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BUSINESS MATHEMATICS & STATISTICS Module 2 Exponents and Radicals Linear Equations (Lectures 7) Investments (Lectures 8) Matrices (Lecture 9) Ratios & Proportions and Index Numbers (Lecture 10) LECTURE 7 Review of lecture 6 Exponents and radicals Simplify algebraic expressions Solve linear equations in one variable Rearrange formulas to solve for any of its contained variables Annuity Value = 4,000 Down payment = 1,000 Rest in 20 installments of 200 Sequence of payments at equal interval of time Time = Payment Interval NOTATIONS R = Amount of annuity N = Number of payments I = Interest rater per conversion period S = Accumulated value A = Discounted or present worth of an annuity ACCUMULATED VALUE S = r ((1+i)^n – 1)/i A = r ((1- 1/(1+i)^n)/i) Accumulated value= Payment x Accumulation factor Discounted value= Payment x Discount factor ACCUMULATION FACTOR (AF) i = 4.25 % n = 18 AF = ((1 + 0.0425)^18-1) = 26.24 R = 10,000 Accumulated value = 10,000x 26.24 = 260,240 DISCOUNTED VALUE Value of all payments at the beginning of term of annuity = Payment x Discount Factor (DF) DF = ((1-1/(1+i)^n)/i) = ((1-1/(1+0.045)^8)/0.045) = 6.595 ACCUMULATED VALUE = 2,000 x ((1-1/(1+0.055)^8)/0.055) = 2,000 x11.95 =23,900.77 Algebraic x(2x2 –3x – 1) Operations Algebraic Expression …indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES Algebraic x(2x2 –3x – 1) Operations Terms …the components of an Algebraic Expression that are separated by ADDITION or SUBTRACTION signs x(2x2 –3x – 1) Algebraic x(2x2 –3x – 1) Operations Terms Monomial Binomial 1 Term 2 Terms 3x2 3x2 + xy Trinomial 3 Terms 3x2 + xy – 6y2 Polynomial …any more than 1 Term! Algebraic x(2x2 –3x – 1) Operations Term …each one in an Expression consists of one or more FACTORS separated by MULTIPLICATION or DIVISION sign …assumed when two factors are written beside each other! xy = x*y …assumed when one factor is written under an other! 36x2y 60xy2 Also Algebraic x(2x2 –3x – 1) Operations Term FACTOR Numerical Coefficient Literal Coefficient 3x2 3 x2 Algebraic x(2x2 –3x – 1) Operations Algebraic Expression Terms Monomial Binomial Trinomial Polynomial FACTORS Numerical Coefficient Literal Coefficient Division by a Monomial Example Step 1 Step 2 Identify Factors in the numerator and denominator 36 x2y 60 xy2 FACTORS 36x2y 60xy2 Cancel Factors in the numerator and denominator = 3(12)(x)(x)(y) 5(12)(x)(y)(y) 3x = 5y Division by a Monomial Example Divide each TERM Step in the numerator by the denominator 1 Step 2 Cancel Factors in the numerator and denominator 48a2 – 32ab 8a 48a2/8a – 32ab/8a or 6 4 = 48(a)(a) - 32ab 8a 8a = 6a – 4b Multiplying Polynomials Example -x(2x2 What is this Expression called? – 3x – 1) Multiply each term in the TRINOMIAL by (–x) = ( -x )( 2x2 ) + ( -x )( -3x ) + ( -x ) ( -1 ) The product of two negative quantities is positive. = -2x3 + 3x2 + x Exponents Rule of Base 34 32 *33 3 = 32 + 3 5 =3 Exponent 3 4 i.e. 3*3*3*3 Power = 81 = 243 (1 + i)20 (1 + i)8 =(1+ = (1+ i)20-8 i) 12 (32)4 = 32*4 = 38 = 6561 Exponents Rule of 3x6y3 x2z3 Square each factor Simplify inside the brackets first X4 3x6y3 2 3x4y3 = 2 3 xz z3 2 2 = 4*2 3* 2 2 3x y 3*2 Z Simplify = 6 8 9x y z 6 Solving Linear Equations in one Unknown Equality in Equations A+9 Expressed as: 137 A + 9 = 137 A = 137 – 9 A = 128 Solving Linear Equations in one Unknown Solve for x from the following: x = 341.25 + 0.025x Collect like Terms x = 341.25 + 0.025x x - 0.025x = 341.25 1 – 0.025 0.975x = 341.25 Divide both sides by 0.975 x = 341.25 0.975 x = 350 BUSINESS MATHEMATICS & STATISTICS for the Unknown Barbie and Ken sell cars at the Auto World. Barbie sold twice as many cars as Ken. In April they sold 15 cars. How many cars did each sell? Algebra Barbie sold twice as many cars as Ken. In April they sold 15 cars. How many cars did each sell? Unknown(s) Cars Variable(s) 2C C Barbie Ken 2C + C = 15 3C = 15 C = 5 Barbie = 2 C = 10 Cars Ken = C = 5 Cars Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively. What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest? Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively. Colleen 7800 Heather : 5200 Mark : 6500 Equivalent ratio (each term divided by 100) 78 : 52 : : 1 : notation format 65 Equivalent ratio with lowest terms 1.5 Expressed In colon 1.25 Divide 52 into each one The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800. What will be the sales of product Y if the sales of the two products maintain the same ratio? A 560 bed hospital operates with 232 registered nurses and 185 other support staff. The hospital is about to open a new 86-bed wing. Assuming comparable staffing levels, how many more nurses and support staff will need to be hired? The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800. Since X : Y = 4 : 3, then $1800 : Y = 4 : 3 Cross - multiply $1800 = 4 Y 3 Divide both sides of 4Y = 1800 * 3 the equation by 4 Y = 1800 * 3 4 = $1350 A 560 bed hospital operates with 232 registered nurses and 185 other support staff. The hospital is about to open a new 86-bed wing. 560 : 232 : 185 R = 86 : RN : SS N 560 86 232 = RN 560RN = 232*86 560RN = 19952 RN = 19952 / 560 Hire 35.63 or 36 RN’s 560 185 = 86 SS 560SS = 185*86 560SS = 15910 SS = 15910 / 560 S Hire 28.41 or 29 SS LO 2. & 3. A punch recipe calls for fruit juice, ginger ale and vodka in the ratio of 3:2:1. If you are looking to make 2 litres of punch for a party, how much of each ingredient is needed? A punch recipe calls for mango juice, ginger ale and orange juice in the ratio of 3:2:1. M J GA O Total Shares 3+2+1 = 6 333 ml per share 2 litres / 6 = 333 ml per share * 3 * 2 * 1 = 1 litre = 667 mls = 333 mls A punch recipe calls for mango juice, ginger ale and orange juice in the ratio of 3:2:1. If you have 1.14 litres of orange juice, how much punch can you make? Total Shares 3+2+1 = 6 1 1.14 Cross - multiply = Punch 6 Punch = 6 * 1.14 litres = 6.84 litres You check the frige and determine that someone has been drinking the orange juice. You have less than half a bottle, about 500 ml. How much fruit juice and ginger ale do you use if you want to make more punch using the following new punch recipe? Mango juice: ginger ale: orange juice = 3 : 2 : 1.5 How much fruit juice and ginger ale do you use if you want to make more punch using the following new punch recipe?: Mango juice: ginger ale: Orange juice = 3 : 2 : 1.5 M J 3 MJ = 0.5 1.5 500 ml Cross - multiply Mango Juice = 3 * 0.5 /1.5 = 1 litre GA 2 GA = 1.5 0.5 Ginger Ale = 2 * 0.5 /1.5 = .667 litre = 667 ml.