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) = 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 Operations x(2x2 –3x – 1) Algebraic Expression …indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES

Algebraic Operations x(2x2 –3x – 1) x(2x2 –3x – 1) Terms …the components of an Algebraic Expression that are separated by ADDITION or SUBTRACTION signs x(2x2 –3x – 1)

Algebraic Operations Monomial Polynomial Binomial Trinomial x(2x2 –3x – 1) Algebraic Operations Terms Monomial Polynomial Binomial Trinomial …any more than 1 Term! 1 Term 2 Terms 3 Terms 3x2 3x2 + xy 3x2 + xy – 6y2

Algebraic Operations x(2x2 –3x – 1) 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! …assumed when one factor is written under an other! xy = x*y 36x2y 60xy2 Also

Numerical Coefficient x(2x2 –3x – 1) Algebraic Operations Term FACTOR Numerical Coefficient Literal Coefficient 3x2 x2 3

Numerical Coefficient x(2x2 –3x – 1) Algebraic Operations Algebraic Expression Terms Monomial Binomial Trinomial Polynomial FACTORS Numerical Coefficient Literal Coefficient

Division by a Monomial 36 x2y 60 xy2 Example Identify Factors in the numerator and denominator Step 1 FACTORS 36x2y 3(12)(x)(x)(y) = 60xy2 5(12)(x)(y)(y) Step 2 Cancel Factors in the numerator and denominator = 3x 5y

= - Division by a Monomial = 6a – 4b 48a2/8a – 32ab/8a or 6 4 48(a)(a) Example 48a2/8a – 32ab/8a or Divide each TERM in the numerator by the denominator Step 1 6 4 = 48(a)(a) 32ab - 8a 8a Step 2 Cancel Factors in the numerator and denominator = 6a – 4b

What is this Expression called? Multiplying Polynomials What is this Expression called? Example -x(2x2 – 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

4 8 12 34 32 *33 (32)4 3 = 32 + 3 = 32*4 3 = 3 5 = 3 Exponents Rule of (1 + i)20 (1 + i)8 34 32 *33 (32)4 Base 3 = 32 + 3 = 32*4 =(1+ i)20-8 Exponent 3 4 = 3 5 = 3 8 i.e. 3*3*3*3 = (1+ i) 12 Power = 81 = 243 = 6561

Exponents Rule of 3x6y3 2 x2z3 Simplify inside the brackets first Step 1 Square each factor Step 2 Simplify Step 3 X4 3x6y3 2 x2z3 = 3x4y3 2 z3 32x4*2y3* 2 9x8y6 = = z6 Z3*2

Solving Linear Equations in one Unknown Equality in Equations A + 9 = ? 137 Expressed as: A + 9 = 137 A = 137 – 9 A = 128

x = 350 Step Step Solving Linear Equations in one Unknown Solve for x from the following: x = 341.25 + 0.025x Collect like Terms Step x = 341.25 + 0.025x x - 0.025x = 341.25 1 – 0.025 0.975x 0.975x = 341.25 Divide both sides by 0.975 Step x = 341.25 0.975 x = 350

BUSINESS MATHEMATICS & STATISTICS

for the Unknown Solving Word Problems Data Barbie and Ken sell cars at the Auto World. Barbie sold twice as many cars as Ken. Data In April they sold 15 cars. How many cars did each sell?

Review and Applications Algebra Barbie sold twice as many cars as Ken. In April they sold 15 cars. How many cars did each sell? Unknown(s) Cars Barbie Ken 2 C C Variable(s) 2C + C = 15 3C = 15 Barbie = 2 C = 10 Cars Ken = C = 5 Cars C = 5

Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively. Q What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest?

Expressed In colon notation format 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 Heather Mark Expressed In colon notation format 7800 : 5200 : 6500 Equivalent ratio (each term divided by 100) : : 78 52 65 Equivalent ratio with lowest terms Divide 52 into each one 1.5 : 1 : 1.25

Expressing a Ratio in Equivalent Forms Q 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?

Expressing a Ratio in Equivalent Forms Q 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?

Divide both sides of the equation by 4 Expressing a Ratio in Equivalent Forms 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 $1800 4 = Cross - multiply Y 3 Divide both sides of the equation by 4 4Y = 1800 * 3 Y = 1800 * 3 = $1350 4

S Expressing a Ratio in Equivalent Forms 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 = 86 : RN : SS S R N 560 86 560 86 Hire 35.63 or 36 RN’s = Hire 28.41 or 29 SS = 185 232 RN SS 560RN = 232*86 560SS = 185*86 560RN = 19952 560SS = 15910 RN = 19952 / 560 SS = 15910 / 560

Expressing a Ratio in Equivalent Forms Q LO 2. & 3. Q 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 G A 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

If you have 1.14 litres of orange juice, how much punch can you make? A punch recipe calls for mango juice, ginger ale and orange juice in the ratio of 3:2:1. Q If you have 1.14 litres of orange juice, how much punch can you make? 3+2+1 = 6 Total Shares 1 1.14 = Cross - multiply 6 Punch Punch = 6 * 1.14 litres = 6.84 litres

Q 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

= 1 litre = .667 litre = 667 ml. M J = = 500 ml M J G A 3 2 G A 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 : 1.5 500 ml M J G A 3 M J 2 G A = = Cross - multiply Cross - multiply 1.5 0.5 1.5 0.5 Mango Juice = 3 * 0.5 /1.5 Ginger Ale = 2 * 0.5 /1.5 = 1 litre = .667 litre = 667 ml.