johnny 4 steps at a time. If they started on the bottom step at the same, on which step will all 3 land together the first time? [] 7. Heidi helps out at her mum’s stall every 9 days while her sister every 3 days. When will they be together if they last helped out on June 16, 2008? 8. A group of students can be further separated into groups of 5, 13 and 17. What is the smallest possible total number of students? 9. Jesslyn goes to the market every 64 days. Christine goes to the same market every 72 days. They met each other one day. How many days later will they meet each other again? [] 10. A Polytechnic choir coordinator wants to divide the choir into smaller groups. There are 24 sopranos, 60 altos and 36 tenors. Each group will have the same number of each type of voice. 10.1 What is the greatest number of can be formed? groups that 10.2 How many sopranos, altos and in each group? tenors will be FUNDAMENTAL OPERATIONS ON WHOLE NUMBERS Directed Numbers To add two directed numbers with the same sign, find the sum of the numbers and give the answer the same sign. Examples 3 (5) 8 7 (3) 10 9.1 (3.1) 12.2 2 (1) (5) 8 To add two directed numbers with different signs, find the difference between the numbers and give the answer the sign of the larger number. Examples 7 (3) 7 3 4 9 (12) 9 12 3 8 (4) 8 4 4 To subtract a directed number, change its sign and add. Examples 7 (5) 7 5 2 7 (5) 7 5 12 8 (4) 8 4 12 9 (11) 9 11 2 MULTIPLICATION (+75) x (-88) = (-97) x (-93) = (-27) x (+49) = (-78) x (-33) = (+31) x (-52) = (-44) x (+22) = Integer Division (+4437) ÷ (-87) = - (-7644) ÷ (-98) = + (-2560) ÷ (+64) = - (+2376) ÷ (+54) = (+360) ÷ (+36) = + (0) ÷ (-6) = (-240) ÷ (-3) = (+15) ÷ (+1) = Rules for multiplications Pos. number x pos. number = pos. number Neg. number x neg. number = pos. number Neg. number x pos. number = neg. number Pos. number x neg. number = neg. number BASIC ARITHMETIC Rules of Arithmetic BEDMAS (brackets, exponents, division, multiplication, addition and subtraction) BODMAS (brackets, powers, division, multiplication, addition and subtraction) 1. Work out brackets 2. work out powers (exponents) 3. Divide and multiply 4. Add and subtract 1. 3(4 1) 35 15 2. 4 3 2 4 6 3. 3 4(5 2) 8 2 5 4. 412 6 6 5. 12 22 (4 23) 6. 32 3 3 22 7 411 7. 2 4 23 8 4 (7) 2 8. 2 4 2 21 4 Simplify each of the following 1. 2(4 23) 8 4 2. 4 6 3 4 2 6 3. (2 3)3 6 2 4 7 4. 2 3 7 2 5 6 2 3 5. 4 2 6 4 4 12 10 6. 25 3 2 7 (8) (5 6 44) 7. 2 5 2 5 3 8 8. 5 3 4 7 6 6 9 9. 813 209 54 10. 72 2(2 4) 24 24 (3 5) 17 3 7 2 5 × 6 2 3 11. 2 12. 13. 14. 4 2 6 4 + 4 ÷ 12 10 3 9 6 3 4 12 7 5 3 4 7 6 6 9 Vulgar Fractions – Concepts and operations In a fraction 7, 8 7 is a numerator and 8 is a denominator. 7 is referred to as a proper fraction and 8 8 is referred to as an improper. 7 2 7 is referred to as a mixed number. 8 Equivalent fractions: 246 8 3 6 9 12 OPERATIONS Evaluate and simplify your answer. 1. 1 2 1 1 2 2 2 1 5 5 2 4 3 3 2 2. 1 2 1 2 4 3 1 3 5 4 3 4 2 3. 113 2 2 1 1 2 5 1 3 3 2 5 3 2 Work out and simplify. 1. 31 8 5 7 2 10 3 3. 2 1 3 1 3 5 4 3 2. 3 1 7 2 8 5 10 3 4. 11 2 2 4 11 2 3 3 5. 11 2 2 7 7 2 3 9 6. 2 2 11 5 3 3 2 7. 2 3 2 3 7 5 7 4 8 8 5. From a group of athletes, 1 of the athletes are chosen for long jump and 1 of the 8 4 remaining athletes were chosen for javelin. One hundred and five athletes remained and they were all chosen for relay race. 5.1 How many athletes were chosen for long A. 75 B. 105 jump? C. 20 D. 35 5.2 How many athletes were chosen for A. 160 6. B. 40 javelin? C. 35 Jane earns a salary every month. She spends accommodation and for other D. 30 N $6800 which is 1 of her salary on 5 N $3 400 on food. What purposes? fraction of her salary is left A. 7. 1 3 B. 2 5 In 2007, a number of auctioneer sold number of C. 7 10 D. 5 200 vehicles were sold 3 of the vehicles. In the next 5 3 5 at an auction. In the first 3 hours the 2 hours, he sold vehicles. In the last hour the auctioneer sold 1 of the remaining 5 1 of the original number of 5 vehicles. 7.1 How many vehicles were sold during the first A. 7.2 749 B. 1456 C. 333 1 4 5 12 C. 3120 hours? D. 1040 decide to buy a car. Alex pays 1 of the cost and Charles pays 3 B. D. 7 12 1 of the 4 the rest. D. 2 7 N $12 000 more than Alex. Calculate the cost of the car. B. N $144 000 A. N $58 000 Brenda pays C. 9. 416 C. What fraction of the cost does Charles Pay? A. 8.2 4160 Three friends, Alex, Brenda and Charles cost, Brenda pays 8.1 B. How many vehicles were sold in the last three A. 8. 3536 five hours? N $102 000 D. N $60 000 Frieda earns a salary every month. She spends accommodation and for other purposes? N $3 400 which is 2 on 5 N $1 700 on food. What fraction of her salary is left A. 10. 1 3 B. 2 3 In 2007, a number of auctioneer sold number of C. 1 5 5 200 vehicles were sold 2 of the vehicles. In the next 5 vehicles. In the last hour the auctioneer sold D. 2 5 at an auction. In the first 3 hours the 2 hours, he sold 1 of the remaining 4 1 of the original number of 4 vehicles. 11.1 How many vehicles were sold during the A. 11.2 B. 780 C. How many vehicles were sold in total? A. 12. 2 860 4160 B. 585 C. 3 380 4680 Mr. Kakololo accumulated a number of shares. He sold cousin and 5 340 which is best friend, Lucas. 2 of 15 first five hours? D. D. 2 080 1 300 1 5 1 of his shares to his brother, 1 to 5 3 the original number of shares to his 12.1 What fraction of shares remained with Mr. Kakololo? 12.2 How many shares did Mr. Kakololo sell in total? Work out each of the following and simplify 13. Mr. Titus had 640 shares. He sold out one third of them to a trading company and 2 of 5 the remainder to another company. How many shares remained with Titus? 14. Andrew sold half of his cows; gave his younger cows. How 15. many cows brother 1 of the original number of 4 does Andrew now have if he had 88 originally? The Simon’s family spends 2 of their income on 5 rent, 1 on food, and 1 on clothes. 5 4 If they are left with N$390.00 each month, find: 16. (a) the fraction which is left. (b) their monthly income A man spends left with 42. 2 of their income on rent, 1 on 5 4 N$390.00 food, and 1 on clothes. If they are 5 each month, find: (a) The total fraction, spent. (b) The amount, which is left There are 60 000 soccer supporters at a game and the police estimated that 5 of 8 them support the 43. home team. Estimate the number who A man saves N$240 every month. This is supports the away team. 4 of his monthly salary. Calculate his 25 monthly salary. 44. The company decided to donate went to the orphans association, 25000 shares to some institutions. 2 of the shares 5 1 went to the cancer association and the remaining part was 4 donated to a church ‘Praise The Lord’ 45. (a) How many shares did church ‘Praise The (b) What fraction of the shares did ‘Praise The Lord’ receive? A farmer takes 250 chickens to be sold at a market. In the first hour he sells chickens. In the second hour he sells 46. Lord’ receive? 3 of those he has left. 5 (a) How many chickens were sold in the second (b) How many chickens has he sold in total? Frieda earns a salary every month. She spends accommodation and 2 of his 5 hour? N $3400 which is 2 on 5 N $1700 on food. What fraction of her salary is left for other purposes? Madam Ecka shares her monthly salary with her children as follows: Maria receives mother’s salary and Tom receives 2 of her 7 1 . If Tom receives N $950 , how much does Maria 5 receive? 47. Three friends, Alex, Brenda and Charles decide to buy a car. Alex pays Brenda pays 1 of the cost and Charles pays the rest. 3 47.1 What fraction of the cost does Charles Pay? 47.2 Brenda pays cost of the car. N $12 000 more than Alex. 1 of the cost, 4 Calculate the Decimal Fractions Converting vulgar Fractions to Decimal Fractions 1. 7 7 8 0.875 8 2. 1 0.3333 3 3. 9 0.9 10 4. 1 7 1.7 10 5. 2 2 2.285714285714 7 6. 3 19 3.19 100 Change decimals to Vulgar fractions and simplify 1. 0.35 35 7 100 20 2. 0.7 7 10 3. 3.2 3 2 31 10 5 4. 0.007 7 1000 5. 0.00011 11 100 000 6. 2.35 2 35 2 7 100 20 7. 0.625 625 5 1000 8 Types of Decimals 1. e.g. 2. Terminating Decimals 7 1.4 or 5 1 0.5 2 Recurring or repeating decimals e.g. 3. e.g. 2 0.66666 or 3 Non-recurring – non terminating decimals 12 0.214285714 56 Use your calculator to evaluate 1. 2. 3. 4. 0.6 0.27140.00589 2.41670.000717 6.51 0.1114 7.24 1.653 4.7 1.6 11.4 3.61 9.7 9.6 1.5 2.4 0.74 2 4.2 1 1 5.5 7.6 5. APPROXIMATIONS Decimal Places and Significant Figures A significant figure is a first non-zero digit. (a) 7.8126 7.81 to 3 sf (b) 0.078126 0.0781 to 3 sf (c) 3596 3600 to 2 sf and 0.078 to 3 dp SIGNIFICANT FIGURES AND DECIMAL PLACES 1. (a) (b) (c) Write the following numbers correct to: Three significant figures Two significant figures Two decimal places 1. 8.174 2. 3. 20.041 4. 0.814 52 5. 311.14 7. 0.007 47 19.617 6. 8. 15.62 0.275 9. 900.12 10. 3.555 11. 5.454 12. 20.961 13. 0.0851 14. 0.5151 15. 3.071 2. Express the following numbers correct to the indicated number of significant figures. 40,283 (2 s.f.) a) b) 0,0275 (2 s.f.) c) 4090,01249 (3 s.f.) d) 20,17 (3 s.f.) e) 38 (4 s.f.) f) 1017,2 (3 s.f.) 4099,789 (3 s.f.) g) h) 3.4104 (1 s.f.) i) 4.2578103 j) 4.2578104 ( 3 s.f.) (3 s.f.) STANDARD FORM The number integer. To write a10n is in standard form when 1 a 10 and n is a positive or negative 3 000 in standard form 3 1000 3103 150 1.5100 1.5102 0.0004 4 1 4104 10 000 Write the following numbers in standard form 1. 46 000 3. 46 900 000 4. 0.007 5. 0.421 6. 0.000 055 7. 564 000 8. 0.0004 9. 19 millions 10. A hydrogen atom weighs 0.000 000 000 000 000 000 001 67 grams. Write this weight in standard form. 11. The population of China is estimated at 1 100 000 000. Write this in standard form. 12. to 2 The population of China is estimated at 1 100 significant figures. 000 000. Write this in standard form 2. POWERS AND ROOTS. REAL NUMBERS xn signifies that x is multiplied by itself n times. x is referred to as the base and n is termed as an exponent or index. Given a positive integer, By convention an exponent of 1 is not expressed. Square numbers are 1, 4, 9, 16, 36 etc. They are called perfect squares The square of a is a2 which is a a . A cube number is the result of multiplying a number by itself three times. 222 23 8 and 38 2 Cube numbers are e.g. 1, 8, 27, 64 etc 64 means 6 multiplied itself four times 64 66666 1296 Remember: and (7)2 77 49 4 1296 6 but 72 (77) 49 Taking a square root of a positive number gives two possible answers, - or + Work out: 1. 0.36 2. 25 3. 0.09 4. 5 32 5. 3 27 81 6. 5 167.9421 leave answer to 3dp 5.67193 7. 3 2.312 0.92 2.319.81 (2sf) 1 5103 8. 4 9. 812 802 10. 11. 2.311.01 (3sf) 3 2 0.79 0.041 6 64 1331 (standard form) (4sf) ALGEBRAIC EXPRESSIONS Terms, Constants, Coefficients and Variables An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operation, and so forth. TERMS AND COEFFICIENTS The terms of an algebraic expression are the parts of the expression that are connected by plus and minus signs. An expression 150 px 25 py 80 pz , is an algebraic expression. Let us consider the algebraic expression below. 2 x 2 3 x 6 xy 4 y 5 The expression above has 5 terms, namely, 2x2 , 3 x, 6 xy , 4 y , and 5 An expression containing only one term, such as 3ab, is called a monomial (mono means one). A binomial contains two terms; e.g. 2r + by. A trinomial consists of three terms. Any expression containing two or more terms may also be called by the general name, polynomial (poly means many). See the following term: Examples Given the two algebraic expressions below, identify each variable and its coefficient. Also state the constants. (a) (b) 3b 5 4 xy x3 x2 4x 12 BASIC ALGEBRAIC EXPRESSIONS AND OPERATIONS 1. Terms, Constants, Variables and Coefficients 2. Simplification (Addition, Subtraction) 3. Expansion of algebraic expressions Brackets and simplifying Two brackets (Expansion) 4. Factorization 5. Addition, Subtraction, Multiplication and Division of Algebraic expressions Simplification: ‘Like terms’ are terms that contain the same variables raised to the same power. When we add or subtract algebraic expressions, we simply collect the like terms. The process of collecting the like terms is called simplification. 4 x 7 y 5x , simply identify the like terms. In this case 4 x and 5x are like terms hence we can add the two terms to become 1x which we normally write as x since the coefficient is one. There is only one term with y variable which is 7 y . Since there is no other like term, we keep the term as it is. If we then simplify the whole expression then we have x 7 y If we are to simplify the expression Algebraic Manipulation Addition and Subtraction of polynomials A monomial is the product of non-negative integer powers of variables. Consequently, a monomial has NO variable in its denominator. It has one term. (mono implies one) 13, 3x, -57, x², 4y², -2xy, or 520x²y² (note: no negative exponents, no fractional exponents) A binomial is the sum of two monomials. It has two unlike terms. (bi implies two) 3x + 1, x² - 4x, 2x + y, or y - y² A trinomial is the sum of three monomials. It has three unlike terms. (tri implies three) x2 + 2x + 1, 3x² - 4x + 10, 2x + 3y + 2 A polynomial is the sum of one or more terms. (poly implies many) x2 + 2x, 3x3 + x² + 5x + 6, 4x - 6y + 8 Polynomials are in simplest form when they contain no like terms. x2 + 2x + 1 + 3x² - 4x when simplified becomes 4x2 - 2x + 1 Polynomials are generally written in descending order. Descending: 4x2 - 2x + 1 exponents of variables decrease from left to right Polynomials A polynomial can be one monomial or a bunch of monomials hooked together with plus/minus signs. Examples of Polynomials are: 2x3 5x2 8x 1, y 2 7 y 6, 4z 3 4x2 is a monomial because it has one term. An expression 4 x2 3x is a binomial because it has two terms. An expression 4x2 3x 5 is a trinomial because it has three terms. An expression All three expressions can be called polynomials. The degree of the polynomial is the highest power in the variable. An expression of degree one and it is called a linear expression. 2x1 is a polynomial 2x2 3x 1 is a polynomial of degree two and it is called a quadratic expression. An expression 2 x3 3x is a polynomial of degree three and it is called a cubic expression. An expression 4 x4 2 x3 1is a polynomial of degree four and it is called a quartic expression. An expression All monomials and polynomials are algebraic expressions. FOR ADDITION AND SUBTRACTION: LIKE TERMS CONTAIN THE SAME VARIABLES WITH THE SAME EXPONENT REMEMBER ONLY LIKE TERMS MAY BE ADDED OR SUBTRACTED GROUP LIKE TERMS, THEN PERFORM ADDITION OR SUBTRACTION OPERATION BY: ADDING / SUBTRACTING COEFFICIENT/S OF LIKE TERMS. POWER/S REMAINS UNCHANGED HINT: TO AVOID ERRORS, STRIKE OUT TERMS WHICH YOU HAVE ADDED OR SUBTRACTED. Simplify each of the following expressions as much as possible. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 11x 12 y 7 x 5 y 21a2 12a 2a a2 4 7 x3 115x2 9x3 22xy2 17 x2 y 17 yx2 23 y2 x 5 ad 2x x 11ad 2 5xt 5x2t (10tx2) (7 xt 2) 8 35xy 20xy2 19xy 19xy2 10xy 1 55x y 60x y x y x y x y 11 21 11 2 2 2 2 25x5 y3z2 xyz2 44x5 y3z xyz2 26x5 y3z2 xy 4ws xy 2ws 2ws 7 571 x x 2 47 1 2 x y x y nmnm 4 3 2 3 (3 y)2 x2 (2 y)2 15. 16. 3 5 6 x2 x 2 x 1 xy2 2 y2x 5xy 5 5xy 5 3 1. Simplify: (a) (b) (c) (d ) 12x 3 y 4x 2cd 2 5cd 2 6d 2c 4c2d 7dc2 dc2 x2 y 2xy2 3x2 y 15a2 2ab 7a2 11bc 8a2 2ba 1 (e) 7a2 2a 8b c 4a2 12c 2a2 9a3 5c 2b 6a2 8b2 13a3 5ab ( f ) 22a2 17a2 5,7a 2,8b 5,2a2 6b 0,4a ( g ) 4a2 2,3b3 8a3 3ab 1,8b3 3,6a2 12,5ab 14a3 2a2 (h) 34a 22b 12b2 6c 2abc Expansion of Algebraic Expressions Expand and simplify 1. 3x 2( x 1) 2. 9 2(3x 1) 3. 3ab 2a(b 2) 4. 7 x ( x 3) 5. 7(2x 2) 3(2x 2) 6. 3( x 2) 3( x 2) 7. 5(6a 8) 4(2a 4) 8. x( x 1) x(2 3x) 9. 5n(4n 2n2 6) 3(4 n2) 10. xy(2x 2) 5(2x xy) 11. 3x 4(2x 3) 12. x( x 2) 3x( x 4) 13. 4x3 y2( xy) 2x 8x4 y3 4( x y) 3x 14. 1 x2(2 y) 2 x2 y 1 x2 3 3 4 15. 23 xy 3 xy x( y 4) 1 x 8 4 TWO BRACKETS Remove the brackets and simplify: 1. ( x y) ( x y) 2. x y 2 3. ( x y) ( x2 2x y2) 4. (a 7)2 a b 5. a b c a b c 6. k 12 k 12 7. 4 2 y 1 3 y 2 8. 3 y y 2 y 3 2 9. 3( x 2)2 x 4 2 2 10. 2 x 1 x 2 x( x 3) 11. 4 ( x 1)2 12. (2x 1)2 ( x 3)2 13. 3( x 2)2 ( x 4)2 14. ( y 3)2 ( y 2)2 Real problems in science or in business occur in ordinary language. To do such problems, we typically have to translate them into algebraic language. Problem 7. Write an algebraic expression that will symbolize each of the following. a) Six times a certain number. 6n, or 6x, or 6m. Any letter will do. b) Six more than a certain number. x + 6 c) Six less than a certain number. x − 6 d) A certain number less than 6. 6 − x e) A number repeated as a factor three times. x· x· x = x3 f) A number repeated as a term three times. x + x + x g) The sum of three consecutive whole numbers. The idea, for example, g) of 6 + 7 + 8. [Hint: Let x be the first number.] g) x + (x + 1) + (x + 2) h) Eight less than twice a certain number. 2x − 8 i) One more than three times a certain number. 3x + 1 Now an algebraic expression is not a sentence, it does not have a verb, which is typically the equal sign =. An algebraic statement has an equal sign. FACTORIZATION OF ALGEBRAIC EXPRESSIONS In the previous section we expanded expressions such as reverse of this process is called factorizing. To factorize linear expressions: x(3x1) to give 3x2 x.The We can check the answer by multiplying out the brackets: 2(2x+3) = 4x+6 Example Factorise 4x² + 6x. In this case 2x is the highest factor of both 4x² and 6x, so 2x will go outside the brackets. The remaining factors of each term are left inside the brackets, where they are recombined. We can check the answer by multiplying out the brackets: 2x(2x+3) = 4x²+6x Example Factorise 3xy² + 12x²y. In this case 3xy is the highest factor of both 3xy² and 12x²y, so 3xy will go outside the brackets. The remaining factors of each term are left inside the brackets, where they are recombined. Hence, To factorise a polynomial: - identify the HCF of the coefficients identify any variable(s) which appear commonly (with lowest power) in given terms. Take out the common factor We can also see it in this way: 4 xy 2 xz , The HCF of the coefficients is 2 and the common variable is x therefore we can factor out 2x . Now from the term 4xy if 2x is a factor then 4xy 2 y and 2xz z 2x 2x then 4 xy 2 xz can be factorized as 2 x(2 y z) In expression, 12ax 18x2 42bx The HCF of the coefficients is 6 and the common variable is x therefore we can factor out 6x . Now from the term 12ax if 6x is a 12ax 2a and from 18x2 3x and 42bx 7b factor then 6x 6x 6x thus 12ax 18x2 42bx can be factorized as 6 x(2a 3x 7b) In expression, Factorize the following expressions: 1. 21a 7 2. 3xy 6x 3. xy2 7 xy3 4.12x3 y3 x4 y2 4x2 y 5. aby 2aby aby2 6. ax bx 2cx 7. x2 y y3 z3 y 8. 3a2b 2ab2 9. ax2 ay 2ab 10. ax2 y 2ax2z 11. abx 6ky 4kz 12. x2 y y3 z2 y 13. 3a2b 2ab2 14. 6a2 4ab 2ac 15. 2a2e 5ae2 16. 2abx 2ab2 2a2b 17. ayx yx3 2 y2 x2 To factorise algebraic expressions involved four terms: Factorise - ah ak bh bk Divide into pairs (in each pair must have a variable in common) e.g. ah ak bh bk here a is common to the first pair and b is common to the second pair, therefore, we factorise each factor as follows: a(h k ) b(h k ) . Since (h k ) is common to both terms, thus we have (h k ) (a b) . We refer to the process as factorization by grouping. Factorise the following expressions 1. 6mx 3nx 2my ny 2. xh xk yh yk 3. ay az by bz 4. as ay xs xy 5. 2ax 6ay bx 3by 6. 6ax 2bx 3ay by 7. 2ax 2ay bx by 8. ms 2mt 2 ns 2nt 2 9. am bm an bn 10. xs xt ys yt 6. km 4m kn 4n 7. 4x2 6xy 6xk 9 yk 8. 20x2 y3 8xp2 6 p2 15xy3 (9) ax 3x 2a 6 (10) xa 2xb ya 2 yb (11) ab2 b3 ad 2 bd 2 (12) 6a2 ab 2b2 2a b ARITHMETIC OF FRACTIONAL ALGEBRAIC EXPRESSIONS Addition and Subtraction of algebraic fractions To simplify, write as a single fraction. e.g. 1. 2 3 8 9 17 3 4 12 12 12 2. 2 3 the LCM of x and y is xy x y 2x 3y 2xyy 3xyx 3x 2 yxy 3. 2 x x 1 2 x 3 2(2 x 3) x( x 1) ( x 1) (2 x 3) 2 4x 6 x x x 1 2x 3 2 x 5x 6 ( x 1) (2 x 3) 4. 3 4 x2 x 5. 2 5 x 3 x 1 Simplify the following algebraic expressions: 1. 5 3 x 2 2x 1 answer 13x 1 ( x 2)(2 x 1) x 2 x 10 (2 x 5)(5 x) 2. x 2 2x 5 5 x answer 3. 5 2 3 2 x 1 x 1 x 2 x2 7x 9 answer ( x 1)( x 1)( x 2) 4. 2 5 x3 x4 answer 5. x 1 x 1 x 1 x 1 answer 6. 2x 7 1 7 2x x 1 answer 4 x 3 4 x 2 35 x 49 14 x( x 1) 7. 4 3 x 1 x 1 answer 3x 7 x2 1 8. x x6 x 2 2x 3 answer x2 x 1 9. x 3x 10 x2 4 answer x5 x2 10. 3x 2 9 x x 2 4x 3 answer 3x x 1 11. 6x 2 2x 12 x 2 4 x answer 1 2 12. x 2 4 x 21 x 2 5 x 14 answer x3 x2 2 7x 7 ( x 3)( x 4) 4x x 1 2 Multiplication of Algebraic Fractions To multiply two algebraic fractions, we simply multiply the numerator to get the numerator of the product, and multiply the denominators to get the denominator of the product. e.g. To multiply: x 3 x3 3x 2 x 1 x 1 2 x 1 x 1 2 x 1 x 1 Linear Equations 1. Solving linear equations in one variable. 2. Simple word problems involving linear equations An equation has to have an equals sign, as in 3x + 5 = 11 . bx c 0 where b 0 bx c 0 has the solution x c b A linear equation has the general form; c b c 0 c Checking the solution by substituting for x in the equation: b b c c 0 Example: 4 3x 2 4 2 3x 4 2 3x 2 3x 2x 3 x terms on both sides, collect them on one side. 2 x 7 5 3x 2 x 3x 5 7 5x 12 x 12 2 2 5 5 If there is a fraction in the x term, multiply out to simplify the equation. If there are 2x 10 3 2x 30 x 30 15 2 Solve the following equations: 1. 2x 5 11 2. 3x 7 20 3. 2 x 6 20 4. 5 x 10 60 5. 6. 3x 7 10 7 x 2 7. x 1 2 3 8. 3 2x 4 3 10. 3 3x 4 5 2x 4 x 3 11. 2y 1 4 3y 12. 7 3x 5 2 x 13. x 16 16 2 x 9. 14. 15. x 1 1 2 4 3 x 1 x 5 10 5 5 16. x 2(x 1) 1 4(x 1) 17. x 3( x 1) 2 x 18. 4(1 2 x ) 3( 2 x ) 19. 7 ( x 1) 9 ( 2 x 1) 20. 3( 2 x 1) 2( x 1) 23 21. 4( y 1) 3( y 2) 5( y 4) 22. 10(2x 3) 8(3x 5) 5(2x 8) 0 23. 10 ( x 4) 9( x 3) 1 8( x 3) 24. 6(3 x 4) 10 ( x 3) 10( 2 x 3) 25. 1 6 x 30( x 12) 2 x 1 2 26. 6(2x 1) 9( x 1) 8 x 11 4 27. 10 ( 2.3 x ) 0.1(5 x 30 ) 0 28. 8 2 1 x 3 1 (1 x) 1 4 4 2 2 29. (6 x ) ( x 5) ( 4 x ) 30. 101 x (10 x) 1 (10 x) 0.05 10 100 x 2 Example: 2 2 x 3 x 2 32 ( x 3)( x 3) ( x 2)( x 2) 9 x2 6 x 9 x2 4 x 4 9 6 x 9 4 x 13 2x 4 x2 Solve the following equations 1. x 2 4 ( x 1)( x 3) 2. x 3x 1 x 2 5 3. x 2x 3 x 7x 7 4. x 2 x 1 2 x 1 x 4 5. x 1x 3 x 12 2 xx 4 6. 7. 2 2 3 x 2 x 3 3x 11 2 2 2 x 1 x 2 x x 3 When solving equations involving fractions, multiply both sides of the equation by a suitable number to eliminate the fractions. x 4 2x 1 4 3 x 4 12 2 x 1 12 4 3 3 x 4 42 x 1 3 x 12 8 x 4 16 5 x 16 x 5 1 x3 5 Solve the following equations: 1. x3 x4 2 5 2. x 2 3x 6 7 5 3. x x 2 3 4 4. 5 10 x 1 x 5. 5 15 x5 x7 6. 4 7 x 1 3x 2 7. x 1 x 1 1 2 3 6 8. 1 x 2 1 (3 x 2 ) 3 5 9. 1 x 1 1 x 1 0 2 6 10. 1 x 5 2 x 0 4 3 11. x 1 2x 3 1 4 5 20 12. 4 3 1 x 1 x Problems solved by linear equations: 1. The sum of three consecutive whole numbers is 78. Find the numbers. 2. The sum of four consecutive numbers is 90. Find the numbers. 3. Find three consecutive even numbers which add up to 1524. 4. When a number is doubled and then added to 13, the result is 38. 5. When 7 is subtracted from three times a certain number, the result is 28. What is the number? 6. The sum of two numbers is 50. The second number is five times the first. Find the numbers. 7. The difference between two numbers is 9. Find the numbers, if their sum is 46. 8. The product of two consecutive even numbers is 12 more than the square of the smaller number. Find the numbers. 9. The sum of three numbers is 66. The second number is twice the first and six less than the third. Find the numbers. 10. David weighs 5kg less than John. John weighs 8kg less than Paul. If their total weight is 197kg, how heavy is each person? 11. Brian is 2 years older than bob who is 7 years older than mark. If their combined age is 61 years, find the age of each person. 12. Richard has four times as many marbles as John. If Richard gave 18 to John they would have the same number. How many marbles has each? 13. Stella has five times as many books as Tina. If Stella gave 16 books to Tina, they would each have the same number. How many books did each girl have? 14. A tennis racket costs N$12 more than a hockey stick. If the price of the two is N$31, find the cost of the tennis racket. Answ.N$21.50 1. One half of Mari’s age two years from now plus one-third of her age three years ago is twenty years. If we let Mari’s age be x, which of the equations below give the correct mathematical translation of the statement? 1 1 A. x 2 x 3 20 2 3 1 1 B. x 2 ( x 3) 20 2 3 1 1 C. ( x 2) ( x 3) 20 2 3 1 1 D. ( x 2) x 3 20 2 3 How old was maria three years ago? How old will maria be in 2yrs from now? How old will maria be in 10 years time? 2. During the class period, the number of girls is 10 less than 2 times the number of boys. 2.1 Formulate a mathematical equation to express the number of girls in terms of boys, given that g represent the number of girls and b represent number of boys. 2.2 If the total number of learners in that class period were 80, use the equation to formulate the above statement to determine the number of boys and girls in that class period. 3. During the Global Leadership Convention, it is discovered that the number of men who are attending the convention is nine hundred and forty less than four times the number of women in attendance. 3.1 From the statement above, formulate a mathematical equation expressing the number of men in terms of women, given that represent the number of women and 3.2 represent the number of men. Given that the total number of people who are attending the Global Leadership Convention are twenty thousand five hundred and sixty, use the equation you formulated in 3.1 to determine the number of men who are attending the convention. 3.3 Fruit & Veg. shop in Windhoek sells 5l of water bottles. 3.3.1 On Wednesday Fruit & Veg shop received N $2 530 from selling 5l bottles of water at N $11.50 . How many bottles of water were sold? 3.3.2 On Thursday, the shop received N$ x by selling bottles of water at N $11.50 each. In terms of x, how many bottles of water were sold? 3.3.3 On Friday the shop received N $( x 20) by selling bottles of water at N $9 each. In terms of x, how many bottles of water were sold? 3.4 If the length of a rectangular play field is double the width and the area of the play field is 24 200 square metres, calculate the perimeter of the play field . 3.5 I am 41 years old and my son is 5 years old. After x years, my son’s age will be half my age. What is the value of x? 3.6 You had a sum of money. Two hundred dollars have just been added on to it. What you now have is four hundred dollars more than half of what you originally had. How much did you originally have? 3.7 John has N$6000 to invest. He invests part of it at 5% and the rest at 8%. How much should be invested at each rate to yield 6% on the total amount? 3.8 A retailer incurs a fixed cost of N$330 when purchasing sugar for his stock. He pays N$15 per packet which he resells at N$18 per packet. How many packets should he purchase and sell in order to break even? 3.9 The sum of four consecutive numbers is 20 more than the sum of the second and the forth numbers. Find the consecutive numbers. SETS and SET THEORY A set is a collection of distinct objects, e.g.symbols, numbers, names etc. considered as an object in its own right. “A collection of well-defined objects". The objects in a set are called the members of the set or the elements of the set. A set should satisfy the following: 1) The members of the set should be distinct.(not be repeated) 2) The members of the set should be well-defined.(well-explained) Sets are one of the most fundamental concepts in mathematics. There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description: For Example A is the set whose members are the first four positive integers. B is the set of colors of the French flag. The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in brackets: C = {4, 2, 1, 3} D = {blue, white, red} Unlike a multiset, every element of a set must be unique; no two members may be identical. NOTATIONS: 1. intersection Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} . If set Set A 1, 2, 3, 4, 5, 7, 9, 10 and Set B 2, 3, 4, 6, 8, 11, 12then A B is 2, 3, 4 A B is shaded 2. union Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} . If set Set A 1, 2, 3, 4, 5, 7, 9, 10 and Set B 2, 3, 4, 6, 8, 11, 12then A B is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 A B is shaded 3. is a subset of A set is a subset of another set when all the elements in the first set are also a member of the second set. By definition, all sets are subsets of themselves and by convention, the null set is a subset of all sets. If A 1, 2, 3 and 4. B 1, 2, 3, 4, 5 then A is a subset of B. A B is a member of or belongs to. If A 1, 2, 3, 4 then 3 is a member of A. 3 A and 5 A 5. universal set or S The totality of all sets. The universe (usually represented as ) is a set containing all possible elements If set Set A 1, 2, 3, 4, 5, 7, 9, 10 and Set B 2, 3, 4, 6, 8, 11, 12then 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 6. A complement of or not in A Complement of set A relative to set U, denoted Ac, is the set of all members of U that are not members of A. The complement of a set is the set containing all elements of the universe which are not elements of the original set. This terminology is most commonly employed when U is a universal set, as in the study of Venn diagrams. This operation is also called the set difference of U and A, denoted U \ A. The complement of {1,2,3} relative to {2,3,4} is {4} , while, conversely, the complement of {2,3,4} relative to {1,2,3} is {1} . If A 1, 2, 3, 4 and 1, 2, 3, 4, 5, 6,7, 8 then A 5, 6, 7, 8 This compliment contains all those elements of that are not in A. 7. n( A) the number of elements in set A If A 1, 2, 3, 4 then n ( A) 4 8. A x : x is an int eger, 2 x 9 A is the set of elements x such that x is an integer and 2 x 9. The set A is 2, 3, 4, 5, 6, 7, 8, 9 This is an example of property definition method 9. empty set Note an 10. for any set A. Difference and Symmetric difference Difference: If A 1, 2, 3, 4, 5, 6, 7 and B 1, 5, 6, 7, 8, 9, 10 then A difference B which is denoted by A-B or A\B, is the set of all those elements of A which are not in B. A B 2, 3, 4 Symmetric difference: Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B). The symmetric difference between two sets A and B is defined as the set of all those elements that belong to A or to B but NOT to both A and B. A B 2, 3, 4, 8, 9, 10 A B ( A B) ( A B) 11. Power set of a set A is the set whose members are all possible subsets of example, the powerset of {1, 2} is { {}, {1}, {2}, {1,2} } . A. For Example: 1, 2, 3...,12, (a ) (b) (c ) A 2, 3, 4, 5, 6 and B 2, 4, 6, 8, 10 A B 2, 3, 4, 5, 6, 8, 10 A B 2, 4, 6 A 1, 7, 8, 9, 10, 11, 12 ( d ) n A B 7 (e) B A 3, 5 1. If X 1, 2, 3, ..., 10, Y 2, 4, 6, ..., 20 and Z x : x is an int eger ,15 x 25 Find: (a) (d) 2. X Y (b) Y Z (c) X Z n( X Y ) (e) n(Z ) (f) n( X Z ) If A a, b, c, d , e B a, b, d , f , g C b, c, e, g , h D d , e, f , g , h (a) A ( B D) (b) ( A D ) B (e) ( A D ) C ( f ) (C A) D (c ) B C D (d ) B (C D ) (g) ( A C) B ( h) p ( A C ) Find: From the Venn diagram above find: (a) M N (b) N M ( f ) M N (c ) ( M N ) (d ) M N ( e) N M In a school with a student population of 204 it was found that the number of girls in that school is 105. It was also discovered that there are 117 students who can swim, 97 students who are lefthanded, 80 girls who can swim, 65 girls who are left-handed, 62 left-handed students who can swim and 50 left-handed girls who can swim. Draw a Venn diagram and present the information given on that Venn diagram and answer the following questions. (a) How many left-handed children are there? (b) How many girls cannot swim? © How many boys can swim? (d) How many girls are left-handed? (e) How many boys are left-handed? (f) How many left-handed girls can swim? (g) How many boys are there in the school? SHADING Venn diagrams 1. Draw Venn diagrams and shade the following areas. (a) A ( B C ) (b) ( A B ) C (e) A ( B C ) ( f ) (B C) A (i ) ( A C ) ( B C ) (c ) A B (d ) B ( A C ) ( g ) C ( A B) ( h) ( A C ) B ( j ) A ( B C ) Application of Venn Diagram n( A B ) n( A) n( B ) n( A B ) n( M N Q ) n( M ) n( N ) n(Q ) n( M N ) n( M Q ) n( N Q ) n( M N Q ) 1. A survey on regular payment of municipal bills was carried out on 140 house owners. It was found that 60 pay electricity (E) bills regularly and 45 pay water (W) bills regularly. Further, 20 pay both bills regularly. Use a Venn diagram to find the number of house owners who (a) (b) (c) 2. pay at least one of the bills regularly. pay exactly one of the two bills regularly do not pay either bill regularly. In a class of 30 girls, 18 play netball and 14 play hockey, whilst 5 play neither. Find the number who play both netball and hockey. Let girls in the class N girls who play netball H girls who play hockey x the number of girls who play both netball and hockey The number of girls in each portion of the universal set is shown in the Venn diagram. n() 30 18 x x 14 x 5 30 Since 37 x 30 x7 7 girls play both netball and hockey 3. In the Venn diagram n ( A) 10, n( B ) 13, n( A B ) x and n( A B ) 29. (a) Write in terms of x the number of elements in A but not in B. (b) Write in terms of x the number of elements in B but not in A. © Add together the number of elements in the three parts of the diagram to obtain the equation 10 x x 13 x 18 (d) 4. Hence find the number of elements in both A and B. The sets M and N intersect such that n ( M ) 31, n( N ) 18 and n( M N ) 35. How many elements are in both M and N? 5. In a school, students must take at least on of these subjects: Maths, Physics or Chemistry. In a group of 50 students, 7 take all three subjects, 9 take physics and Chemistry only, 8 take Maths and Physics only and 5 take Maths and Chemistry only. Of these 50 students, x take Math only, x take physics only and x 3 take Chemistry only. Draw a Venn diagram, find x, and hence find the number taking Maths. 6. All of 60 different vitamin pills contain at least one of the vitamins A, B and C. Twelve have A only, 7 have B only, and 11 have C only. If 6 have all three vitamins and there are x having A and B only, B and C only and A and C only, how many pills contain vitamin A? 7. In a street of 150 houses, three different newspapers are delivered. T, G, and M. Of these, 40 receive T, 35 receive G, and 60 receive M, 7 receive T and G, 10 receive G and M and 4 receive T and M, 34 receive no paper at all. How many receive all three? 8. In a survey conducted on 2000 officers in an establishment, 48% prefer coffee ©, 54% like tea (T), and 64% do smoke (S). Further, 28% use C and T, 32 use T and S, and 39% use C and S. Only 6% use none of these. Find: How many use all three How many use T and S but not C How many use C only 9. In a survey of 60 people, it was found that 25 read the Namibian, 26 read the Republikein and 23 read the New Era, Also 9 read both the Namibian and the New Era, 11 read the Namibian and the Republikein, 8 read the Republikein and the New Era. All three papers are read by 3 people. Draw a Venn diagram to represent the given information Find the number of people in the survey who read: only the Namibian only the Republikein only the New Era the Namibian and the Republikein but not the new Era only one of the paper none of the papers 1.1 Given S 1, 2, 3, 4, 5, 6 1.1.1 Find A B A. 2, 3, 4, 6 1.2 B. 1, 5 A 1, 3, 4, 5, C. 1, 3, 4, 6 B 1, 2, 5, D. 2, 3, 4, In a survey of 200 households regarding the ownership of desktop and laptop computers, the following information was obtained: 120 households own only desktop computers, 10 households own only laptop computers and 40 households own neither desktop nor laptop computers. How many households own both desktop and laptop computers? A. 70 B. 30 C. 40 Which of the following statements is false? D. 170 A. a, b, c c, a, b 2.2 B. C. a, b a, b, c D. A A Given that A x : x is a whole number between 0 and 4 and B x : x is a negative int eger greater than 4 , find A B. A. 0 B. C. 0, 4, 4 D. The values of p, q and r in the Venn diagram below are: 2.6 A. p 160, q 200 and r 200 B. p 130, q 200 and r 320 C. p 90, q 110 and r 220 D. p 90, q 110 and r 320 From the Venn diagram below, describe the region shaded. A. A B 3.3 B. ( A B ) C C. ( A B ) C D. ( A B ) C Out of 240 students interviewed, it was found that 120 students speak Spanish (S), 60 students speak neither Spanish nor Portuguese. Further more ( x 10 ) students speak Portuguese (P) only and x speak both languages. 3.3.1 Draw a Venn diagram and show the information as given above on the Venn diagram. 3.3.2 Solve for x . 3.3.3 Find the number of students who speak Spanish only. 1.5 The values of p, q and r in the Venn diagram below are: 1.3 The values of p, q and r in the Venn diagram below are: A. p 160, q 200 and r 200 B. p 130, q 200 and r 320 C. p 90, q 110 and r 220 D. p 90, q 110 and r 320 In a survey of 240 households regarding the ownership of desktop and laptop following information was obtained: computers, the 130 households own only desktop computers, 25 households own only laptop computers and 36 households own neither desktop nor laptop computers. How many households own both desktop and laptop computers? A. 179 B. 74 C. 49 D. 104 The values of p, q and r in the Venn diagram below are: A. p 50, q 270 and r 500 B. p 500, q 50 and r 230 C. p 320, q 50 and r 592 D. p 320, q 50 and r 642 3.3 Out of 360 students interviewed, it was found that 185 students speak Spanish (S), 55 students speak neither Spanish nor Portuguese. Further more ( x 7 ) students speak Portuguese (P) only and x speak both languages. 3.3.1 Draw a Venn diagram and show the information as given above on the Venn diagram. 3.3.2 Solve for x . 3.3.3 Find the number of students who speak Spanish only. , If Ω , , , , , , , , , , , , , , , , ′ ∩ ′ ′ ∪ . . , , , , , , , , , , , , , work out the following sets: ∩ ∩ 1.2 In a survey conducted on 3400 officers in an establishment, 48% prefer coke (C), 54% like juice (J) while 64% like milk (M), Furthermore 28% drink coke and juice, 32% drink juice and milk and 30% drink coke and milk. Only 6% use none of these. 1.2.1 Draw a Venn diagram to represent this information. 1.2.2 How many officers use neither coke nor juice? 1.2.2 How many drink milk only? 1.3 Draw a Venn diagram and shade the region, ∪ ′ ∩ EXERCISES 1. Given that of 380 soccer players, 210 drink tea and coffee, 260 drink coffee and 60 drink neither tea nor coffee. How many golf players drink tea only? A. 210 B. 60 C. 120 D. 270 E. 200 2. Out of 120 students interviewed, it was found that 60 students speak Spanish (S), 30 students speak neither Spanish nor Portuguese. Further more x 8 students speak Portuguese (P) only and x speak both languages. 2.1 Draw a Venn diagram and show the information as given above on the Venn diagram. 2.2 Solve for x . 2.3 Find the number of students who speak Spanish only. 3. A survey on regular payment of municipal bills was carried out on 140 house owners. It was found that 60 pay electricity (E) bills regularly and 45 pay water (W) bills regularly. Further, 20 pay both bills regularly. How many house owners pay at least one of the bills regularly? A. 40 B.20 C. 65 D. 85 4. The values of p, q and r in the Venn diagram below are: 5. A. p 160, q 200 and r 200 B. p 130, q 200 and r 320 C. p 90, q 110 and r 220 D. p 90, q 110 and r 320 In a group of 155 students, it was discovered that 70 students are male (M ) , 90 students are first year students (Y1 ) and 15 are neither male nor first year students. 5.1 Present this information in a Venn diagram. 5.2 How many female students were first year? 5.3 How many male students were first year? 6. (H). A team of athletes was selected to compete in long jump (L), javelin (J) and high jump The Venn diagram is a complete representation of the distribution of the selected athletes. From the above Venn diagram find the total number of athletes in: 6.1 ( L H ) J A. 51 6.2 8. C. 102 D. 131 B. 51 C. 18 D. 21 (L H ) J A. 29 7. B. 22 From the Venn diagram below, describe the region shaded. In a group of 255 students, it was discovered that 140 students are male (M ) , 110 students are first year students (Y1 ) and 35 are neither male nor first year students. 8.1 Present this information in a Venn diagram. 8.2 How many female students were first year? 8.3 How many male students were first year? 9. The values of p, q and r in the Venn diagram below are: A. p 160, q 200 and r 200 B. p 130, q 200 and r 320 C. p 90, q 110 and r 220 D. p 90, q 110 and r 320 10. Out of 360 students interviewed, it was found that 185 students speak Spanish (S), 55 students speak neither Spanish nor Portuguese. Further more ( x 7 ) students speak Portuguese (P) only and x speak both languages. 10.1 Draw a Venn diagram and show the information as given above on the Venn diagram. 10.2 Solve for x . 10.3 Find the number of students who speak Spanish only. 11. A survey shows that 71% of Indians like to watch cricket, whereas 64% like to watch hockey. What percentage of Indians like to watch both cricket and hockey? (Assuming every Indian watches at least one of these games) that A. 135%% B. 36% C. 7% D. 35% 12. In a class of 85 boys, there are 60 boys who play chess and 35 play table tennis. 12.1 How many boys play chess only? A. 45 B. 100 How many play table tennis only? C. 15 2.7.2 A. 50 C. 15 B. 35 D. 25 D. 25 MATRIX ALGEBRA 1. VECTORS 1.1 ROW AND COLUMN VECTOR A vector is a special type of matrix that has only one row (called a row vector) or one column (called a column vector). A vector of the general form is called an n-component row vector. a , a ,..., an 1 2 a 1 a 2 . A vector of the general form is called an n-component column vector. . . a n EXAMPLES: A vector 3; 4; 5 is a 3 – component row vector. A vector 5; 0; 1; 4; 9 is a 5 – component row vector. 1 3 is a 4 – component column vector. A vector 10 5 A vector 0 0 is a 2 – component column vector. It is called a null / zero vector since all the components are zeros. The zero vector has zero magnitude and no direction. 1.2 ADDITION AND SUBTRACTION OF VECTORS Two or more vectors can be added or subtracted if they are both row vectors with the same number of components or if they are both column vectors with the same number of components. It is customary to denote vectors by bold, lower case EXAMPLES letters (e.g., a) Given: a 2; 1; 3 , Work out: 3 2 b 2; 3; 9 , c 1 , d 0 5 4 1. a b 2. b a 3. b c 4. d c 1.3 SCALAR MULTIPLES OF VECTORS If is a vector and k is a constant (number) then, we refer to the vector (k ) as a scalar multiple of where k is the scalar. It is customary to denote scalars by italicized, lower case letters (e.g., k). a a a If a a1, a2,...,an , then ka ka1, ka2,...,kan . If k 0, k only changes the magnitude (size) of the vector a . If k 0, k changes both the magnitude (size) and direction of the vector a. EXAMPLES a 2; 1; 3 , Work out: 1. 2a 2. 5c 3. 1 b 3 2 3 b 2; 3; 9 , c 1 , d 0 5 4 4. 2a 4b 1. MATRICES In this unit, we shall focus on how to: Express data correctly in matrix format Perform matrix operations (addition, subtraction and multiplications) In our discussion, we are limited to 2 2 matrices. A matrix is a collection of numbers ordered by rows and columns. It is a rectangular array of numbers arranged in rows and columns. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces, hence the definition of a matrix. The array of numbers below is an example of a matrix. 21 62 44 95 The number of rows and columns that a matrix has is called its dimension or its order. By convention, rows are listed first; and columns, second. Thus, we would say that the dimension (or order) of the above matrix is 2 x 2, meaning that it has 2 rows and 2 columns. Numbers that appear in the rows and columns of a matrix are called elements of the matrix. For example, the following is a matrix: A 5 9 8 7 3 2 The matrix A has two rows and three columns, so it is referred to as a “2 by 3” matrix. The order (size) of a matrix depends on firstly the number of rows it has and secondly the number of columns it has. The matrix A above has order 23 . 1.1 Matrix Notation Statisticians use symbols to identify matrix elements and matrices. Matrix elements. Consider the matrix below, in which matrix elements are represented entirely by symbols. This is what we refer to as general representation of matrices. a ij a a 11 12 A a a 21 22 By convention, first subscript refers to the row number; and the second subscript, to the column number. Thus, the first element in the first row is represented by a 11 The second element in the first row is represented by a 12 There are several ways to represent a matrix symbolically. The simplest is to use a boldface letter, such as A, B, or C. EXAMPLE Given: 2 1 2 1 0 1 A 1 2 11 3 Entries 4 9 1 5 a 2 , a , 11 12 a , 32 a 23 1.3 Matrix Addition and Subtraction Addition and subtraction of a matrix of order 2 x 2. If a 11 A a 21 a 12 a 22 (a ; a ) and (a ; a ) It also has two 11 12 21 22 a 12 and a 22 The matrix has two rows namely columns namely a 11 a 21 Matrices of the same order are added (or subtracted) by adding (or subtracting) the corresponding elements in each matrix. To add two matrices, they both must have the same number of rows and they both must have the same number of columns. The elements of the two matrices are simply added together, element by element (corresponding elements), to produce the results. So we can add matrices 1 4 5 3 1 4 7 9 to get 2 0 12 6 . 1.4 Scalar Multiple of a matrix To multiply matrix A a by a scalar, Thus, ij k , we multiply each and every element of A by k . kA k aij ka . ij EXAMPLE Given: 3 4 B , 0 5 Find: 3 4 6 8 2B 2 0 5 0 10 Multiplication by another matrix For 2 x 2 matrices b w d y a c x aw by z cw dy ax bz cx dz The same process is used for matrices of other orders. To perform the following multiplication: (a) 3 4 2 2 1 1 1 (3 2) (21) 5 (4 2) (11) (31) (25) 6 2 41 (15) 8 1 3 10 8 4 5 9 13 9 Matrices may be multiplied only if they are compatible. The number of columns in the left-hand matrix must equal the number of rows in the-hand matrix. Matrix multiplication is not commutative, i.e. for square matrices A and B, the product AB does not necessarily equal the product BA. Exercises 2 A 3 1 0 B 4 1 5 4 c 2 1 1. A B 2. B c 3. 2 B 4. 3 A B 5. 2C 3 A 6. 2 A B 7. C B A 8. AB 9. 2( BC ) 10. C 2 Find the value of the letters. 2 1. y x 4 7 3 y x 2 z 9 9 3 2 x 2. 1 w a 3. c x 4. 2 2 5. 0 p 6. q 2 x y 8 2 y 3 x 3 v 5 w b 2 5 1 2 0 3 d b 3 2 5 y 1 0 0 m 10 3 n 1 2 2 1 5 1 2 2q 10 3 0 y z 6 3 x 4 0 8 w 3 7. 2 3z 6 3 3y 8. 2z 8 w 2 y 4x 1 8 e 2 3 9. k 3 2 3 a 0 4 10. 1 z w 8 a 1 6 1 p 20 0 n 12 m 2 q 0 1 0 0 1 x 11. if A , B , and AB BA, find x 2 3 3 1 3 3 13. B 1 1 (a) Find k if B 2 kB SIMPLE INTERESTS P = Money borrowed or invested i = Interest on P r = annual interest rate t = time in years A = the amount due after t year Simple Interest is calculated on a on-time investment at the end of the investment period. It does not generate any interest itself. Formulas to be used in calculating simple interests: I prt A P 1 rt I r pt and I p rt or or A P I I t pr p A I or p A 1 rt EXAMPLES 1. Find the simple interest payable on a loan of N$2 500 at years. 25% p.a. at the end of 3 I prt I 2500 25%3 I N $1875 2. Find the simple interest payable on a loan of N$2 500 at 18 months. 12 1 % p.a. at the end of 2 I prt I 250012.51.5 100 I N $468.75 3. For how long should an amount of N$5000 be invested at 5% p.a. to generate an interestN$750? t ? p 5000 r 5% I 750 I t pr 750 50000.05 t 3 t 4. John wants to buy a car after 10 years. He wants to have N$75 000 at the time of purchase. How much should he invest in a savings account that pays simple interest at 12% t = 10yrs, A = 75 00 r = 12% = 0.12 p=? A p(1 rt ) or A p N $34 090.91 (1 0.1210) 5. Andrew invested N$12 550 for 5 years. After 5 yrs he received a total amount of N$22 500 from his investment. Calculate the annual rate at which interest was paid. r=? p = 12 550 A = 22 500 t = 5yrs A p 1 rt 22500 125501 r 5 22500 1 5r 12550 22500 1 5r 12550 r 0.158565737 r 0.16 16% 6. Find the simple interest on N$8 500 loan at an annual p = 8 500 r = 12% interest rate of 12% for 2yrs. t = 2yrs I prt I 85000.12 2 N $2040 7. Calculate the maturity value of an investment of N$30 000 due in 5yrs when the annual simple interest rate is 16%. r = 0.16 t=5 p = 30 000 A=? A p 1 rt A 30000(1 50.16) A N $54000 8. Benson wishes to take a loan at an annual simple interest rate of 14.5% for 7 months. He is told that he will have to pay back the sum of N$5422.92 at the end of the 7th month. Calculate the loan Benson wishes to take. r = 0.145 t= 7 12 A = 5422.92 p=? A p(1 rt) 5422.92 p(1 7 0.145) 12 5422.92 p(1.084583333) p N $5000 9. The maturity value of a loan of N$30 000.00 is N$54 000.00. (a) Calculate the annual simple interest if the loan takes 5 yrs to mature. (b) Calculate the time the loan takes to mature if the annual simple interest rate is 16% (a) I prt (there is no r, therefore find r first) A p(1 rt ) 54000 30000(1 5r) 54000 30000 150000r r 54000 30000 150000 r 24000 .16 16% 150000 I prt I 300000.165 I N $24000.00 (b) I t pr 24000 30000.16 t 24000 5 yrs 4800 t MORE EXERCISES 1. How much would you have to invest for nine years at a simple interest rate of 17.25% per annum in order to receive N $250 840.00 at the end of the ninth year? 1.2million in her estate account. This amount is to be invested in the estate for 3 years at simple interest rate of 12.5% per annum. After 3 years 2. Madam Henk left N$ the maturity value will be distributed amongst her 3 sons in the ratio of their age. Mark will be 24 years old, Paul will be 36 years old and Cyril will be 60 years old. 2.1 The maturity value after 3 years will be; A. N $1708 593.75 C. N $1650 000 D. B. N $1200 000 N $450 000 3. Dora invested N$40 000 for 10 years. After 10 years she received a total amount of N$52 000 from her investment. Calculate the annual simple interest rate at which interest was paid. 4. Find the simple interest payable on a loan of the end of 9 N $170 000 at 6.75% p.a. at years. 5. Benson wishes to take a loan at an annual simple interest rate of 14.5% for 7 months. He is told that he will have to pay back the sum of N$5422.92 at the end of the 7th month. Calculate the loan Benson wishes to take? 6. The maturity value of a loan of N$30 000.00 is N$54 000.00. (a) Calculate the annual simple interest if the loan takes 5 yrs to mature. (b) is 12.75% 7. Calculate the time the loan takes to mature if the annual simple interest rate Dora invested N$40 000 for 10 years. After 10 years she received a total amount of N$52 000 from her investment. Calculate the annual rate at which interest was paid. COMPOUND INTEREST SI is calculated once on a once-off investment at the end of the investment period. Compound Interest is calculated periodically (within the investment period). p = capital or investment A = amount at the end of investment period i = interest rate per compounding period n = number of compounding periods Formulas: A p(1 i)n 1. and P A n 1 i Calculate the amount payable for a loan of N$1000 for 3yrs at the rate of 10% p.a. compounding annually. p = 1000 r = 0.1 n=3 A p 1 i n A 1000(1 0.1)3 A N $1331 2. Calculate the amount payable for a loan of N$1000 for 3yrs at the rate of 10% p.a. compounded quarterly. p = 1000 i= 0.1 4 n=12 A p(1 i)n A 1000(1 0.025)12 A 1000(1.025)12 A N $1344.89 3. Jane inherited a sum of money from her father. She wants to invest part of the inherited money so that after 10 years, she could get N$250 000 from the investment. The bank has accepted to pay interest at 7 (a) (b) 1 % p.a. compounded semi-annually. 2 How much should Jane invest? How much interest would her investment generate? (a) A=250 000 A p(1 i) n A p (1 i ) n 250000 P 20 0.075 1 2 p N $119723.09 I= 7.5% 0.0375 2 n = 20 (b) I A p 250000 119723.09 N $130276.91 4. A trust fund is expected to grow from 360 000 to N$500 000 in 4 years when the interest rate is compounded monthly. At what annual interest rate is the trust expected to grow? p = 360 000 A= 500 000 n = 12 x 4yrs = 48 i=? A p(1 i ) n 500000 (1 i ) 48 360000 48 1.388888889 1 i 1.006867307 1 i i 0.00687 annual int erest rate is 0.00687 12 0.0824 8.24% 5. Determine the compound amount if N$5000 is invested for 10 years at 5%p.a. compounded annually. p = 5000 n = 10 i = 5% A p (1 i ) n A 5000(1 0.05)10 A N $8144.47 6. Tony invested a sum of money for 2 years at 8%p.a. compounded annually. At the end of the 2 years he received a total amount of N$1166.40. How much did Tony invest? t = 2yrs N$1166.40 r = 8% n=2 A= A (1 i) n 1166.40 p (1 0.08) 4 1166.40 p N $1000 1.1664 p 7. Determine the sum to be invested for 4 yrs at *% p.a. compounded semi-annually to amount to N$3 500 at the end of the investment period. p=? p p A = 3 500 i= 8% 0.08 2 2 n=4x2=8 A (1 i ) n 3500 1 0.048 P N $2557.42 8. If N$750 amounts to N$1200 in 3years, determine the nominal rate converted monthly. A = 1200 p = 750 n = 3 x12 I=? A p (1 i ) n 1200 36 1 i 750 1.6 (1 i ) 36 36 1. 6 1 i 36 1. 6 1 i 0.013141253 i i 0.013141253 12 0.157695036 15.8% 9. Fifty-five years old Tate Paul invested N$80000 in a savings account that paid 10% p.a. compounded semi-annually. After 5 years, the interest rate increased by 2%. The compounding period also changed to quarterly. Tate Paul made no withdrawal from this savings account until he was seventy years old. How much was in Tate Paul’s savings account at the age of seventy? For the 1st part, at the end of the 1st five years: p = 80000 I= 10% 0.05 2 n = 5 x 2 = 10 A p (1 i ) n A 800001 0.05 A N $130311.57 10 For the 2nd part, at the end of the next 10 years: p = 130311.57 i= 12% 0.03 4 n = 10 x 4 = 40 A p (1 i ) n A 130311.571 0.03 A N $425081.27 40 10. Miss Ndapandula wishes to save for her wedding day, which comes up exactly two and a half years from now. She has N$6000 to invest in a savings account that pays interest at 10% p.a. compounded every two months. How much will she have to borrow to add to her investment amount if her wedding budget stands at N$12500 on the day of her wedding? p = 6000 i= 10% 0.016666666 6 1 n = 2 6 2 A p (1 i ) n 0 .1 A 60001 6 A N $7688.29 15 N $12500 N $7688.29 N $4811.71 MORE EXERCISES 1. Leon left N$ 800 000 in his estate account. This amount is to be invested in the estate for 6 years at the interest rate of 12.75% p.a. compounded monthly. After 6 years the maturity value will be distributed amongst his 4 daughters in the ratio of their age. Maria will be 15 years old, Jolene will be 22 years old, Rolna will be 28 years old and Tina will be 8 years old. 1.1 The maturity value after 6 years will be: A. N $1 643 574 .11 N $1 712 271 .66 B. N $1 412 000 C. N $4 523 573 243 D. 2. to Determine the sum to be invested for 4 years at 7.5% per annum compounded quarterly amount to N$45 000 at the end of the investment. 3. The sum to be invested for four years at 8% p.a. compounded semi-annually to to N $3 500 at the end of the investment period is: A. N $2 651 .52 B. N $4 761 .71 C. N $2 572 .60 amount D. N $2 557.42 4. Determine the sum to be invested for 4 years at 4.5% per annum compounded monthly to amount to N$25 000 at the end of the investment. 5. Kavita has N $30 000 .00 to invest in an account that pays interest at 12.75% p.a. for five years. He has two options: Option A: Investment at simple interest. Option B: Investment with interest compounded quarterly. By showing full calculations, determine which interest option is better for Kavita 6. Determine the sum to be invested for 4 years at 7% per annum compounded semiannually to amount to N$55 000 at the end of the investment. 7. A trust fund is expected to grow from 360 000 to N$500 000 in 4 years when the interest rate is compounded monthly. At what annual interest rate is the trust expected to grow? 8. Fifty-five years old Tate Paul invested N$80000 in a savings account that paid 10% p.a. compounded semi-annually. After 5 years, the interest rate increased by 2%. The compounding period also changed to quarterly. Tate Paul made no withdrawal from savings account until he was seventy years old. How much was in Tate Paul’s savings account at the age of seventy? this 9. to Determine the sum to be invested for 4 years at 12.5% per annum compounded monthly amount to N$65 000 at the end of the investment. 10. Determine the sum to be invested for 4 years at 7.5% per annum compounded quarterly to amount to N$45 000 at the end of the investment.