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SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BBA120 BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS Lecture 2 Moses Mwale e-mail: [email protected] BBA 120 Business Mathematics Contents Unit 2: Equations and Inequalities 3 2.0 Arithmetic Operations................................................................................................. 3 2.1. Addition and subtraction .................................................................................. 3 Examples ............................................................................................... 4 2.2 Multiplication and Division............................................................................... 4 Examples ............................................................................................... 5 2.3 Order of Operations ........................................................................................... 5 Examples ............................................................................................... 5 2.4 Fractions ............................................................................................................ 6 2.4.1 Addition and Subtraction of Fractions .................................................. 7 Example ................................................................................................. 7 2.4.2 Multiplication and Division of fractions ............................................... 7 Examples ............................................................................................... 8 2.5 Decimal Representation of Numbers ................................................................ 8 2.5.1 Converting Decimals to Fractions ........................................................ 9 2.5.2 Converting repeating decimals to fractions .......................................... 9 Example #1: ......................................................................................... 10 Example #2: ......................................................................................... 11 2.5.2 Significant Figures .............................................................................. 12 Example ....................................................................................................... 13 BBA 120 Business Mathematics Unit 2: Equations and Inequalities 2.0 Arithmetic Operations Arithmetic or arithmeticβs is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. 2.1. Addition and subtraction Adding: If all the signs are the same, simply add all the terms and give the answer with the common overall sign. Subtracting: When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term. If a and b are any two numbers, then we have the following rules π + (βπ) = π β π, π β (+π) = π β π, π β (βπ) = π + π. Thus we can regard β(βπ) as equal to +π. We consider a few examples: 4 + (β1) = 4 β 1 = 3, and 3 β (β2) = 3 + 2 = 5. The last example makes sense if we regard 3 β (β2) as the difference between 3 and β2 on the number line. Note that a β b will be negative if and only if a < b. For example, β2 β (β1) = β2 + 1 = β1 < 0. 3 4 Unit 2: Equations and Inequalities Examples Add/subtract with numbers, mostly Add/subtract with variable terms 5 + 8 + 3 = 16 5π₯ + 8π₯ + 3π₯ = 16π₯ 5 + 8 + 3 + π£ = 16 + π¦ 5π₯ + 8π₯ + 3π₯ + π£ = 16π₯ + π¦ (The y-term is different, so it cannot be 5π₯π£ + 8π₯π£ + 3π₯π£ + π£ = 16π₯π£ + π¦ added to the others) 2.2 Multiplication and Division Multiplication If a and b are any two positive numbers, then we have the following rules for multiplying positive and negative numbers: π × (βπ) = β(π × π), Remember (βπ) × π = β(π × π), It is useful to remember that a minus sign is a -1, (βπ) × (βπ) = π × π. so -5 is read as β1 × 5 Also 0 × (any real number) = 0 0 ÷ (any real number) = 0 But you cannot divide by 0 Brackets are used for grouping terms together in maths for: (i) Clarity (ii) Indicating the order in which a series of operations should be carried out So multiplication of two numbers of the same sign gives a positive number, while multiplication of two numbers of different signs gives a negative number. For example, to calculate 2 × (β5), we multiply 2 by 5 and then place a minus sign before the answer. Thus, 2 × (β5) = β10. These multiplication rules give, for example, (β2) × (β3) = 6, (β4) × 5 = β20, 7 × (β5) = β35. Division The same rules hold for division because it is the same sort of operation as multiplication, since π 1 =π× π π So the division of a number by another of the same sign gives a positive number, while division of a number by another of the opposite sign gives a negative number. BBA 120 Business Mathematics Examples a) 5 × 7 = 35 b) β5 × β7 = 35 c) β5 × 7 = β35 d) (β15) ÷ (β3) = 5 e) (β16) ÷ 2 = β8 f) 1 2 ÷ (β4) = β 2 g) (βπ₯)(βπ¦) = π₯π¦ h) 2(π₯ + 4) = 2π₯ + 8 i) (π₯ β 3)(π₯ + 4) = π₯(π₯ + 4) β 3(π₯ + 4) = π₯ 2 + 4π₯ β 3π₯ β 12 = π₯ 2 + π₯ β 12 2.3 Order of Operations The order in which operations in an arithmetical expression are performed is important. Consider the calculation 12 + 8 ÷ 4. Different answers are obtained depending on the order in which the operations are executed. If we first add together 12 and 8 and then divide by 4, the result is 5. However, if we first divide 8 by 4 to give 2 and then add this to 12, the result is 14. Note: Other texts use the Acronyms BODMAS or PEMDAS. Therefore, the order in which the mathematical operations are performed is important and the convention is as follows: Brackets, exponents, division, multiplication, addition, and subtraction. This convention has the acronym BEDMAS. However, the main point to remember is that if you want a calculation to be done in a particular order, you should use brackets to avoid any ambiguity. Examples Evaluate the expressions a) 3 β ( 5 + 8 ) β 22 ÷ 4 + 3 Brackets or Parenthesis first: 5 + 8 = 13 3 β 13 β 22 ÷ 4 + 3 5 6 Unit 2: Equations and Inequalities Exponent next: square the 2 or 22 = 4 3 β 13 β 4 ÷ 4 + 3 Multiplication and Division next (3 β 13) (4 ÷ 4) left to right: 39 β 1 + 3 Addition and Subtraction next left to right: 39 β 1 + 3 = 41 b) 23 × 3 + (5 β 1). c) 4 β 3[4 β 2(6 β 3)] ÷ 2. 2.4 Fractions π A fraction is a number that expresses part of a whole. It takes the form π where a and b are any integers except that b β 0. Note that in the π fraction π a can be greater than b The integers a and b are known as the numerator and denominator of the fraction, respectively. Examples of statements that use fractions are, ο· ο· 3 5 1 3 of students in a lecture may be female or of a personβs income may be taxed by the government. Fractions may be simplified to obtain what is known as a reduced fraction or a fraction in its lowest terms. This is achieved by identifying any common factors in the numerator and denominator and then cancelling those factors by dividing both numerator and denominator by them. 27 For example, consider the simplification of the fraction 45. Both the numerator and denominator have 9 as a common factor since 27 = 9 × 3 and 45 = 9 × 5 and therefore it can be cancelled: 27 3 × 9 3 = = 45 9 × 5 5 We say that fraction. 27 45 3 and 5 are equivalent fractions and that 3 5 is a reduced Comparing two fractions To compare the relative sizes of two fractions and also to add or subtract two fractions, we express them in terms of a common denominator. The common denominator is a number that each of the denominators of the respective fractions divides, i.e., each is a factor of the common denominator. BBA 120 Business Mathematics Least Common Multiple (LCM) The least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM(a, b) is defined to be zero. For example the LCM of 4 and 6 is 12 Suppose we wish to determine which is the greater of the two fractions 4 5 and 11. 9 The common denominator is 9 × 11 = 99. Each of the denominators (9 and 11) of the two fractions divides 99. We follow a similar procedure when we want to add or subtract two fractions. 2.4.1 Addition and Subtraction of Fractions To add or subtract fractions we use the following method; Step 1: Take a common denominator, that is, a number or term which is divisible by the denominator of each fraction to be added or subtracted. A safe bet is to use the product of all the individual denominators as the common denominator. Step 2: For each fraction, divide each denominator into the common denominator, then multiply the answer by the numerator. Step 3: Simplify your answer if possible. Example Simplify a) Greatest Common Divisor (gcd) The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4. b) c) 13 5 1 +3β5 β 16 24 7 π₯ 7 + 2 2π₯ 3 4 β 4π₯ 5 2.4.2 Multiplication and Division of fractions To multiply together two fractions, we simply multiply the numerators together and multiply the denominators together: π c a × b ac × = = π d c × b bd To divide one fraction by another, we multiply by the reciprocal of the π divisor where the reciprocal of the fraction a/b is defined to be provided π a, b β 0. That is π π a × d ππ ÷ = = π π c × b ππ 7 8 Unit 2: Equations and Inequalities Examples 2 5 (2)(5) 10 a) ( ) ( ) = = 3 7 (3)(7) 21 2 7 b) (β ) ( ) = 3 5 2 3 (β2)(7) (3)(5) 2 =β 14 15 (3)(2) 6 1 c) 3 × = ( ) ( ) = = =1 5 1 5 (1)(5) 5 5 3 (π₯+3) 3(π₯+3) 3π₯+9 11 22 d) ( ) = = 2 π₯ (π₯β5) π₯(π₯β5) π₯ β5π₯ e) 2 3 5 ( ) 11 f) 2π₯ π₯+π¦ 3π₯ 2(π₯βπ¦) ( ) 2 = (3) ( 5 ) = 15 = = 2π₯ 2(π₯βπ¦) π₯+π¦ 4π₯(π₯βπ¦) 3π₯ 4(π₯βπ¦) = 3(π₯+π¦) 3π₯(π₯+π¦) 2.5 Decimal Representation of Numbers A fraction or rational number may be converted to its equivalent decimal representation by dividing the numerator by the denominator. 3 For example, the decimal representation of is found by dividing 3 by 4 4 to give 0.75. This is an example of a terminating decimal since it ends after a finite number of digits. The following are examples of rational numbers that have a terminating decimal representation: BBA 120 Business Mathematics 1 8 In some text books, a bar notation is used instead of a dot e.g. 52.71656565β¦ (65 repeating infinitely often) may be written Μ Μ Μ Μ 52.7165 = 0.125, and 3 25 = 0.12. Some fractions do not possess a finite decimal representation β they go 1 on forever. The fraction 3 is one such example. Its decimal representation is 0.3333... where the dots denote that the 3βs are repeated and we write 1 = 0. 3Μ 3 where the dot over the number indicates that it is repeated indefinitely. This is an example of a recurring decimal. All rational numbers have a decimal representation that either terminates or contains an infinitely repeated finite sequence of numbers. Another example of a recurring decimal is the decimal representation of 1/13: 1 = 0.0769230769230 β¦ = 0.07Μ69230Μ 13 where the dots indicate the first and last digits in the repeated sequence. 2.5.1 Converting Decimals to Fractions To convert a decimal to a fraction, you simply have to remember that the first digit after the decimal point is a tenth, the second a hundredth, and so on. For example, 0.2 = 2 1 = 10 5 and 0.375 = 375 3 = 1000 8 2.5.2 Converting repeating decimals to fractions When converting repeating decimals to fractions, just follow the two steps below carefully. Step 1: Let x equal the repeating decimal you are trying to convert to a fraction Step 2: Examine the repeating decimal to find the repeating digit(s) Step 3: Place the repeating digit(s) to the left of the decimal point Step 4: Place the repeating digit(s) to the right of the decimal point 9 10 Unit 2: Equations and Inequalities Step 5: Subtract the left sides of the two equations. Then, subtract the right sides of the two equations As you subtract, just make sure that the difference is positive for both sides Now let's practice converting repeating decimals to fractions with two good examples Example #1: What rational number or fraction is equal to 0.55555555555 Step 1: x = 0.5555555555 Step 2: After examination, the repeating digit is 5 Step 3: To place the repeating digit ( 5 ) to the left of the decimal point, you need to move the decimal point 1 place to the right Technically, moving a decimal point one place to the right is done by multiplying the decimal number by 10. When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced Thus, 10x = 5.555555555 Step 4: Place the repeating digit(s) to the right of the decimal point Look at the equation in step 1 again. In this example, the repeating digit is already to the right, so there is nothing else to do. x = 0.5555555555 Step 5: Your two equations are: 10x = 5.555555555 x = 0.5555555555 10x - x = 5.555555555 β 0.555555555555 9x = 5 Divide both sides by 9 x = 5/9 BBA 120 Business Mathematics Example #2: What rational number or fraction is equal to 1.04242424242 Step 1: x = 1.04242424242 Step 2:After examination, the repeating digit is 42 Step 3:To place the repeating digit ( 42 ) to the left of the decimal point, you need to move the decimal point 3 place to the right Again, moving a decimal point three place to the right is done by multiplying the decimal number by 1000. When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced Thus, 1000x = 1042.42424242 Step 4:Place the repeating digit(s) to the right of the decimal point In this example, the repeating digit is not immediately to the right of the decimal point. Look at the equation in step 1 one more time and you will see that there is a zero between the repeating digit and the decimal point To accomplish this, you have to move the decimal point 1 place to the right This is done by multiplying both sides by 10 10x = 10.4242424242 Step 5:Your two equations are: 11 12 Unit 2: Equations and Inequalities 1000x = 1042.42424242 10x = 10.42424242 1000x - 10x = 1042.42424242 β 10.42424242 990x = 1032 Divide both sides by 990: x = 1032/990 To master this lesson about converting repeating decimals to fractions, you will need to study th2e two examples above carefully and practice with other examples 2.5.2 Significant Figures Sometimes we are asked to express a number correct to a certain number of decimal places or a certain number of significant figures. Suppose that we wish to write the number 23.541638 correct to two decimal places. To do this, we truncate the part of the number following the second digit after the decimal point: 23.54 | 1638. Then, since the first neglected digit, 1 in this case, lies between 0 and 4, then the truncated number, 23.54, is the required answer. If we wish to write the same number correct to three decimal places, the truncated number is 23.541 | 638, and since the first neglected digit, 6 in this case, lies between 5 and 9, then the last digit in the truncated number is rounded up by 1. Therefore, the number 23.541638 is 23.542 correct to three decimal places or, for short, βto three decimal placesβ. To express a number to a certain number of significant figures, we employ the same rounding strategy used to express numbers to a certain number of decimal places but we start counting from the first non-zero digit rather than from the first digit after the decimal point. For example, 72,648 = 70,000 (correct to 1 significant figure) = 73,000 (correct to 2 significant figures) = 72,600 (correct to 3 significant figures) BBA 120 Business Mathematics = 72,650 (correct to 4 significant figures), and 0.004286 = 0.004 (correct to 1 significant figure) = 0.0043 (correct to 2 significant figures) = 0.00429 (correct to 3 significant figures). Note that 497 = 500 correct to 1 significant figure and also correct to 2 significant figures. Example a) A teacher may ask for an exact answer to the problem βWhat is the length of the diagonal of a square whose sides have length 2?β The exact answer is β8. An approximate answer is 2.8284. b) A somewhat more complicated example is 3227/555 = 5.8144144144β¦, here the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" indefinitely. 13